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Research article

On the zero forcing number and propagation time of oriented graphs

  • Received: 07 October 2020 Accepted: 25 November 2020 Published: 01 December 2020
  • MSC : 05C20, 05C15, 05C57

  • Zero forcing is a process of coloring in a graph in time steps known as propagation time. These graph-theoretic parameters have diverse applications in computer science, electrical engineering and mathematics itself. The problem of evaluating these parameters for a network is known to be NP-hard. Therefore, it is interesting to study these parameters for special families of networks. Perila et al. (2017) studied properties of these parameters for some basic oriented graph families such as cycles, stars and caterpillar networks. In this paper, we extend their study to more non-trivial structures such as oriented wheel graphs, fan graphs, friendship graphs, helm graphs and generalized comb graphs. We also investigate the change in propagation time when the orientation of one edge is flipped.

    Citation: Sakander Hayat, Hafiz Muhammad Afzal Siddiqui, Muhammad Imran, Hafiz Muhammad Ikhlaq, Jinde Cao. On the zero forcing number and propagation time of oriented graphs[J]. AIMS Mathematics, 2021, 6(2): 1833-1850. doi: 10.3934/math.2021111

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  • Zero forcing is a process of coloring in a graph in time steps known as propagation time. These graph-theoretic parameters have diverse applications in computer science, electrical engineering and mathematics itself. The problem of evaluating these parameters for a network is known to be NP-hard. Therefore, it is interesting to study these parameters for special families of networks. Perila et al. (2017) studied properties of these parameters for some basic oriented graph families such as cycles, stars and caterpillar networks. In this paper, we extend their study to more non-trivial structures such as oriented wheel graphs, fan graphs, friendship graphs, helm graphs and generalized comb graphs. We also investigate the change in propagation time when the orientation of one edge is flipped.


    In his memoir [21], Lyapunov stated several theorems about analytic families of periodic solutions of some analytic autonomous vector fields near equilibrium point. Below, we distinguish three of them.

    First, consider an analytic planar vector field with a singular point having a pair of pure imaginary eigenvalues. Thus, we can assume an autonomous differential system of the form

    ˙x=ωy+, ˙y=ωx+. (1.1)

    with real analytic right-hand sides.

    Here, one looks for a first integral in the form of a formal power series:

    F=ω2(x2+y2)+. (1.2)

    In general, such a formal first integral does not exist; one finds

    ˙F=c2k+1(x2+y2)k+1+, (1.3)

    where c2k+10 is the so-called Poincaré–Lyapunov focus quantity and we have a weak focus of order 2k1 (see Section 2 below). Otherwise, we have the following result attributed to Lyapunov [21, Section 39] and Poincaré [27, Chapter 11].

    Theorem 1. (Lyapunov–Poincaré) If all focus quantities vanish, then the formal integral (1.2) is, in fact, analytic, and all solutions near x=y=0 are periodic.

    In [21, Section 40] the latter situation is generalized to the following system:

    ˙x=Ax+, (1.4)

    x(Rn,0), where the matrix A has a pair λ1,2=±iω of pure imaginary eigenvalues and all of its other eigenvalues lie in the half-plane {Reλ<0}. Again, one looks for a formal first integral of the form

    F=ω2(x21+x22)+, (1.5)

    where x1,2 are the variables associated with λ1,2, and there are again obstacles to such an integral occurring.

    Theorem 2. (Lyapunov) If all of these obstacles vanish, then the first integral is analytic and system (1.4) has an analytic 1–parameter family of periodic solutions.

    The last situation considered in [21, Section 42]* is the case when the matrix A in system (1.4) has the pure imaginary eigenvalues

    *In fact, Lyapunov considered a more general situation where, besides ±iωj, there are also eigenvalues in the left half-plane.

    ±iω1,,±iωm, ωj>0,

    and one of the frequencies, say, ω1, is such that none of the other frequencies is an integer multiple of it; thus,

    ωj/ω1Z,  j1. (1.6)

    Theorem 3. (Lyapunov) In this case, there exists a family of periodic solutions

    x=ϕ(t;c), c(R+,0),

    of period T(c)2π/ω1, depending analytically on c and such that ϕ(t;0)0.

    The latter theorem has attracted the attention of specialists in Hamiltonian dynamics.

    I have somehow overlooked this result and have learned about it only recently on the occasion of reviewing the Ph.D. thesis of D. Strzelecki in the Nicolas Copernicus University in Toruń.

    Recall that an autonomous analytic Hamiltonian system with m=n2 degrees of freedom takes the form

    ˙qj=Hpj, ˙pj=Hqj,  j=1,,n; (1.7)

    the corresponding vector field is usually denoted as

    XH=Hpi/qiHpi/pi.

    Assume that it has equilibrium point q=p=0 with the eigenvalues ±iω1,,±iωm,ωj>0. Assuming H(0)=0, the leading part of the Taylor expansion of the Hamiltonian function is

    H2=12ϵjωj(q2j+p2j), (1.8)

    where qj,pj are suitable canonical variables, i.e., with the Poisson brackets {pi,qj}=δij, and ϵj=±1 are well-defined signs.

    We have ˙f={f,H} in the case of a general Hamiltonian system with the symplectic structure defined by a Poisson bracket.

    In particular, for zj=qj+ipj and vj=qjipj, we have {zj,zk}={vj,vk}=0 and {zj,vk}=2iδjk. Thus ˙zj={zj,vk}H/vk=2iH/vk.

    Also, the resonant monomials g=zkvl that form the Birkhoff theorem satisfy {g,H2}=0.

    In the case of (1.8) the corresponding linear system is diagonalizable in the complex variables. The non-diagonalizable case is somewhat special and we consider it only in Example 7 in Section 5.

    Schmidt [28] studied 1–parameter families of periodic solutions for such systems with two degrees of freedom in the cases of resonant frequencies ω1 and ω2. His analysis was mainly focused on definite Hamiltonians, i.e., when ϵ1=ϵ2; but, he also considered the situation near the Lagrangian libration point in the restricted three-body problem, where the Hamiltonian function is indefinite. He did not refer to the third Lyapunov theorem. Anyway, his results agree with what is presented below.

    Weinstein [35] has applied the Lusternik–Schnirelmann category to prove the following.

    Theorem 4. (Weinstein) Assume that H2 is positive definite, i.e., all ϵj=1 in Eq. (1.8). Then, system (1.7) has at least m=n2 1–parameter families of periodic solutions.

    Next, Moser tried in [23, Theorem 4] to specify Weinstein's result by assuming that H2 is positive on a linear subspace E associated with one frequency ω, i.e., all solutions in E of the linear system have the period 2π/ω and these are the only such solutions. He claimed that there exists at least 12dimE of periodic solution of the period 2π/ω of the full Hamiltonian system. But his own example [23, Example 2] contradicts his statement; see also Example 4 below. Note also that the change HH means the reversion of the time, and it does not influence the periodic property of solutions.

    We have the following specification of Theorem 4. Again assume an analytic Hamiltonian system (1.6) with the eigenvalues ±iωj,ωj>0.

    Following [35, Proof of Theorem 2.1] we consider the following equivalence relation on the set {ω1,,ωm} of frequencies

    ωiωj iff ωi/ωjQ. (1.9)

    For each equivalence class Cν we have a linear subspace Eν that is invariant for the linear part of the system.

    Choose some equivalence class C1 and let E1 be the corresponding linear subspace. Let us order the frequencies from C1 as follows:

    ˜ω1:=ω1==ωk1>˜ω2:=ωk1+1==ωk1+k2>>˜ωr:=ωk1++kr1+1==ωk1++kr. (1.10)

    Theorem 5. Let thinspace H be analytic and such that H2|E1 is definite (say, positive), i.e., ϵ1==ϵk1++kr=1 in Eq. (1.8); then, there exist the following:

    at least k1 1–parameter families of periodic solutions to system (1.7) with periods 2π/˜ω1,

    at least k1+k2 1–parameter families of periodic solutions to system (1.7) with periods 2π/˜ω1 or 2π/˜ω2,

    at least k1++kr 1–parameter families of periodic solutions to system (1.7) with periods 2π/˜ω1, or 2π/˜ω2,, or 2π/˜ωr.

    Next, this subject was raised up by specialists in the nonlinear functional analysis, including Szulkin [32] and several other groups [10,15,25,26]. The principal result here is that we assume the ordering (1.10) of all frequencies (not only from C1).

    Theorem 6. (Szulkin) If, for a given l,

    (i) ˜ωj/˜ωlN for all jl in Eq. (1.10), and

    (ii) ϵk1++kl1+1++ϵk1++kl0 for ϵi's from Eq. (1.8), then there exists a sequence {γn(t)} of periodic solutions to the equation ˙x=XH(x) of non-constant periodic solutions tending to γ(t)0 of periods tending to 2π/˜ωl.

    We see that the first assumption is like in the third Lyapunov theorem. But, the assumption (ii) is new; it replaces the definiteness assumption of H2 from Theorems 4 and 5.

    Finally, we recall the following non-Hamiltonian version of the third Lyapunov theorem due to Moser [23, Theorem 2].

    Theorem 7. (Moser) Assume that an analytic system (1.4) with only purely imaginary eigenvalues has an analytic first integral F(x) such that D2F(0) is positive definite. Then, this system has at least one 1–parameter family of periodic solutions.

    The aim of this paper is to take a somehow different look at this center problem. Our approach is rather qualitative and less topological, like in [1] and [18]. It turns out that much can be deduced from the Poincaré–Dulac normal form and its Hamiltonian version, i.e., the Birkhoff normal form (Schmidt also used the Birkhoff normal form). Important is the use of some versions of the standard Poincaré return map combined with the analyticity assumption of the vector fields. We will reprove most of above theorems and we shall prove a new result.

    Assume that the matrix A in system (1.4) has pure imaginary eigenvalues: ±iω1,,±iωm, ωj>0. We can reduce it, via a formal change of variables, to the Poincaré–Dulac normal form (see Theorem 9 in the next section). It may happen that the latter formal system admits a formal 1–parameter family of periodic solutions with the period 2π/ωl.

    More precisely, one can consider the situation with only one equivalence class for the frequencies, i.e., ωj/ωk=pj/pk, with relatively prime positive integers pi, and that ω1ωl. Let Zj be the formal complex variables associated with the Poincaré–Dulac normal form. For the new variables RR,WjC and ΘS1, defined by Zl=ReiΘ and Zj=WjeipjΘ/pl, one obtains a formal differential system:

    dRdΘ=Π(R,W,ˉW), dWjdΘ=Λj(R,W,ˉW)

    (called the return system). The analogue of the center conditions in this case means that the latter system has a formal curve Γ of non-isolated equilibrium points. All of these notions are made precise in Section 7 (Eqs. (7.1)–(7.3), Lemma 3 and Definition 4).

    The next statement can be regarded as a fourth Lyapunov theorem.

    Theorem 8. In this case, there exists a continuous family of periodic solutions to system (1.4) with the period 2π/ωl.

    The paper is organized as follows. In Section 2, we define the Poincaré –Dulac and Birkhoff normal forms, with their principal properties. In Section 3, we present and discuss theorems about invariant manifolds. Section 4 is devoted to various proofs of the three Lyapunov theorems; they are included in this paper (which is treated as a sort of review) because some of the Lyapunov theorems are not widely known, and because their proofs are (we hope) illuminating. In Section 5, we present some examples of Hamiltonian systems with explicit families of periodic solutions. In Section 6, we prove the existence of some additional invariant submanifolds following from the normal forms. In Section 7, we prove Theorem 8. Section 8 is devoted to the discussion of situations with a first integral; there, we give proofs of Theorems 4, 5, 7 and an introduction to the functional analytic method. In Section 9, we present our approach, based only on the Birkhoff normal form, of two examples from the celestial mechanics: geostationary orbits of satellites and libration points in the restricted four-body problem. The last section contains appendices about the topology of weighted projective spaces and the Lusternik–Schnirelmann category.

    Consider the analytic system (1.4), but in (Cn,0) (not in (Rn,0). Thus, we can assume that the matrix A is in the Jordan form with eigenvalues λ1,,λn.

    Definition 1. We say that the latter system of eigenvalues satisfies the resonant relation of type (j;k),j=1,,n,k=(k1,,kn)Zn0, if

    λj=(k,λ)=k1λ1++knλn. (2.1)

    Theorem 9. (Poincaré–Dulac [13,27]) There exists a change xX, defined by formal power series, which transforms system (1.4) to the following Poincaré–Dulac normal form:

    ˙Xj=λjXj+kaj;kXk,j=1,,n, (2.2)

    where the summation runs over multi-indices k such that the resonant relations of type (j;k) hold and Xk=Xk11Xknn.

    The proof can be found in [3,17,37].

    Remark 1. In the case when there are multiple eigenvalues λj and the corresponding Jordan cell is not diagonal, we have the resonant relations (j;k), where k=(0,,0,1,0,0) with 1 in the (j+1)th entry. They imply non-diagonal entries in A.

    Remark 2. In the so-called Poincaré domain, i.e., when the convex hull of the set {λ1,,λn}C is separated from 0C, the Poincaré–Dulac normal form is polynomial and the normalizing change is analytic. We refer the reader to [30] for a short proof.

    Remark 3. In the general, situation the Poincaré–Dulac normal form is divergent. Indeed, in the example

    ˙x1=x21, ˙x2=x2x21, (2.3)

    due to Euler (see [37]), the Poincaré–Dulac normal form is

    ˙X1=X21+a3X31+, ˙X2=X2(1+b1X1+b2X21+).

    But, the so-called center manifold Wc, corresponding to the zero eigenvalue and formally defined by X2=0, in the original system (2.3) takes the form

    Wc={x2=(m1)!xm1}.

    Consider now an autonomous analytic Hamiltonian system (1.7) with m=n2 degrees of freedom near an equilibrium point q=p=0 with the eigenvalues ±iω1,,±iωm,ωj>0. Assuming H(0)=0, the leading part of the Taylor expansion of the Hamiltonian function is like in Eq. (1.8), i.e., H2=12ϵjωj(q2j+p2j), where qj,pj are suitable canonical variables, i.e., with the Poisson brackets {pi,qj}=δij, and ϵj=±1 are well-defined signs.

    It is natural to introduce the complex variables

    zj=qj+ipj, vj=qjipj, (2.4)

    with the Poisson brackets {zj,vk}=2iδjk. Then, we have

    H2=12ϵjωjzjvj (2.5)

    and

    ˙zj=λjzj+, ˙vj=λjvj+,  λj=iϵjωj. (2.6)

    Of course, in the real domain (where pj and qk are real), we have vj=ˉzj.

    The following result from [8] is a Hamiltonian analogue of the Poincaré–Dulac theorem.

    Theorem 10. (Birkhoff) There exists a formal canonical change (z,v)(Z,V) which reduces the Hamiltonian function to the following one:

    H2(Z,V)+ak;lZkVl,

    where summation runs over the pairs (k;l)=(k1,,kn;l1,,ln)Zn0×Zn0 such that the resonant relations

    (kl,λ)=(kjlj)λj=0

    hold and |k|+|l|3.

    Definition 2. The singular point x=0 of system (1.4) is hyperbolic if none of the eigenvalues of the matrix A is imaginary. Thus, the eigenvalues are divided into two groups:

    {λ1,,λp} with Reλj<0 and {λp+1,,λn} with Reλj>0; (3.1)

    i.e., separated by the imaginary axis.

    The principal result about invariant manifolds is the Hadamard–Perron theorem for hyperbolic singular points. These manifolds are the local stable manifold (of dimension p)

    Ws={x:gt(x)0 as t+} (3.2)

    and the local unstable manifold (of dimension q=np)

    Wu={x:gt(x)0 as t}, (3.3)

    where {gt} is the phase flow generated by the vector field (1.4); gt(x0) is the solution x(t) obeying the initial condition x(0)=x0. In the case of a finitely smooth vector field, the proof of the Hadamard–Perron theorem is rather involved. But, in the analytic case, it is quite easy.

    Theorem 11. (Holomorphic Hadamard–Perron) Assume that the vector field (1.4) is analytic. Assume also that the set {λ1,,λn}C of eigenvalues of the matrix A can be separated by a straight line through λ=0 into two groups: {λ1,,λp} on one side of and {λp+1,,λn} on the other side. Then, there exist local analytic invariant manifolds for (1.4) tangent to the corresponding linear subspaces invariant for A.

    It follows that, if the singular point x=0 is hyperbolic, then the invariant manifolds Ws and Wu are real analytic.

    Proof. We present a rather novel proof, following a method from [30]. Another proof can be found in [17]; note also that, in the case n=2 and p=1, this result was proved by Briot and Bouquet [9] (see also [37]).

    Let EsCp and EuCq be the linear subspaces associated with the division of the eigenvalues set, and let (x,y) be the coordinates associated with the splitting Cn=EsEu. We can assume that the matrix A has the form AsAu, where As=diag(μ1,,μp)+As1 and Au=diag(ν1,,νq)+Au1, where As,u1 are off-diagonal with small entries. We have

    ˙x=Asx+ϕ(x,y),  ˙y=Auy+ψ(x,y).

    We look for the manifold Ws as a graph of a map F:EsEu,Ws={(x,F(x))}; the case of Wu is treated analogously. The invariance condition, i.e.,

    ˙yDF˙x|y=F(x)0,

    leads to the equation

    AuF(x)DF(x)Asx=DF(x)ϕ(x,F(x))ψ(x,F(x)). (3.4)

    This equation takes the form

    LF=T(F),

    where L and T are operators, linear and nonlinear. We treat these operators as acting on the Banach space F consisting of power series F=j,kfj,kxkej with the norm

    where \rho is the 'radius of convergence' (suitably chosen). We have \boldsymbol{L} = \boldsymbol{L}_{0}+\boldsymbol{L}_{1}, where

    \begin{equation*} \boldsymbol{L}_{0}x^{k}e_{j} = \left[ \nu _{j}-\left( k,\mu \right) \right] x^{k}e_{j} \end{equation*}

    and \boldsymbol{L}_{1} is small with respect to \boldsymbol{L}_{0}. Because \nu _{j} and \mu _{i} are separated by the line \ell, the operator \boldsymbol{L} is invertible.

    Moreover, if \mathcal{F}_{d} = \left\{ F = \sum_{j, \left\vert k\right\vert = d}f_{j; k}x^{k}e_{j}\right\} are the subspaces of homogeneous maps of degree d\geq 2 , then

    \begin{equation*} \left\Vert \boldsymbol{L}^{-1}|_{\mathcal{F}_{d}}\right\Vert < C_{1}/d \end{equation*}

    for some constant C_{1}. On the other hand, the operators \boldsymbol{T}{| }_{\mathcal{F}_{d}}:\mathcal{F}_{d}\longmapsto \mathcal{F}_{ > d} have the Lipschitz constants bounded by C_{2}d\rho, where the factor d arises from DF and C_{2} is some constant. It follows that the operator

    \begin{equation*} \boldsymbol{P} = \boldsymbol{L}^{-1}\boldsymbol{T} \end{equation*}

    is Lipschitz-continuous with a small Lipschitz constant. Therefore, the fixed-point problem F = \boldsymbol{P}\left(F\right) has a unique solution.

    Assume now the non-hyperbolic situation. Thus, we have the splittings \mathbb{R}^{n} = E^{s}\oplus E^{u}\oplus E^{c} and A = A^{s}\oplus A^{u}\oplus A^{c} , where the eigenvalues of A^{s} (respectively, A^{u}) lie in the left (respectively, right) half-plane and the eigenvalues of A^{c} lie on the imaginary axis. The next theorem is usually cited without proof (which is quite technical, see [3,17]).

    Theorem 12. (Center Manifold) There exist smooth invariant manifolds W^{s}, W^{u} and W^{c} tangent to the subspaces E^{s}, \; E^{u} and E^{c} , respectively. The manifolds W^{s} and W^{u} are analytic; W^{c} is only infinitely smooth in general, but its Taylor series is defined uniquely.

    Remark 4. The Euler example from Remark 3 demonstrates that the center manifold can be non-analytic. If W^{c} = \left\{ y = F(x)\right\} , then the analogue of Eq. (3.4) is

    \begin{equation*} \boldsymbol{L}F = x^{2}\frac{\mathrm{d}}{\mathrm{d}x}F+x^{2}, \end{equation*}

    where \boldsymbol{L} = Id. Here \boldsymbol{L} , although invertible, is not dominating in comparison to x^{2}\frac{\mathrm{d}}{\mathrm{d}x}; we have \left\Vert x^{2}\frac{\mathrm{d}}{\mathrm{d}x}x^{k}\right\Vert /\left\Vert x^{k}\right\Vert = k\rho.

    In [17] and [37], one can find an explanation of the non-analyticity of W^{c} in terms of the so-called Stokes phenomena.

    Next, the center manifold can be non-unique. Indeed, the real Euler example is the so-called saddle-node singularity; on the left from the unstable manifold W^{u} = \left\{ x = 0\right\} , we have two hyperbolic sectors separated by the left part of W^{c} (which is unique), but, on the right of W^{u} , we have a parabolic sector, and any of its phase curves can serve as the right part of W^{c}. This non-uniqueness is explained by the method of the proof of the existence of W^{c}; there, one extends the vector field to the whole \mathbb{R}^{n} and W^{c} (which is a fixed point of some functional operator) depends on this extension.

    Below, we present other examples with non-analytic invariant manifolds.

    Example 1. Let z = x_{1}+\mathrm{i}x_{2}, \; v = x_{1}-\mathrm{i}x_{2} and y be coordinates in \mathbb{C}^{3}. The system is as follows:

    \begin{equation} \dot{z} = \mathrm{i}z+z^{2}v,\text{ }\dot{v} = -\mathrm{i}v+zv^{2},\text{ }\dot{y } = -y+zv. \end{equation} (3.5)

    One finds that

    \begin{equation*} W^{c} = \left\{ y = \frac{1}{2}\sum \left( n-1\right) !\left( 2zv\right) ^{n}\right\} . \end{equation*}

    The non-analyticity of the center manifold is explained via the nonlinear Stokes phenomena as follows. In the variables u = zv and y , we have

    \begin{equation*} \dot{u} = 2u^{2},\text{ }\dot{y} = -y+u, \end{equation*}

    i.e., a saddle-node similar to the Euler example. In a sectorial domain around \left\{ \left(u, y\right) :\mathrm{Im}\left(u\right) = 0, \text{ } \mathrm{Re}\left(u\right) > 0\right\} , the phase portrait near the saddle, with a unique center manifold and unique stable manifold; but, in a sectorial domain near \{\left(u, y\right) :\mathrm{Im}\left(u\right) = 0, \; \mathrm{Re}\left(u\right) < 0\} , the phase portrait is like near the node with many center-ype manifolds.

    Example 2. Let z_{1} = x_{1}+\mathrm{i}x_{2}, \; v_{1} = x_{1}-\mathrm{i}x_{2} and z_{2} = x_{3}+\mathrm{i}x_{4}, \; v_{2} = x_{3}-\mathrm{i}x_{4} be coordinates in \mathbb{C}^{4}. The system is as follows:

    \begin{equation} \dot{z}_{1} = \mathrm{i}\omega _{1}z_{1}+z_{1}^{2}v_{1},\text{ }\dot{v}_{1} = - \mathrm{i}\omega _{1}v_{1}+z_{1}v_{1}^{2},\text{ }\dot{z}_{2} = \mathrm{i} \omega _{2}z_{2}+z_{1}v_{1},\text{ }\dot{v}_{2} = -\mathrm{i}\omega _{2}v_{2}+z_{1}v_{1}. \end{equation} (3.6)

    It has two invariant planes:

    \begin{equation*} W_{1} = \left\{ z_{1} = v_{1} = 0\right\} \end{equation*}

    (analytic) and

    \begin{equation*} W_{2} = \left\{ z_{2} = -\frac{1}{2}\sum \left( n-1\right) !\left( \frac{ 2z_{1}v_{1}}{\mathrm{i}\omega _{2}}\right) ^{n},\text{ }v_{2} = -\frac{1}{2} \sum \left( n-1\right) !\left( \frac{2\mathrm{i}z_{1}v_{1}}{\omega _{2}} \right) ^{n}\right\} , \end{equation*}

    which is non-analytic.

    We finish this section with another theorem about invariant manifolds.

    Definition 3. Let f be a smooth diffeomorphism of a manifold M , and let L\subset M be a smooth invariant submanifold for f, \; f\left(L\right) \subset L. We say that f is normally hyperbolic at N if there is a splitting

    \begin{equation*} N_{L} = T_{M}/T_{N}\simeq E^{s}\oplus E^{u} \end{equation*}

    of the normal bundle to L into sub-bundles E^{s} and E^{u} that are invariant with respect to Df and the following estimates hold:

    the bundle E^{s} is contracted more strongly than the bundle TL , i.e.,

    \begin{equation*} \sup\limits_{L}\left\Vert Df|_{E^{s}}\right\Vert < \left\{ \inf\limits_{L}\left\Vert \left( Df|_{TL}\right) ^{-1}\right\Vert \right\} ^{-1}; \end{equation*}

    the bundle E^{u} is expanded more strongly than the bundle TL , i.e.,

    \begin{equation*} \left\{ \inf\limits_{L}\left\Vert \left( Df|_{E^{u}}\right) ^{-1}\right\Vert \right\} ^{-1} > \sup\limits_{L}\left\Vert Df|_{TL}\right\Vert . \end{equation*}

    We refer the reader to [16] for the proof of the following statement.

    Theorem 13. (Normal Hyperbolicity) If f is normally hyperbolic on L , then there exists a neighborhood \mathcal{U} of f in the functional space of C^{1} –diffeomorphisms such that any g\in \mathcal{U} has a unique invariant submanifold L_{g} close to L at which it is normally hyperbolic.

    Eq. (1.1) with the variables z = x+\mathrm{i}y, \; v = x-\mathrm{i}y takes the form

    \begin{equation} \dot{z} = \mathrm{i}\omega z+\ldots ,\text{ }\dot{v} = -\mathrm{i}\omega v+\ldots . \end{equation} (4.1)

    The Poincaré–Dulac normal form here is as follows:

    \begin{equation} \dot{Z} = Z\left\{ \mathrm{i}\omega +\sum \left( a_{k}+\mathrm{i}b_{k}\right) \left( ZV\right) ^{k}\right\} ,\text{ }\dot{V} = V\left\{ -\mathrm{i}\omega +\sum \left( a_{k}-\mathrm{i}b_{k}\right) \left( ZV\right) ^{k}\right\} . \end{equation} (4.2)

    It is easy to find that the first nonzero coefficient a_{k} is related to the first nonzero Poincaré–Lyapunov quantity:

    \begin{equation} c_{2k+1} = \omega a_{k}. \end{equation} (4.3)

    Next, after passing to the polar coordinates r, \theta (in the real domain), i.e., with z = r\mathrm{e}^{\mathrm{i}\theta } and v = \bar{z} = r \mathrm{e}^{-\mathrm{i}\theta }, we get the equation

    \begin{equation*} \frac{\mathrm{d}r}{\mathrm{d}\theta } = f\left( r,\theta \right) , \end{equation*}

    where f is a convergent series in powers of r with trigonometric polynomials in \theta as coefficients. We define the Poincaré return map

    \begin{equation} \mathcal{P}\left( r_{0}\right) = \phi \left( 2\pi ;r_{0}\right) , \end{equation} (4.4)

    where \phi \left(\theta; r_{0}\right) is the solution to the latter equation dr/d\theta with the initial condition \phi \left(0;r_{0}\right) = r_{0}. The Poincaré map is analytic and has a Taylor expansion of the form

    \begin{equation*} \mathcal{P}\left( r\right) = r+d_{2k+1}r^{2k+1}+\ldots , \end{equation*}

    where

    \begin{equation} d_{2k+1} = \left( 2\pi /\omega \right) a_{k} \end{equation} (4.5)

    provided the Poincaré–Lyapunov quantity c_{2k+1}\not = 0.

    The assumption of Theorem 1 means formally that either

    (i) the Poincaré–Dulac normal form equals

    \begin{equation} \dot{Z} = \mathrm{i}Z\cdot M\left( ZV\right) ,\text{ }\dot{V} = -\mathrm{i} V\cdot M\left( ZV\right) , \end{equation} (4.6)

    where M = \omega +\sum b_{k}\left(ZV\right) ^{k} is a formal integrating factor and F = ZV is the formal first integral; or (equivalently),

    (ii) formally,

    \begin{equation} \mathcal{P} = Id. \end{equation} (4.7)

    Moreover, the assumptions of the holomorphic Hadamard–Perron theorem for system (4.1) are satisfied. Therefore, we have two holomorphic invariant lines, W^{+} and W^{-}, formally defined by \left\{ V = 0\right\} and \left\{ W = 0\right\} , respectively.

    Below, we present two geometrical proofs of Theorem 1.

    First proof. (Note that this proof was not found in the literature). Since the Poincaré map is analytic and formally obeys Eq. (4.7), it is the identity. Therefore, the real phase portrait is of the center type and the equilibrium point is surrounded by a family of closed phase curves. Moreover, we have a first integral \widetilde{G}\left(r, \theta \right) defined via the initial condition \widetilde{G}\left(r, \theta \right) = r_{0}^{2} if r = \phi \left(\theta; r_{0}\right).

    But, this function is not correct because its value for negative r 's should be compatible with the 'square root' of the Poincaré map \mathcal{Q}:\left(\mathbb{R}, 0\right) \longmapsto \left(\mathbb{R}, 0\right), \; \mathcal{Q} (r_{0}) = -r_{0}+\ldots , such that \mathcal{P} = \mathcal{Q}\circ \mathcal{Q} on \left(\mathbb{R}_{+}, 0\right). But, we can take a function defined initially on \left(\mathbb{R}_{+}, 0\right) as G\left(r_{0}\right) = \frac{1}{2}\left(r_{0}^{2}+\mathcal{Q}^{2}\left(r_{0}\right) \right). Thus,

    \begin{equation} G\left( r,\theta \right) = G\left( r_{0}\right) \end{equation} (4.8)

    if r = \phi \left(\theta, r_{0}\right). This first integral is holomorphic near the circle \left\{ r = 0, 0\leq \theta \leq 2\pi \right\} \subset \mathbb{ C}\times R , where R is a ring around the circle \mathbb{R}/2\pi \mathbb{Z }\subset \mathbb{C}/2\pi \mathbb{Z} .

    In fact, we can extend it to a holomorphic first integral in Z, V. For this, one first defines it as G\left(x\right) = \frac{1}{2}\left(x^{2}+ \mathcal{Q}^{2}\left(x\right) \right) on the disc \Sigma = \left\{ \left(x, 0\right) :x\in \mathbb{C}, \left\vert x\right\vert < \varepsilon \right\} \subset \mathbb{C}_{x, y}^{2}, or \Sigma = \left\{ \left(z, \bar{z}\right) :z\in \mathbb{C}, \left\vert z\right\vert < \varepsilon \right\} \subset \mathbb{C}_{z, v}^{2}, where \mathcal{Q} is the extension of the square root of the return map. Outside of the disc \Sigma , this function is defined by the condition of being constant on the complex phase curves (Riemann surfaces). In particular, it is single-valued (see the second proof) and equals zero on the complex separatrices of the singular point.

    Second proof (due to Moussu [24]). Using the holomorphic Hadamard–Perron theorem, we can assume that the coordinate lines are invariant, i.e.,

    \begin{equation*} \dot{z} = z\left( \mathrm{i}\omega +\ldots \right) ,\text{ }\dot{v} = v\left( - \mathrm{i}\omega +\ldots \right) . \end{equation*}

    Then, the phase curves outside of W^{-} = \left\{ z = 0\right\} are graphs of functions,

    \begin{equation*} v = \varphi \left( z;v_{0}\right) , \end{equation*}

    which satisfy the initial value problem

    \begin{equation*} \frac{\mathrm{d}v}{\mathrm{d}z} = \frac{v\left( -\mathrm{i}\omega +\ldots \right) }{z\left( \mathrm{i}\omega +\ldots \right) },\text{ }\varphi \left( z_{0};v_{0}\right) = v_{0,} \end{equation*}

    where z_{0}\not = 0 is fixed. The analytic continuation of solutions \varphi along the circle \left\{ z = \mathrm{e}^{\mathrm{i}\alpha }z_{0}:0\leq \alpha \leq 2\pi \right\} defines the monodromy map \mathcal{M}:D\longmapsto D, \; D = \left\{ z_{0}\right\} \times \left\{ \left\vert v_{0}\right\vert < \varepsilon \right\}, as

    \begin{equation} \mathcal{M}\left( v_{0}\right) = \varphi \left( \mathrm{e}^{2\pi \mathrm{i} }z_{0};v_{0}\right) . \end{equation} (4.9)

    This map is holomorphic and we have

    \begin{equation} \mathcal{M}\left( v_{0}\right) = v_{0}+e_{k}v_{0}^{k+1}+O\left( v_{0}^{k+2}\right) ,\text{ }e_{k} = \left( 4\pi z_{0}/\omega \right) a_{k}. \end{equation} (4.10)

    The center conditions from Theorem 1 mean that \mathcal{M} = Id.

    Now, we can define a first integral F\left(z, v\right) by putting

    \begin{equation} F\left( z_{0},v_{0}\right) = z_{0}v_{0} \end{equation} (4.11)

    on D and continuing it analytically as constant on the phase curves. Because the monodromy is trivial, the function F is single-valued outside of the line \left\{ z = 0\right\} . It is also bounded there. Hence, it is holomorphic.

    The function F has a Morsean critical point at z = v = 0. By the Morse lemma [37], there exist analytic coordinates Z = z+\ldots, \; V = V+\ldots such that F = ZV.

    The vector field is parallel to the analytic Hamiltonian vector field X_{F} = F_{V}^{\prime }\frac{\partial }{\partial Z}-F_{Z}^{\prime }\frac{ \partial }{\partial V}, i.e., it equals \Phi \cdot X_{F}, where \Phi = \mathrm{i}\omega +\ldots is an analytic function (the orbital factor).

    Finally, one can reduce \Phi to an analytic function of one variable: \Phi \left(Z, V\right) = \mathrm{i}M\left(ZV\right); we refer the reader to [30] for the proof.

    Final remark. The integrating factor M from Eq. (4.6) is also analytic. If it is constant, then the center is isochronous and the period does not depend on the periodic solution.

    Recall that the spectrum of the linear part consists of \lambda _{1, 2} = \pm \mathrm{i}\omega and of \mu _{1}, \ldots, \mu _{m} ( m\geq 1) in the left half-plane. Therefore, the resonant relations from Definition 1 are as follows:

    \begin{eqnarray*} \lambda _{1} & = &\left( p+1\right) \lambda _{1}+p\lambda _{2},\text{ }\lambda _{2} = q\lambda _{1}+\left( q+1\right) \lambda _{2}, \\ \mu _{j} & = &p\lambda _{1}+p\lambda _{2}+\left( l,\mu \right) ,\text{ } \left\vert l\right\vert \geq 1. \end{eqnarray*}

    In the Poincaré–Dulac normal form, we have

    \begin{equation*} \dot{Z} = Z\left\{ \mathrm{i}\omega +\sum \left( a_{p}+\mathrm{i}b_{p}\right) \left( ZV\right) ^{p}\right\} \end{equation*}

    and the corresponding equation for \dot{V}. Moreover,

    \begin{equation*} \dot{Y}_{j} = \sum Y_{r}F_{r}\left( Y,ZV\right) ,\text{ }j = 1,\ldots ,m. \end{equation*}

    We see that the variables Z, V are formally separated from Y_{j} and the center condition from Theorem 2 reads as

    \begin{equation*} a_{p} = 0,\text{ }p = 1,2,\ldots . \end{equation*}

    Moreover, we have the stable invariant manifold W^{s} = \left\{ Z = V = 0\right\} (which is analytic) and a formal center manifold W^{c} = \left\{ Y = 0\right\}, about its analytic properties we cannot judge at this moment (see Example 1 above).

    But, the eigenvalue \lambda _{1} = \mathrm{i}\omega is separated from the remaining eigenvalues \mu _{1}, \ldots, \mu _{m}, \; \lambda _{2} by a line in \mathbb{C} through 0 , and, analogously, \lambda _{2} is separated from the other eigenvalues. By the analytic Hadamard–Perron theorem, we have analytic invariant hypersurfaces: W^{+} corresponding to \lambda _{2}, \mu _{1}, \ldots, \mu _{m} and W^{-} corresponding to \lambda _{1}, \mu _{1}, \ldots, \mu _{m}; of course, W^{s} = W^{+}\cap W^{-}. We also have two invariant lines L^{+} and L^{-} corresponding to the eigenvalues \lambda _{1} and \lambda _{2} , respectively; W^{c} is 'spanned' by L^{+} and L^{-} .

    We are ready to present the following:

    Proof of Theorem 2. We would like to follow the second proof of Theorem 1.

    Thus, we can assume the following analytic system:

    \begin{equation*} \dot{z} = z\left( \mathrm{i}\omega +\ldots \right) ,\text{ }\dot{v} = v\left( - \mathrm{i}\omega +\ldots \right) ,\text{ }\dot{y} = Y\left( z,v,y\right) ; \end{equation*}

    so, W^{+} = \left\{ z = 0\right\} and W^{-} = \left\{ v = 0\right\}. The phase curves outside of W^{+} are graphs of functions of z.

    Let us choose a hypersurface of initial conditions with the form

    \begin{equation*} \Sigma = \left\{ z_{0}\right\} \times \left\{ \left\vert v\right\vert < \varepsilon \right\} \times \left\{ \left\vert y\right\vert < \varepsilon \right\} \end{equation*}

    (point \times disc \times ball) of complex dimension 1+m\geq 2. The solutions to the equations for \frac{\mathrm{d}v}{\mathrm{d}z} and for \frac{\mathrm{d}y}{\mathrm{d}z} with the initial conditions v\left(z_{0}\right) = v_{0} and y\left(z_{0}\right) = y_{0} are of the form

    \begin{equation*} v = \varphi \left( z;v_{0},y_{0}\right) = z_{0}v_{0}/z+\ldots ,\text{ }y = \chi \left( z;v_{0},y_{0}\right) . \end{equation*}

    We also have the monodromy map \mathcal{M}:\Sigma \longmapsto \Sigma :

    \begin{equation*} \mathcal{M}\left( v_{0},y_{0}\right) = \left( \varphi \left( \mathrm{e}^{2\pi \mathrm{i}}z_{0};v_{0},y_{0}\right) ,\chi \left( \mathrm{e}^{2\pi \mathrm{i} }z_{0};v_{0},z_{0}\right) \right) . \end{equation*}

    In contrast to the Poincaré–Lyapunov case, this map is far from the identity. It has the form

    \begin{equation*} \mathcal{M}:\left( v,y\right) \longmapsto \left( v+\ldots ,By+\ldots \right), \end{equation*}

    where the spectrum of the matrix B lies in the open unit disc; B is a strong contraction. Therefore, we need an additional argument.

    On the one hand, we can use the normal hyperbolicity theorem, which ensures the existence of a 2- dimensional invariant manifold V^{c} = V_{z_{0}}^{c} , which is the center manifold for \mathcal{M}. For this, we treat \mathcal{M} as a perturbation of the monodromy map \mathcal{M}_{0} associated with a polynomial truncation of the Poincaré–Dulac normal form (when V^{c} = \left\{ y = 0\right\}. The manifold V^{c} is unique.

    Let us construct the first integral F, beginning with its definition F_{0} = F|_{\Sigma } on \Sigma. This function should have the property

    \begin{equation*} \mathcal{M}^{\ast }F_{0} = F_{0}; \end{equation*}

    then, we will use the analytic continuation. Let

    \begin{equation*} \partial \Sigma = \left\{ z_{0}\right\} \times \left\{ \left\vert v\right\vert < \varepsilon \right\} \times \left\{ \left\vert y\right\vert = \varepsilon \right\} \end{equation*}

    (point \times disc \times sphere). We put

    \begin{equation*} F_{0}|_{\partial \Sigma } = z_{0}v. \end{equation*}

    We define the following family of maps h^{\alpha }:\Sigma \longmapsto \Sigma, \; \alpha \geq 0:

    \begin{equation*} h^{\alpha }\left( v_{0},y_{0}\right) = \left( \mathrm{e}^{\mathrm{i}\alpha }\varphi \left( \mathrm{e}^{\mathrm{i}\alpha }z_{0};v_{0},y_{0}\right) ,\chi \left( \mathrm{e}^{\mathrm{i}\alpha };v_{0},y_{0}\right) \right) . \end{equation*}

    Since \varphi \left(\mathrm{e}^{\mathrm{i}\alpha }z_{0};v; y\right) \approx \mathrm{e}^{-\mathrm{i}\alpha }, the maps h^{\alpha } do not change much for v. But, they are contractions when acting on y. Moreover,

    \begin{equation*} h^{2\pi } = \mathcal{M}. \end{equation*}

    If \left(v, y\right) = h^{\alpha }\left(v_{0}, y_{0}\right), \; \left(v_{0}, y_{0}\right) \in \partial \Sigma, then we put

    \begin{equation*} F_{0}\left( v,y\right) = z_{0}v_{0}. \end{equation*}

    This defines a correct analytic function on \Sigma \diagdown V^{c}, which is extended to an analytic function on \Sigma. Next, the latter function F_0 is extended to a neighborhood of the singular point, i.e., we get the first integral

    \begin{equation*} F\left( z,v,y\right) = zv+\ldots . \end{equation*}

    with the critical locus along the stable manifold W^{s} = \left\{ z = v = 0\right\} . This follows from the center conditions.

    By the parametric Morse lemma [37], there exist analytic coordinates Z = z+\ldots, \; V = v+\ldots, \; Y = y+\ldots such that

    \begin{equation*} F = ZV. \end{equation*}

    From the center conditions, it also follows that

    (i) \mathcal{M}|_{V^{c}} = Id|_{V^{c}},

    (ii) W^{c} = \bigcup\limits_{z_{0}}V_{z_{0}}^{c} is an analytic center manifold, W^{c} = \left\{ Y = 0\right\} , which supports a family of periodic solutions

    \begin{equation*} Z\left( t\right) = c\mathrm{e}^{\mathrm{i}t},\text{ }V\left( t\right) = d \mathrm{e}^{-\mathrm{i}t},\text{ }Y\left( t\right) \equiv 0. \end{equation*}

    The corresponding family

    \begin{equation*} z\left( t\right) = c\mathrm{e}^{\mathrm{i}t}+\ldots ,\text{ }v\left( t\right) = d\mathrm{e}^{-\mathrm{i}t}+\ldots ,\text{ }y\left( t\right) = \ldots \end{equation*}

    depends analytically on the parameters c, d (defined by the initial conditions). In the real domain, we have the family z\left(t\right) = r \mathrm{e}^{\mathrm{i}t}+\ldots, \; \bar{z}\left(t\right) = r\mathrm{e}^{- \mathrm{i}t}+\ldots, \; y\left(t\right) = \ldots, with the real parameter r; this is the convergent family from [21].

    Remark 5. We see that the existence of an analytic family of periodic solutions, under the assumptions of Theorem 2 (and also of Theorem 3), implies the uniqueness and analyticity of the center manifold W^{c}\simeq \left(\mathbb{C}^{2}, 0\right). This surface is parametrized by complex parameters c, d and \zeta = \left(1+\delta \right) \mathrm{e}^{\mathrm{i} t} (from a neighborhood of the unit circle):

    \begin{equation*} z = c\zeta +\ldots ,\text{ }v = d\zeta ^{-1}+\ldots ,\text{ }y = \ldots . \end{equation*}

    (But, this situation is somewhat special In Example 5\ below, we encounter a 1–parameter family of periodic solutions spreading non-smooth surfaces.)

    Next, one can show that the Poincaré–Dulac normal form

    \begin{equation*} \dot{Z} = \mathrm{i}Z\cdot M\left( ZV\right) ,\dot{V} = -\mathrm{i}V\cdot M\left( ZV\right) ,\text{ }\dot{Y} = \Omega \left( Y,ZV\right) , \end{equation*}

    where M is a real series and \Omega is polynomial in Y (with series in VZ as coefficients), is analytic.

    Finally, note that, in Example 1, the formal center manifold does not support a family of periodic solutions; formally, it has an unstable focus.

    Recall that we have pure imaginary eigenvalues \pm \mathrm{i}\omega _{1}, \ldots, \pm \mathrm{i}\omega _{m}, \; \omega _{j} > 0. Let \dot{z}_{j} = \mathrm{i}\omega _{j}z_{j}+\ldots, \; \dot{v}_{j} = -\mathrm{i}\omega _{j}v_{j}+\ldots.

    Let us describe the Poincaré–Dulac (P–D) normal form for such systems. Assume first that the frequencies \omega _{j} are independent over \mathbb{Q} . Then, the only resonant relations are

    \begin{eqnarray*} \mathrm{i}\omega _{j} & = &k_{1}\mathrm{i}\omega _{1}+k_{1}\left( -\mathrm{i} \omega _{1}\right) +\ldots +\left( 1+k_{j}\right) \left( \mathrm{i}\omega _{j}\right) +k_{j}\left( -\mathrm{i}\omega _{j}\right) \\ &&+\ldots +k_{m}\mathrm{i}\omega _{m}+k_{m}\left( -\mathrm{i}\omega _{m}\right) , \\ -\mathrm{i}\omega _{j} & = &k_{1}\mathrm{i}\omega _{1}+k_{1}\left( -\mathrm{i} \omega _{1}\right) +\ldots +k_{j}\mathrm{i}\omega _{j}+\left( 1+k_{j}\right) \left( -\mathrm{i}\omega _{j}\right) \\ &&+\ldots +k_{m}\mathrm{i}\omega _{m}+k_{m}\left( -\mathrm{i}\omega _{m}\right) . \end{eqnarray*}

    It follows that the P–D normal form is

    \begin{equation} \dot{Z}_{j} = Z_{j}\left( \mathrm{i}\omega _{j}+f_{j}\left( Z_{1}V_{1},\ldots ,Z_{m}V_{m}\right) \right) ,\text{ }\dot{V}_{j} = V_{j}\left( -\mathrm{i} \omega _{j}+g_{j}(Z_{1}V_{1},\ldots ,Z_{m}V_{m})\right) , \end{equation} (4.12)

    j = 1, \ldots, m. If there are relations, like

    \begin{equation} p\omega _{1} = q\omega _{2},{ \ }\gcd \left( p,q\right) = 1, \end{equation} (4.13)

    then the P–D normal form will contain additional monomial terms, like

    \begin{equation*} Z_{j}\left( Z_{1}^{q}V_{2}^{p}\right) ^{r}\frac{\partial }{\partial Z_{j}}, \text{ }V_{j}\left( V_{1}^{q}Z_{2}^{p}\right) ^{r}\frac{\partial }{\partial V_{j}},\text{ }V_{1}^{rq-1}Z_{2}^{rp}\frac{\partial }{\partial Z_{1}} \end{equation*}

    (and some other in the case of (4.13)).

    Anyway, the sublattice (in \mathbb{Z}^{m}) of relations \left(k, \omega \right) = 0 is finitely generated and the P–D normal form has finitely many series (formal functional moduli) depending on finitely many monomials.

    Assume now that the frequency \omega _{1} is such that no other frequency \omega _{j} is a multiple of it:

    \begin{equation} \omega _{j}/\omega _{1}\not\in \mathbb{Z},\text{ }j\geq 2. \end{equation} (4.14)

    Lemma 1. Under the which assumption, the P–D normal form is such that the equations

    \begin{equation*} Z_{2} = V_{2} = \ldots = Z_{m} = V_{m} = 0 \end{equation*}

    define a formal invariant manifold (invariant surface) W_{1} .

    In this case, the P–D normal form restricted to W_{1} is like in Eq. (4.2) with the center conditions a_{k} = 0, \; k = 1, 2, \ldots.

    Proof. The obstacles to the existence of such an invariant surface are implied by the following terms:

    \begin{equation*} Z_{1}^{k}V_{1}^{l}\frac{\partial }{\partial Z_{j}},\text{ }Z_{1}^{k}V_{1}^{l} \frac{\partial }{\partial V_{j}},\text{ }\left( j\geq 2\right) . \end{equation*}

    They correspond to the resonant relations

    \begin{equation*} \mathrm{i}\omega _{j} = \left( k-l\right) \mathrm{i}\omega _{1},\text{ }- \mathrm{i}\omega _{j} = \left( k-l\right) \mathrm{i}\omega _{1}. \end{equation*}

    By condition (4.14), there are no such relations.

    Below we give two proofs of Theorem 3. Let us being with Lyapunov's argument. For this, the following standard is needed.

    Lemma 2. Consider the linear equation

    \begin{equation*} \frac{\mathrm{d}x}{\mathrm{d}\theta } = \lambda x+a\left( \theta \right) , x\in \mathbb{C}, \end{equation*}

    where a\left(\theta \right) is a periodic function with period 2\pi. If \lambda is not an integer multiple of \mathrm{i} = \sqrt{-1}, then this equation has a unique periodic solution with the period 2\pi defined by

    \begin{equation} x = \phi \left( \theta \right) = \left( \mathrm{e}^{-2\pi \lambda }-1\right) ^{-1}\int_{\theta -2\pi }^{\theta }\mathrm{e}^{\lambda \left( \theta -\alpha \right) }a\left( \alpha \right) \mathrm{d}\alpha . \end{equation} (4.15)

    Proof. The solutions of the initial value problem with the initial conditions x\left(0\right) = x_{0} are affine in x_{0}. After the period of 2\pi , these solutions define the monodromy map x_{0}\longmapsto \mathcal{M}\left(x_{0}\right) , which is also affine:

    \begin{equation*} \mathcal{M}\left( x_{0}\right) = \Lambda x_{0}+A, \end{equation*}

    where \Lambda = \mathrm{e}^{2\pi \lambda }. By the assumption, \Lambda \not = 1. Hence, the map \mathcal{M} has a unique fixed point corresponding to a unique 2\pi - periodic solution.

    If \mathrm{Re}\left(\lambda \right) < 0 , then this solution is as follows:§

    §This formula was used by Lyapunov in his proof of Theorem 2.

    \begin{equation*} x = \phi \left( \theta \right) = \int_{-\infty }^{\theta }\mathrm{e}^{\lambda \left( \theta -\alpha \right) }f\left( \alpha \right) \mathrm{d}\alpha . \end{equation*}

    After partition of the integration domain (-\infty, \theta ] into segments of length 2\pi , we get Eq. (4.15), which is valid for all \lambda \in \mathbb{C}\diagdown 2\pi \mathbb{Z}.

    Lyapunov's proof of Theorem 3. Let z_{1} = r\mathrm{e}^{\mathrm{i} \theta } and v_{1} = \bar{z}_{1} = r\mathrm{e}^{-\mathrm{i}\theta } , where r, \theta are polar coordinates. The phase curves of our system are defined by the equations

    \begin{equation} \begin{array}{lll} r^{\prime } & = & f\left( r,\theta ,\tilde{z},\tilde{v}\right) , \\ z_{j}^{\prime } & = & \mathrm{i}\left( \omega _{j}/\omega _{1}\right) z_{j}+g_{j}\left( r,\theta ,\tilde{z},\tilde{v}\right) , \\ v_{j}^{\prime } & = & -\mathrm{i}\left( \omega _{j}/\omega _{1}\right) v_{j}+h_{j}\left( r,\theta ,\tilde{z},\tilde{v}\right) , \end{array} \end{equation} (4.16)

    j = 2, 3, \ldots, where \tilde{z} = \left(z_{2}, \ldots, z_{m}\right), \; \tilde{v} = \left(v_{2}, \ldots, v_{m}\right) and ^{\prime } = \mathrm{d}/ \mathrm{d}\theta . We can assume that f begins with quadratic terms in r, \tilde{z}, \tilde{v}. For simplicity, assume that the linearization matrix of the complex system is diagonal; the triangular case requires only slight modification of the proof. Thus, also g_{j} and h_{j} begin with quadratic terms.

    We look for 2\pi - periodic solutions to system (4.16) as power series with periodic coefficients:

    \begin{equation} \begin{array}{lll} r & = & \phi ^{\left( 1\right) }\left( \theta \right) c+\phi ^{\left( 2\right) }\left( \theta \right) c^{2}+\ldots \\ z_{j} & = & \psi _{j}^{\left( 2\right) }c^{2}+\psi _{j}^{\left( 3\right) }c^{3}+\ldots , \\ v_{j} & = & \chi _{j}^{\left( 2\right) }c^{2}+\chi _{j}^{\left( 3\right) }c^{3}+\ldots . \end{array} \end{equation} (4.17)

    We set the following initial conditions:

    \begin{equation} \phi ^{\left( 1\right) }\left( 0\right) = 1,\text{ }\phi ^{\left( 2\right) }\left( 0\right) = 0,\phi ^{\left( 3\right) }\left( 0\right) = 0,\ldots ; \end{equation} (4.18)

    thus, we have that r\left(0\right) = c and \phi ^{\left(1\right) }\left(\theta \right) \equiv 1 (which is 2\pi - periodic). Substituting the Ansatz (4.17) into system (4.18), we obtain a recurrent system of equations for \phi ^{\left(k\right) }, \; \psi ^{\left(k\right) } and \chi ^{\left(k\right) } .

    We have

    \begin{equation*} \mathrm{d}\phi ^{\left( k\right) }/\mathrm{d}\theta = \Phi ^{\left( k\right) }\left( \theta \right) ,\text{ }k\geq 2, \end{equation*}

    where the functions \Phi ^{\left(k\right) }\left(\theta \right) are defined inductively and should be 2\pi - periodic. Thus,

    \begin{equation} \phi ^{\left( k\right) }\left( \theta \right) = \int_{0}^{\theta }\Phi ^{\left( k\right) }\left( \alpha \right) \mathrm{d}\alpha , \end{equation} (4.19)

    where \int_{0}^{2\pi }\Phi ^{\left(k\right) }\left(\alpha \right) \mathrm{ d}\alpha should equal zero. The latter vanishing conditions are guaranteed by the assumptions of the theorem, i.e., the center conditions.

    The equations for \psi _{j}^{\left(k\right) } take the following form:

    \begin{equation*} \mathrm{d}\psi _{j}^{\left( k\right) }/\mathrm{d}\theta = \lambda _{j}\psi _{j}^{\left( k\right) }+\Psi _{j}^{\left( k\right) }\left( \theta \right) , \end{equation*}

    where \lambda _{j} = \mathrm{i}\omega _{j}/\omega _{1} and \Psi _{j}^{\left(k\right) }\left(\theta \right) are defined inductively and are 2\pi - periodic; analogous equations hold for \chi _{j}^{\left(k\right) }, but with \lambda _{j} replaced with -\lambda _{j}. The latter equations d \psi_{j}^{(k)}/d\theta are like the equation from Lemma 2 with \lambda = \lambda _{j}\not\in \mathrm{i}\mathbb{Z} . So, they have unique 2\pi - periodic solutions defined in Eq. (4.15); no center conditions are needed here.

    The series in Eq. (4.17) is convergent due to the analyticity of the right-hand sides of the equations of Eq. (4.16), the compactness of the domain \left[ 0, 2\pi \right] of definition of this system and the simplicity of the integral formulas (4.15) and (4.19). In [21], suitable estimates are given.

    Second proof of Theorem 3. Recall that we have the formal invariant surface W_{1}, defined in Lemma 2. We will show that it is analytic.

    For this, we use the Poincaré return map \mathcal{P} defined by solutions after the period of 2\pi of system (4.16). We have

    \begin{equation} \mathcal{P}\left( r,z_{2},\ldots ,z_{m}\right) = \left( \rho ,\zeta _{1},\ldots ,\zeta _{m}\right) , \end{equation} (4.20)

    where \rho = \rho \left(r, z_{2}, \ldots, z_{m}\right) = r+\ldots, \; \zeta _{j} = \zeta _{j}(r, z_{2}, \ldots, z_{m}) = \mu _{j}z_{j}+\ldots, \; \mu _{j} = \mathrm{e}^{2\pi \mathrm{i}\omega _{j}/\omega _{1}}\not = 1. The periodic orbits of the period \approx 2\pi /\omega _{1} correspond to the fixed points of the Poincaré map. The set of fixed points of \mathcal{P} lies in the curve

    \begin{equation} \zeta _{2}-z_{2} = \ldots = \zeta _{m}-z_{m} = 0, \end{equation} (4.21)

    where \zeta _{j}-z_{j} = \mu _{j}z_{j}+\ldots with \mu _{j}\not = 0. By the implicit function theorem, Eq. (4.21) defines a real analytic curve.

    By the center conditions, this curve is \mathcal{P} –invariant and defines an analytic smooth surface supporting a family of periodic solutions.

    The following example demonstrates the importance of the condition (4.14).

    Example 3. The system

    \begin{equation*} \dot{z}_{1} = \mathrm{i}z_{1},\text{ }\dot{v}_{1} = -\mathrm{i}v_{1},\text{ } \dot{z}_{2} = 2\mathrm{i}z_{2}+z_{1}^{2},\text{ }\dot{v}_{2} = -2\mathrm{i} v_{2}+v_{1}^{2} \end{equation*}

    has a general solution with the form

    \begin{equation*} z_{1} = c_{1}\mathrm{e}^{\mathrm{i}t},\text{ }v_{1} = d_{1}\mathrm{e}^{-\mathrm{i }t},\text{ }z_{2} = \left( c_{2}+c_{1}^{2}t\right) \mathrm{e}^{2\mathrm{i}t}, \text{ }v_{2} = \left( d_{2}+d_{1}^{2}t\right) \mathrm{e}^{-2\mathrm{i}t}. \end{equation*}

    Only the solutions with c_{1} = d_{1} = 0 are periodic, with a period of \pi (not 2\pi).

    Remark 6. (a) We see that the existence of a 1–parameter family of periodic solutions can be deduced from the P–D normal form. We need the following:

    (i) formal invariant surface, and

    (ii) vanishing of the series of Poincaré–Lyapunov coefficients after restriction of the P–D normal form to this surface.

    The following questions arise naturally in this context.

    1. Are there other 1–parameter families of periodic solutions?

    2. Are there many–parameter families of periodic solutions?

    Concerning the first question, there should exist other formal invariant surfaces with vanishing corresponding Poincaré–Lyapunov quantities.

    The answer to the second question can be positive in some resonant cases. For example, this holds for the following system:

    \begin{equation} \dot{z}_{j} = \mathrm{i}p_{j}z_{j}\cdot M,\text{ }\dot{v}_{j} = -\mathrm{i} p_{j}v_{j}\cdot M, \end{equation} (4.22)

    with the integer p_{j} and the integrating factor M = \omega +\ldots . Here, the typical solution is periodic with the period \frac{2\pi }{p\omega }\left(1+\ldots \right), where p = \gcd \left(p_{1}, \ldots, p_{m}\right).

    But, when

    \begin{equation} \dot{z}_{j} = \mathrm{i}p_{j}z_{j}\cdot M_{j},\text{ }\dot{v}_{j} = -\mathrm{i} p_{v}v_{j}\cdot M_{j}, \end{equation} (4.23)

    where the factors M_{j} = \omega +\ldots are independent, then the motions take place on the invariant 'tori' \left\{ z_{1}v_{1} = C_{1}, \ldots, z_{m}v_{m} = C_{m}\right\}. In the real domain, where v_{j} = \bar{z}_{j}, the motion on the generic torus is quasi-periodic, or periodic with a very long period.

    (b) Note that, in the case considered in this subsection, the system has two analytic invariant manifolds, W^{+} and W^{-} , corresponding to the partition of the eigenvalues' set into two subsets \left\{ \mathrm{i}\omega _{1}, \ldots, \mathrm{i}\omega _{m}\right\} and \left\{ -\mathrm{i}\omega _{1}, \ldots, -\mathrm{i}\omega _{m}\right\}. The system restricted to either of these submanifolds has analytic P–D normal form (the systems are in the Poincaré domain).

    Here, we discuss Theorem 3 (as well as the Weinstein- and Moser-type theorems) in the context of Hamiltonian systems. Again, we assume the imaginary eigenvalues \pm \mathrm{i}\omega _{1}, \ldots, \pm \mathrm{i}\omega _{m}, \; \omega _{j} > 0 , where m = \frac{n}{2} is the number of degrees of freedom. The Hamiltonian functions considered in this section are of the form

    \begin{equation} H = H_{2}+H_{3}+\ldots , \end{equation} (5.1)

    where H_{2} is like in Eq. (2.5) and the functions H_{j} are homogeneous of degree j. Such an H_{2} is obvious in the case that all frequencies \omega _{j} are pairwise different. The opposite case is discussed below.

    Recall that, in the third Lyapunov theorem, there are two assumptions: the existence of a formal invariant surface W_{1} associated with the eigenvalues \pm \mathrm{i}\omega _{1}, and the vanishing of the series of the Poincaré–Lyapunov focus quantities after restriction of the system to W_{1} (see Remark 6 above). But, in the Hamiltonian case, the situation is simpler.

    Proposition 1. If there exists a formal invariant surface W_{1} , then all of the focus quantities for the restriction of the system to W_{1} vanish.

    Proof. Indeed, H restricted to W_{1} is a formal first integral for the restricted system. So, in the real domain, all phase curves of that system are closed.

    People have tried to improve the third Lyapunov theorem in the Hamiltonian case. Recall that Weinstein ([35, Theorem 2.1] and Theorem 4 above) proved that, if H_{2} is positive definite, i.e., all \epsilon _{j} = 1, then there exist at least m = \frac{n}{2} families of periodic solutions.

    Moreover, Moser, in [23, Theorem 4], claimed that, if H_{2} restricted to the maximal linear invariant subspace E supporting periodic solutions of period T = 2\pi /\omega of the linearized system is positive definite, then 'on each energy surface H = \varepsilon ^{2} , the number of periodic orbits is at least \frac{1}{2}\mathrm{dim}E '. Recall that that result is wrong, but that some version of Moser's statement is valid provided that H_{2} is positive definite (Theorem 5).

    Of course, the number m from the Weinstein theorem cannot be improved because the Hamiltonian system

    \begin{equation*} \dot{z}_{j} = \mathrm{i}\omega _{j}z_{j}\cdot M_{j}\left( \left\vert z_{j}\right\vert ^{2}\right) ,\text{ }j = 1,\ldots ,m, \end{equation*}

    with different factors M_{j}, has m families of periodic solutions with independently varying periods T_{j} = T_{j}\left(\left\vert z_{j}\right\vert ^{2}\right) \approx 2\pi /\omega _{j}. Compare with Remark 6 above.

    Recall also Example 3 above, where we observe 1:2 resonance between the frequencies and only one family of periodic solutions; but, there, the system is not Hamiltonian.

    Example 4. In the case of the Hamiltonian

    In [35], we find another Hamiltonian H = \frac{1}{2}\left\vert z_{1}\right\vert ^{2}-\left\vert z_{2}\right\vert +\frac{1}{2}\mathrm{Im} \left(z_{1}^{2}z_{2}\right) (due to Siegel [29]), also with only one family of periodic solutions.

    \begin{equation} H = \frac{1}{2}\left\{ \left\vert z_{1}\right\vert ^{2}-k\left\vert z_{2}\right\vert ^{2}+z_{1}^{k}z_{2}+\bar{z}_{1}^{k}\bar{z}_{2}\right\} , \text{ }k\geq 2, \end{equation} (5.2)

    from [23, Example 2], we deal with the k:1 resonance, but with indefinite quadratic part. The corresponding equations for z_{j} are

    \begin{equation*} \dot{z}_{1} = \mathrm{i}z_{1}+\mathrm{i}k\bar{z}_{1}^{k-1}\bar{z}_{2},\text{ } \dot{z}_{2} = -k\mathrm{i}z_{2}+\mathrm{i}\bar{z}_{1}^{k}. \end{equation*}

    Of course, the plane W_{2} = \left\{ z_{1} = v_{1} = 0\right\} is invariant and supports a family of periodic solutions with the period 2\pi /k.

    But, the Lyapunov-type function L = \frac{1}{2\mathrm{i}}\left(z_{1}^{k}z_{2}- \bar{z}_{1}^{k}\bar{z}_{2}\right) satisfies

    \begin{equation*} \dot{L} = \left\vert z_{1}\right\vert ^{2(k-1)}\left( \left\vert z_{1}\right\vert ^{2}+k^{2}\left\vert z_{2}\right\vert ^{2}\right) , \end{equation*}

    which is positive in the domain \left\{ z_{1}\not = 0\right\}. It prevents the existence of other periodic solutions.

    Surprisingly, this example contradicts Moser's original statement [23, Theorem 4]. Indeed, we have two invariant subspaces of the linear system supporting periodic solutions: E_{2} = W_{2} = \left\{ \left(0, z_{2}\right) \right\} and E_{1} = \left\{ \left(z_{1}, 0\right) \right\}. We have that H_{2}|_{E_{2}} = -\frac{k}{2}\left\vert z_{2}\right\vert ^{2}\leq 0 and H_{2}|_{E_{1}}\geq 0. But, only periodic solutions from E_{2} are 'extended' to periodic solutions of the whole system.

    We complete this example by analysis of some return map associated with the polar coordinates in the z_{1}- plane. Thus, we put

    \begin{equation} z_{1} = r\mathrm{e}^{\mathrm{i}\theta },\text{ }z_{2} = w_{2}\mathrm{e}^{- \mathrm{i}k\theta }. \end{equation} (5.3)

    We have

    \begin{eqnarray*} \dot{r} & = &\frac{1}{r}\mathrm{Re}\left( \dot{z}_{1}\bar{z}_{1}\right) = \frac{ 1}{r}\mathrm{Re}\left( \mathrm{i}r^{2}+\mathrm{i}kr^{k}\bar{w}_{2}\right) = kr^{k-1}\mathrm{Im}w_{2}, \\ \dot{\theta} & = &\mathrm{Im}\left( \dot{z}_{1}/z_{1}\right) = 1+kr^{k-2} \mathrm{Re}w_{2}. \end{eqnarray*}

    Next, \dot{w}_{2}-\mathrm{i}kw_{2}\dot{\theta} = -\mathrm{i}kw_{2}+\mathrm{i} r^{k} gives

    \begin{equation*} \dot{w}_{2} = \mathrm{i}r^{k-2}(r^{2}+k^{2}w_{2}\mathrm{Re}w_{2}). \end{equation*}

    We get the system

    \begin{equation} \frac{\mathrm{d}r}{\mathrm{d}\theta } = k\frac{r^{k-1}\mathrm{Im}w_{2}}{ 1+kr^{k-2}\mathrm{Re}w_{2}},\text{ }\frac{\mathrm{d}w_{2}}{\mathrm{d}\theta } = \mathrm{i}r^{k-2}\frac{r^{2}+k^{2}w_{2}\mathrm{Re}w_{2}}{1+kr^{k-2}\mathrm{ Re}w_{2}}, \end{equation} (5.4)

    which we call the return system. The twisted Poincaré map \left(r, w_{2}\right) \longmapsto \mathcal{P}\left(r, w_{2}\right) is defined via the evaluation of solutions at the 'time' \theta = 2\pi with the initial value \left(r, w_{2}\right) at the 'time' \theta = 0. The periodic solutions with the period \approx 2\pi and outside of W_{2} = \left\{ r = 0\right\} correspond to the singular points of the latter system (5.4) with r\not = 0. Thus, we have

    \begin{equation*} \mathrm{Im}w_{2} = 0, \end{equation*}

    i.e., w_{2} = \mathrm{Re}w_{2}, and

    \begin{equation} r^{2}+k^{2}w_{2}^{2} = 0. \end{equation} (5.5)

    The latter equation does not have nontrivial solutions.

    The following example demonstrates that, in the case of a positively definite Hamiltonian, the situation is quite different.

    Example 5. (a) Consider the positive definite Hamiltonian

    \begin{equation} H = \frac{1}{2}\left\{ k\left\vert z_{1}\right\vert ^{2}+\left\vert z_{2}\right\vert ^{2}+z_{1}\bar{z}_{2}^{k}+\bar{z}_{1}z_{2}^{k}\right\} , \end{equation} (5.6)

    generating the system

    \begin{equation} \dot{z}_{1} = \mathrm{i}kz_{1}+\mathrm{i}z_{2}^{k},\text{ }\dot{z}_{2} = \mathrm{ i}z_{2}+\mathrm{i}kz_{1}\bar{z}_{2}^{k-1}. \end{equation} (5.7)

    Of course, the plane W_{1} = \left\{ z_{2} = 0\right\} is invariant and has a a family of periodic solutions with the period 2\pi /k.

    (b) To look for periodic solutions with the period \approx 2\pi , use the polar-type coordinates

    \begin{equation} z_{2} = r\mathrm{e}^{\mathrm{i}\theta },\text{ }z_{1} = w_{1}\mathrm{e}^{\mathrm{ i}k\theta }; \end{equation} (5.8)

    compare with Eq. (5.3). We get the return system

    \begin{equation} \frac{\mathrm{d}r}{\mathrm{d}\theta } = -k\frac{r^{k-1}\mathrm{Im}w_{1}}{ 1+kr^{k-2}\mathrm{Re}w_{1}},\text{ }\frac{\mathrm{d}w_{1}}{\mathrm{d}\theta } = \mathrm{i}r^{k-2}\frac{r^{2}-k^{2}w_{1}\mathrm{Re}w_{1}}{1+kr^{k-2}\mathrm{ Re}w_{1}}. \end{equation} (5.9)

    Again, the closed phase curves of the period \approx 2\pi outside of W_{1} = \left\{ r = 0\right\} correspond to the singular points of the vector field given by Eq. (5.9). We find that

    \begin{equation} w_{1} = \mathrm{Re}w_{1} = \pm r/k. \end{equation} (5.10)

    Therefore, we have three families of periodic solutions. The first family \{z_{1} = c\mathrm{e}^{\mathrm{i}kt}, \; z_{2} = 0\} spans the invariant plane W_{1}, but none of the two other families of closed phase curves \left\{ z_{1} = \pm \left(r/k\right) \mathrm{e}^{\mathrm{i}k\theta }, z_{2} = r\mathrm{e} ^{\mathrm{i}t\theta }\right\} span a smooth surface. These real surfaces (in \mathbb{R}^{4}) take the form

    \begin{equation*} kz_{1}\left\vert z_{2}\right\vert ^{k-1} = \pm z_{2}^{k} \end{equation*}

    and are singular; in \mathbb{C}^{4} , we get one complex singular surface \{k^{2}z_{1}^{2}v_{2}^{k-1} = z_{2}^{k+1}, \; k^{2}v_{1}^{2}z_{2}^{k-1} = v_{2}^{k+1}\}.

    The number of families of periodic solutions is greater than in the Weinstein theorem.

    (c) To explain the this difference we use the fact that the Hamiltonian system in the Birkhoff normal form is invariant with respect to the \mathbb{ S}^{1}- action. In the present case, the action is as follows:

    \begin{equation} \left( z_{1},z_{2};\phi \right) \longmapsto \left( \mathrm{e}^{\mathrm{i} k\phi }z_{1},\mathrm{e}^{\mathrm{i}\phi }z_{2}\right) ; \end{equation} (5.11)

    it is symplectic and generated by a corresponding Hamiltonian function (the momentum map), namely, F = H_{2} = \frac{1}{2}\left(k\left\vert z_{1}\right\vert ^{2}+\left\vert z_{2}\right\vert ^{2}\right) (which is a first integral for system (5.7)). The symplectic reduction procedure (see [4]) says that, for each level surface M_{f} = \left\{ F = f\right\} (which is \mathbb{S}^{1}- invariant), the quotient space N_{f} = M_{f}/\mathbb{S}^{1} with respect to the \mathbb{S}^{1}- action acquires a symplectic structure and we get a Hamiltonian vector field Y_{f} in this quotient space. The singular points of the latter vector field correspond to periodic solutions to system (5.7).

    In this case, the hypersurface is a 3–sphere M_{f}\simeq \mathbb{S}^{3}, \; f > 0, and the quotient space N_{f} turns out to be the weighted projective space, N_{f}\simeq \mathbb{P}_{k, 1} (see Section 10.1). The weighted projective space \mathbb{P}_{k, 1} is homeomorphic with the standard complex projective line \mathbb{P}_{k, 1}\simeq \mathbb{P}^{1}. The natural charts in N_{f} are \zeta = z_{2}^{k}/z_{1} and \eta = 1/\zeta = z_{1}/z_{2}^{k}; thus, z_{1} = \eta z_{2}^{k} and z_{2} = \left(\zeta z_{1}\right) ^{1/k} .

    In these charts, we get the systems

    \begin{equation} \dot{\zeta} = \mathrm{i}k^{2}\left\vert \zeta \right\vert ^{2(k-1)/k}\left\vert z_{1}\right\vert ^{2\left( k-1\right) /k}-\mathrm{i} \zeta ^{2},{ \ }\dot{\eta} = \mathrm{i}\left( 1-k^{2}\left\vert z_{2}\right\vert ^{2\left( k-1\right) }\eta ^{2}\right) . \end{equation} (5.12)

    Note that, in the \zeta - chart on N_{f} , we have that k\left\vert z_{1}\right\vert ^{2}+\left\vert \zeta \right\vert ^{2/k}\left\vert z_{1}\right\vert ^{2/k} = 2f; so, for \zeta \rightarrow 0 , we find that \left\vert z_{2}\right\vert is separated from zero and the first term on the right-hand side of the equation for \dot{\zeta} is dominating. It follows that the latter system(s) (5.12) has three singular points: \zeta = 0 (with index 0) and \eta = \pm 1/k\left\vert z_{2}\right\vert ^{k-1} (both with index 1) . As the Euler characteristic \chi \left(\mathbb{P} ^{1}\right) = 2 , this agrees with the Poincaré–Hopf formula.

    Note also that the equilibrium points of Y_{f} correspond to the critical points of H as a function on N_{f}. This amounts to critical circles of the function G = H-F = \frac{1}{2}\left(z_{1}\bar{z}_{2}^{k}+\bar{z} _{1}z_{2}^{k}\right) , restricted to M_{f} . With \lambda and the Lagrange multiplier, we get the equations

    \begin{equation} \bar{z}_{2}^{k} = \lambda k\bar{z}_{1},{ \ }k\bar{z}_{1}z_{2}^{k-1} = \lambda \bar{z}_{2}. \end{equation} (5.13)

    One solution is z_{2} = 0 , \lambda = 0 and k\left\vert z_{1}\right\vert ^{2} = 2f. For the other solutions, we have \lambda ^{2} = \left\vert z_{2}\right\vert ^{2\left(k-1\right) }; thus, \lambda = \pm \left\vert z_{2}\right\vert ^{k-1}, \; z_{1} = \pm \frac{1}{k}\left(z_{2}/\left\vert z_{2}\right\vert \right) ^{k-1}z_{1} and \left\vert z_{2}\right\vert ^{2} = 2f/\left(1+1/k^{2}\right).

    (d) Finally, the Hamiltonian system (5.7) is completely integrable, with two independent first integrals (in involution). By the Liouville–Arnold theorem [3], the common level surfaces \left\{ F = f, \text{ } G = g\right\}, which are compact, are tori (if smooth). There are also the action–angle variables, which are our objectives.

    Let z_{1, 2} = r_{1, 2}\mathrm{e}^{\mathrm{i}\alpha _{1, 2}} and \beta = k\alpha _{2}-\alpha _{1}. We have the differential equations

    \begin{eqnarray*} \dot{r}_{1} & = &-r_{2}^{k}\sin \beta ,{ \ \ \ \ \ \ \ \ }\dot{r} _{2} = kr_{1}r_{2}^{k-1}\sin \beta , \\ {\ }\dot{\alpha}_{1} & = &k+r_{1}^{-1}r_{2}^{k}\cos \beta ,\text{ }\dot{ \alpha}_{2} = 1+kr_{1}r_{2}^{k-1}\cos \beta ,\text{ }. \end{eqnarray*}

    The levels of the first integrals take the form

    \begin{equation*} kr_{1}^{2}+r_{2}^{2} = 2f,{ \ }r_{1}r_{2}^{k}\cos \beta = g. \end{equation*}

    This leads to r_{1} = g/\left(r_{2}^{k}\cos \beta \right), \; \cos ^{2}\beta = kg^{2}/r_{2}^{2k}\left(2f-r_{2}^{2}\right),

    \begin{equation*} \sin ^{2}\beta = \frac{2f\rho ^{k}-\rho ^{k+1}-kg^{2}}{\rho ^{k}\left( 2f-\rho \right) }, \end{equation*}

    where \rho = r_{2}^{2}. Finally, we arrive at the system

    \begin{eqnarray} \dot{\rho} & = &\pm 2k^{-1/2}\sqrt{2f\rho ^{k}-\rho ^{k+1}-kg^{2}}, \end{eqnarray} (5.14)
    \begin{eqnarray} \dot{\alpha}_{2} & = &1+kg/\rho . \end{eqnarray} (5.15)

    We have an oval \Gamma = \left\{ k\sigma ^{2} = 4\left(2f\rho ^{k}-\rho ^{k+1}-kg^{2}\right) \right\} in the \left(\rho, \sigma \right) = \left(\rho, \dot{\rho}\right) - plane defined by Eq. (5.14). This is one of the circles generating the invariant torus. The corresponding Liouville–Arnold angle is

    \begin{equation} \phi _{1}\left( \rho ,\sigma \right) = \frac{2\pi }{T_{1}}\int_{\gamma }\frac{ \mathrm{d}\rho }{\sigma }, \end{equation} (5.16)

    where \gamma = \gamma \left(\rho, \sigma \right) is a path in \Gamma from \left(\rho _{0}, \sigma _{0}\right) = \left(\rho _{0}, 0\right) to \left(\rho _{0}, \sigma \right) ( \rho _{0} is the left root of the equation 2f\rho ^{k}-\rho ^{k+1} = kg^{2}) and T_{1} = \oint\limits_{\Gamma } \frac{\mathrm{d}\rho }{\sigma } is the corresponding period. It is the period of the solution \rho = R\left(t\right) to Eq. (5.14).

    Next, the second generating circle is parametrized by the angle \alpha _{1}. The corresponding solution to Eq. (5.15) takes the form \alpha _{1} = A\left(t\right) = \alpha _{0}+t+kg\int_{0}^{t}\frac{\mathrm{d}s}{ R\left(s\right) } = \alpha _{1}^{\left(0\right) }+t+kg\int_{\gamma }\frac{ \mathrm{d}\rho }{\rho \sigma }. We have the second period T_{2} = A(T_{1})-A(0) = T_{1}+kg\oint\limits_{\Gamma }\frac{\mathrm{d}\rho }{ \rho \sigma }. The second Liouville–Arnold angle equals

    \begin{equation} \phi _{2} = \frac{T_{1}}{T_{2}}\phi _{1}+\frac{2\pi }{T_{2}}\left( \alpha _{1}-\int_{\gamma }\left( 1-\frac{kg}{\rho }\right) \frac{\mathrm{d}\rho }{ \sigma }\right) . \end{equation} (5.17)

    One can find corresponding action variables I_{j} = I_{j}\left(F, G\right) from the condition \mathrm{d}I_{1}\wedge \mathrm{d}\phi _{1}+\mathrm{d} I_{2}\wedge \mathrm{d}\phi _{2} = \mathrm{d}p_{1}\wedge \mathrm{d}q_{1}+ \mathrm{d}p_{2}\wedge \mathrm{d}q_{2}. For this, one has to solve some first order PDE; we omit the details.

    Consider now the situation with 1:1 resonance and, more generally, with 1:1:\cdots :1 resonances. Here, non-diagonal cells for the linear systems associated with homogeneous quadratic Hamiltonians H_{2} are expected. Canonical forms of such Hamiltonians are given in [3, Appendix 6].

    Probably, the following easy statement was a motivation of Moser's work [23].

    Proposition 2. In the case of 1:1:\ldots :1 resonance, i.e., with \omega _{1} = \ldots = \omega _{m}, and of definite H_{2} (positive definite or negative definite), the corresponding linearization matrix is diagonalizable.

    Proof. If there existed a pair of Jordan cells, then the general solution of the corresponding linear system would be unbounded. But, this would contradict the compactness of the level hypersurfaces \left\{ H_{2} = h\right\}.

    It is natural to look for families of periodic solutions in the cases with a positive definite H_{2}, i.e., with a diagonalizable linear part. How many such families should exist?

    Consider first the positive definite H_{2} and a homogeneous perturbation in the Birkhoff normal form. Thus, we have

    \begin{equation} H = \frac{1}{2}\left\vert z\right\vert ^{2}+H_{2d} = \frac{1}{2}z\cdot v+H_{2d}, \end{equation} (5.18)

    where z = (z_{1}, \ldots, z_{m})\in \mathbb{C}^{m} and H_{2d} is a homogeneous polynomial depending only on the monomials z_{j}\bar{z}_{k} , j, k = 1, \ldots, m. Here, we have some freedom. Namely, we can apply linear changes of the variables z_{j} which leave H_{2} invariant. The corresponding group is U(m) ; this group preserves the Hermitian product \left(z, \zeta \right) \longmapsto z\cdot \bar{\zeta}, whose real part is the scalar product and whose imaginary part defines the symplectic form. Using these changes, one can reduce the our situation to the case with m invariant complex planes.

    (Note that, in the case with the indefinite quadratic part, H_{2} = \frac{1}{2} \left\vert x\right\vert ^{2}-\frac{1}{2}\left\vert y\right\vert ^{2} , where x = \left(x_{1}, \ldots, x_{p}\right) \in \mathbb{C}^{p} and y = \left(y_{1}, \ldots, y_{q}\right) \in \mathbb{C}^{q} , the corresponding group is U\left(p, q\right) and the Birkhoff resonant terms depend on x_{j}\bar{x} _{k}, y_{j}\bar{y}_{k}, \; x_{j}y_{k} and \bar{x}_{j}\bar{y}_{k}. Also, here, m = p+q invariant planes can be found for H_{2}+H_{2d}.)

    Example 6. A general quartic Hamiltonian with the form of Eq. (5.16) for m = 2 with two invariant complex planes is of the form

    \begin{equation} H = \frac{1}{2}\left\vert z\right\vert ^{2}+\frac{a}{4}\left\vert z_{1}\right\vert ^{4}+\frac{b}{2}\left\vert z_{1}\right\vert ^{2}\left\vert z_{2}\right\vert ^{2}+\frac{c}{4}\left\vert z_{2}\right\vert ^{4}+d\cdot \mathrm{Re}z_{1}^{2}\bar{z}_{2}^{2}, \end{equation} (5.19)

    with real constants a, \ldots, d. The corresponding system becomes

    \begin{equation*} \dot{z}_{1} = \mathrm{i}z_{1}\left( 1+a\left\vert z_{1}\right\vert ^{2}+b\left\vert z_{2}\right\vert ^{2}\right) +2\mathrm{i}d\bar{z} _{1}z_{2}^{2},\text{ }\dot{z}_{2} = \mathrm{i}\left( 1+b\left\vert z_{1}\right\vert ^{2}+c\left\vert z_{2}\right\vert ^{2}\right) +2\mathrm{i} dz_{1}^{2}\bar{z}_{2}. \end{equation*}

    We study the twisted Poincaré map using the substitutions z_{1} = r\mathrm{ e}^{\mathrm{i}\theta } and z_{2} = w\mathrm{e}^{\mathrm{i}\theta }. It leads to the equations

    \begin{equation*} \dot{r} = -2dr\mathrm{Im}w^{2},\text{ }\dot{w} = \mathrm{i}\left\{ \left( b-a\right) r^{2}w+\left( c-b\right) \left\vert w\right\vert ^{2}w+2d\left( rw^{2}-w\mathrm{Re}w^{2}\right) \right\} \end{equation*}

    (we do not write equations for \mathrm{d}r/ \mathrm{d} \theta and \mathrm{d}w/ \mathrm{d} \theta).

    The periodic solutions with period the \approx 2\pi outside of r = 0 are given by \mathrm{Im}w^{2} = 0 and \dot{w} = 0. Thus, either (1) w = \mathrm{Re}w, \; \left\vert w\right\vert ^{2} = w^{2}; or (2) w = \mathrm{i\cdot Im}w, \; \left\vert w\right\vert ^{2} = -w^{2}. Let \lambda = w/r.

    In case (1), we get the equation

    \begin{equation*} r^{2}w\left[ b-a+2d\lambda +\left( c-b-2d\right) \lambda ^{2}\right] = 0, \end{equation*}

    and, in case (2), we get

    \begin{equation*} r^{2}w\left[ b-a+2d\lambda +\left( b-c-2d\right) \lambda ^{2}\right] = 0. \end{equation*}

    It follows that, for generic values of the parameters, besides the solutions r = 0 and w = 0 (corresponding to periodic solutions of the Hamiltonian system in the invariant complex planes), we have four additional solutions. Thus, the total number of periodic solutions in the generic case is six.

    On the other hand, by analogy with the previous example, the symplectic reduction, with F = H_{2} and G = H_{2d}, leads to Hamiltonian vector field Y_{f} on N_{f}\simeq \mathbb{P}^{1} with \chi \left(N_{f}\right) = 2. One can check that the additional two pairs of singular points of Y_{f} have opposite indices.

    The above agrees with the following result of van Straten [34].

    Theorem 14. We have

    \begin{equation} \sum\limits_{m\geq 1}M\left( m-1,d\right) T^{m-1} = \left( 1-T\right) ^{-3/2}\left[ 1-\left( 2d-1\right) ^{2}T\right] ^{-1/2}, \end{equation} (5.20)

    where, on the left-hand side stands the generating function for the numbers M(m-1, d) of periodic modes for Hamiltonian systems with a generic Hamiltonian function in the Birkhoff normal form given by Eq. (5.18).

    For d = 2 , the right-hand side of Eq. (5.20) becomes 1+6T+\ldots, which agrees with Example 6; in fact, there are no such examples in [34].

    Finally, we have the following Hamiltonian systems without periodic solutions for the expected period.

    Example 7. Assume the following Poisson brackets for the complex variables z_{j} :

    \begin{equation*} \left\{ z_{i},z_{j}\right\} = \left\{ \bar{z}_{i},\bar{z}_{j}\right\} = 0, { \ }\left\{ z_{1},\bar{z}_{2}\right\} = \left\{ \bar{z} _{1},z_{2}\right\} = 2. \end{equation*}

    The Hamiltonian function equals

    \begin{equation*} H = \frac{1}{2}\left\{ \left\vert z_{1}\right\vert ^{2}+\mathrm{i}\left( z_{1} \bar{z}_{2}-\bar{z}_{1}z_{2}\right) +a\left\vert z_{2}\right\vert ^{4}\right\} \end{equation*}

    and generates the system

    \begin{equation*} \dot{z}_{1} = \mathrm{i}z_{1}+4az_{2}\left\vert z_{2}\right\vert ^{2},{ \ }\dot{z}_{2} = \mathrm{i}z_{2}-z_{1}. \end{equation*}

    Note that, for a = 0 , we get a nilpotent linear system with invariant plane \left\{ z_{1} = 0\right\} that supports a family of periodic solutions; this Hamiltonian can be found also in [3,7]. But, we assume that a < 0 ; hence, the planes \left\{ z_{1} = 0\right\} and \left\{ z_{2} = 0\right\} are not invariant.

    One should look for periodic solutions for the period \approx 2\pi. Hence, we make the natural change (look at the previous examples): z_{1} = r\mathrm{e}^{ \mathrm{i}\theta }, \; z_{2} = w\mathrm{e}^{\mathrm{i}\theta } with r\geq 0. We get the equations

    \begin{equation*} \frac{\mathrm{d}r}{\mathrm{d}\theta } = \frac{4ar\left\vert w\right\vert ^{2} \mathrm{Re}w}{r+4a\left\vert w\right\vert ^{2}\mathrm{Im}w},{ \ }\frac{ \mathrm{d}w}{\mathrm{d}\theta } = -\frac{r^{2}+4\mathrm{i}a\left\vert w\right\vert ^{2}\mathrm{Im}w}{r+4a\left\vert w\right\vert ^{2}\mathrm{Im}w}. \end{equation*}

    The equations for the equilibrium points are given by \mathrm{Re}w = 0 (i.e., w = \mathrm{i}\zeta) and r^{2}+4 i a\left\vert w\right\vert ^{2} \mathrm{Im}w = r^{2}-4a\zeta ^{4} = 0.

    We refer also to previous works [12,33] devoted to bifurcations of Hamiltonian systems in the 1:1 resonant case.

    In the last example from [22, Example 9.2], the linear part is diagonal.

    Example 8. Let

    \begin{equation*} H = \frac{1}{2}\left\{ \left\vert z_{1}\right\vert ^{2}-\left\vert z_{2}\right\vert ^{2}+2\left\vert z\right\vert ^{2}\mathrm{Re}\left( z_{1}z_{2}\right) \right\} . \end{equation*}

    It generates the system

    \begin{equation*} \dot{z}_{1} = \mathrm{i}z_{1}\left( 1+2\mathrm{Re}\left( z_{1}z_{2}\right) \right) +\mathrm{i}\left\vert z\right\vert ^{2}\bar{z}_{2},{ \ }\dot{z} _{2} = -\mathrm{i}z_{2}\left( 1-2\mathrm{Re}\left( z_{1}z_{2}\right) \right) + \mathrm{i}\left\vert z\right\vert ^{2}\bar{z}_{1}. \end{equation*}

    One finds that

    \begin{equation*} \mathrm{dIm}\left( z_{1}z_{2}\right) /\mathrm{d}t = 2\left[ \mathrm{Re}\left( z_{1}z_{2}\right) \right] ^{2}+\left\vert z\right\vert ^{4}, \end{equation*}

    which excludes the existence of nontrivial periodic solutions.

    Again, assume an analytic system (1.4) with the eigenvalues \pm \mathrm{i} \omega _{j}, \; \omega _{j} > 0. Recall the equivalence relation (1.9) on the set \left\{ \omega _{1}, \ldots, \omega _{m}\right\} of frequencies. For each equivalence class C_{\nu } , we have a linear subspace \mathcal{E}_{\nu } that is invariant for the linear part \dot{x} = Ax of system (1.4). In fact, we can say more.

    Proposition 3. For each class C_{\nu } , system (1.4) has an invariant analytic submanifold \mathcal{V}_{\nu } tangent to \mathcal{E}_{\nu } at the origin.

    Proof. First, it is rather obvious that there exists such an invariant manifold at the formal level. To prove its analyticity, we use the Poincaré return map (4.20), introduced in the second proof of Theorem 3.

    Let z_{1}, \bar{z}_{1}, \ldots, z_{k}, \bar{z}_{k} be the linear eigenfunctions associated with one equivalence class, say, C_{1}, and let z_{k+1}, \bar{z}_{k+1}, \ldots be the remaining eigenfunctions.

    We put z_{1} = r\mathrm{e}^{\mathrm{i}\theta } and get Eq. (4.16) for the phase curves, as graphs of functions of \theta . We have that \omega _{j} = \left(p_{j}/q_{j}\right) \omega _{1} and \mathrm{gcd} (p_{j}, q_{j}) = 1 . Let \Theta = 2\pi \mathrm{gcm}\left(q_{2}, \ldots, q_{k}\right). The solutions to Eq. (4.16) after 'time' \theta = \Theta define the Poincaré return map \mathcal{P}:\left(r, z_{2}, \ldots, z_{m}\right) \longmapsto \left(\rho, \zeta _{2}, \ldots, \zeta _{m}\right) (like in Eq. 4.20)).

    The linear part of this map has the form

    \begin{equation*} \left( r,z_{2},\ldots ,z_{m}\right) \longmapsto \left( r,z_{2},\ldots ,z_{k},\mu _{k+1}z_{k+1},\ldots ,\mu _{m}z_{m}\right), \end{equation*}

    \mu _{j} = \mathrm{e}^{\mathrm{i}\Theta _{j}}\not = 1 for j > k; in fact, \mu _{j} are is not a root of unity.

    Then, the equations \zeta _{k+1}-z_{k+1} = \ldots = \zeta _{m}-z_{m} = 0 define an analytic submanifold \widetilde{V}_{\nu} . It generates the submanifold V_{\nu } = \bigcup_{\theta \in [0, 1\pi]} g_{0}^{\theta}(\widetilde{V}_{\nu}) , where \{ g_{\vartheta}^{\theta}\} is the 2–parameter family of diffeomorphisms defined by solutions to Eq. (4.6).

    Therefore, our problem is reduced to the case when there is only one equivalence class, i.e., that all of the frequencies are rationally related. Let us order these frequencies like in Eq. (1.10), i.e.,

    \begin{equation*} \omega _{1} = \ldots = \omega _{k_{1}} > \omega _{k_{1}+1} = \ldots = \omega _{k_{1}+k_{2}} > \ldots > \omega _{k_{1}+\ldots +k_{r-1}+1} = \ldots = \omega _{k_{1}+\ldots +k_{r}}. \end{equation*}

    Proposition 4. In this situation, there exists a series V_{1}\subset V_{2}\subset \ldots \subset V_{r-1}\subset \left(\mathbb{R}^{2m}, 0\right) of analytic invariant submanifolds (of dimensions k_{1}, k_{1}+k_{2}, \ldots, tangent at the origin to linear subspaces) associated with corresponding groups of frequencies.

    Proof. It essentially repeats the previous proof. We find V_{r-1}; other submanifolds are obtained inductively.

    Let z_{1}, \bar{z}_{1}, \ldots, z_{l}, \bar{z}_{l}, \; l = k_{1}+\ldots +k_{r-1} be the eigenfunctions associated with the eigenvalues \pm \mathrm{i}\omega _{1}, \ldots \pm \mathrm{i}\omega _{l} and z_{l+1}, \bar{z}_{l+1}, \ldots, z_{m}, \bar{z}_{m} be the remaining eigenvalues (associated with \pm \mathrm{i}\omega _{m}). We put z_{l} = r\mathrm{e}^{\mathrm{i}\theta } and get the equations

    \begin{equation*} \frac{\mathrm{d}r}{\mathrm{d}\theta } = \ldots ,{ \ }\frac{\mathrm{d}z_{j} }{\mathrm{d}\theta } = \mathrm{i}\frac{\omega _{j}}{\omega _{l}}z_{j}+\ldots , \end{equation*}

    j\not = l, for phase curves. Their solutions after the 'time' \theta = 2\pi define the Poincaré map

    \begin{eqnarray*} \mathcal{P} &:&\left( r,z_{1},\ldots ,z_{l-1},z_{l+1},\ldots z_{m}\right) \longmapsto \left( \rho ,\zeta _{1},\ldots ,\zeta _{l-1},\zeta _{l+1},\ldots \zeta _{m}\right) \\ & = &\left( r+\ldots ,z_{1}+\ldots ,z_{l-1}+\ldots ,\mu z_{l+1}+\ldots ,\ldots ,\mu z_{m}+\ldots \right) , \end{eqnarray*}

    where the coefficient \mu \not = 1 is a root of unity. The equations \zeta _{l+1}-z_{l+1} = \ldots = \zeta _{m}-z_{m} = 0 define a submanifold which is spread to the submanifold V_{r-1}.

    From now on, we focus our attention on the case with only one equivalence class for the frequencies. Thus, we can write

    \begin{equation} \omega _{j} = p_{j}\omega _{0},{ \ }p_{j}\in \mathbb{N},{ \ }\gcd \left( p_{1},\ldots ,p_{m}\right) = 1. \end{equation} (6.1)

    Proposition 5. In the case of Eq. (6.1), the Poincaré–Dulac normal form is invariant with respect to the following action of the circle \mathbb{S}^{1} :

    \begin{equation} z = \left( z_{1},\ldots ,z_{m}\right) \longmapsto \sigma ^{\phi }\left( z\right) = \left( \mathrm{e}^{\mathrm{i}p_{1}\phi }z_{1},\ldots ,\mathrm{e}^{ \mathrm{i}p_{m}\phi }z_{m}\right) , \ 0\leq \phi \leq 2\pi . \end{equation} (6.2)

    Proof. Recall that a term U = z_{1}^{k_{1}}\cdots z_{m}^{k_{m}}\bar{z }_{1}^{k_{1}}\cdots \bar{z}_{m}^{l_{m}}\partial /\partial z_{j} is resonant in the P–D normal form if and only if the resonant relation \left(k_{1}-l_{1}\right) \omega _{1}+\ldots \left(k_{m}-l_{m}\right) \omega _{m} = \omega _{j} holds. But, the result of change, given by Eq. (6.2), on U is as follows:

    \begin{equation*} U\longmapsto \left( \rho ^{\phi }\right) ^{\ast }U = \exp \left\{ \mathrm{i} \left[ \left( k_{1}-l_{1}\right) p_{1}+\ldots \left( k_{m}-l_{m}\right) p_{m}-p_{j}\right] \right\} U = U. \end{equation*}

    Only such terms are \mathbb{S}^{1}- invariant.

    Recall that the assumptions of the three Lyapunov's theorems are two-fold. One has an assumption about the eigenvalues of the linear part; it implies the existence of a formal invariant smooth surface (it is rather irrelevant in Theorem 1). The second assumption means that the focus quantities of the system restricted to the formal invariant surface vanish; it amounts to the existence of a family of periodic solutions at a formal level. We underline that this vanishing condition admits a precise definition in terms of the P–D normal form; but, there are other approaches.

    To provide a precise assumption of Theorem 8, we definitely need the P–D theorem. It roughly states that the P–D normal form predicts the existence of a 1–parameter family of periodic solutions for the period \approx 2\pi /\omega _{l} at a formal level. Consider the equivalence class of frequencies which contains \omega _{l} . By Proposition 3, we can focus our attention on this class, i.e., that we have the situation of Eq. (1.10). By Proposition 4, there is an invariant submanifold associated with the frequencies \omega _{j} such that \omega _{j}\geq \omega _{l}. Therefore, we can assume that

    \begin{equation} \omega _{1}\geq \omega _{2}\geq \ldots \geq \omega _{l}, \end{equation} (7.1)

    i.e., m = l, besides property (1.10).

    Assume the Poincaré–Dulac normal form (with the coordinates Z_{j}) . Following Examples 4 and 5, introduce the change

    \begin{equation} Z_{l} = R\mathrm{e}^{\mathrm{i}\Theta },{ \ }Z_{j} = W_{j}\mathrm{e}^{ \mathrm{i}p_{j}\Theta /p_{l}},{ \ }(j < l). \end{equation} (7.2)

    Lemma 3. We have the following return system for phase curves:

    \begin{equation} \frac{\mathrm{d}R}{\mathrm{d}\Theta } = \Pi \left( R,W,\overline{W}\right) = \ldots , \ \frac{\mathrm{d}W_{j}}{\mathrm{d}\Theta } = \Lambda _{j}\left( R,W,\overline{W}\right) = \ldots , (j < l), \end{equation} (7.3)

    where the right-hand sides do not depend on \Theta and the dots mean nonlinear terms.

    Proof. We have that \dot{\Theta} = \omega _{l}+\ldots, \; \dot{R} = \ldots and \dot{W}_{j} = \mathrm{i}\omega _{j}W_{j}+\ldots, \; (j < l), where the right-hand sides do not depend on \Theta, due to the P–D normal form.

    Definition 4. We say that the vector field given by (1.4) satisfies a center condition if system (7.3) has a formal curve \Gamma, staring at \left(R, W\right) = \left(0, 0\right), of (non-isolated) equilibrium points.

    Theorem 8 states that, in this case, there exists a genuine continuous 1–parameter family of periodic solutions to system (1.4).

    Proof of Theorem 8. Analogously to the change given by Eq. (7.2), introduce the change

    \begin{equation*} z_{m} = r\mathrm{e}^{\mathrm{i}\theta },{ \ }z_{j} = w_{j}\mathrm{e}^{ \mathrm{i}p_{j}\theta /p_{m}},\text{ }(j < m). \end{equation*}

    We get a system of equations \frac{\mathrm{d}r}{\mathrm{d}\theta } = \ldots, \frac{\mathrm{d}w_{j}}{\mathrm{d}\theta } = \ldots (the return system) with analytic right-hand sides. The periodic solutions of the period \approx 2\pi /\omega _{m} correspond to the fixed points of the twisted Poincaré map \mathcal{P}, which is defined via solutions to the latter system dr/d\theta and dw_j /d\theta after the 'time' \theta = 2\pi.

    The equation \mathcal{P}\left(r, w\right) -\left(r, w\right) = 0 has a zero locus along an analytic curve \gamma which is formally defined as the curve \Gamma from Definition 4 (after the change \left(R, W\right) \longmapsto \left(r, w\right)).

    Remark 7. The statement of Theorem 8 can be generalized to the case when system (7.3) has a formal singular locus \Gamma of higher dimension, say, d . Then, the equation \mathcal{P}-Id = 0 has an analytic zero locus \gamma of dimension d; it corresponds to a d- parameter family of periodic solutions for the period \approx 2\pi /\omega _{m}.

    The assumptions of Theorems 1, 2, 3 and 8 involve an infinite number of conditions, related to either the vanishing of all focus quantities after restriction to a formal invariant plane or the existence of a curve of non-isolated singular points for system (7.3). These phenomena have infinite codimension.

    In this section, we avoid the infinite number of conditions by imposing an additional restriction to the class of considered vector fields and the existence of an analytic first integral with a regular leading part. This restriction is also of infinite codimension, but it is natural in some applications.

    Theorem 15. Assume that analytic system (1.4) has pure imaginary eigenvalues \pm \mathrm{i}\omega _{j}, \; \omega _{j} > 0, and that it has a first integral F such that D^{2}F is definite on a linear subspace E_{j_{0}} (of dimension \geq 2) corresponding to one pair of eigenvalues \pm \mathrm{i} \tilde{\omega}_{j_{0}} such that, for any other frequency \tilde{\omega} _{j}\not = \tilde{\omega}_{j_{0}} , one has \tilde{\omega}_{j}/\tilde{\omega} _{j_{0}}\not\in \mathbb{Z}. Then, there exists at least one 1–parameter family of periodic solutions for the period \approx 2\pi /\tilde{\omega} _{j_{0}}.

    Proof. By Proposition 4, we reduce the situation to one equivalence class of frequencies containing \omega _{j_{0}}; thus, we can assume that the class C_{1} and \tilde{\omega}_{j_{0}} = \tilde{\omega}_{1} in Eq. (1.10). By Proposition 5, we can assume E_{1} = V_{1} = \mathbb{R}^{2m}, \; q = m and there is a definite first integral F\left(z, \bar{z}\right) , say, with positive definite D^{2}F\left(0, 0\right) .

    Assume first that the P–D normal form is analytic. So, let the system be in this form; denote the corresponding vector field X (with variables z_{j}, \; \bar{z}_{j}). Recall that such a system is invariant with respect to the corresponding \mathbb{S}^{1}- action \left\{ \rho ^{\phi }\right\} (Proposition 6). By averaging,

    \begin{equation} \frac{1}{2\pi }\int_{0}^{2\pi }\left( \sigma ^{\phi }\right) ^{\ast }F \mathrm{d}\phi; \end{equation} (8.1)

    we get that the first integral can be chosen also as \mathbb{S}^{1}- invariant: F\circ \sigma ^{\phi } = F.

    The level hypersurfaces M_{f} = \left\{ F(z, \bar{z}) = 0\right\}, \; f > 0, are diffeomorphic with a \left(2m-1\right) - dimensional sphere. They are invariant for the vector field X. Like in [23], we obtain vector fields Y_{f} on the quotient varieties N_{f} = M_{f}/\mathbb{S}^{1} . The vector fields Y_f ' are diffeomorphic with the complex projective space:

    \begin{equation*} N_{f}\simeq \mathbb{P}^{m-1}. \end{equation*}

    So, the Euler characteristic

    \begin{equation*} \chi \left( N_{f}\right) = \chi \left( \mathbb{P}^{m-1}\right) = m\not = 0. \end{equation*}

    By the Poincaré–Hopf theorem [14], \sum i_{y_{j}}Y_{f} = \chi \left(N_{f}\right) (where i_{y_{j}}Y_{f} denotes the indices of Y_{f} at the isolated singular points y_{j}) , and the vector field Y_{f} has at least one singular point (if its singular points are isolated). This amounts to the existence of a periodic solution of X on M_{f}. This implies the existence of a 1- parameter family of periodic solutions like in the thesis of Theorem 15.

    Let us present our approach without the assumption of analyticity of the P–D normal form.

    Suppose that there are no families of periodic solutions as above. We shall construct a real vector field on the projective space \mathbb{P}^{m-1} , which is a P–D approximation \widetilde{V}^{\mathrm{ap}} of the vector field V.

    We take an \mathbb{S}^{1}- invariant approximation F^{\mathrm{ap}} of the first integral F; we take the average given by Eq. (8.1).

    We project \widetilde{V}^{\mathrm{ap}} onto the tangent spaces of the level hypersurfaces M_{f}^{\mathrm{ap}} = \left\{ F^{\mathrm{ap}} = f\right\}, f > 0, along the radii, such that the projection is \mathbb{S}^{1}- equivariant; we get a regular vector field V^{\mathrm{ap}} in the P–D normal form.

    Next, we quotient M_{f}^{\mathrm{ap}} and V^{\mathrm{ap}} by \mathbb{S}^{1} and obtain a vector field Y^{\mathrm{ap}} on N_{f}^{ \mathrm{ap}} = M_{f}^{\mathrm{ap}}/\mathbb{S}^{1}\simeq \mathbb{P}^{m-1} without singular points. This contradicts the Poincaré–Hopf theorem.

    In the proof of Theorem 15 (in the previous section), we have used, essentially, the Poincaré–Hopf formula. But, that formula does not give us much information about the singular points of the quotient vector field Y_{f} on the quotient variety N_{f} = M_{f}/\mathbb{S}^{1}; it is because there is no restriction for the values of the indices. In particular, we cannot get any bound for the number of singular points of Y_{f}.

    Probably, the only topological tool that could be used in this problem is the Lusternik–Schnirelmann category. The corresponding theorem (Theorem 17 from Section 10.2) provides the estimate from below for the number of critical points of a sufficiently smooth function on a manifold based on the category of this manifold.

    Weinstein [35] used this idea in the case of germs of Hamiltonian vector fields X_{H} generated by Hamiltonian functions H\left(z, \bar{z} \right) such that D^{2}H\left(0, 0\right) are definite, and in fact, positive definite. He skillfully constructed a function on the level hypersurface L_{h} = \left\{ H\left(z, \bar{z}\right) = h\right\}, \; h > 0, which has a critical locus at the set of periodic phase curves of X_{H} in L_{h}. His construction is not direct and involves many technical details.

    Below, we propose a more direct proof.

    Proof of Theorem 4. Recall that we have a Hamiltonian vector field X_{H} in \left(\mathbb{R}^{2m}, 0\right) with nonzero pure imaginary eigenvalues of the linear part and with a positive definite D^{2}H. Thus, H = H_{2}+H_{3}+\ldots, where

    \begin{equation*} H_{2} = \frac{1}{2}\sum \omega _{j}\left\vert z_{j}\right\vert ^{2} \end{equation*}

    with \omega _{j} > 0 (compare with Eq. (2.5)). Here, we can reduce H to the Birkhoff normal form, which is an analogue of the P–D normal form. Like in the proof of Theorem 15 (in the previous section), we can reduce the situation to one equivalence class of frequencies, i.e., we assume Eq. (7.1).

    Assume first that the Birkhoff normal form is analytic. So, let H be in this form.

    Proposition 6. The corresponding action \left\{ \sigma ^{\phi }\right\} of \mathbb{S}^{1} on \left(\mathbb{R}^{2m}, 0\right) defines a symmetry of the vector field X_{H}. This action is symplectic; in fact, it is Hamiltonian with

    \begin{equation*} F = H_{2} \end{equation*}

    as the momentum map.

    Proof. We have to prove the second statement. Note that the phase flow generated by the X_{F} is g^{t}\left(z\right) = \left(\mathrm{e}^{ \mathrm{i}\omega _{1}t}z_{1}, \ldots, \mathrm{e}^{\mathrm{i}\omega _{m}t}z_{m}\right) and coincides with the action \left\{ \sigma ^{\phi }\right\}.

    Now, we apply the symplectic reduction. Recall that, in the case of the general (non-abelian) symmetry group, the symplectic reduction concerns only the zero level of the momentum map (see [4]). Here, the group \mathbb{S}^{1} is abelian and the symplectic reduction works for any level of the momentum function; in fact, it is the simplest case of the symplectic reduction.

    So, we take the manifolds

    \begin{equation*} M_{f} = \left\{ F(z,\bar{z}) = f\right\} ,{ \ }f > 0, \end{equation*}

    which are diffeomorphic with \mathbb{S}^{2m-1} , and their quotients

    \begin{equation*} N_{f} = M_{f}/\mathbb{S}^{1} \end{equation*}

    of dimension 2m-2. The varieties N_f are equipped with a natural symplectic structure and support \pi_{*}H = \pi_{*}F+\pi_{*}G fields Y_{f} obtained from X_{H}. Each vector field Y_{f} is Hamiltonian with the Hamiltonian function \pi ^{\ast }H = \pi ^{\ast }F+\pi ^{\ast }G, where \pi :M_{f}\longmapsto N_{f} is the projection and

    \begin{equation*} G = H-H_{2} \end{equation*}

    contains higher-order terms.

    Proposition 7. The periodic orbits of the X_{H} in M_{f} of period \approx 2\pi /\omega _{k}, for some 1\leq k\leq m, correspond to the critical points of the function \pi ^{\ast }G on N_{f}.

    Proof. The periodic orbits from this proposition are somewhat distinguished. They are defined via the return vector fields U_{k} of type (7.3), i.e., they are associated with a fixed choice of the frequency \omega _{k}; \; z_{k} = r\mathrm{e}^{\mathrm{i}p_{k}\theta } and z_{j} = w_{j}\mathrm{e}^{ \mathrm{i}p_{j}\theta } for j\not = k. (We do not consider possible orbits of a very long period.) The singular points of U_{k} correspond to such periodic orbits.

    But, the singular points of U_{k} , when considered on M_{f} , are the equilibrium points of Y_{f}. They are the critical points of \pi_{*}F , i.e., of \pi_{*}G , on N_{f}. Here, there is some subtlety associated with the fact that N_{f} is singular, but one can use the local charts \left[ w_{1}^{\left(k\right) }:\ldots :r^{\left(k\right) }:\ldots :w_{m}^{\left(k\right) }\right] in N_{f} defined in Eq. (8.2) below.

    Since M_{f} is a sphere, its quotient N_{f} is the weighted projective space \mathbb{P}_{p_{1}, \ldots, p_{m}}. By the Lusternik–Schnirelmann theorem (Theorem 16 in Section 10.2), the number of critical points of the function \pi ^{\ast }G is bounded from below by the category of N_{f}. But, \mathbb{P}_{p_{1}, \ldots, p_{m}} is homologically 'similar' to \mathbb{P} ^{m-1} (see Section 10.1) and \mathrm{cat}\mathbb{P}_{p_{1}, \ldots, p_{m}}\geq m (see Theorem 17 from Section 10.2).

    The category of quotient spaces was studied also by Weinstein in [36].

    This gives the estimate \geq m for the number of isolated periodic orbits.

    Now, we present our argumentation without the assumption of the analyticity of the Birkhoff normal form. We have essentially two tools at our disposal (we have used them in the proof of Theorem 8). Again, we assume only one equivalence class of the frequencies.

    The first tool is the Birkhoff normal form, which we assume only to be formal.

    The other is the Poincaré return map (as defined in the second proof of Theorem 3, see Eq. (4.20)), or, rather, the twisted Poincaré map (as defined in the proof of Theorem 8 and in Examples 4–6).

    In fact, we will use a collection of twisted Poincaré maps \mathcal{P} ^{\left(k\right) }, \; k = 1, \ldots, m, via the substitutions

    \begin{equation*} z_{k} = r^{\left( k\right) }\mathrm{e}^{\mathrm{i}\theta },{ \ } z_{j} = w_{j}^{\left( k\right) }\mathrm{e}^{\mathrm{i}p_{j}\theta /p_{k}}\text{ }\left( j\not = k\right) , \end{equation*}

    as well as the solutions after the 'time' \theta = 2\pi of corresponding differential systems. The variables r^{\left(k\right) }\in \left(\mathbb{R }, 0\right) and w_{j}^{\left(k\right) }\in \left(\mathbb{C}, 0\right) parametrize corresponding twisted Poincaré sections.

    As before, we assume that \omega _{j} = p_{j}\omega _{0}, with relatively prime positive integers such that p_{1}\geq p_{2}\geq \ldots \geq p_{m}. The twisted Poincaré map differs from the standard Poincaré map (where z_{j} = w_{j}) in the coordinates w_{j} such that \omega _{j} > \omega _{k}. In the standard case we have that z_{j}\longmapsto \mathrm{e}^{\mathrm{i} p_{j}/p_{k}}z_{j}+\ldots and, in the twisted case, we have that w_{j}\longmapsto w_{j}+\ldots.

    If the system (1.7) were in an analytic Poincaré–Dulac–Birkfoff normal form with an \mathbb{S}^{1}- invariant first integral F , then the variables r^{\left(k\right) } and w_{j}^{\left(k\right) }, when restricted to M_{f} = \left\{ F = f\right\} , would form a local chart

    \begin{equation} \left[ w_{1}^{\left( k\right) }:\ldots :r^{\left( k\right) }:\ldots :w_{m}^{\left( k\right) }\right] \end{equation} (8.2)

    in the quotient variety N_{f} = M_{f}/\mathbb{S}^{1}, or an affine chart w_{1}^{\left(k\right) }/r^{\left(k\right) }, \ldots, w_{m}^{\left(k\right) }/r^{\left(k\right) }, \; r^{\left(k\right) } > 0.

    Recall that the 1–parameter families of periodic solutions of system (1.7), whose corresponding vector field we denote by X_{H}, of period \approx 2\pi /\omega _{k}, correspond to 1–dimensional curves of fixed points of the twisted Poincaré map \mathcal{P}^{\left(k\right) } (see the above proof of Theorem 7). They are analytic curves \gamma defined by the analytic equation \mathcal{P}_{k}-Id = 0. We assume that this equation has a zero locus of dimension 1; otherwise, we have infinitely many 1–parameter families of such periodic solutions.

    From the analytic geometry, we know that (germs of) analytic sets are analytically equivalent (diffeomorphic) with algebraic sets. This holds in the case of complex analytic sets and in the case of real analytic sets. For example, a germ of an analytic curve is defined by a corresponding Puiseux series. The proof uses the Weierstrass preparation theorem (see [37]). Moreover, if such a germ is defined by a system of germs of analytic functions, then an approximation of these functions by polynomials (sufficiently long jets) define an algebraic set locally diffeomorphic with the original germ.

    We apply this to the germs \gamma of curves defined by the equation \mathcal{P}^{\left(k\right) }-Id = 0. These equations admit the approximation \mathcal{P}_{\varepsilon }^{\left(k\right) }-Id = 0, where the twisted Poincaré map \mathcal{P}_{k}^{\left(\varepsilon \right) } is defined via a Hamiltonian vector field X_{H^{\varepsilon }} generated by a polynomial approximation H^{\varepsilon } of H in the Birkhoff normal form. Then, the curve \gamma will correspond to a curve \gamma _{\varepsilon }\subset \left\{ \mathcal{P}_{\varepsilon }^{\left(k\right) }-Id = 0\right\}. The number of the latter curves is estimated like above.

    Recall also that, in the original version of the work [23, Theorem 4], Moser assumed only that H_{2}|_{E\diagdown 0} > 0, where E is the linear subspace associated with one frequency for the linear system, but his example from [23, Example 2] (or our Example 4 above) provides a contradiction to this restricted version.

    It seems that the crucial point in Moser's approach is his lemma (Lemma 1) from [23]. There, he somehow associates with any small vector \xi \in E\diagdown 0 a solution to the Hamiltonian system and claims that some of them are periodic; probably, therein lies the flaw of his argument. Finally, he uses the Lusternik–Schnirelmann category for a suitable function on \mathbb{P}^{r-1}.

    Our approach is more direct. In part, it repeats the above proof of the Weinstein theorem.

    Proof of Theorem 5. Much of his proof repeats the corresponding part of the proof of Theorem 4 in the previous section. In particular, we can assume that the Hamiltonian function is in an analytic Birkhoff normal form. Recall that we assume only one equivalence class among the frequencies.

    Thus, we have the additional first integral F = H_{2}, which defines the momentum map for the action of the group \mathbb{S}^{1}.

    Recall that, by Proposition 5, we have the filtration V_{1}\subset \ldots \subset V_{r-1}\subset V_{r} = \left(\mathbb{R}^{2m}, 0\right) of submanifolds that are invariant for the vector field X_{H} such that V_{j}\diagdown V_{j-1} is associated with one frequency. This filtration induces corresponding filtrations in M_{f} = \left\{ F = f\right\} and in N_{f} = M_{f}/\mathbb{S}^{1}:

    \begin{equation*} V_{1}\cap M_{f}/\mathbb{S}^{1} = \mathbb{P}^{\left( 1\right) }\subset \ldots \subset V_{r}\cap M_{f}/\mathbb{S}^{1} = \mathbb{P}^{\left( r\right) }, \end{equation*}

    where \mathbb{P}^{\left(j\right) } denotes weighted projective spaces with complex dimensions \mathrm{dim}V_{j}-1.

    We have to show that G = H-F, as a function on N_{f}, has at least \frac{1}{2}\mathrm{dim}V_{j} = \frac{1}{2}(k_{1}+\ldots k_{j}) critical points in \mathbb{P}^{\left(j\right) }. But, this follows from Theorems 16 and 17 in Section 10.2.

    Topological theorems of nonlinear functional analysis are well suited to deal with nonlinear PDEs. But, some specialists began to use them to prove (or reprove) some qualitative results in the theory of ODEs; see [22] for an example.

    In particular, Szulkin [32] developed a functional analytic approach to the question of the accumulation of small-amplitude periodic solutions to Hamiltonian systems. His method was followed in papers by Rybicki with various collaborators [10,15,25,26,31].

    The first trick in the analysis of periodic solutions for the period T = 2\pi /\omega, which usually varies with variation of the solution, is to normalize this period to 2\pi. This is done by replacing the Hamiltonian system \dot{x} = X_{H}\left(x\right) with the following family of Hamiltonian systems:

    \begin{equation} \dot{y} = \lambda X_{H}\left( y\right) , \end{equation} (8.3)

    where \lambda > 0 is a parameter chosen (for a given periodic solution x = \varphi \left(t\right)) such that the corresponding solution y = \varphi \left(\lambda t\right) to Eq. (8.3) has the period 2\pi . One says that a family y = \psi _{\lambda }\left(t\right) of 2\pi - periodic solutions to Eq. (8.3) emanates from \left(\lambda, y\right) = \left(\lambda _{0}, 0\right) , \lambda _{0} = 1/\omega _{j}, when x = \psi _{\lambda }\left(t/\lambda \right) , with \lambda close to \lambda _{0} , denotes solutions to \dot{x} = X_{H} with the period \approx 2\pi /\omega _{j}.

    Next, it is not difficult to show that the 2\pi - periodic solutions to system (8.3), where y = \left(q, p\right), correspond to the stationary paths of the following functional (see also [23]):

    \begin{equation} \Phi \left( \gamma \right) = \Phi _{\lambda }\left( \gamma \right) = \int_{0}^{2\pi }p\mathrm{d}q+\lambda H\mathrm{d}t. \end{equation} (8.4)

    This functional is defined on the 2\pi - periodic path \gamma :\mathbb{S} ^{1}\longmapsto \mathbb{R}^{2m}, \; \gamma \left(t\right) = \left(q(t), p(t)\right). The author considers \Phi as a functional on the Sobolev space \mathcal{H} = H^{1/2}\left(\mathbb{S}^{1}, \mathbb{R} ^{2m}\right); using complex vector-variables z = q+\mathrm{i}p and \bar{z} = q- \mathrm{i}p , we have that z = \gamma \left(t\right) = \sum_{k\in \mathbb{Z} }z^{\left(k\right) }\mathrm{e}^{\mathrm{i}kt}, \; z^{\left(k\right) } = \left(z_{1}^{\left(k\right) }, \ldots, z_{m}^{\left(k\right) }\right) and \left\Vert \gamma \right\Vert ^{2} = \pi \left\{ \left\vert z^{\left(0\right) }\right\vert ^{2}+\sum k\left\vert z^{\left(k\right) }\right\vert ^{2}\right\}.

    The first part of the functional (8.4) takes the form

    \begin{equation*} \langle \boldsymbol{L}\gamma ,\gamma \rangle = \int_{0}^{2\pi }p \mathrm{d}q = \frac{1}{2}\mathrm{Im}\int_{0}^{2\pi }z\mathrm{d}\bar{z} = \pi \sum k\left\vert z^{\left( k\right) }\right\vert ^{2}. \end{equation*}

    With the splitting \mathcal{H} = \mathcal{H}_{-}\oplus \mathcal{H}_{0}\oplus \mathcal{H}_{+} = \left\{ \left(\gamma _{-}, \gamma _{0}, \gamma _{+}\right) \right\} of the above Hilbert space into subspaces generated by negative, zero and positive modes, we have that \langle \boldsymbol{L}\gamma, \gamma \rangle = \left\Vert \gamma _{+}\right\Vert ^{2}-\left\Vert \gamma _{-}\right\Vert ^{2}. In particular, the subspace \mathcal{H}_{0, \pm } is invariant for \boldsymbol{L}.

    Assume that H = H_{2}+\ldots, with H_{2} = \frac{1}{2}\left(Az, \bar{z}\right). We have the quadratic functional

    \begin{equation*} \langle \boldsymbol{B}\gamma ,\gamma \rangle = \int H_{2}\mathrm{d} t; \end{equation*}

    if H_{2} = \frac{1}{2}\sum_{j}\epsilon _{j}\omega _{j}\left\vert z_{j}\right\vert ^{2}, \omega _{j} > 0, \epsilon _{j} = \pm 1, like in Eq. (1.8), then \langle \boldsymbol{B}\gamma, \gamma \rangle = \pi \left\{ \sum_{j, k}\epsilon _{j}\omega _{j}\left\vert z_{j}^{\left(k\right) }\right\vert ^{2}\right\}. It turns out (see [32]) that the variational derivative \frac{\delta }{\delta \gamma }\int H\mathrm{d}t is a compact operator in \mathcal{H} , and that the functional \Phi satisfies the so-called Palais–Smale condition (which we do not define here).

    Consider the operator \boldsymbol{L}+\lambda \boldsymbol{B} and its restrictions T^{\left(k\right) } = \left(\boldsymbol{L}+\lambda \boldsymbol{ B}\right) |_{\mathcal{H}^{\left(k\right) }} to the subspace \mathcal{H} ^{\left(k\right) } = \mathrm{span}\left\{ \mathrm{e}^{\mathrm{i}kt}, \mathrm{e} ^{-\mathrm{i}kt}\right\} \simeq \mathbb{R}^{2m}, \; k > 0 , which are as follows:

    \begin{equation*} T^{\left( k\right) }(z^{\left( k\right) },z^{\left( -k\right) }) = \left( \mathrm{i}\left( kI_{m}+\lambda A\right) z^{\left( k\right) },\mathrm{i} \left( -kI_{m}+\lambda A\right) z^{\left( -k\right) }\right) . \end{equation*}

    We see that the operator T^{\left(k\right) } = T^{\left(k\right) }\left(\lambda A\right) is non-invertible if and only if \lambda = k/\omega _{j} for some j; this is the case when the linear system \dot{x} = X_{H_{2}}\left(x\right) has a family of periodic solutions for the period 2\pi /\omega _{j}.

    Consider the splitting \mathcal{H} = \mathcal{F}\oplus \widetilde{\mathcal{F}}, where \mathcal{F} = \mathcal{H}_{-} and \widetilde{\mathcal{F}} = \mathcal{ H}_{0}\oplus \mathcal{H}_{+}. The subspace \mathcal{F} is invariant for \boldsymbol{L} and \boldsymbol{L}|_{\mathcal{F}} < 0 has a quadratic form. The operator \boldsymbol{B} is a bounded self-adjoint operator in \mathcal{H} . Let \mathcal{E}_{n} = \mathrm{span}\left\{ \mathrm{e}^{\mathrm{i }kt}:k = 1, \ldots, n\right\} \oplus \widetilde{\mathcal{F}} be subspaces of \mathcal{H} (invariant for \boldsymbol{L}) and \boldsymbol{P}_{n}: \mathcal{H}\longmapsto \mathcal{E}_{n} be corresponding orthogonal projections. We have the Morse indices M^{-}\left(\left(\boldsymbol{L} +\lambda \boldsymbol{P}_{n}\boldsymbol{B}\right) |_{\mathcal{E}_{n}}\right), i.e., the numbers of negative eigenvalues. One defines another index

    \begin{equation*} M^{-}\left( \lambda \boldsymbol{B}\right) = \lim\limits_{n\rightarrow \infty }\left\{ M^{-}\left( \left( \boldsymbol{L}+\lambda \boldsymbol{P}_{n} \boldsymbol{B}\right) |_{\mathcal{E}_{n}}\right) -n\right\} . \end{equation*}

    (It has a cohomological interpretation; some cohomology groups H_{\mathcal{F }}^{q}\left(W, W^{-}\right) , which we do not define, vanish for q\not = M^{-}\left(\lambda \boldsymbol{B}\right) and are equal to \mathbb{R} otherwise; see [32, Proposition 2.3].)

    An abstract statement, i.e., [32, Theorem 2.3], says the following:

    Proposition 8. If (i) \boldsymbol{L}+\lambda \boldsymbol{B} is invertible for \lambda \in I\diagdown \lambda _{0}, where I is a neighborhood of \lambda _{0}, and (ii) the function \lambda \longmapsto M^{-}\left(\lambda \boldsymbol{B}\right) has a jump at \lambda _{0}, then either (a) the path \gamma \left(t\right) \equiv 0 is a non-isolated critical path of \Phi _{\lambda _{0}} , or, (b) for all \lambda 's in I\diagdown \lambda _{0} from one side of \lambda _{0} , the functional \Phi _{\lambda } has a critical path \gamma _{\lambda }\left(t\right) \not\equiv 0 near \gamma \left(t\right) \equiv 0.

    The quantity M^{-}\left(\lambda \boldsymbol{B}\right) is explicitly calculated in terms of the matrix \lambda A; we have

    \begin{equation*} M^{-}\left( \lambda \boldsymbol{B}\right) = i^{-}\left( \lambda A\right) : = M^{-}\left( -\lambda A\right) +\sum\limits_{k > 0}\left\{ M^{-}\left( T^{\left( k\right) }\right) -2m\right\} \end{equation*}

    (see [32, Proposition 3.3]).

    Assume the situation like in Eq. (1.10), i.e., that we have r different frequencies, \tilde{\omega}_{1} = \omega _{1} (of multiplicity k_{1}), \; \tilde{\omega}_{2} = \omega _{k_{1}+1} (of multiplicity k_{2}), etc., possibly belonging to different equivalence classes. Let us fix \tilde{\omega} _{l}. It turns out that, if \tilde{\omega}_{j}/\tilde{\omega}_{l}\not\in \mathbb{N} for all j\not = l, then M^{-}\left(\lambda \boldsymbol{B} \right) = i^{-}\left(\lambda A\right) jumps at \lambda = 1/\tilde{\omega} _{l} if and only if M^{-}\left(T^{\left(1\right) }\left(\lambda A\right) \right) jumps.

    Next, M^{-}\left(T^{\left(1\right) }\left(\lambda A\right) \right) jumps at \lambda = 1/\tilde{\omega}_{l} if and only if

    \begin{equation} M^{-}\left( A|_{E_{l}}\right) \not = M^{+}\left( A|_{E_{l}}\right) : = M^{-}\left( -A|_{E_{l}}\right) , \end{equation} (8.5)

    where E_{l} is the invariant subspace for A corresponding to the eigenvalues \pm \mathrm{i}\tilde{\omega}_{l} (see [32, Proposition 3.6]). Condition (8.5) means that, if H_{2} is like in Eq. (1.8) and \tilde{\omega}_{l} = \omega _{k_{1}+\ldots +k_{s}+1} , then \epsilon _{k_{1}+\ldots +k_{l-1}+1}+\ldots +\epsilon _{k_{1}+\ldots +k_{l}}\not = 0 for \epsilon _{i} from Eq. (1.8).

    This yields the Szulkin's Theorem 6 from Introduction.

    We see that the first assumption of Theorem 6, i.e., \tilde{\omega}_{j}/ \tilde{\omega}_{l}\not\in \mathbb{N}, is like in the third Lyapunov theorem. But, the assumption, i.e., inequality (8.5), is new; it replaces the definiteness assumption of H_{2} from Theorems 4 and 5. Note also that, in this topological approach, one does not assume the analyticity of the Hamiltonian function. But, when one additionally assumes the analyticity of the Hamiltonian function, then the statement of Theorem 6 can be strenghtened: there exists at least one continuous family of periodic solutions.

    At this moment, the method to reprove this result using the proposed approach is unknown by the author. But, in all examples considered by the author, my tools seem to be more effective. In fact, examples of where only the Szulkin theorem works are not known by the author.

    Geostationary orbits of celestial objects, like GPS satellites, are determined by the property by which they are fixed with respect to an Earth's observer. The following definitions were taken from Strzelecki's thesis [31].

    Regarding the position and momentum framework of q = \left(x, y, z\right) , p = \left(p_{x}, p_{y}, p_{z}\right) associated with the moving object, we get a Hamiltonian system with the following Hamiltonian:

    \begin{equation} H = \frac{1}{2}\left\vert p\right\vert ^{2}+\nu \left( xp_{y}-yp_{x}\right) +V, \end{equation} (9.1)

    where \nu is the Earth's angular velocity and the gravity potential (generated by an ellipsoid of rotation) equals

    \begin{equation*} V = V\left( \rho ,z\right) = -r^{-1}\left( 1-2cr^{-2}P_{2}\left( z/r\right) \right) = -r^{-1}-cr^{-3}+3cz^{2}r^{-5}, \end{equation*}

    where P_{2}\left(\lambda \right) = \frac{1}{2}\left(3\lambda ^{2}-1\right) is the quadratic Legendre polynomial, r = \left\vert q\right\vert = \sqrt{ \rho ^{2}+z^{2}} and c > 0 is a constant evaluated experimentally. The corresponding Hamiltonian equations

    \begin{equation} \dot{x} = p_{x}-\nu y,\text{ }\dot{y} = p_{y}+\nu x,\text{ }\dot{z} = p_{z},\text{ }\dot{p}_{x} = -V_{x}^{\prime }-\nu p_{y},\text{ }\dot{p}_{y} = -V_{y}^{\prime }+\nu p_{x},\text{ }\dot{p}_{z} = -V_{z}^{\prime } \end{equation} (9.2)

    imply the following second-order Newtonian equations:

    \begin{equation*} \ddot{x} = -V_{x}^{\prime }-2\nu \dot{y}+\nu ^{2}x,\text{ }\ddot{y} = -V_{y}^{\prime }+2\nu \dot{x}+\nu ^{2}y,\text{ }\ddot{z} = -V_{z}^{\prime }; \end{equation*}

    here, \left(-2\nu \dot{y}, 2\nu \dot{x}, 0\right) and \left(\nu ^{2}x, \nu ^{2}y, 0\right) are the Coriolis and the centrifugal forces caused by the rotation of the coordinate frame.**

    **Such systems were already considered by Lyapunov [21, Section 45]

    The function (9.1) is invariant with respect to the simultaneous rotations in the \vec{\rho} = \left(x, y\right) and \vec{\sigma} = \left(p_{x}, p_{y}\right) planes. This \mathbb{S}^{1}- action is symplectic, and the corresponding momentum map is the vertical component of the classical angular momentum, F = xp_{y}-yp_{x} = \vec{\rho}\times \vec{\sigma}. The symplectic reduction (see Section 10.1) means the introduction of the coordinates \rho = |\vec{\rho}|, \phi, z, \zeta = p_{3}, where \phi is the angle between the vectors \vec{\rho} and \vec{\sigma}. Thus, F = \rho \sigma \mathrm{sin}\phi , \sigma = |\vec{\sigma}| . On the hypersurface M_{f} = \left\{ F = f\right\} , where f\not = 0 , we have

    \begin{equation*} \sigma = f/\left( \rho \mathrm{sin}\phi \right) . \end{equation*}

    Next, \frac{\mathrm{d}}{\mathrm{d}t}\vec{\rho} = \vec{\sigma}+\nu (-q_{2}, q_{1}) implies \dot{\rho} = \frac{1}{\rho }\vec{\rho}\cdot \vec{ \sigma} = \sigma \mathrm{cos}\phi and \dot{\sigma} = -V_{\rho }^{\prime } \mathrm{cos}\phi. So, we get the reduced differential system on N_{f} = M_{f}/\mathbb{S}^{1} :

    \begin{equation} \dot{\rho} = f\rho ^{-1}\mathrm{cot}\phi ,\text{ }\dot{\phi} = -f\rho ^{-2}+\rho V_{\rho }^{\prime }\mathrm{sin}^{2}\phi /f,\text{ }\dot{z} = \zeta ,\text{ } \dot{\zeta} = -V_{z}^{\prime }. \end{equation} (9.3)

    The equilibrium points are found as follows. First, \zeta _{0} = z_{0} = 0 since V depends on z^{2}. We have \mathrm{cos}\phi = 0, i.e., \phi = \pm \frac{\pi }{2} , and we choose \phi _{0} = \frac{\pi }{2}. Next, from Eq. (9.2), we get \sigma _{0} = \nu \rho _{0} and f = \nu \rho _{0}^{2}. Thus, \dot{\phi} = 0 implies -\nu +\frac{1}{\nu \rho _{0}}\left(\rho _{0}^{-2}+3c\rho _{0}^{-4}\right) = 0 , and we get the following equation for \rho _{0} :

    \begin{equation*} \nu ^{2}\rho _{0}^{5}-\rho _{0}^{2}-3c = 0; \end{equation*}

    for c > 0, it has a unique positive solution.

    Let us linearize system (9.3) at this equilibrium point. First, \mathrm{ cot}^{\prime }\left(\pi /2\right) = -1, Next, \beta = \partial \dot{\phi} /\partial \rho = 2f\rho _{0}^{-3}+\frac{2c}{f}\rho _{0}^{-3} > 0. Finally, for a small z , we have that V\approx -\rho _{0}^{-1}\left(1-z^{2}/2\rho _{0}^{2}\right) -c\rho _{0}^{-3}\left(1-3z^{2}/2\rho _{0}^{2}\right) +3cz^{2}\rho _{0}^{-5} = \mathrm{const}+\frac{1}{2}\left(\rho _{0}^{-3}+15\rho _{0}^{-5}\right) z^{2} = \mathrm{const}+\gamma z^{2}, \; \gamma > 0. So, the linearization, with \rho _{1} = \rho -\rho _{0} , \rho_0 = \rho -\rho_0, \ldots , is as follows:

    \begin{equation*} \dot{\rho}_{1}\approx -\alpha \phi _{1},\text{ }\dot{\phi}_{1}\approx \beta \rho _{1},\text{ }\dot{z}_{1} = \zeta _{1},\text{ }\dot{\zeta}_{1}\approx \gamma z_{1}, \end{equation*}

    with positive constants \alpha = f/\rho _{0}, \beta, \gamma. It follows that there is a pair of pure imaginary eigenvalues \pm \mathrm{i}\omega = \; \pm \mathrm{i}\sqrt{\alpha \beta } and a pair of real eigenvalues \pm \sqrt{\gamma } . Since the system is Hamiltonian, we have a 1–parameter family of periodic solutions for the period \approx 2\pi /\omega on the 2–dimensional center manifold W^{c}\approx \left\{ z_{1} = \zeta _{1} = 0\right\} .

    But, for the Hamiltonian system with three degrees of freedom (before the symplectic reduction), we have, in fact, invariant tori, with the additional angle \psi = \mathrm{arg}\left(x+\mathrm{i}y\right). The movement on these tori is quasi-periodic; we have

    \begin{equation*} \dot{\psi}\approx \delta , \end{equation*}

    \delta = f\rho _{0}^{-2}.

    In Strzelecki's approach [31], the same result was obtained using the so-called G- equivariant Conley index.

    In the restricted four-body problem, we deal with three heavy bodies, S (from the sun), J (from Jupiter) and P (from a planet), and one light body A (an asteroid). The heavy bodies lie in the vertices of an equilateral triangle which rotates with constant angular velocity (equal \nu = -1) about their center of mass and in a fixed plane. The asteroid moves in the gravity field generated by the principal bodies and does not influence their movement. ††

    ††Recall that periodic solutions near the triangular libration point L_{4} for the restricted three-body problem, i.e., with only S, \, J and A, were studied by Schmidt [28]. I would like to thank one of the reviewers for this reference

    As in the previous section, one passes to the coordinate–momentum frame wherein the heavy bodies rest. Thus, we have the Hamiltonian function like in Eq. (9.1), but with \nu = 1 and

    \begin{equation} V\left( q\right) = -\frac{m_{S}}{\left\vert q-q_{S}\right\vert }-\frac{m_{J}}{ \left\vert q-q_{J}\right\vert }-\frac{m_{P}}{\left\vert q-q_{P}\right\vert }; \end{equation} (9.4)

    here, q = 0 is the center of mass of the system. We assume two degrees of freedom (the case with three degrees of freedom is only slightly more complex; see [15]), i.e., that A lies in the plane SJP; so, q = \left(x, y\right) and p = \left(p_{x}, p_{y}\right).

    It is useful to rewrite the Hamiltonian as

    \begin{equation*} H = \frac{1}{2}\left( p_{x}-y\right) ^{2}+\frac{1}{2}\left( p_{y}+x\right) ^{2}+W\left( x,y\right) , \end{equation*}

    where

    \begin{equation*} W = -\left( x^{2}+y^{2}\right) /2+V \end{equation*}

    is the effective potential. The critical points of H are defined by p_{x} = y_{j} and p_{y} = -x_{j}, where \left(x_{j}, y_{j}\right) are critical points of W. By putting q_{1} = x-x_{j}, \; q_{2} = y-y_{j}, \; p_{1} = p_{x}+y_{j}, \; p_{2} = p_{y}-x_{j} (and \tilde{q} = \left(q_{1}, q_{2}\right), \; \tilde{p} = \left(p_{1}p_{2}\right)) , the quadratic part of the expansion of the Hamiltonian at such a critical point becomes

    \begin{equation*} H_{2}\left( \tilde{q},\tilde{p}\right) = \frac{1}{2}\left\{ \left( p_{1}-q_{2}\right) ^{2}+\left( p_{2}+q_{1}\right) ^{2}-aq_{1}^{2}-2bq_{1}q_{2}-cq_{2}^{2}\right\} . \end{equation*}

    One finds (see [20]) that the characteristic polynomial of the corresponding linear system becomes

    \begin{equation*} \begin{array}{c} P\left( \lambda \right) = \lambda ^{4}+a_{2}\lambda ^{2}+a_{4}, \\ a_{2} = 4-a-c,\text{ }a_{4} = ac-b^{2}. \end{array} \end{equation*}

    Note that the characteristic polynomial of the Hessian matrix D^{2}W\left(x_{j}, y_{j}\right) = \left(\begin{array}{cc} -a & -b \\ -b & -c \end{array} \right) equals

    \begin{equation*} R\left( \beta \right) = \beta ^{2}+\left( a+c\right) \beta +ac-b^{2}. \end{equation*}

    If Q\left(\mu \right) = \mu ^{2}+a_{2}\mu +a_{4}, then we have the following relations between zeros of the polynomials Q and R:

    \begin{equation} \mu _{1}\mu _{2} = \beta _{1}\beta _{2},\text{ }\mu _{1}+\mu _{2} = -\left( \beta _{1}+\beta _{2}+4\right) . \end{equation} (9.5)

    We have the following generic possibilities for the zeros of P (in [15], these cases are expressed in terms of \beta _{1}, \beta _{2}):

    (i) \mu _{1} = \bar{\mu}_{2}\in \mathbb{C}\diagdown \mathbb{R}, i.e., a_{4} > 0 and \Delta = a_{2}^{2}-4a_{4} < 0; then, \lambda _{j}\in \mathbb{C} \diagdown \mathbb{R}\diagdown \mathrm{i}\mathbb{R}.

    (ii) \mu _{1}, \mu _{2} > 0, i.e., a_{4} > 0, \; a_{2} < 0 and \Delta > 0; then, \lambda _{1, 2} = \pm \sqrt{\mu _{1}} and \lambda _{3, 4} = \pm \sqrt{ \mu _{2}}.

    (iii) \mu _{1} < 0 < \mu _{2}, i.e., a_{4} < 0 and \Delta > 0; then, \lambda _{1, 2} = \pm \mathrm{i}\sqrt{\left\vert \mu _{1}\right\vert } and \lambda _{3, 4} = \pm \sqrt{\mu _{2}}.

    (iv) \mu _{1} < \mu _{2} < 0, i.e., a_{4} > 0, \; a_{2} > 0 and \Delta > 0; then, \lambda _{1, 2} = \pm \mathrm{i}\sqrt{\left\vert \mu _{1}\right\vert } and \lambda _{3, 4} = \pm \mathrm{i}\sqrt{\left\vert \mu _{2}\right\vert } .

    We note also the following non-generic case of 0:1 resonance (considered in [15]):

    (v) \mu _{1} < 0 = \mu _{2}, i.e., a_{4} = 0 and a_{2} > 0; then, \lambda _{1, 2} = \pm \mathrm{i}\sqrt{\left\vert \mu _{1}\right\vert } and \lambda _{3, 4} = 0.

    Using only the above information about the eigenvalues \lambda _{j} , we can state the following.

    (1) In cases (i) and (ii), there are no families of periodic solutions.

    (2) In case (iii), there is one 1–parameter family of periodic solutions for the period \approx 2\pi /\sqrt{\left\vert \mu _{1}\right\vert }.

    (3) In case (iv), there exists at least one 1–parameter family of periodic solutions for the period \approx 2\pi /\sqrt{ \left\vert \mu _{1}\right\vert }; moreover, if \sqrt{\left\vert \mu _{1}\right\vert /\left\vert \mu _{2}\right\vert } > 1 is not an integer, then there is another 1–parameter family for the period \approx 2\pi / \sqrt{\left\vert \mu _{2}\right\vert }.

    Analogues of these statements can be found in [15, Theorem 3.1]. But, in the same theorem, we find the following statement:

    If in case (v), additionally, the index i_{\left(x_{j}, y_{j}\right) }\nabla W (the authors call it the Brouwer index) is nonzero, then there exists a branch of closed orbits.

    We shall justify this using a refined version of the Birkhoff normal form in the 0:1 resonant case:

    \begin{equation*} H = \frac{1}{2}\left\{ \left\vert Z\right\vert ^{2}\Omega \left( Y,\left\vert Z\right\vert ^{2}\right) -X^{2}+\Phi \left( Y\right) \right\} , \end{equation*}

    where \Omega = \omega +\ldots and \Phi = a_{k}Y^{k}+\ldots, \; a_{k}\not = 0. In the case of one degree of freedom, this result is due to Baider and Sanders [5] (some coefficients in the series \Phi \left(Y\right) vanish); the above case is a generalization of [7, Eq. (3.4)]. The corresponding differential system becomes

    \begin{equation*} \dot{Z} = \mathrm{i}\left( \omega +\ldots \right) Z,\text{ }\dot{X} = ka_{k}Y^{k-1}\left( 1+O\left( Y\right) \right) +\left\vert Z\right\vert ^{2}\Psi \left( Y,\left\vert Z\right\vert ^{2}\right) ,\text{ }\dot{Y} = X. \end{equation*}

    As usual, when looking for periodic phase curves, we make the substitution Z = r\mathrm{e}^{\mathrm{i}\theta } . We get the system

    \begin{equation*} \mathrm{d}r/\mathrm{d}\theta = 0,\text{ }\mathrm{d}X/\mathrm{d}\theta = cY^{k-1}\left( 1+\ldots \right) +r^{2}\Xi \left( Y,r^{2}\right) ,\text{ } \mathrm{d}Y/\mathrm{d}\theta = X/\left( \omega +\ldots \right) , \end{equation*}

    c = ka_{k}/\omega \not = 0, with equilibrium points defined by X = 0 and cY^{k-1}\left(1+\ldots \right) +r^{2}\Xi \left(Y, r^{2}\right) = 0. The sufficient condition for the existence of a nontrivial solution to the this system is that the exponent k-1 is odd. But, this amounts to the property that the index of X_{H} at this singular point is nonzero; equivalently, the index of \nabla W at \left(x_{j}, y_{j}\right) is nonzero.

    In the case of the potential given by Eq. (9.4) for the planar restricted four-body problem, the authors of [15] referred to works of other authors when looking for libration points (see [2,6,19] for example). One assumes m_{S}+m_{J}+m_{P} = 3\sqrt{3} (normalization); then, the sides of the equilateral triangle \mathcal{T} equal \sqrt{3}.

    One draws three circles of radius \sqrt{3} with centers at S, \; J and P and three straight lines which extend the sides of \mathcal{T} . Then, one obtains 16 bounded regions, but only seven of them contain libration points (critical points of W). These (open) domains are as follows: the triangle \mathcal{T} ; three circular sectors \mathcal{D}_{S}, \; \mathcal{D}_{J} and \mathcal{D} _{P} with vertices at S, \; J and P , respectively; three circular triangles \mathcal{O}_{SJ}, \; \mathcal{O}_{JP} and \mathcal{O}_{PS} opposite to the corresponding sides of \mathcal{T} (see [15, Theorem 4.1(i)]).

    Moreover, in the case of equal masses, there are 10 critical points of W, all nondegenerate: the minimum x = y = 0 and three saddle points in \mathcal{T} , one local maximum in each \mathcal{O}_{\#\#} and one saddle in each \mathcal{D}_{\#} (see [15, Lemma 4.2]). In the case of non-equal masses, one calculates the indices of \nabla W along the boundaries of the above domains (called the Brouwer degree) and finds that, in each of the above seven domains, there exists a libration point with a nonzero index of \nabla W.

    By the above analysis of the eigenvalues \lambda _{j} = \pm \sqrt{\mu _{l}}, using the relations of Eq. (9.5) between \mu _{k} and the eigenvalues \beta _{m} of D^{2}W , one concludes the existence of a 1–parameter family of periodic solutions for each of the seven above libration points.

    It would be interesting to study periodic solutions in the 1:1 resonance case, i.e., when a_{2} > 0 and \Delta = 0 ( \mu _{1} = \mu _{2} < 0).

    Finally, we note that the restricted four-body problem was studied from the points of view of the Lyapunov stability of the libration points, i.e., for case (iv); the existence of a family of periodic solutions does not imply it. Here, one uses the KAM theorem and fourth-order Birkhoff normal form. In [20], the generic Lyapunov stability for case (iv) was established for the following situations:

    when m_{S} and m_{J}\ dominate over m_{P} and the libration points lie near P;

    when m_{S} = m_{J} and the libration points lie in a symmetry line of \mathcal{T} ;

    when m_{S} dominates over m_{J} and m_{P} and the libration points lie near the circle with its center at S and radius \left\vert SJ\right\vert.

    Let p_{1}, \ldots, p_{m} be relatively prime positive integers, called the weights. Consider the following action of \mathbb{C}^{\ast } on \mathbb{C}^{m}:

    \begin{equation*} \left( \lambda ,z\right) \longmapsto \lambda \cdot z = \left( \lambda ^{p_{1}}z_{1},\ldots ,\lambda ^{p_{m}}z_{m}\right) . \end{equation*}

    By definition, the quotient of \mathbb{C}^{m}\diagdown 0 by \mathbb{C} ^{\ast } is the weighted projective space,

    \begin{equation} \mathbb{P}_{p_{1},\ldots ,p_{m}} = \left( \mathbb{C}^{\ast }\diagdown 0\right) /\mathbb{C}^{\ast }. \end{equation} (10.1)

    Its complex dimension is m-1 and real dimension is 2m-2.

    From the algebraic geometry point of view (see [11]), it is the projective spectrum \mathrm{Proj}S \left(P\right), \; P = \left(p_{1}, \ldots, p_{m}\right), of the graded polynomial algebra S \left(P\right) = \mathbb{C}\left[ T_{1}, \ldots, T_{m}\right] such that \mathrm{deg} T_{j} = p_{j}. These spaces are important from the complex geometry point of view. In particular, the morphism z_{j}\longmapsto T_{j} = z_{j}^{p_{j}} defines the formal isomorphism

    \begin{equation*} \mathbb{P}_{p_{1},\ldots ,p_{m}}\simeq \mathbb{P}^{m-1}/\mathbb{Z} _{p_{1}}\times \ldots \times \mathbb{Z}_{p_{m}}. \end{equation*}

    But, we are more interested in their topological properties. Consider the standard sphere \mathbb{S}^{2m-1} = \left\{ \sum \left\vert z_{j}\right\vert ^{2} = 1\right\}. Each orbit \mathbb{C}^{\ast }\cdot z, \; z\not = 0, intersects this sphere along a circle. Therefore, we have

    \begin{equation*} \mathbb{P}_{p_{1},\ldots ,p_{m}} = \mathbb{S}^{2m-1}/\mathbb{S}^{1}, \end{equation*}

    where the action of the circle \mathbb{S}^{1} = \left\{ \mathrm{e}^{\mathrm{i} \phi }\right\} is like in Eq. (6.2), i.e., \sigma ^{\phi }\cdot z = \left(\mathrm{e}^{\mathrm{i}p_{1}\phi }z_{1}, \ldots, \mathrm{e}^{\mathrm{i} p_{m}\phi }z_{m}\right).

    Let us find the representation of our weighted projective space as a kind of CW complex (see [14]). This complex consists of 'cells' of (real) dimensions 2m-2, 2m-4, \ldots, 2, 0.

    The 'cell' of maximal dimension \sigma ^{2m-2} is defined by \left\{ z_{m}\not = 0\right\} in \mathbb{S}^{2m-1}. We rotate the variable z_{m} to get \mathrm{arg}z_{m} = 0, i.e., z_{m} = r_{m} > 0. We have the following description of this 'cell':

    \begin{equation*} \sigma ^{2m-2} = D^{2m-2}/\Delta , \end{equation*}

    where D^{2m-2} = \left\{ r_{m} = \sqrt{1-\left\vert z_{1}\right\vert ^{2}-\ldots -\left\vert z_{m-1}\right\vert ^{2}}\right\} and \Delta \subset \mathbb{S}^{1} is the stabilizer of the set \left\{ \mathrm{arg} z_{m} = 0\right\}. The above upper semi-sphere is diffeomorphic with the standard \left(2m-2\right) - dimensional unit ball, and its quotient is singular but with the singular locus of (real) codimension \geq 4 (complex codimension is \geq 2). This cell defines the fundamental cycle of \mathbb{P}_{p_{1}, \ldots, p_{m}}.

    The boundary of \sigma ^{2m-2} is the weighted projective space \mathbb{P} _{p_{1}, \ldots, p_{m-1}}. Note that now the weights can be not relatively prime; but, if p_{j} = r\tilde{p}_{j} , then the spaces \mathbb{P} _{p_{1}, \ldots, p_{m-1}} and \mathbb{P}_{\tilde{p}_{1}, \ldots, \tilde{p} _{m-1}} are isomorphic (see [11]).

    In this way, we successively construct the lower-dimensional cells, as well as cells in the singular locus of the action of the group \Delta on D^{2m-2}. By the dimensional argument, the boundaries of these 'cells' are zero in the homological sense; so, the only nonzero homology groups have even dimensions. We distinguish the \left(2m-4\right) - dimensional cycles \left[ \sigma _{1}^{2m-4}\right], \ldots, \left[ \sigma _{m}^{2m-4}\right] corresponding to the hyperplane sections \left\{ z_{1} = 0\right\}, \ldots, \left\{ z_{m} = 0\right\} respectively. The intersection of any collection of m-1 of the above cycles is nonempty and, hence, \mathrm{cat}\mathbb{P} _{p_{1}, \ldots, p_{m}}\geq m (see Theorem 17 below).

    Recall [14] that the homological length of a manifold M is the maximal number k such that there exist homology classes \rho_{1}, \ldots, \rho_{k}\in H_{\ast }\left(M, \mathbb{Z}\right), \mathrm{dim}\rho_{j} < \mathrm{dim}M, such that their intersection is a nonzero cycle.‡‡

    ‡‡In fact, in [14], one finds the definition of the cohomological length (or the cup length) as the maximal number k such that there exist cohomology classes a_{1}, \ldots, a_{k}, \; \mathrm{dim}a_{j} > 0, such that a_{1}\cup \ldots \cup a_{k}\not = 0. These cohomology classes are Poincaré-dual to the cycles \rho _{1}, \ldots, \rho _{k}.

    But, in the proof of Theorem 17 below given in [14], only intersections of the cycles are used.

    Finally, we note that, in our analysis, we can replace the standard sphere with any manifold of the form

    \begin{equation*} \left\{ c_{1}\left\vert z_{1}\right\vert ^{2}+\ldots +c_{m}\left\vert z_{m}\right\vert ^{2} = 1\right\} \end{equation*}

    for positive constants c_{j}.

    By definition (see [14]), the Lusternik–Schnirelmann category \mathrm{cat}_{X}A of A with respect to X is the smallest number k for which there exist closed subsets A_{1}, \ldots, A_{k} of X such that

    (i) A = \bigcup A_{j} and

    (ii) each A_{j} is contractible in X. If X is connected, then we define \mathrm{cat}_{X}X = \mathrm{cat}X.

    In this work, we have used the following properties of this notion.

    Theorem 16. (Lusternik–Schnirelmann) Let M be a compact closed (without boundary) connected manifold, and let f be a smooth function on M with isolated critical points (or bifurcational points). Then, k\geq \mathrm{cat} M.

    Theorem 17. If M is a manifold, then \mathrm{cat}M is bounded from below by 1 plus the homological length of M.

    For the proofs, we refer the reader to a previous monograph [14].

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    Supported by Polish National Science Centre OPUS Grant No. 2017/25/B/ST1/00931.

    The authors declare that there is no conflict of interest.



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