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Research article

An example in Hamiltonian dynamics

  • Received: 11 November 2023 Revised: 03 April 2024 Accepted: 08 May 2024 Published: 11 June 2024
  • Primary 05C38, 15A15; Secondary 05A15, 15A18

  • We present an example of a three-degrees-of-freedom polynomial Hamilton function with a critical point characterized by indefinite quadratic part with a Morse index 2. This function generates a Hamiltonian system wherein all eigenvalues equal ±i, but it lacks small-amplitude periodic solutions with a period 2π.

    Citation: Henryk Żoła̧dek. An example in Hamiltonian dynamics[J]. Communications in Analysis and Mechanics, 2024, 16(2): 431-447. doi: 10.3934/cam.2024020

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  • We present an example of a three-degrees-of-freedom polynomial Hamilton function with a critical point characterized by indefinite quadratic part with a Morse index 2. This function generates a Hamiltonian system wherein all eigenvalues equal ±i, but it lacks small-amplitude periodic solutions with a period 2π.



    In this paper, we study the Hamiltonian vector field (denoted also by XH)

    ˙qj=Hpj, ˙pj=Hqj, (1.1)

    j=1,,m, generated by the following Hamilton function:

    H=12(|z1|2+|z2|2|z3|2)+(|z2|2+|z3|2)Re(z2z3)+|z1|2Re(ˉz1(z2+εˉz3)), (1.2)

    where zj=qj+ipj, i=1, m=3, and ε is a complex parameter.

    Note that the linear part of system (1.1) at the origin z=0 has eigenvalues ±i, each with multiplicity 3, but with trivial Jordan cells. Thus, all solutions of the corresponding linear system are 2πperiodic. Moreover, the quadratic part

    F=12(|z1|2+|z2|2|z3|2) (1.3)

    of H is indefinite, with the Morse index (the number of negative terms) equal 2.

    Theorem 1. System (1.1), with the Hamilton function (1.2), does not have periodic solutions near z=0 of period 2π for ε in a neighborhood of the origin in CR2 minus a finite collection of real analytic curves.

    Let us remind ourselves of the history of the problem of small-amplitude periodic solutions to autonomous differential systems. It has begun with Lyapunov theorems (see [1,2]); we present the last of them.

    Consider an autonomous differential system

    ˙x=Ax+, x(Rn,0), (1.4)

    with analytic right-hand sides such that the matrix A has pure imaginary eigenvalues

    ±iω1,,±iωm, ωj>0, (1.5)

    m=n2, and one of the frequencies, say ω1, is such that none of the other frequencies is is an integer multiple of it; thus

    ωj/ω1Z,  j2. (1.6)

    One can define formal obstructions to the existence of a 1–parameter family of periodic solutions of period 2π/ω1. For this, one uses the so-called Poincaré–Dulac normal form. There exists a formal invariant surface V tangent to the invariant plane E for A with the eigenvalues ±iω1 and one defines so-called Poincaré–Lyapunov focus quantities, which constitute obstructions to the existence of a formal first integral on V.

    Lyapunov proved that:

       in this case, if all the focus quantities vanish, then there exists a family of periodic solutions x=ϕ(t;c), c(R+,0), of period T(c)2π/ω1 as c0, depending analytically on c, and such that ϕ(t;0)0. In the case n=2, this result was independently proved by Poincaré [3] and is known as the Lyapunov–Poincaré theorem.

    The Lyapunov theorem has attracted the attention of specialists in the Hamiltonian dynamics. Note that, in the Hamiltonian case, the above-mentioned obstructions are absent, because H restricted to the invariant plane V is a suitable first integral. Therefore, inequalities (1.6) are the only assumption of the Hamiltonian version of the Lyapunov theorem.

    Assume that system (1.1) 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 Hamilton function is

    F=12ϵjωj(q2j+p2j), (1.7)

    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 ˉzj=qjipj, we have {zj,zk}={ˉzj,ˉzk}=0 and {zj,ˉzk}=2iδjk. Thus ˙zj={zj,ˉzj}H/ˉzj=2iH/ˉzj.

    Also the resonant monomials g=zαˉzβ form the Birkhoff theorem satisfy {g,F}=0.

    D. Schmidt [4] 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 Hamilton function is indefinite. He did not refer to the Lyapunov theorem.

    A. Weinstein [5] has applied the Lusternik–Schnirelmann category to prove that:

       if the quadratic part F of H is positive definite, i.e., all ϵj=1 in Eq. (1.7), then the vector field XH has at least m=n2 1–parameter families of periodic solutions.

    In [5], it was stated that each hypersurface {H=c}, c>0, contains at least m periodic trajectories; but, in the analytic case, these trajectories form families parametrized by c. The positive definiteness of the F condition is important, because we have the following example from the book [6,Example 9.2] by J. Mawhin and J. Willem.

    Example 1. Let

    H=12(|z1|2|z2|2)+|z|2Re(z1z2). (1.8)

    It generates the system

    ˙z1=iz1(1+2Re(z1z2))+i|z|2ˉz2, ˙z2=iz2(12Re(z1z2))+i|z|2ˉz1.

    One finds

    ddtIm(z1z2)=2[Re(z1z2)]2+|z|4,

    which excludes the existence of nontrivial periodic solutions.

    Next, J. Moser has somehow specified Weinstein's result, but his statements were not precise and his own example [7,Example 2] was confusing; see also my discussion of Moser's approach in Remark 1 in the next section.

    Next, this subject was brought up by specialists in nonlinear functional analysis. In particular, A. Szulkin [8] considered the case when one of the frequencies, say ω1, of multiplicity k, is such that condition (1.6) holds for all ωjω1 and

    ωj=ω1ϵj0, (1.9)

    i.e., the Morse index m(F1) ( = the number of minuses) of the quadratic form F1=F restricted to the invariant subspace E1 associated with the eigenvalues ±iω1 differs from m+(F1)=m(F1). He claimed that:

       then there exists a sequence {γn(t)} of non-constant periodic solutions to the system ˙x=XH(x) tending to γ(t)0 of periods tending to 2π/ω1.

    His (not very long) proof is quite technical, i.e., with many homological groups in an infinite dimensional context.

    There is also another paper [9] by E. N. Dancer and S. Rybicki, where the Szulkin's statement is confirmed and generalized. The corresponding 'bifurcations' are investigated in the context of S1 invariant Hamiltonian systems. There some S1–equivariant indices and degrees are defined and used.

    In the case of the quadratic part of the Hamiltonian (1.2), we have m(F)=2 and m+(F)=4; hence, our Theorem 1 contrasts with the Szulkin's claim.

    Szulkin's statement was used by other specialists in this field: E. P érez-Chavela, A. Gołȩbiewska, S. Rybicki, D. Strzelecki, and A. Ure ña, see [10,11] for example. Fortunately, those results are correct, because the corresponding Hamiltonians restricted to the center manifold are positive-definite.

    Finally, it worth mentioning the paper of S. van Straten [12], where the number of 1–parameter families is calculated in the cases of Hamiltonians of the form

    H=12|z|2+H2d(z,ˉz),

    where H2d is a generic homogeneous polynomial of degree 2d in the Birkhoff normal form.

    In the next section, we recall some tools developed in [2], e.g., we replace our dynamical problem with a suitable algebraic problem, and in the third section, we complete the proof of Theorem 1.

    The aim of this section is to reformulate the problem of small-amplitude periodic solutions to some algebraic problem.

    Proposition 1. The small-amplitude periodic solutions of system (1.1) of period 2π with H given in Eq. (1.2) correspond to small solutions of the following system:

    r2Im(u+εˉv)=0uRe(2uv3r(u+εˉv))+(|u|2+|v|2)ˉv+r3=0vRe(2uv+3r(u+εˉv))+(|u|2+|v|2)ˉu+εr3=0 (2.1)

    for

    r>0

    and complex u and v.

    Before proving this statement, we present some tools introduced in [2].

    Recall that the Hamiltonian system generated by a Hamilton function with the quadratic part (1.7) takes the form

    ˙zj=λjzj+=iϵjωjzj+, ˙vj=λjvj+=iϵjωjvj+,

    where zj=qj+ipj and vj=qjipj; in the real domain, we have vj=ˉzj. We assume that the right-hand sides are analytic.

    G. Birkhoff [13] proved that there exists a formal symplectic change (z,v)(Z,V) which reduces the Hamilton function to the following Birkhoff normal form:

    F(Z,V)+ak;lZkVl,

    where the summation runs over the pairs (k;l)=(k1,,km;l1,,lm)Zm0×Zm0 such that the resonant relations

    (kl,λ)=(kjlj)λj=0

    hold, and |k|+|l|3.

    Note that the Hamiltonian (1.1) is in the Birkhoff normal form.

    From the dynamical point of view, the property of H being in the Birkhoff normal form means that the Hamiltonian flow {gtXH} commutes with the Hamiltonian flow {gsXF} generated by the quadratic part F of H; thus, {H,F}=0. The second flow is the following:

    zjeiϵjωjszj,j=1,,m.

    Assume firstly that

    ω1=ω2==ωm=ω, (2.2)

    i.e., all solutions to the corresponding linear system defined by F are 2π/ωperiodic, and

    ϵ1=1.

    Moreover, assume that the Birkhoff normal form is analytic (or that the analytic Hamilton function is in the Birkhoff normal form). We look for periodic solutions close to periodic solutions of the linear approximation of period 2π/ω.

    Such a periodic solution, of period T=2π/η with ηω, is of the form zj(t)cjeiϵjηt, where cj are small constants, not all of which equal zero. Assume that c10. The angle θ=argz1 is of the form θ(t)θ0+ϵ1ηt+θ1(t) (where θ1(t) is small, 2π/η periodic) and varies along the whole S1=R/Z. Therefore, the phase curves are graphs of functions of θ:

    |z1|=r=r(θ), zj=zj(θ), (j>1),

    which are 2πperiodic.

    We define the Poincaré return map

    P:SS,

    where S=(R+,0)×(Cm1,0) is the Poincaré section. This amounts to putting z1=reiθ, writing down equations for dr/dθ=˙r/˙θ and for dzj/dθ, j>1, (elimination of time), and evaluating the solution after the new time 2π of a corresponding initial value problem. Since ˙θϵ1ω0, the return time to the section S is 2π/ω, and the map P is well defined.

    The fixed points of this map correspond to periodic orbits of the Hamiltonian system of period 2π/ω. Other periodic orbits of P correspond to periodic orbits of XH of period being approximately a multiple of 2π/ω. Moreover, due to the analyticity of the right-hand sides of the differential equations for r and zj's, the equilibrium points of the return system are of two types:

       equal (r,z)=(0,0), i.e., there are no nontrivial periodic solutions; or

       a real analytic subvariety of (R+×Cm1,(0,0)) of positive dimension, usually a 1–dimensional curve.

    Now we slightly change our point of view. We apply the following change:

    z1=reiθ, zj=wjeiϵjθ, (j>1), (2.3)

    where r0 and wj(C,0). This is the same as the action of the flow {gsXF} generated by the quadratic part F of H (see the previous section). In fact, the variables r and wj are invariant for the flow {gsXF}.

    The property of commuting of the two Hamiltonian flows (as H is in the Birkhoff normal form) implies that corresponding differential equations, after elimination of the time, take the form

    drdθ=Π(r,w,ˉw), dwjdθ=Λj(r,w,ˉw), (2.4)

    i.e., the right-hand sides do not depend on θ. We call system (2.4) the return system. We define the twisted Poincaré map Ptw via solutions of Eqs. (2.4) after the new time 2π.

    Of course, the fixed points of the twisted Poincaré map are the fixed points of the twisted Poincaré map. Among them are the equilibrium points of the return system, i.e., defined by the equations Π=Λ2==Λm=0.

    Assume now that the hyperplane {z1=0} is invariant; this means that the right-hand side of the equation for ˙z1 lies in the ideal generated by z1 and ˉz1. For the return system, it means that ˙r is divided by r and ˙θ=ω+; thus, Π(r,w)=r˜Π(r,w), with analytic ˜Π. (The opposite case is more complicated.)

    We claim that:

       Under the invariance of the hyperplane {z1=0} assumption, the fixed points, being the equilibrium points of the return system, are the only fixed points of the twisted Poincaré map in a neighborhood of the origin.

    Indeed, other fixed points of Ptw would correspond to (nontrivial) closed phase curves of the return vector field; moreover, of period 2π/k for an integer k. In fact, such periodic curves should lie in analytic families, {γc}c(R+,0) (by the analyticity of the right-hand sides). The period of γc is calculated as follows:

    T(c)=γcdθ=γcdrΠ(r,w).

    But Π(r,w)=r˜Π(r,w) and is of high order, 2. If the variables r, wj, ˉwj at γc are of given orders of c, e.g., rcα0, then T(c)cβ as c0.

    By the way, such solutions would not approximate the corresponding solutions of the linear system (like in the Weinstein theorem).

    Recall that the solutions to the system Π=Λ2==Λm=0 are of two types:

       (i) equal (r,z)=(0,0); or

       (ii) a real analytic variety of positive dimension, usually a 1–dimensional curve.

    Example 2. (Example 1 revisited). The Hamiltonian (1.8) is in Birkhoff normal form. The change (2.3) means

    z1=reiθ, z2=weiθ.

    We have the following return system:

    drdθ=r(r2+|w|2)Imwr+(3r2+|w|2)Rew, dwdθ=i(5r2+|w|2)wRew+r2(r2+|w|2)r+(3r2+|w|2)Rew.

    Its equilibrium points are defined by: either r=0, and then w=0; or Imw=0, i.e., Rew=w, and hence (5r2+|w|2)|w|2+r2(r2+|w|2)=0, and then again r=w=0. So, there are no nontrivial equilibrium points.

    But the differential system from Example 1 does not have invariant planes; so, one needs an additional argument provided in Example 1.

    Recall that the Hamiltonian (1.1) is in Birkhoff normal form. It generates the system

    ˙z1=i{z1(1+2Reˉz1(z2+εˉz3))+|z1|2(z2+εˉz3)},˙z2=i{z2(1+2Re(z2z3))+(|z2|2+|z3|2)ˉz3+|z1|2z1},˙z3=i{z3(1+2Re(z2z3))+(|z2|2+|z3|2)ˉz2+ε|z1|2ˉz1}. (2.5)

    One can see that neither of the coordinate 4–spaces {z2=0}, {z3=0} nor of the coordinate planes {zj=zk=0} is invariant, but the subspace {z1=0} is invariant.

    However, system (2.5) restricted to the subspace {z1=0}, i.e., the last two equations, is the same as the system from Example 1, which is without periodic solutions.

    Therefore, we can introduce the variables r0, θ, u, v (analogues of the variables (2.3)) via the formulas

    z1=reiθ, z2=ueiθ, z3=veiθ,

    or r=|z1|, θ=argz1, u=ˉz1z2/|z1|, v=z1z3/|z1|. In fact, we have r>0. We get

    ˙θ=1+3rRe(u+εˉv) (2.6)

    and

    ˙r=r2Im(u+εˉv),˙u=i{uRe(2uv3r(u+εˉv))+(|u|2+|v|2)ˉv+r3},˙v=i{vRe(2uv+3r(u+εˉv))+(|u|2+|v|2)ˉu+εr3}. (2.7)

    By the arguments given in Section 2.2 the small-amplitude periodic solutions to our Hamiltonian system of period 2πare in one-to-one correspondence with the equilibrium points of the above system.

    But the right-hand sides of system (2.7) are the left-hand sides of Eqs. (2.1).

    Note also that the period of a periodic solution z1(t)r0eit, z2(t)u0eit, z2(t)v0eit, corresponding to an eventual equilibrium point (r0,u0,v0), equals

    T=2π0dθ1+3r(θ)Re(u(θ)+εˉv(θ))=2π1+3r0Re(u0+εˉv0),

    where r(θ)r0, u(θ)u0 and v(θ)v0 are solutions to the corresponding return system with the initial condition r(0)=r0,u(0)=u0,v(0)=v0.

    Finally, since the right-hand sides of Eqs. (2.7) are homogeneous, eventual equilibrium points are not isolated. They should form 1–dimensional straight semi-lines, corresponding to 1–parameter families of periodic solutions to the Hamiltonian system.

    We complete this section with a short discussion of the additional elements of the novel approach to the problem started in [2], which are potentially interesting to the reader.

    In [2,Proposition 6], it was proved that a Hamiltonian in the Birkhoff normal form, under assumptions (2.2), is invariant with respect to the following action of circle S1:

    z=(z1,,zm)σϕ(z)=(eiϵ1ϕz1,,eiϵmϕzm), 0ϕ2π. (2.8)

    Action (2.8) is symplectic, it is a periodic phase flow generated by the Hamilton function F(z,ˉz), i.e., the homogeneous quadratic part of H.

    We deal with the classical phenomenon called symplectic reduction (see [14]). The function F, called momentum mapping, is the first integral for the Hamiltonian vector field XH system. So, we take the invariant manifolds

    Mf={F(z,ˉz)=f}, (2.9)

    and their quotients

    Nf=Mf/S1 (2.10)

    of dimension 2m2. The latter varieties are smooth and equipped with a natural symplectic structure and support vector fields Yf obtained from the Hamiltonian vector field XH. Each vector field Yf is Hamiltonian with the Hamilton function πH=πF+πG, where π:MfNf is the projection and

    G=HF (2.11)

    contain higher order terms.

    The variables (r,w)=(r(1),w(1)) from Eqs. (2.3) form a local chart in the quotient variety Nf. Other local charts, (r(l),w(l)), l>1, are defined via the formulas

    zl=r(l)eiθ, zj=w(l)jeiϵjθ/ϵl, (jl),

    where r(l)0 and w(l)j(C,0). With each such chart, we associate a corresponding return system, like system (2.4), whose equilibrium locus either reduces to (r(l),w(l))=(0,0) or is a real analytic set of the positive dimension (corresponding to some families of periodic solutions with period 2π/ω).

    In [2,Proposition 8], it was proved that the periodic orbits of XH in Mf of period 2π/ω correspond to the critical points of the function πG on Nf.

    In the case of a definite (say positive definite) momentum map F, the quotient varieties Nf, f>0, are complex projective spaces Pm1. Here, the number of critical points of πG on Nf is estimated from below using the Schnirelmann–Lusternik category; this is the estimate from the Weinstein theorem. Moreover, we also have the Poincar é–Hopf formula at our disposal.

    Weinstein skillfully constructed a function on the level hypersurface Lh={H(z,ˉz)=h}, h>0, which has critical locus at the set of periodic phase curves of XH in Lh. His construction is not direct and involves many technical details. The approach from [2] is more direct.

    But in the case of indefinite Hamiltonians, one cannot use the above topological tools. Here, the varieties Nf are non-compact, and it is easy to find vector fields without singular points on them.

    Following [5,Proof of Theorem 2.1], we consider the following equivalence relation on the set {ω1,,ωm} of frequencies:

    ωiωj iff ωi/ωjQ.

    Assume firstly that there is only one equivalent class and H is in analytic Birkhoff normal form.

    So, we can write

    ωj=pjω0, pjN, gcd (2.12)

    In [2,Proposition 6], it was proved that in this case the Birkhoff normal form is invariant with respect to the following action of the circle (generalization of action (2.8)):

    (2.13)

    This action is also symplectic and its periodic phase flow is generated by the Hamilton function , the homogeneous quadratic part of

    Again, we deal with the symplectic reduction. The function , called momentum mapping, is the first integral for the vector field . So, we take the invariant manifolds

    and their quotients

    of dimension The latter varieties are equipped with a natural symplectic structure and support vector fields obtained from the Hamiltonian vector field Each vector field is Hamiltonian with the Hamilton function where is the projection and

    contain higher order terms.

    But now the quotient varieties may be singular, but with with normal singularities (quotients of by an action of a finite group). For example, in the cases of positive definite , the sets are diffeomorphic with and their quotients are the weighted projective spaces. Nevertheless, due to normality, the corresponding functions , and the vector field are well defined.

    We have the local charts defined by

    and corresponding return systems whose equilibrium points correspond to periodic orbits of with period

    Those equilibrium points correspond to the critical points of the function on Again, in the case of compact the number of critical points of on is estimated from below using the Schnirelmann–Lusternik category. In the non-compact case we do not have such tools.

    Consider now the case of several equivalent classes for the collection of frequencies.

    For each equivalence class we have a linear subspace invariant for the linear part of the system, but we can say more. In [2,Proposition 3], it was proved that for each such class, there exists a formal invariant submanifold tangent to at the origin.

    Remark 1. Here I would like to comment on Moser's statement mentioned in the introduction. Let us recall it (compare [7,Theorem 4]):

    'Assume that , where and are invariant subspaces of the matrix defining the linear part of , such that all solutions in of the linear system have the same period , while no nontrivial solution in has this period. Assume also that the quadratic part of restricted to is positive definite. Then, on each energy surface and small, the number of periodic orbits of is at least .'

    In [2], I have expressed the opinion that this statement must be wrong. My argument relied upon an analysis of Moser's example ([7,Example 2]) with the Hamiltonian , which is not in the Birkhoff normal form and leads to wrong statements. Recently I realized that the latter example was given with a mistake, and the correct Hamiltonian is

    where is an integer; note that it is in the Birkhoff normal form. We have

    The quadratic part is positive definite on the plane with periodic solutions for the linear system of period other periodic solutions in the plane have period But the corresponding nonlinear system has only periodic solutions in

    Next, I have assumed that the period in Moser's theorem is the minimal period, but the reviewer of this work has pointed out that I could be wrong. Plausibly, Moser had in mind an invariant subspace corresponding to one of Weinstein's equivalence classes of frequencies. Otherwise, he could not claim the consequence of Weinstein's theorem from his statement.

    Indeed, consider the case with three frequencies: , and Then we have two invariant linear subspaces: associated with and and associated with and all solutions of the linear system in have period , and all solutions in have period One can show that there exist corresponding formal invariant subspaces and for moreover, we have the invariant subspaces and From the original statement of Moser's theorem it follows that there exist many periodic orbits in and in but all of them could lie in (provided ).

    Finally, I want to note that in [2,Theorem 5], I have specified the Weinstein theorem. It is associated with the ordering of the different frequencies in one of Weinsten's equivalence classes.

    Remark 2. (a correction) I would like to use this opportunity to make a correction to my previous paper [2]. Namely, Propositions 3 and 4 in Section 6, about the analytic property of the invariant submanifolds, are not true, at least without an additional assumption. That assumption is the center condition, i.e., that there is a family of periodic solutions at the formal level. Of course, everything is OK when the Poincaré–Dulac–Birkhoff normal forms are analytic.

    In the general case, in the proofs of Theorem 4 (in Section 8.2), Theorem 5 (in Section 8.3), and Theorem 7 (in Section 8.1), one should first use the approximation argument. It relies on the fact that the general (topological) properties of an analytic curve, defined by the fixed-point equation for the twisted Poincaré map, are determined by the polynomial approximation of this map. Therefore, we can approximate the corresponding system by a truncated Poincar é–Dulac or Birkhoff normal form. Then the corresponding invariant manifolds become analytic.

    Moreover, in [2,Eq. (6.2)], the action of (in the case of one equivalence class) was defined incorrectly; the correct formula is Eq. (2.13)

    We shall show that system (2.1) does not have nontrivial solutions.

    Since Eqs. (2.1) are homogeneous and we assume , we put

    (dehomogenization); thus, we replace with and with . We get the algebraic system

    (3.1)

    We treat Eqs. (3.1) as a system of five real algebraic equations on six real variables , i.e., in . It defines an algebraic variety . The projection of the variety to the plane is an semi-algebraic variety consisting of those parameters for which there exists a parameter family of periodic solutions to the perturbed Hamiltonian system near . For 's outside , there are no such periodic solutions. Our goal is to prove that the intersection of with a neighborhood of is 1–dimensional, a union of germs of irreducible curves (its components).

    For this, it is enough to show that the part of the variety above a neighborhood of is a dimensional algebraic curve, a union of irreducible local curves (components).

    Assume firstly that

    (3.2)

    Lemma 1. Under assumption (3.2), system (3.1) has three solutions:

    ;

    Proof. Indeed, by the first of Eqs. (3.1) we can assume that is real. Then the third of these equations factorizes,

    If then the second of Eqs. (3.1) gives the solution .

    Otherwise, is also real and we have

    Then the second equation gives , i.e., , with the additional values But only for the quantity is positive and gives form the thesis of the lemma.

    Let now

    (3.3)

    We claim that no solution to system (3.1) bifurcates as approaches zero along a generic ray.

    Lemma 2. No solution bifurcates from infinity. Namely, there exist and such that system (3.1), with does not have solutions in

    Proof. Let us sum up the left-hand side of the second of Eqs. (3.1) multiplied by and the third multiplied by We get

    where the dots mean lower-degree terms. This expression is separated from for large and small .

    Therefore, any eventual solution could bifurcate only from one of the points

    Lemma 3. No solution bifurcates from , as varies in a neighborhood of outside a finite number of real analytic curves. Namely, for any there exists and a finite collection of germs of real analytic curves such that, if and , then system (3.1) does not have solutions in

    Proof. Putting

    (3.4)

    with we can treat Eqs. (3.1), near as a system of five real analytic equations in six real variables and , i.e., in .

    One solves system (3.1) by successive approximations; first, considering the linear approximation of system (3.1), we find initial (linear) terms of the Puiseux type expansions of a corresponding component . Next, one finds several further terms using higher-order terms in system (3.1) and the linear approximation of etc. In the expansion of the curves and , the corresponding coefficients will not be exact, only approximate, like the values of and in Lemma 1.

    In Case we arrive at the following system:

    where the dots in the second equation mean quadratic and cubic terms containing and the dots in the third equation mean cubic terms with The linear parts of these equations define the plane

    note that and have proportional linear parts. This plane is parametrized by and we can assume that and are functions of

    Taking into account nonlinear terms in the second and third equations, with and we get

    (3.5)

    where now the dots mean higher-order terms in Then the first equation implies

    (3.6)

    Eqs (3.5)–(3.6) define the curve . It has three components: one is defined by (and Eqs. (3.5)), one by and one by . The projection of has at most three components, defined by:

    (3.7)

    Consider Case By abuse of notation, we put

    (3.8)

    and replace , and , by

    (3.9)

    respectively.

    The notation above and below will mean that the coefficients are not exact, only approximate. Dealing with the exact values given in Lemma 1 would lead only to much more complicated expressions without affecting the conclusions.

    We have where are homogeneous polynomials of degree . Namely,

    First, we solve the corresponding linear equations. From , we express as linear function of , and substitute it to then we get as a linear function of , and also becomes expressed via Finally, we substitute these and to we get a linear complex equation for by comparing the real and imaginary parts, we get

    Then we find

    We see that and become functions of and

    Moreover, this suggests that in a linear approximation. If were not equal for all then the condition would imply a linear restriction for 's.

    Next, we expand the solutions and in powers of modulo For this, we firstly express , and via , and then we repeat calculations for , and We get

    Next, we find

    We see that the real parts of and agree, but the imaginary parts disagree:

    (3.10)

    This suggests that the projected curve has two components, one defined by and, the other defined by also, the curve has two components. (If also the real parts of and disagreed, then we would have another restriction on 's.)

    Consider now Case We follow the method from the previous case.

    Using notations analogous to (3.8)–(3.9), we get

    The solution of the linear equations gives

    and

    Again, we find that and become functions of and and that in linear approximation.

    Substituting the above values into quadratic parts of our equations, we find

    Then the solutions up to are the following:

    We see that the real parts of and agree, but

    (3.11)

    Like in the previous case, the curve and its projection have two components.

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

    I would like to thank the reviewer for useful comments and suggestions.

    The author declares there is no conflict of interest.



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