An example in Hamiltonian dynamics

1.   Introduction

In this paper, we study the Hamiltonian vector field (denoted also by $X_{H}$)

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

where $z_{j} = q_{j}+\mathrm{i}p_{j},$ $\mathrm{i} = \sqrt{-1}$, $m = 3$, and $\varepsilon$ is a complex parameter.

Note that the linear part of system (1.1) at the origin $z = 0$ has eigenvalues $\pm \mathrm{i}$, each with multiplicity 3, but with trivial Jordan cells. Thus, all solutions of the corresponding linear system are $2\pi -$periodic. Moreover, the quadratic part

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 $\approx 2\pi$ for $\varepsilon$ in a neighborhood of the origin in $\mathbb{C\simeq R}^{2}$ 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

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

$m = \frac{n}{2},$ and one of the frequencies, say $\omega _{1},$ is such that none of the other frequencies is is an integer multiple of it; thus

One can define formal obstructions to the existence of a 1–parameter family of periodic solutions of period $\approx 2\pi /\omega _{1}.$ For this, one uses the so-called Poincaré–Dulac normal form. There exists a formal invariant surface $\mathcal{V}$ tangent to the invariant plane $\mathcal{E}$ for $A$ with the eigenvalues $\pm \mathrm{i}\omega _{1}$ and one defines so-called Poincaré–Lyapunov focus quantities, which constitute obstructions to the existence of a formal first integral on $\mathcal{V}$.

Lyapunov proved that:

   in this case, if all the focus quantities vanish, then there exists a family of periodic solutions $x = \phi \left(t; c\right),$ $c\in \left(\mathbb{R}_{+}, 0\right),$ of period $T\left(c\right) \rightarrow 2\pi /\omega _{1}$ as $c\rightarrow 0,$ depending analytically on $c$, and such that $\phi \left(t; 0\right) \equiv 0.$ 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 $\mathcal{V}$ 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 $\pm \mathrm{i}\omega _{1}, \ldots, \pm \mathrm{i}\omega _{m},$ $\omega _{j} > 0.$ Assuming $H\left(0\right) = 0$ the leading part of the Taylor expansion of the Hamilton function is

where $q_{j}, p_{j}$ are suitable canonical variables, i.e., with the Poisson brackets $\left\{ p_{i}, q_{j}\right\} = \delta _{ij},$ and $\epsilon _{j} = \pm 1$ are well defined signs. ^*

^*We have $\dot{f} = \left\{ f, H\right\}$ in the case of a general Hamiltonian system with the symplectic structure defined by a Poisson bracket.

In particular, for $z_{j} = q_{j}+\mathrm{i}p_{j}$ and $\bar{z}_{j} = q_{j}- \mathrm{i}p_{j},$ we have $\left\{ z_{j}, z_{k}\right\} = \left\{ \bar{z}_{j}, \bar{z}_{k}\right\} = 0$ and $\left\{ z_{j}, \bar{z}_{k}\right\} = 2\mathrm{i} \delta _{jk}.$ Thus $\dot{z}_{j} = \left\{ z_{j}, \bar{z}_{j}\right\} \partial H/\partial \bar{z}_{j} = 2\mathrm{i\cdot }\partial H/\partial \bar{z}_{j}.$

Also the resonant monomials $g = z^{\alpha }\bar{z}^{\beta }$ form the Birkhoff theorem satisfy $\left\{ g, F\right\} = 0.$

D. Schmidt ^[4] studied 1–parameter families of periodic solutions for such systems with two degrees of freedom in the cases of resonant frequencies $\omega _{1}$ and $\omega _{2}.$ His analysis was mainly focused on definite Hamiltonians, i.e., when $\epsilon _{1} = \epsilon _{2},$ 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 $\epsilon _{j} = 1$ in Eq. (1.7), then the vector field $X_{H}$ has at least $m = \frac{n}{2}$ 1–parameter families of periodic solutions.

In ^[5], it was stated that each hypersurface $\left\{ H = c\right\},$ $c > 0,$ 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

It generates the system

One finds

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 $\omega _{1}$, of multiplicity $k$, is such that condition (1.6) holds for all $\omega _{j}\not = \omega _{1}$ and

i.e., the Morse index $m^{-}\left(F_{1}\right)$ ( = the number of minuses) of the quadratic form $F_{1} = F$ restricted to the invariant subspace $\mathcal{E}_{1}$ associated with the eigenvalues $\pm \mathrm{i}\omega _{1}$ differs from $m^{+}\left(F_{1}\right) = m^{-}\left(-F_{1}\right).$ He claimed that:

   then there exists a sequence $\left\{ \gamma _{n}\left(t\right) \right\}$ of non-constant periodic solutions to the system $\dot{x} = X_{H}\left(x\right)$ tending to $\gamma \left(t\right) \equiv 0$ of periods tending to $2\pi /\omega _{1}$.^†

^†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 $\mathbb{S}^{1}-$ invariant Hamiltonian systems. There some $\mathbb{S}^{1}$–equivariant indices and degrees are defined and used.

In the case of the quadratic part of the Hamiltonian (1.2), we have $m^{-}\left(F\right) = 2$ and $m^{+}\left(F\right) = 4;$ 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

where $H_{2d}$ 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.

2.   Birkhoff normal form, twisted Poincaré map and symplectic reduction

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 $\approx 2\pi$ with $H$ given in Eq. (1.2) correspond to small solutions of the following system:

for

and complex $u$ and $v.$

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

2.1.   Birkhoff normal form

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

where $z_{j} = q_{j}+\mathrm{i}p_{j}$ and $v_{j} = q_{j}-\mathrm{i}p_{j};$ in the real domain, we have $v_{j} = \bar{z}_{j}.$ We assume that the right-hand sides are analytic.

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

where the summation runs over the pairs $\left(k; l\right) = \left(k_{1}, \ldots, k_{m}; l_{1}, \ldots, l_{m}\right) \in \mathbb{Z}_{\geq 0}^{m}\times \mathbb{Z}_{\geq 0}^{m}$ such that the resonant relations

hold, and $\left\vert k\right\vert +\left\vert l\right\vert \geq 3.$

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 $\left\{ g_{X_{H}}^{t}\right\}$ commutes with the Hamiltonian flow $\left\{ g_{X_{F}}^{s}\right\}$ generated by the quadratic part $F$ of $H;$ thus, $\left\{ H, F\right\} = 0.$ The second flow is the following:

2.2.   Return system

Assume firstly that

i.e., all solutions to the corresponding linear system defined by $F$ are $2\pi /\omega -$periodic, and

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 $\approx 2\pi /\omega.$

Such a periodic solution, of period $T = 2\pi /\eta$ with $\eta \approx \omega,$ is of the form $z_{j}\left(t\right) \approx c_{j}\mathrm{e}^{\mathrm{i} \epsilon _{j}\eta t},$ where $c_{j}$ are small constants, not all of which equal zero. Assume that $c_{1}\not = 0.$ The angle $\theta = \arg z_{1}$ is of the form $\theta \left(t\right) \approx \theta _{0}+\epsilon _{1}\eta t+\theta _{1}\left(t\right)$ (where $\theta _{1}(t)$ is small, $2\pi /\eta -$ periodic) and varies along the whole $\mathbb{S}^{1} = \mathbb{R}/\mathbb{Z}$. Therefore, the phase curves are graphs of functions of $\theta :$

which are $2\pi -$periodic.

We define the Poincaré return map

where $S = \left(\mathbb{R}_{+}, 0\right) \times \left(\mathbb{C} ^{m-1}, 0\right)$ is the Poincaré section. This amounts to putting $z_{1} = r\mathrm{e}^{\mathrm{i}\theta },$ writing down equations for $\mathrm{d }r/\mathrm{d}\theta = \dot{r}/\dot{\theta}$ and for $\mathrm{d}z_{j}/\mathrm{d }\theta$, $j > 1,$ (elimination of time), and evaluating the solution after the new time $2\pi$ of a corresponding initial value problem. Since $\dot{\theta }\approx \epsilon_{1} \omega \neq 0$, the return time to the section $S$ is $\approx 2\pi/\omega$, and the map $P$ is well defined.

The fixed points of this map correspond to periodic orbits of the Hamiltonian system of period $\approx 2\pi /\omega.$ Other periodic orbits of $P$ correspond to periodic orbits of $X_{H}$ of period being approximately a multiple of $2\pi /\omega.$ Moreover, due to the analyticity of the right-hand sides of the differential equations for $r$ and $z_{j}$'s, the equilibrium points of the return system are of two types:

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

   a real analytic subvariety of $\left(\mathbb{R}_{+}\times \mathbb{C} ^{m-1}, \left(0, 0\right) \right)$ of positive dimension, usually a 1–dimensional curve.

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

where $r\geq 0$ and $w_{j}\in \left(\mathbb{C}, 0\right)$. This is the same as the action of the flow $\left\{ g_{X_{F}}^{s}\right\}$ generated by the quadratic part $F$ of $H$ (see the previous section). In fact, the variables $r$ and $w_{j}$ are invariant for the flow $\left\{ g_{X_{F}}^{s}\right\}$.

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

i.e., the right-hand sides do not depend on $\theta.$ We call system (2.4) the return system. We define the twisted Poincaré map $P^{\mathrm{tw}}$ via solutions of Eqs. (2.4) after the new time $2\pi.$

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 $\Pi = \Lambda _{2} = \ldots = \Lambda _{m} = 0.$

Assume now that the hyperplane $\{ z_1 = 0 \}$ is invariant; this means that the right-hand side of the equation for $\dot{z}_1$ lies in the ideal generated by $z_1$ and $\bar{z}_1$. For the return system, it means that $\dot{r}$ is divided by $r$ and $\dot{\theta} = \omega +\ldots$; thus, $\Pi (r, w) = r\widetilde{\Pi}(r, w)$, with analytic $\widetilde{\Pi}$. (The opposite case is more complicated.)

We claim that:

   Under the invariance of the hyperplane $\{ z_1 = 0 \}$ 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 $P^{\mathrm{tw}}$ would correspond to (nontrivial) closed phase curves of the return vector field; moreover, of period $2\pi /k$ for an integer $k.$ In fact, such periodic curves should lie in analytic families, $\left\{ \gamma _{c}\right\} _{c\in (\mathbb{R} _{+}, 0)}$ (by the analyticity of the right-hand sides). The period of $\gamma _{c}$ is calculated as follows:

But $\Pi \left(r, w\right) = r\widetilde{\Pi}(r, w)$ and is of high order, $\geq 2.$ If the variables $r, \ w_j, \ \bar{w}_j$ at $\gamma_c$ are of given orders of $c$, e.g., $r\sim c^{\alpha}\rightarrow 0$, then $\mathcal{T}\left(c\right) \sim c^{-\beta}\rightarrow \infty$ as $c\rightarrow 0.$

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 $\Pi = \Lambda _{2} = \ldots = \Lambda _{m} = 0$ are of two types:

   (i) equal $\left(r, z\right) = \left(0, 0\right)$; 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

We have the following return system:

Its equilibrium points are defined by: either $r = 0$, and then $w = 0;$ or $\mathrm{Im}w = 0$, i.e., $\mathrm{Re}w = w,$ and hence $\left(5r^{2}+\left\vert w\right\vert ^{2}\right) \left\vert w\right\vert ^{2}+r^{2}\left(r^{2}+\left\vert w\right\vert ^{2}\right) = 0$, 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.

2.3.   Proof of Proposition 1

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

One can see that neither of the coordinate 4–spaces $\left\{ z_{2} = 0\right\},$ $\left\{ z_{3} = 0\right\}$ nor of the coordinate planes $\left\{ z_{j} = z_{k} = 0\right\}$ is invariant, but the subspace $\left\{ z_{1} = 0\right\}$ is invariant.

However, system (2.5) restricted to the subspace $\left\{ z_{1} = 0\right\},$ 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 $r\geq 0,$ $\theta$, $u,$ $v$ (analogues of the variables (2.3)) via the formulas

or $r = \left\vert z_{1}\right\vert,$ $\theta = \arg z_{1}$, $u = \bar{z} _{1}z_{2}/\left\vert z_{1}\right\vert,$ $v = z_{1}z_{3}/\left\vert z_{1}\right\vert$. In fact, we have $r > 0.$ We get

and

By the arguments given in Section 2.2 the small-amplitude periodic solutions to our Hamiltonian system of period $\approx 2\pi$are 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 $z_{1}(t)\approx r_{0} \mathrm{e}^{\mathrm{i}t}$, $z_{2}\left(t\right) \approx u_{0}\mathrm{e}^{ \mathrm{i}t},$ $z_{2}\left(t\right) \approx v_{0}\mathrm{e}^{-\mathrm{i}t}$, corresponding to an eventual equilibrium point $\left(r_{0}, u_{0}, v_{0}\right),$ equals

where $r\left(\theta \right) \equiv r_{0},$ $u\left(\theta \right) \equiv u_{0}$ and $v\left(\theta \right) \equiv v_{0}$ are solutions to the corresponding return system with the initial condition $r\left(0\right) = r_{0}, u\left(0\right) = u_{0}, v\left(0\right) = v_{0}.$

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. □

2.4.   Symplectic reduction

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 $\mathbb{S}^{1}:$

Action (2.8) is symplectic, it is a periodic phase flow generated by the Hamilton function $F\left(z, \bar{z}\right),$ 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 $X_{H}$ system. So, we take the invariant manifolds

and their quotients

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

contain higher order terms.

The variables $\left(r, w\right) = \left(r^{\left(1\right) }, w^{\left(1\right) }\right)$ from Eqs. (2.3) form a local chart in the quotient variety $N_{f}.$ Other local charts, $\left(r^{\left(l\right) }, w^{\left(l\right) }\right),$ $l > 1$, are defined via the formulas

where $r^{\left(l\right) }\geq 0$ and $w_{j}^{\left(l\right) }\in \left(\mathbb{C}, 0\right)$. With each such chart, we associate a corresponding return system, like system (2.4), whose equilibrium locus either reduces to $\left(r^{\left(l\right) }, w^{\left(l\right) }\right) = \left(0, 0\right)$ or is a real analytic set of the positive dimension (corresponding to some families of periodic solutions with period $\approx 2\pi /\omega).$

In [2,Proposition 8], it was proved that the periodic orbits of $X_{H}$ in $M_{f}$ of period $\approx 2\pi /\omega$ correspond to the critical points of the function $\pi _{\ast }G$ on $N_{f}.$

In the case of a definite (say positive definite) momentum map $F$, the quotient varieties $N_{f},$ $f > 0,$ are complex projective spaces $\simeq \mathbb{P}^{m-1}$. Here, the number of critical points of $\pi _{\ast }G$ on $N_{f}$ 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 $L_{h} = \left\{ H\left(z, \bar{z}\right) = h\right\},$ $h > 0,$ which has 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. The approach from ^[2] is more direct.

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

2.5.   Generalizations

Following [5,Proof of Theorem 2.1], we consider the following equivalence relation on the set $\left\{ \omega _{1}, \ldots, \omega _{m}\right\}$ of frequencies:

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

So, we can write

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 $\mathbb{S}^{1}$ (generalization of action (2.8)):

This action is also symplectic and its periodic phase flow is generated by the Hamilton function $F\left(z, \bar{z}\right)$, the homogeneous quadratic part of $H.$

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

and their quotients

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

contain higher order terms.

But now the quotient varieties $N_{f}$ may be singular, but with with normal singularities (quotients of $\left(\mathbb{C}^{k}, 0\right)$ by an action of a finite group). For example, in the cases of positive definite $F$, the sets $M_{f},$ $f > 0,$ are diffeomorphic with $\mathbb{S}^{2m-1}$ and their quotients are the weighted projective spaces. Nevertheless, due to normality, the corresponding functions $\pi _{\ast }F,$ $\pi _{\ast }G$, and the vector field $Y_{f}$ are well defined.

We have the local charts defined by

and corresponding return systems whose equilibrium points correspond to periodic orbits of $X_{H}$ with period $\approx 2\pi /\omega _{k}.$

Those equilibrium points correspond to the critical points of the function $\pi _{\ast }G$ on $N_{f}.$ Again, in the case of compact $N_{f},$ the number of critical points of $\pi _{\ast }G$ on $N_{f}$ 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 $C_{\nu }$ we have a linear subspace $\mathcal{E}_{\nu }$ 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 $\mathcal{V}_{\nu }$ tangent to $\mathcal{E}_{\nu }$ 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 $\mathbb{R}^{2m} = \mathcal{E}\oplus \mathcal{F}$, where $\mathcal{E}$ and $\mathcal{F}$ are invariant subspaces of the matrix $A$ defining the linear part of $X_{H}$, such that all solutions in $\mathcal{E} \ $of the linear system have the same period $T > 0$, while no nontrivial solution in $\mathcal{F}$ has this period. Assume also that the quadratic part $F$ of $H$ restricted to $\mathcal{E}$ is positive definite. Then, on each energy surface $\left\{ H = c\right\} \, c > 0$ and small, the number of periodic orbits of $X_{H}\ $is at least $\frac{1}{2}\dim \mathcal{E}$.'

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 $H = \frac{1}{2}\left(\left\vert z_{1}\right\vert ^{2}-j\left\vert z_{2}\right\vert ^{2}\right) + \mathrm{Re}(z_{1}z_{2}^{j})$, 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 $j\geq 2$ is an integer; note that it is in the Birkhoff normal form. We have

The quadratic part is positive definite on the plane $\left\{ z_{2} = 0\right\}$ with periodic solutions for the linear system of period $2\pi /j;$ other periodic solutions in the plane $\left\{ z_{1} = 0\right\}$ have period $2\pi.$ But the corresponding nonlinear system has only $2\pi /j-$periodic solutions in $\left\{ z_{2} = 0\right\}.$

Next, I have assumed that the period $T$ 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 $\mathcal{E}$ 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: $\omega _{1} = 2,$ $\omega _{2} = 3$, and $\omega _{3} = 6.$ Then we have two invariant linear subspaces: $\mathcal{E}_{1}$ associated with $\omega _{1}$ and $\omega _{3}$ and $\mathcal{E}_{2}$ associated with $\omega _{2}$ and $\omega _{3};$ all solutions of the linear system in $\mathcal{E}_{1}$ have period $T_{1} = \pi$, and all solutions in $\mathcal{E}_{2}$ have period $T_{2} = 2\pi /3.$ One can show that there exist corresponding formal invariant subspaces $\mathcal{V} _{1}$ and $\mathcal{V}_{2}$ for $X_{H};$ moreover, we have the invariant subspaces $\mathcal{E}_{3} = \mathcal{E}_{1}\cap \mathcal{E}_{2}$ and $\mathcal{V}_{3} = \mathcal{V}_{1}\cap \mathcal{V}_{2}.$ From the original statement of Moser's theorem it follows that there exist many periodic orbits in $\mathcal{V}_{1}$ and in $\mathcal{V}_{2},$ but all of them could lie in $\mathcal{V}_{3}$ (provided $\dim \mathcal{E}_{3} > 2$).

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 $\mathbb{S}^{1}$ (in the case of one equivalence class) was defined incorrectly; the correct formula is Eq. (2.13)

3.   Proof of Theorem 1

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

Since Eqs. (2.1) are homogeneous and we assume $r\not = 0$, we put

(dehomogenization); thus, we replace $u$ with $ru$ and $v$ with $rv$. We get the algebraic system

We treat Eqs. (3.1) as a system of five real algebraic equations on six real variables $\mathrm{Re}\varepsilon,$ $\mathrm{Im}\varepsilon,$ $\mathrm{Re} u,$ $\mathrm{Im}u,$ $\mathrm{Re}v,$ $\mathrm{Im}v$, i.e., in $\mathbb{R}^{6}$. It defines an algebraic variety $C\subset \mathbb{R}^{6}$. The projection of the variety $C$ to the $\varepsilon -$plane is an semi-algebraic variety $D\subset \mathbb{R}^{2}$ consisting of those parameters for which there exists a $1-$parameter family of periodic solutions to the perturbed Hamiltonian system near $z\left(t\right) \approx \left(r\mathrm{e}^{ \mathrm{i}t}, ru\mathrm{e}^{\mathrm{i}t}, rv\mathrm{e}^{-\mathrm{i}t}\right)$. For $\varepsilon$'s outside $D$, there are no such periodic solutions. Our goal is to prove that the intersection of $D$ with a neighborhood of $\varepsilon = 0$ 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 $C$ above a neighborhood of $\varepsilon = 0$ is a $1-$dimensional algebraic curve, a union of irreducible local curves (components).

Assume firstly that

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

$\tilde{u}_{0} = 0, \tilde{v}_{0} = -1$;

$\tilde{u}_{1, 2} = \pm \sqrt{\frac{3}{8}\left(4\sqrt{6}-9\right) }\approx \pm 0.54702,$ $\tilde{v}_{1, 2} = \frac{1}{4}\left(2-\sqrt{6}\right) \approx -0.11237.$

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

If $u = 0$ then the second of Eqs. (3.1) gives the solution $\left(\tilde{u} _{0}, \tilde{v}_{0}\right)$.

Otherwise, $v$ is also real and we have

Then the second equation gives $3u^{2}\left(v-1\right) +v^{3}+1 = 0$, i.e., $-8v^{3}+9v+1 = -\left(v+1\right) \left(-8v+8v^{2}-1\right) = 0$, with the additional values $v = \frac{1}{4}\left(2\pm \sqrt{6}\right).$ But only for $v = \tilde{v}_{1} = \tilde{v}_{2} = \frac{1}{4}\left(2-\sqrt{6}\right)$ the quantity $-3v\left(v+1\right)$ is positive and gives $u = \tilde{u} _{1, 2} = \pm \sqrt{-3\tilde{v}_{1}\left(\tilde{v}_{1}+1\right) }$ form the thesis of the lemma. □

Let now

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

Lemma 2. No solution bifurcates from infinity. Namely, there exist $\varepsilon _{0} > 0$ and $R > 0$ such that system (3.1), with $\left\vert \varepsilon \right\vert < \varepsilon _{0},$ does not have solutions in $\left\{ R < \left\vert u\right\vert +\left\vert v\right\vert < \infty \right\}.$

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

where the dots mean lower-degree terms. This expression is separated from $0$ for large $\left\vert u\right\vert +\left\vert v\right\vert$ and small $\left\vert \varepsilon \right\vert$. □

Therefore, any eventual solution could bifurcate only from one of the points $\left(\tilde{u}_{j}, \tilde{v}_{j}\right).$

Lemma 3. No solution bifurcates from $\left(\tilde{u}_{j}, \tilde{v}_{j}\right)$, $j = 0, 1, 2,$ as $\varepsilon$ varies in a neighborhood of $\varepsilon = 0$ outside a finite number of real analytic curves. Namely, for any $\delta > 0,$ there exists $\varepsilon _{0} = \varepsilon _{0}\left(\delta \right) > 0$ and a finite collection of germs $\left(D_{k}, 0\right) \subset \left(\mathbb{C}, 0\right)$ of real analytic curves such that, if $0 < \left\vert \varepsilon \right\vert < \varepsilon _{0}$ and $\varepsilon \not\in \bigcup D_{k}$, then system (3.1) does not have solutions in $\bigcup\limits_{j}\left\{ \left\vert u-\tilde{u}_{j}\right\vert +\left\vert v-\tilde{v}_{j}\right\vert < \delta \right\}.$

Proof. Putting

with $\varepsilon = \varepsilon _{1}+\mathrm{i}\varepsilon _{2},$ we can treat Eqs. (3.1), near $\left(\tilde{u}_{j}, \tilde{v}_{j}, 0\right)$ as a system of five real analytic equations in six real variables $u_{1, 2},$ $v_{1, 2}$ and $\varepsilon _{1, 2}$, i.e., in $\left(\mathbb{R}^{6}, 0\right)$.

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 $C_{k}$. Next, one finds several further terms using higher-order terms in system (3.1) and the linear approximation of $C_{k},$ etc. In the expansion of the curves $C_{1}$ and $C_{2}$, the corresponding coefficients will not be exact, only approximate, like the values of $\tilde{u}_{1, 2}$ and $\tilde{v}_{1, 2}$ in Lemma 1.

In Case $\left(\tilde{u}_{0}, \tilde{v}_{0}\right)$ we arrive at the following system:

where the dots in the second equation mean quadratic and cubic terms containing $v_{j}$ and the dots in the third equation mean cubic terms with $v_{j}.$ The linear parts of these equations define the plane

note that $f$ and $\mathrm{Im}h$ have proportional linear parts. This plane is parametrized by $u_{1}$ and $u_{2};$ we can assume that $\varepsilon _{i}$ and $v_{j}$ are functions of $u_{1, 2}.$

Taking into account nonlinear terms in the second and third equations, with $\varepsilon _{1}\approx \frac{1}{4}u_{1}$ and $\varepsilon _{2}\approx u_{2},$ we get

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

Eqs (3.5)–(3.6) define the curve $C_{0}$. It has three components: one is defined by $u_{2} = O\left(u_{1}^{2}\right)$ (and Eqs. (3.5)), one by $u_{2} = \sqrt{44/95}u_{1}+O\left(u_{1}^{2}\right)$ and one by $u_{2} = -\sqrt{44/95} u_{1}+O(u_{1}^{2})$. The projection $D_{0}$ of $C_{0}$ has at most three components, defined by:

Consider Case $\left(\tilde{u}_{1}, \tilde{v}_{1}\right) \approx \left(0.54702, -0.11237\right).$ By abuse of notation, we put

and replace $u,$ $\bar{u},$ $v$, and $\bar{v}$, by

respectively.

The notation $\approx$ 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 $f = f_{1}+f_{2},$ $g = g_{1}+g_{2}+\ldots,$ $h = h_{1}+h_{2}+\ldots,$ where $f_{j}, g_{j}, h_{j}$ are homogeneous polynomials of degree $j$. Namely,

First, we solve the corresponding linear equations. From $2f_{1} = 0$, we express $u$ as linear function of $w, v, t, \varepsilon, \epsilon$, and substitute it to $g_{1} = 0;$ then we get $w$ as a linear function of $v, t, \varepsilon, \epsilon$, and also $u$ becomes expressed via $v, t, \varepsilon, \epsilon.$ Finally, we substitute these $u$ and $w$ to $h_{1} = 0;$ we get a linear complex equation for $v,$ $t = \bar{v};$ by comparing the real and imaginary parts, we get

Then we find

We see that $u$ and $v$ become functions of $\varepsilon _{1}$ and $\varepsilon _{2}.$

Moreover, this suggests that $w = \bar{u}$ in a linear approximation. If $w$ were not equal $\bar{u}$ for all $\varepsilon _{j}$ then the condition $w = \bar{u}$ would imply a linear restriction for $\varepsilon _{j}$'s.

Next, we expand the solutions $v,$ $u$ and $w$ in powers of $\varepsilon _{j}$ modulo $O\left(\left\vert \varepsilon \right\vert ^{3}\right).$ For this, we firstly express $f_{2},$ $g_{2}$, and $h_{2}$ via $\varepsilon _{j}$, and then we repeat calculations for $v,$ $u$, and $w.$ We get

Next, we find

We see that the real parts of $\bar{u}$ and $w$ agree, but the imaginary parts disagree:

This suggests that the projected curve $D_{1}$ has two components, one defined by $\left\{ \varepsilon _{1}\approx 0\right\}$ and, the other defined by $\left\{ \varepsilon _{2}\approx 0\right\};$ also, the curve $C_{1}$ has two components. (If also the real parts of $u$ and $w$ disagreed, then we would have another restriction on $\varepsilon _{j}$'s.)

Consider now Case $\left(\tilde{u}_{2}, \tilde{v}_{2}\right) \approx \left(-0.54702, -0.11237\right).$ 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 $u$ and $v$ become functions of $\varepsilon _{1}$ and $\varepsilon _{2}$ and that $w\approx \bar{u}$ in linear approximation.

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

Then the solutions up to $O\left(\left\vert \varepsilon \right\vert ^{3}\right)$ are the following:

We see that the real parts of $w$ and $\bar{u}$ agree, but

Like in the previous case, the curve $C_{2}$ and its projection $D_{2}$ have two components. □

Use of AI tools declaration

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

Acknowledgments

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

Conflict of interest

The author declares there is no conflict of interest.