Hamiltonian and reversible systems with smooth families of invariant tori

doi:10.1016/j.indag.2020.12.001

Indagationes Mathematicae

Volume 32, Issue 2, April 2021, Pages 406-425

https://doi.org/10.1016/j.indag.2020.12.001 Get rights and content

Abstract

For various values of $n$ , $d$ , and the phase space dimension, we construct simple examples of Hamiltonian and reversible systems possessing smooth $d$ -parameter families of invariant $n$ -tori carrying conditionally periodic motions. In the Hamiltonian case, these tori can be isotropic, coisotropic, or atropic (neither isotropic nor coisotropic). The cases of non-compact and compact phase spaces are considered. In particular, for any $N \geq 3$ and any vector $ω \in R^{N}$ , we present an example of an analytic Hamiltonian system with $N$ degrees of freedom and with an isolated (and even unique) invariant $N$ -torus carrying conditionally periodic motions with frequency vector $ω$ (but this torus is atropic rather than Lagrangian and the symplectic form is not exact). Examples of isolated atropic invariant tori carrying conditionally periodic motions are given in the paper for the first time. The paper can also be used as an introduction to the problem of the isolatedness of invariant tori in Hamiltonian and reversible systems.

Section snippets

Kronecker tori

Finite-dimensional invariant tori carrying conditionally periodic motions are among the key elements of the structure of smooth dynamical systems with continuous time. The importance and ubiquity of such tori stems, in the long run, from the fact that any finite-dimensional connected compact Abelian Lie group is a torus [1], [17], [42]. By definition, given a certain flow on a certain manifold, an invariant $n$ -torus carrying conditionally periodic motions ( $n$ being a non-negative integer) is an

Preliminaries

Given non-negative integers $a$ and $b$ , we designate the identity $a \times a$ matrix as $I_{a}$ and the zero $a \times b$ matrix as $0_{a \times b}$ . In fact, the symbols $I_{0}$ , $0_{0 \times b}$ , and $0_{a \times 0}$ correspond to no actual objects and will only be used for unifying the notation.

Let $s \geq 1$ , $k$ , and $l$ be non-negative integers and consider a skew-symmetric $(2 s + 2 k) \times (2 s + 2 k)$ matrix $J$ of the form $J = (\begin{pmatrix} 0_{s \times s} & - Z^{t} \\ Z & L \end{pmatrix}),$ where $Z$ is an $(s + 2 k) \times s$ matrix of rank $s$ and $L$ is a skew-symmetric $(s + 2 k) \times (s + 2 k)$ matrix (the superscript “t” denotes transposing). If $k = 0$ then

The system

Now let $ζ_{1}, \dots, ζ_{s}$ , $ξ_{1}, \dots, ξ_{l}$ , $η_{1}, \dots, η_{l}$ be arbitrary non-negative real constants and let $h : R^{s} \to R$ be an arbitrary smooth function. Consider the Hamilton function $H (u, p, q) = h (u) + l p_{1} \sum_{i = 1}^{s} ζ_{i} u_{i}^{2} + \sum_{ν = 1}^{l} (ξ_{ν} p_{ν} q_{ν}^{2} + η_{ν} p_{ν}^{3} ∕ 3)$ on the symplectic manifold (5). The term $l p_{1} \sum_{i = 1}^{s} ζ_{i} u_{i}^{2}$ is automatically absent for $l = 0$ . According to (6), the equations of motion afforded by $H$ take the form ${\dot{u}}_{i} = 0,$ ${\dot{φ}}_{α} = \sum_{i = 1}^{s} Z_{α i} (\frac{\partial h (u)}{\partial u_{i}} + 2 l ζ_{i} u_{i} p_{1}),$ ${\dot{p}}_{ν} = - 2 ξ_{ν} p_{ν} q_{ν},$ ${\dot{q}}_{ν} = ξ_{ν} q_{ν}^{2} + η_{ν} p_{ν}^{2} + δ_{1 ν} l \sum_{i = 1}^{s} ζ_{i} u_{i}^{2},$ where $1 \leq i \leq s$ , $1 \leq α \leq s + 2 k$ , $1 \leq ν \leq l$ , and $δ_{1 ν}$ is the Kronecker

Compact phase spaces

Like in the setting of our note [50], the general construction of Sections 2 Preliminaries, 3 The main construction admits an analogue with a compact phase space. Consider the symplectic manifold $\hat{M} = T_{u}^{s} \times T_{φ}^{s + 2 k} \times T_{p}^{l} \times T_{q}^{l}$ with the same structure matrix (4). Of course, now the corresponding symplectic form $Ω$ is always non-exact. The $(s + 2 k)$ -tori (7) with $u^{0} \in T^{s}$ , $p^{0} \in T^{l}$ , $q^{0} \in T^{l}$ are again isotropic for $k = 0$ , are coisotropic for $l = 0$ , and are atropic for $k l > 0$ . For any angular variable $z$ introduce the notation

Reversible analogues

Both the Hamiltonian systems (9), (11) are reversible with respect to the phase space involution $\tilde{G} : (u, φ, p, q) \mapsto (u, - φ, p, - q)$ of type $(s + 2 k + l, s + l)$ , so that $dim Fix \tilde{G} = s + l \geq 1$ , $codim Fix \tilde{G} = s + 2 k + l \geq dim Fix \tilde{G}$ , and $codim Fix \tilde{G} - n = l < dim Fix \tilde{G}$ , where $n = s + 2 k$ . However, $\tilde{G} (T_{u^{0}, p^{0}, q^{0}}) = T_{u^{0}, p^{0}, - q^{0}}$ , so that not all the $n$ -tori (7) are invariant under $\tilde{G}$ . In the case of the system (9), a torus $T_{u^{0}, p^{0}, q^{0}}$ is invariant under $\tilde{G}$ if and only if $q^{0} = 0$ . Consequently, the statement “each torus $T_{u^{0}, p^{0}, q^{0}} \subset M$ is symmetric” is valid if

Compactified reversible analogues

The system (14) can be compactified in the same way as the system (9), cf. [50]. Consider the manifold (3) equipped with the involution $G$ given by the same formula (2) and having the same type $(n + l, m)$ . For any $u^{0} \in T^{m}$ and $q^{0} \in T^{l}$ , consider the $n$ -torus $T_{u^{0}, q^{0}}$ given by the same expression (13). Since $G (T_{u^{0}, q^{0}}) = T_{u^{0}, - q^{0}}$ , a torus $T_{u^{0}, q^{0}}$ is invariant under $G$ if and only if $q^{0} = - q^{0}$ , i.e., if each component of $q^{0}$ is equal to either $0$ or $π$ .

Now let again $ζ_{1}, \dots, ζ_{m}$ , $ξ_{1}, \dots, ξ_{l}$ be arbitrary non-negative real

Acknowledgment

I am grateful to B. Fayad for fruitful correspondence and sending me the breakthrough preprint [21] prior to submission to arXiv.

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View full text

Hamiltonian and reversible systems with smooth families of invariant tori

Abstract

Section snippets

Kronecker tori

Preliminaries

The system

Compact phase spaces

Reversible analogues

Compactified reversible analogues

Acknowledgment

Adv. Math.

J. Differential Equations

Chaos Solitons Fractals

Indag. Math.

Adv. Math.

Lectures on Lie Groups

Mathematical Aspects of Classical and Celestial Mechanics

Encyclopaedia of Mathematical Sciences

Linear Algebra and Projective Geometry

Involutions whose fixed set has three or four components: a small codimension phenomenon

Math. Scand.

Quasimorphisms on contactomorphism groups and contact rigidity

Geom. Topol.

Some instability properties of resonant invariant tori in Hamiltonian systems

Math. Res. Lett.

Unfoldings of quasi-periodic tori in reversible systems

J. Dynam. Differential Equations

Quasi-Periodic Motions in Families of Dynamical Systems. Order Amidst Chaos

Lecture Notes in Mathematics

Unfoldings and Bifurcations of Quasi-Periodic Tori

Mem. Amer. Math. Soc.

On invariant tori in some reversible systems

A tutorial on KAM theory

Involutions fixing many components: a small codimension phenomenon

J. Fixed Point Theory Appl.

Lie Groups

The KAM Story. A Friendly Introduction to the Content, History, and Significance of Classical Kolmogorov–Arnold–Moser Theory

Introduction to the h-Principle

Around the stability of KAM tori

Duke Math. J.

Instabilities for analytic quasi-periodic invariant tori

Superintegrable Hamiltonian systems: geometry and perturbations

Acta Appl. Math.

Lyapunov unstable elliptic equilibria, third version

Some questions around quasi-periodic dynamics

Introduction to the $h$ -Principle