[Paper Review] Uniform hyperbolicity of invariant cylinder

Chong-Qing Cheng|arXiv (Cornell University)|Sep 10, 2015

Quantum chaos and dynamical systems10 references3 citations

TL;DR

This paper establishes uniform hyperbolicity of normally hyperbolic invariant cylinders (NHICs) in three-degree-of-freedom nearly integrable Hamiltonian systems, showing that large NHICs persist along resonant paths except within $\sqrt{\epsilon}^{1+d}$-neighborhoods of finitely many strong double resonant points. The result enables the construction of global diffusion orbits by connecting NHICs across double resonances via Arnold's mechanism, with transition chains existing despite small gaps near strong resonances.

ABSTRACT

For a nearly integrable Hamiltonian systems $H=h(p)+εP(p,q)$ with $(p,q)\in\mathbb{R}^3 imes\mathbb{T}^3$, large normally hyperbolic invariant cylinders exist along the whole resonant path, except for the $\sqrtε^{1+d}$-neighborhood of finitely many double resonant points. It allows one to construct diffusion orbits to cross double resonance.

Motivation & Objective

• To establish the existence of large normally hyperbolic invariant cylinders (NHICs) along resonant paths in nearly integrable Hamiltonian systems with three degrees of freedom.
• To analyze the behavior of NHICs near double resonant points, particularly distinguishing between strong and weak double resonances.
• To demonstrate that diffusion orbits can be constructed across double resonances by connecting NHICs through transition chains.
• To show that the only obstructions to global diffusion are $\sqrt{\epsilon}^{1+d}$-neighborhoods of finitely many strong double resonant points.
• To provide a criterion for weak vs. strong double resonance based on non-degeneracy of minimal periodic orbits and genericity of non-degenerate potentials.

Proposed method

• Uses a symplectic coordinate transformation to diagonalize the resonant structure using integer vectors $k'$, $k''$, and $k_3$ with $\det M = 1$, simplifying the Hamiltonian near double resonant points.
• Applies a time-$2\pi$-map of the Hamiltonian flow generated by $\epsilon F(p,q)$ to solve the homological equation $\langle \partial h/\partial p, \partial F/\partial q \rangle = -P + Z$, eliminating non-resonant Fourier modes.
• Performs a homogenization scaling: $\tilde{y} = (p - p'')/\sqrt{\epsilon}$, $s = \sqrt{\epsilon}t$, to rescale the dynamics and extract a limiting Hamiltonian $\tilde{G}_\epsilon$ with a slow time scale.
• Constructs a function $G_\epsilon(x,y,\theta)$ solving $\tilde{G}_\epsilon = 0$ via implicit function theorem, showing $\partial \tilde{G}_\epsilon / \partial I = 1 + O(\epsilon^\sigma)$, ensuring well-posedness of the dynamics on the energy level set.
• Applies a genericity result (Theorem 6.1) to show that non-degeneracy of minimal points of the potential $V = -Z(p'',x)$ holds uniformly for all $p$ on the resonant path $\Gamma'$, except at finitely many bifurcation points.
• Uses weak KAM theory and Hölder continuity of weak KAM solutions to establish cohomology equivalence and construct transition chains across cylinders, enabling diffusion via Arnold's mechanism.

Experimental results

Research questions

• RQ1 Under what conditions do normally hyperbolic invariant cylinders (NHICs) persist along a resonant path in a nearly integrable Hamiltonian system with three degrees of freedom?
• RQ2 How do double resonant points affect the structure of invariant cylinders, and which ones obstruct global diffusion?
• RQ3 What is the size and nature of the neighborhoods around double resonant points where NHICs fail to exist?
• RQ4 Can diffusion orbits be constructed across double resonances by connecting NHICs through transition chains?
• RQ5 What generic conditions ensure that the minimal points of the effective potential are non-degenerate across the entire resonant path?

Key findings

• NHICs exist uniformly along the entire resonant path $\Gamma'$, except within $O(\sqrt{\epsilon}^{1+d})$-neighborhoods of finitely many strong double resonant points.
• The number of strong double resonant points requiring special treatment is finite and independent of $\epsilon$, due to a genericity condition on the potential $Z_{k'}(p,x)$.
• Non-degeneracy of the minimal periodic orbit of the effective potential $V = -Z(p'',x)$ is uniformly bounded from below for all $p$ on $\Gamma'$, except at finitely many bifurcation points.
• The Aubry set lies within a NHIC for all $c \in \text{int}(\mathbb{C}_g)$ with $d(c, \mathbb{F}_i) > O(\sqrt{\epsilon}^{1+d})$, ensuring local connectivity of invariant sets.
• Transition chains of cohomology equivalence can be constructed across the resonant path, with only finitely many gaps of size $O(\sqrt{\epsilon}^{1+d})$ near strong double resonances.
• The Hölder continuity of weak KAM solutions on the cylinder allows global connection of Aubry sets via Arnold's mechanism, enabling diffusion across double resonances.