arXiv subjects
Publications and source records attributed to Chong-Qing Cheng.
It is well known that under generic $C^r$ smooth perturbations, the phenomenon of global instability, known as Arnold diffusion, exists in a priori unstable Hamiltonian systems. In this paper, by using variational methods, we will prove that under generic Gevrey smooth perturbations, Arnold diffusion still exists in the a priori unstable Hamiltonian systems of two and a half degrees of freedom.
We study the continuation of periodic orbits from various compound of homoclinics in classical system. Together with the homoclinics, the periodic orbits make up a $C^1$-smooth, normally hyperbolic invariant cylinder with holes. It plays a key role to cross multiple resonant point.
In this paper Arnold diffusion is proved to be a generic phenomenon in nearly integrable convex Hamiltonian systems with arbitrarily many degrees of freedom: $$ H(x,y)=h(y)+\eps P(x,y), \qquad x\in\mathbb{T}^n,\ y\in\mathbb{R}^n,\quad n\geq 3. $$ Under typical perturbation $\eps P$, the system admits "connecting" orbit that passes through any finitely many prescribed small balls in the same energy level $H^{-1}(E)$ provided $E>\min h$.
In this paper, we prove that the net of transition chain is $δ$-dense for nearly integrable positive definite Hamiltonian systems with 3 degrees of freedom in the cusp-residual generic sense in $C^r$-topology, $r\ge 6$. The main ingredients of the proof existed in \cite{CZ,C17a,C17b}. As an immediate consequence, Arnold diffusion exists among this class of Hamiltonian systems. The question of \cite{C17c} is answered in Section 9 of the paper.
Let $f$ be an exact area-preserving monotone twist diffeomorphism of the infinite cylinder and $P_{ω,f}(ξ)$ be the associated Peierls barrier. In this paper, we give the Hölder regularity of $P_{ω,f}(ξ)$ with respect to the parameter $f$. In fact, we prove that if the rotation symbol $ω\in (\mathbb{R}\setminus\mathbb{Q})\bigcup(\mathbb{Q}+)\bigcup(\mathbb{Q}-)$, then $P_{ω,f}(ξ)$ is $1/3$-Hölder continuous in $f$, i.e. $$|P_{ω,f'}(ξ)-P_{ω,f}(ξ)|\leq C\|f'-f\|_{C^1}^{1/3} ,~~\forall ξ\in\mathbb{R}$$ where $C$ is a constant. Similar results also hold for the Lagrangians with one and a half degrees of freedom. As application, we give an open and dense result about the breakup of invariant circles.
For typical perturbations of convex integrable Hamiltonian system with three degrees of freedom, a path of diffusion is established to cross strong double resonant point. Together with the uniform hyperbolicity of invariant cylinders got in \cite{C15}, one obtains a transition chain along which one is able to construct diffusion orbits suggested in \cite{A66}.
For a nearly integrable Hamiltonian systems $H=h(p)+εP(p,q)$ with $(p,q)\in\mathbb{R}^3\times\mathbb{T}^3$, 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.
In this paper we construct a certain type of nearly integrable systems of two and a half degrees of freedom: \[H(p,q,t)=h(p)+\epsilon f(p,q,t),\quad (q,p)\in T^{*}\mathbb{T}^2,t\in \mathbb{S}^1=\mathbb{R}/\mathbb{Z}, \] with a self-similar and weak-coupled $f(p,q,t)$ and $h(p)$ strictly convex. For a given Diophantine rotation vector $\vec{\omega}$, we can find asymptotic orbits towards the KAM torus $\mathcal{T}_{\omega}$, which persists owing to the classical KAM theory, as long as $\epsilon\ll1$ sufficiently small and $f\in C^r(T^{*}\mathbb{T}^2\times\mathbb{S}^1,\mathbb{R})$ properly smooth. The construction bases on the new methods developed in {\it a priori} stable Arnold Diffusion problem by Chong-Qing Cheng. As an expansion of that, this paper sheds some light on the seeking of much preciser diffusion orbits.
In this paper, Arnold diffusion is proved to be generic phenomenon in nearly integrable convex Hamiltonian systems with three degrees of freedom: $$ H(x,y)=h(y)+εP(x,y), \qquad x\in\mathbb{T}^3,\ y\in\mathbb{R}^3. $$ Under typical perturbation $εP$, the system admits "connecting" orbit that passes through any two prescribed small balls in the same energy level $H^{-1}(E)$ provided $E$ is bigger than the minimum of the average action, namely, $E>\minα$.
For positive definite Lagrange systems with two degrees of freedom, it is a typical phenomenon that all minimal periodic orbits are hyperbolic.
For an integrable Hamiltonian $H_0=1/2\sum_{i=1}^dy_i^2$ $(d\geq 2)$, we show that any Lagrangian torus with a given unique rotation vector can be destructed by arbitrarily $C^{2d-δ}$-small perturbations. In contrast with it, it has been shown that KAM torus with constant type frequency persists under $C^{2d+δ}$-small perturbations.