Chong-Qing Cheng, Min Zhou
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.
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We prove the existence of normally hyperbolic invariant cylinders in nearly integrable hamiltonian systems.
In the present paper we prove a form of Arnold diffusion. The main result says that for a "generic" perturbation of a nearly integrable system of arbitrary degrees of freedom $n\ge 2$ \[ H_0(p)+\eps H_1(þ,p,t),\quad þ\in \T^n,\ p\in B^n,\ t\in \T=\R/\T, \] with strictly convex $H_0$ there exists an orbit $(þ_{\e},p_{e})(t)$ exhibiting Arnold diffusion in the sens that [\sup_{t>0}\|p(t)-p(0) \| >l(H_1)>0] where $l(H_1)$ is a positive constant independant of $\e$. Our proof is a combination of geometric and variational methods. We first build 3-dimensional normally hyperbolic invariant cylinders…
We prove a form of Arnold diffusion in the a priori stable case. Let H0(p) + $ε$H1($θ$, p, t), $θ$ $\in$ T n , p $\in$ B n , t $\in$ T = R/T be a nearly integrable system of arbitrary degrees of freedom n 2 with a strictly convex H0. We show that for a "generic" $ε$H1, there exists an orbit ($θ$, p)(t) satisfying p(t) -- p(0) {\textgreater} l(H1) {\textgreater} 0, where l(H1) is independent of $ε$. The diffusion orbit travels along a co-dimension one resonance , and the only obstruction to our construction is a finite set of additional resonances. For the proof we use a combination geometric a…
In the present paper we consider a generic perturbation of a nearly integrable system of $n$ and a half degrees of freedom $ H_ε(θ,p,t)=H_0(p)+εH_1(θ,p,t)$, with a strictly convex $H_0$. For $n=2$ we show that at a strong double resonance there exist 3-dimensional normally hyperbolic invariant cylinders going across. This is somewhat unexpected, because at a strong double resonance dynamics can be split into one dimensional fast motion and two dimensional slow motion. Slow motions are described by a mechanical system on a two-torus, which are generically chaotic. The construction of invariant …
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.
We develop a geometric mechanism to prove the existence of orbits that drift along a prescribed sequence of cylinders, under some general conditions on the dynamics. This mechanism can be used to prove the existence of Arnold diffusion for large families of perturbations of Tonelli Hamiltonians on $\mathbb{A}^3$. Our approach can also be applied to more general Hamiltonians that are not necessarily convex. The main geometric objects in our framework are $3$-dimensional invariant cylinders with boundary (not necessarily hyperbolic), which are assumed to admit center-stable and center-unstable m…
We present a computer assisted proof or diffusion in the Planar Elliptic Restricted Three Body Problem. We treat the elliptic problem as a perturbation of the circular problem, where the perturbation parameter is the eccentricity of the primaries. The unperturbed system preserves energy, and we show that for sufficiently small perturbations we have orbits with explicit energy changes, independent from the size of the perturbation. The result is based on shadowing of orbits along transversal intersections of stable/unstable manifolds of a normally hyperbolic cylinder.
In this paper we study existence of Normally Hyperbolic Invariant Laminations (NHIL) for a nearly integrable system given by the product of the pendulum and the rotator perturbed with a small coupling between the two. This example was introduced by Arnold. Using a {\it separatrix map}, introduced in a low dimensional case by Zaslavskii-Filonenko and studied in a multidimensional case by Treschev and Piftankin, for an open class of trigonometric perturbations we prove that NHIL do exist. Moreover, using a second order expansion for the separatrix map from [GKZ], we prove that the system restric…
This paper is the first of a series of three dedicated to a proof of the Arnold diffusion conjecture for perturbations of {convex} integrable Hamiltonian systems on $\mathbb{A}^3=\mathbb{T}^3\times \mathbb{R}^3$. We consider systems of the form $H(θ,r)=h(r)+f(θ,r)$, where $h$ is a $C^κ$ strictly convex and superlinear function on $\mathbb{R}^3$ and $f\in C^κ(\mathbb{A}^3)$, $κ\geq2$. Given $e>\textrm{Min}\,h$ and a finite family of arbitrary open sets $O_i$ in $\mathbb{R}^3$ intersecting $h^{-1}(e)$, a diffusion orbit associated with these data is an orbit of $H$ which intersects each open set…
Computation of a separatrix map and a normally hyperbolic invariant lamination for the RP3BP
本文研究了RP3BP中3:1 Kirkwood间隙的正常双曲不变鳞片的存在性，并展示了其诱导动力学为部分双曲斜移，揭示了小偏心率下小行星的随机扩散行为。
In this paper we propose a model of random compositions of cylinder maps, which in the simplified form is as follows: let $(θ,r)\in \mathbb T\times \mathbb R=\mathbb A$ and \[ f_{\pm 1}: \left(\begin{array}{c}θ\\r\end{array}\right) \longmapsto \left(\begin{array}{c}θ+r+\varepsilon u_{\pm 1}(θ,r) \\ r+\varepsilon v_{\pm 1}(θ,r) \end{array}\right), \] where $u_\pm$ and $v_\pm$ are smooth and $v_\pm$ are trigonometric polynomials in $θ$ such that $\int v_\pm(θ,r)\,dθ=0$ for each $r$. We study the random compositions \[ (θ_n,r_n)=f_{ω_{n-1}}\circ \dots \circ f_{ω_0}(θ_0,r_0), \] where $ω_k =\pm 1$…
In the present paper we prove a strong form of Arnold diffusion. Let $\mathbb{T}^2$ be the two torus and $B^2$ be the unit ball around the origin in $\mathbb{R}^2$. Fix $ρ>0$. Our main result says that for a "generic" time-periodic perturbation of an integrable system of two degrees of freedom \[ H_0(p)+εH_1(θ,p,t),\quad \ θ\in \mathbb{T}^2,\ p\in B^2,\ t\in \mathbb{T}, \] with a strictly convex $H_0$, there exists a $ρ$-dense orbit $(θ_ε,p_ε,t)(t)$ in $\mathbb{T}^2 \times B^2 \times \mathbb{T}$, namely, a $ρ$-neighborhood of the orbit contains $\mathbb{T}^2 \times B^2 \times \mathbb{T}$. Our …
A new fast multiple-shooting method for computing periodic orbits in symplectic maps leveraging simultaneous Floquet vector computation to avoid large linear systems
本文提出了一种新的快速多射击方法，用于计算辛映射中的周期轨道，通过同时计算Floquet向量以提高效率，适用于2.5自由度哈密顿流的脉冲映射情况。
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