本文构造了一类二又二分之一自由度的近可积系统，基于先验稳定 Arnold 扩散问题的新方法，证明了在特定条件下存在趋向于 KAM 环面的渐近轨道，为寻找更精确的扩散轨道提供了启示。
摘要：
本文构造了一类二又二分之一自由度的近可积系统：\\[H(p,q,t)=h(p)+εf(p,q,t),\quad (q,p)\in T^{*}\mathbb{T}^2,t\in \mathbb{S}^1=\mathbb{R}/\mathbb{Z},\\] 其中 $f(p,q,t)$ 自相似且弱耦合，$h(p)$ 严格凸。对于给定的 Diophantine 旋转向量 $\vecω$，只要 $ε\ll1$ 足够小且 $f\in C^r(T^{*}\mathbb{T}^2\times\mathbb{S}^1,\mathbb{R})$ 适当光滑，我们就能找到趋向于因经典 KAM 理论而持续存在的 KAM 环面 $\mathcal{T}_ω$ 的渐近轨道。该构造基于 Chong-Qing Cheng 在先验稳定 Arnold 扩散问题中发展的新方法。作为其扩展，本文为寻求更精确的扩散轨道提供了一些启示。
英文摘要：
In this paper we construct a certain type of nearly integrable systems of two and a half degrees of freedom:
\[H(p,q,t)=h(p)+εf(p,q,t),\quad (q,p)\in T^{*}\mathbb{T}^2,t\in \mathbb{S}^1=\mathbb{R}/\mathbb{Z},
\] with a self-similar and weak-coupled $f(p,q,t)$ and $h(p)$ strictly convex. For a given Diophantine rotation vector $\vecω$, we can find asymptotic orbits towards the KAM torus $\mathcal{T}_ω$, which persists owing to the classical KAM theory, as long as $ε\ll1$ 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 present paper, from the viewpoint of physical intuition we introduce a Hamiltonian system with multiscale rotation, which describes many systems, for example, the forced pendulum with fast rotation, weakly coupled $N$-oscillators with quasiperiodic force and so on. We study the persistence of invariant tori for this Hamiltonian system, and establish some KAM type results including the isoenergetic type. As consequences, we can show that Boltzmann's ergodicity hypothesis is also not true for this Hamiltonian system.
In this paper, we study the persistence and remaining regularity of KAM invariant torus under sufficiently small perturbations of a Hamiltonian function together with its derivatives, in sense of finite smoothness with modulus of continuity, as a generalization of classical Hölder continuous circumstances. To achieve this goal, we extend the Jackson approximation theorem to the case of modulus of continuity, and establish a corresponding regularity theorem adapting to the new iterative scheme. Via these tools, we establish a KAM theorem with sharp differentiability hypotheses, which asserts th…
For the conformally symplectic system \[ \left\{ \begin{aligned} \dot{q}&=H_p(q,p),\quad(q,p)\in T^*\mathbb{T}^n\\ \dot p&=-H_q(q,p)-λp, \quad λ>0 \end{aligned} \right. \] with a positive definite Hamiltonian, we discuss the variational significance of invariant Lagrangian graphs and explain how the KAM torus impacts the $W^{1,\infty}-$convergence speed of the Lax-Oleinik semigroup.
We study the planetary system of $\upsilon$~Andromedæ, considering the three-body problem formed by the central star and the two largest planets, $\upsilon$~And~\emph{c} and $\upsilon$~And~\emph{d}. We adopt a secular, three-dimensional model and initial conditions within the range of the observed values. The numerical integrations highlight that the system is orbiting around a one-dimensional elliptic torus (i.e., a periodic orbit that is linearly stable). This invariant object is used as a seed for an algorithm based on a sequence of canonical transformations. The algorithm determines the no…
We show the following dichotomy for a linear parabolic $\mathbb Z^2$-action $ρ_L$ on the torus with at least one step-2 generator: (i) Any affine $\mathbb Z^2$-action with linear part $ρ_L$ has a $\mathbb Z$-factor that is either identity or genuinely parabolic, and is thus not KAM-rigid, or (ii) Almost every affine $\mathbb Z^2$-action with linear part $ρ_L$ is KAM-rigid under volume preserving perturbations.
We consider the KAM theory for rotational flows on an $n$-dimensional torus. We show that if its frequencies are diophantine of type $n-1$, then Moser's KAM theory with parameters applies to small perturbations of weaker regularity than $C^n$. Derivatives of order $n$ need not be continuous, but rather $L^2$ in a certain strong sense. This disproves the long standing conjecture that $C^n$ is the minimal regularity assumption for KAM to apply in this setting while still allowing for Herman's $C^{n-ε}\!$-counterexamples.
We prove that there is an invariant torus with given Diophantine frequency vector for a class of Hamiltonian systems defined by an integrable large Hamiltonian function with a large non-autonomous Hamiltonian perturbation. As for application, we prove that a finite network of Duffing oscillators with periodic exterior forces possesses Lagrangian stability for almost all initial data.
Rotation Sets of Billiards with N Obstacles on a Torus
对于在环面上有$N$障碍物的台球，我们研究了其特定类型的轨迹\emph{the so called admissible trajectories}的行为。利用\cite{1}中的方法，证明了\emph{admissible rotation set}是凸的，且容许型周期轨线在容许旋转集中是稠密的.此外，我们还证明了容许旋转集是一般旋转集的一个真子集。
In a previous work [Asymptotically quasiperiodic solutions for time-dependent Hamiltonians, arXiv preprint arXiv:2211.06623 (2022)], we consider time-dependent perturbations of a Hamiltonian having an invariant torus supporting quasiperiodic solutions. Assuming the perturbation decays polynomially fast as time tends to infinity, we prove the existence of an asymptotic KAM torus. That is a time-dependent family of embedded tori converging as time tends to infinity to the quasiperiodic invariant torus of the unperturbed system. In this paper, the dynamic on the invariant torus associated with th…
In this paper, we develop an approach to the problem of closing lemma based on KAM normal form. The new approach differs from existing $C^1$ perturbation approach and spectral approach, and can handle the high regularity, high dimensional cases and even Riemannian metric perturbations. Moreover, the proof is constructive and effective. We apply the method to the original nearly integrable setting of Poincaré and confirm several old and new conjectures with weak formulations. First, for Poincaré's original setting of nearly integrable systems, we prove that for typical perturbations, periodic o…
We provide evidence of the existence of KAM quasi-periodic attractors for a dissipative model in Celestial Mechanics. We compute the attractors extremely close to the breakdown threshold. We consider the spin-orbit problem describing the motion of a triaxial satellite around a central planet under the simplifying assumption that the center of mass of the satellite moves on a Keplerian orbit, the spin-axis is perpendicular to the orbit plane and coincides with the shortest physical axis. We also assume that the satellite is non-rigid; as a consequence, the problem is affected by a dissipative t…
Parallelized computation of quasi-periodic solutions for finite element problems: A Fourier series expansion-based shooting method
State Key Laboratory for Strength and Vibration, Xi’an Jiaotong University（西安交通大学 强度与振动国家重点实验室）；Department of Mechanics and Aerospace Engineering, Southern University of Science and Technology（南方科技大学 力学与航空航天工程系）
