Variational approach to Arnold diffusion

Cheng, Chuan; Xue, Jinxin

doi:10.1007/s11425-019-1553-3

Cited by 3 publications

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“…The genericity in the C r -differentiable topology is now well understood. However, as pointed out in a recent survey [14], it remains a deep open problem to prove Arnold diffusion in the analytic category. This issue is of great interest since many Hamiltonians with physical significance are analytic.…”

Section: Introductionmentioning

confidence: 99%

Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems

Chen

Llave

2022

Nonlinearity

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The genericity of Arnold diffusion in the analytic category is an open problem. In this paper, we study this problem in the following a priori unstable Hamiltonian system with a time-periodic perturbation H ε ( p , q , I , φ , t ) = h ( I ) + ∑ i = 1 n ± 1 2 p i 2 + V i ( q i ) + ε H 1 ( p , q , I , φ , t ) , where ( p , q ) ∈ R n × T n , ( I , φ ) ∈ R d × T d with n, d ⩾ 1, V i are Morse potentials, and ɛ is a small non-zero parameter. The unperturbed Hamiltonian is not necessarily convex, and the induced inner dynamics does not need to satisfy a twist condition. Using geometric methods we prove that Arnold diffusion occurs for generic analytic perturbations H 1. Indeed, the set of admissible H 1 is C ω dense and C 3 open (a fortiori, C ω open). Our perturbative technique for the genericity is valid in the C k topology for all k ∈ [3, ∞) ∪ {∞, ω}.

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Section: Introductionmentioning

confidence: 99%