The genericity of Arnold diffusion in nearly integrable Hamiltonian systems

In this paper, we prove that the net of transition chain is $\delta$-dense for nearly integrable positive definite Hamiltonian systems with $3$ degrees of freedom in the cusp-residual generic sense in $C^r$-topology, $r \geq 6$. The main ingredients of the proof existed in [CZ, C17a, C17b]. As an immediate consequence, Arnold diffusion exists among this class of Hamiltonian systems. The question of [C17c] is answered in Section 9 of the paper.

Keywords

dynamical instability, Arnold diffusion

2010 Mathematics Subject Classification

37D05, 37J40

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This work is supported by NNSF of China (No.11790272 and No.11631006) and the program PAPD of Jiangsu Province, China.

Received 1 September 2017

Accepted 8 December 2017

Published 9 July 2019