The genericity of Arnold diffusion in nearly integrable Hamiltonian systems

TL;DR

This paper demonstrates that Arnold diffusion is generically dense in nearly integrable Hamiltonian systems with three degrees of freedom, confirming the widespread occurrence of this phenomenon in such systems.

Contribution

It proves the density of transition chains in a generic class of Hamiltonian systems, establishing the genericity of Arnold diffusion in this context.

Findings

Net of transition chain is δ-dense in the system class.

Arnold diffusion exists among these Hamiltonian systems.

Answers an open question from previous research.

Abstract

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, $r\ge 6$. 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.