On linear stability of KAM tori via the Craig-Wayne-Bourgain method
Xiaolong He, Jia Shi, Yunfeng Shi, Xiaoping Yuan
TL;DR
This paper combines KAM and CWB methods to prove the persistence and linear stability of invariant tori in Hamiltonian systems, providing detailed exposition of the CWB approach without relying on the second Melnikov condition.
Contribution
It introduces a novel combination of KAM and CWB techniques to establish linear stability of perturbed tori, simplifying the CWB method for broader accessibility.
Findings
Proves Melnikov's persistence theorem using combined methods
Establishes linear stability of perturbed invariant tori
Provides detailed exposition of the CWB method
Abstract
In this paper, we prove the Melnikov's persistency theorem by combining the traditional Kolmogorov-Arnold-Moser (KAM) technique and the Craig-Wayne-Bourgain (CWB) method. The aim of this paper is twofold. One is to establish the linear stability of the perturbed invariant tori by using the CWB method without the second Melnikov condition. The other one is to illustrate the CWB method in detail and make the CWB method more accessible.
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Taxonomy
TopicsQuantum chaos and dynamical systems · Numerical methods for differential equations · Nonlinear Dynamics and Pattern Formation