Lower dimensional invariant tori for multi-scale Hamiltonian systems

2023/09/07 by Qian, Weichao, Ji, Shuguan, Li, Yong
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2309.03463

Abstract

The ``Fundamental Theorem" given by Arnold in [2] asserts the persistence of full dimensional invariant tori for 2-scale Hamiltonian systems. However, persistence in multi-scale systems is much more complicated and difficult. In this paper, we explore the persistence of lower dimensional invariant tori for multi-scale Hamiltonian systems, which play an important role in dynamics of resonant Hamiltonian systems. Moreover, using the corresponding results we give a quasi-periodic Poincaré Theorem for multi-scale Hamiltonian systems, i.e., at least 2^m₀ families resonant tori survive small perturbations, where the integer m₀ is the multiplicity of resonance.

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