Response Solutions for Completely Degenerate Oscillators Under Arbitrary Quasi-Periodic Perturbations

• Published in: Communications in mathematical physics Vol. 402; no. 1; pp. 1 - 33
• Main Authors: Ma, Zhichao, Xu, Junxiang
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.08.2023; Springer Nature B.V
• Subjects: Classical and Quantum Gravitation; Complex Systems; Equivalence; Hamiltonian functions; Mathematical and Computational Physics; Mathematical Physics; Oscillators; Periodic variations; Perturbation; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Theoretical; Toruses
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-023-04719-4

In this paper, we consider a one-dimensional completely degenerate oscillator subjected to an analytically ϵ -dependent quasi-periodic perturbation, whose frequencies satisfy a Diophantine condition. By the KAM method, we show that one of the following results holds true: 1. For all sufficiently small ϵ and all initial values φ ∈ T 1 , there exists a family of analytically ( ϵ , φ ) -parameterized response solutions, which corresponds to the persistence of the resonant Lagrangian torus of the equivalent Hamiltonian system. 2. For all sufficiently small ϵ , there exists a response solution, moreover, for an uncountable number of sufficiently small ϵ , there exists another response solution. In this case, the resonant Lagrangian torus of the equivalent Hamiltonian system is destroyed and it splits into a hyperbolic or hyperbolic-type degenerate lower dimensional torus for all sufficiently small ϵ , and another (possibly elliptic) lower dimensional torus for an uncountable number of sufficiently small ϵ .