Abstract
We analyze the isochronous island chains that appear in the Poincaré sections of near integrable twist systems. When the system presents just one resonant perturbation with a winding number, the number of chains is constant and it is completely determined by the perturbation. However, for systems that are perturbed by an infinite number of resonant perturbations with the same winding number, the number of isochronous chains depends on the superposition of the perturbations and it is a function of the parameters. Considering a system that describes wave-particle interaction, we show that the number of island chains increases without limit when the wave period or wave number are increased.
Highlights
The Poincaré-Birkhoff Fixed Point Theorem states that the resonances of a near integrable system present an even number of periodic points [1,2,3,4,5,6]
The winding number is a monotonic function of the action variables [5, 6, 9], which makes it impossible for the system to present isochronous chains in different regions of the Poincaré sections
We showed that the winding number is not enough to determine all the features of a resonance because an infinite number of resonant perturbations have the same winding number
Summary
The Poincaré-Birkhoff Fixed Point Theorem states that the resonances of a near integrable system present an even number of periodic points [1,2,3,4,5,6]. The winding number is a monotonic function of the action variables [5, 6, 9], which makes it impossible for the system to present isochronous chains in different regions of the Poincaré sections. The Poincaré-Birkhoff Fixed Point Theorem does not make any claim on the number of isochronous chains [1,2,3], we generally observe in the literature twist systems that present only one chain [2].
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