Nonlinear resonances in the $ABC$-flow

TL;DR

This paper investigates resonances in the $ABC$-flow near the integrable limit, providing analytical conditions and numerical evidence for the existence of multiple resonances and their energy levels.

Contribution

It offers new analytical conditions for resonance existence and demonstrates the simultaneous presence of multiple resonances in a Hamiltonian system with 3/2 degrees of freedom.

Findings

Largest $n:1$ resonances exist and match theoretical energies

Two branches of the largest resonances are identified

Resonance conditions are analytically derived and numerically confirmed

Abstract

In this paper we study resonances of the $ABC$-flow in the near integrable case ($C\ll 1$). This is an interesting example of a Hamiltonian system with 3/2 degrees of freedom in which simultaneous existence of two resonances of the same order is possible. Analytical conditions of the resonance existence are received. It is shown numerically that the largest $n:1$ ($n=1,2,3$) resonances exist, and their energies are equal to theoretical energies in the near integrable case. We provide analytical and numerical evidences for existence of two branches of the two largest $n:1$ ($n=1,2$) resonances in the region of finite motion.