Nonlinear resonances in the $ABC$-flow

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.