On the Theory of Nonlinear Resonance

The paper presents a brief review of results on the theory of nonlinear resonance in systems with $\frac{3}{2}$ degrees of freedom. We discuss the averaged systems near resonance levels, distinguish nondegenerate and degenerate resonances, and describe typical phase portraits of the corresponding first- and second-approximation systems. Conditions for the existence of resonant periodic solutions are formulated, and the appearance of invariant two-dimensional tori is discussed. Special attention is paid to resonance zones near degenerate energy levels. Two illustrative examples are presented, including the occurrence of a degenerate resonance inside a nondegenerate resonance zone and the appearance of vortex pairs in the Poincaré map. The authors dedicate the article to V.S. Afraimovich, an outstanding specialist in dynamical systems, on the occasion of his 80th birthday.