Isochronous Behavior and Diophantine Relations for Complex-Valued Nonlinear Systems

Abstract

Isochronous systems are not rare in dynamical systems. Three complex-valued nonlinear systems (quadratic and cubic nonlinearity, van der Pol, gyroscopic oscillator) are investigated by an asymptotic perturbation method based on Fourier expansion and time rescaling. Four coupled equations for the amplitude and the phase of solutions are derived. Approximate solutions are obtained and their stability is discussed. We find that in the first two cases the motion is periodic, while in the third case the motion is periodic only if appropriate Diophantine relations are satisfied. Analytic approximate solutions are checked by numerical integration.