Isochronous and Unexpected Behavior for Complex-Valued Non-linear Oscillators with Parametric Excitation

Abstract

Usually oscillators with periodic excitations show a periodic motion with frequency equal to the forcing one. A complex-valued nonlinear oscillator under parametric excitation is investigated by an asymptotic perturbation method based on Fourier expansion and time rescaling. Four differential equations for two nonlinearly coupled oscillators are derived. Approximate solutions are obtained and their stability is discussed. We found that the resulting motion is periodic with a frequency equal to the forcing one, if appropriate inequalities are satisfied and then for a large parameter range. The system is isochronous because periodic solutions are possible in a well-defined phase region and not only for certain discrete values. Moreover, we demonstrate that if we insert a gyroscopic term the motion can be always periodic for a well-defined parameter range but with a frequency different from the forcing frequency. Analytic approximate solutions are checked by numerical integration.