Isochronous and Unexpected Behavior for Complex-Valued Nonlinear 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 satisfiedand 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.
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