Dynamical behavior analysis of the Van der Pol oscillator with sine nonlinearity subjected to non-sinusoidal periodic excitations by the bifurcation structures

Abstract

Abstract The present work studies the dynamical behavior of the Van der Pol oscillator with sine nonlinearity subjected to the effects of non-sinusoidal excitations through the analysis of bifurcation structures. In this work, the classical Van der Pol oscillator is modified by replacing the linear term x with the nonlinear term sin n x . The idea is to see the impact of the strength of this function on the appearance of chaotic dynamics for small and large values of the nonlinear dissipation term ε. To do this, studies using numerical simulation and analogue simulation are proposed. Firstly, using nonlinear analysis tools, a similarity in the bifurcation sequences is observed despite a difference in the ranks at which the particular behaviours and bifurcation points are obtained. This study is confirmed by their respective maximum Lyapunov exponents. Secondly, a real implementation using microcontroller technology, motivated by the use of an artificial pacemaker, is carried out. For a more practical case, the excitation signal is generated by another microcontroller. The study shows a similarity with the results obtained numerically. Thirdly, in order to show that the model can be derived from mathematical modelling, an electronic simulation using OrCAD-Pspice software is also proposed. The results obtained are in qualitative agreement.