Control of chaotic behavior in the dynamics of generalized Bonhoeffer-van der Pol system: Effect of asymmetric parameter

Abstract

The effect of the asymmetric parameter in the dynamics, both of the forced Duffing van der Pol (DVdP) system and of the forced generalized Bonhoeffer-van der Pol (BVdP) system is investigated, namely the possibility for these systems to execute dynamics chaos even for large values of this asymmetric parameter. We have shown that the dynamics of the associated non dissipative and unforced system can be interpreted by means of an effective energy potential which may exhibit a two-hump or a two-well configuration according to the magnitude of the parameters of the system. In the two-well configuration, a pair of pulses of different amplitudes describing homoclinic orbits useful in the prediction of irregular dynamics for the associated forced system are analytically derived as well as the expression of the single pulse describing the single homoclinic orbit in the case of two-hump configuration and which leads to a kink-antikink pair (heteroclinic orbits) in the absence of the asymmetric parameter. By means of the Melnikov theory, the conditions for the existence of the transverse intersection of stable and unstable orbits or dynamics chaos are also derived. In the particular case for which the system can exhibit a pair of homoclinic orbits, the presence of the asymmetric parameter induces two types of domains among which a more favorable domain for the existence of chaotic behavior. In both cases, the asymmetric parameter reduces the domain range in which the system can execute regular dynamics and consequently increases its domain of a possible dynamics chaos. The accuracy of these analytical results is checked through the bifurcation diagrams and the corresponding Lyapunov exponents resulting from numerical simulations. It appears that the background properties of the unforced system, namely its capacity to exhibit autonomous oscillations and the stability of its equilibrium points which are closely connected to the magnitude of the asymmetric parameter, influence the accuracy of the analytical predictions.