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

Abstract

A generalized form of the autonomous Bonhoeffer-van der Pol (BVdP) system described by a second-order dynamical system with six independent parameters consistent with its optimal mathematical modeling, instead of three usually used, is investigated. Through its equivalent form, the generalized asymmetric van der Pol-Duffing (GAVdPD) system and the steady states of this system are derived. The analysis show that the system may exhibit one or three steady states when it is driven by an external constant impulse taken as a main control parameter. Domain ranges in which the system can function as well as monostable system as a bistable system are derived. In addition, by means of the theory of Hopf Bifurcation, it appears that there are large possibilities for the system to work as self-sustained oscillator, forced oscillator or other possibilities for which the system does not operate, indicating the richness of this generalized form of the BVdP system. Limit cycle solutions are derived at the neighboring of the Andronov-Hopf Bifurcation points even for large values of the asymmetric parameter. All these results are checked through numerical simulations. Applying these analytical investigations to the electronic circuit executing the dynamics of the basic BVdP system, two distinct working regimes are highlighted, depending on the magnitude of the capacitance with respect to a threshold value function of the characteristic parameters both of the self and of the nonlinear resistance. Through PSPICE simulations the accuracy of these analytical and numerical investigations have been confirmed.