Numerical Studies of the Periodically Forced Bonhoeffer van der Pol System

Abstract

The Bonhoeffer van der Pol system, with an applied constant forcing, was invoked by Fitzhugh [1961] as a two-dimensional representation of the four-dimensional Hodgkin–Huxley system [1952], a well-known physiological model representing the electrical behavior across a nerve membrane. The system has been analyzed within a particular parameter regime relevant to the physiology (see [Fitzhugh, 1961]) and for the full parameter space with emphasis on the prediction of periodic solutions (see [Barnes & Grimshaw, 1995]). In this paper the system is considered with a time dependent sinusoidal forcing term in which form it represents a nonlinear, nonautonomous system of differential equations with five parameters. The study is motivated by physiological experiments with neurons subjected to periodic stimuli (see, e.g. [Hayashi et al., 1982]). A few aspects pertaining to the system behavior have been explored by others for particular fixed parameter combinations (and with different purposes from those here), for example [Wang, 1989; Rajasekar & Lakshmanan, 1988, 1993; Yasin et al., 1993; Braaksma & Grasman, 1993; Rabinovitch et al., 1994]). In this paper we present the numerical results of simulations for a more general parameter space and propose theoretical interpretations for a broad range of the behavior states incurred. Further, we present numerical results describing how the system dynamics change as each of the five parameters is varied, and thus predict regions where periodic, quasiperiodic or chaotic behavior can be expected.