Control Techniques for Synchronizing the States of Two Coupled Van der Pol Oscillators

Abstract

Mathematical models of nonlinear oscillators are used to describe a wide variety of physical and biological phenomena that exhibit self-sustained oscillatory behavior. When these oscillators are strongly driven by forces that are periodic in time, they often exhibit a remarkable ‘‘mode-locking’’ that synchronizes the nonlinear oscillations to the driving force. Oscillation is the repetitive variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states and is characterized by their amplitude and their phase. Their interactions can result in a systematic process of synchronization which is the adjustment of rhythms of oscillating objects due to an interaction and is quite distinct from a simple stimulus response pattern. Oscillators respond to stimuli at some times in their cycle and may not respond at others. Many important physical, chemical and biological systems are composed of coupled nonlinear oscillators. The Van der Pol equation has been used to model a number of biological processes such as the heartbeat, circadian rhythms, biochemical oscillators, and pacemaker neurons. Two such resistively coupled Van der Pol oscillators are analyzed and the phenomenon of synchronization between the states of the coupled oscillators is explored. Several control techniques to achieve synchronization are designed, implemented and performance evaluation carried out by simulation using MATLAB Software.