Analog circuit implementation and synchronization of a system consisting of a van der Pol oscillator linearly coupled to a Duffing oscillator

This paper deals with the analog circuit implementation and synchronization of a model consisting of a van der Pol oscillator coupled to a Duffing oscillator. The coupling between the two oscillators is set in a symmetrical way that linearly depends on the difference of the systems solutions (i.e., elastic coupling). The primary motivation of our investigations lays in the fact that coupled attractors of different types might serve as a good model for real systems in nature (e.g., electromechanical, physical, biological, or economic systems). The stability of fixed points is examined. The bifurcation structures of the system are analyzed with particular emphasis on the effects of nonlinearity. An appropriate electronic circuit (analog simulator) is proposed for the investigation of the dynamical behavior of the system. Correspondences are established between the coefficients of the system model and the components of the electronic circuit. A comparison of experimental and numerical results shows a very good agreement. By exploiting recent results on adaptive control theory, a controller is designed that enables both synchronization of two unidirectionally coupled systems and the estimation of unknown parameters of the drive system.