Dynamic Analysis, Electronic Circuit Realization of Mathieu-Duffing Oscillator and Its Synchronization with Unknown Parameters and External Disturbances

Abstract

This chapter deals with dynamic analysis, electronic circuit realization and adaptive function projective synchronization (AFPS) of two identical coupled Mathieu-Duffing oscillators with unknown parameters and external disturbances. The dynamics of the Mathieu-Duffing oscillator is investigated with the help of some classical nonlinear analysis techniques such as bifurcation diagrams, Lyapunov exponent plots, phase portraits as well as frequency spectrum. It is found that the oscillator experiences very rich and striking behaviors including periodicity, quasi-periodicity and chaos. An appropriate electronic circuit capable to mimic the dynamics of the Mathieu-Duffing oscillator is designed. The correspondences are established between the parameters of the system model and electronic components of the proposed circuit. A good agreement is obtained between the experimental measurements and numerical results. Furthermore, based on Lyapunov stability theory, adaptive controllers and sufficient parameter updating laws are designed to achieve the function projective synchronization between two identical drive-response structures of Mathieu-Duffing oscillators. The external disturbances are taken into account in the drive and response systems in order to verify the robustness of the proposed strategy. Analytical calculations and numerical simulations are performed to show the effectiveness and feasibility of the method.