Synchronization And Control Of Nonlinear Oscillatory Systems

By using differential inequalities to construct two auxiliary autonomous systems and using active control technique, we present synchronization with error bound between two non-identical forced oscillators with the parameter mismatch of the external harmonic excitations. When the external harmonic excitations are the same, a new active control technique is introduced to make the two non-identical forced oscillators achieve complete synchronization. The new active control technique is simpler and easier to implement. These methods can control a chaotic system to be a periodic system or a periodic system to be a chaotic system. The master-slave synchronization schemes consisted of a chaotic oscillator and a periodic oscillator as numerical examples verify the effectiveness of these techniques.Based on the approximate solution of the forced Van der Pol equation, the virtual trajectories of the master system can be obtained. By substituting the virtual trajectories into the approximate error system between the master and the slave systems, we derive the relation between synchronization time and feedback gain by the multiple scale method. A fairly good agreement is obtained between the analytical and numerical results.Algebraic criteria for periodic oscillatory synchronization of the generalized Lienard system are derived based respectively on the stability theory of linear time-varied systems, Lyapunov direct method and Gerschgorin disc theorem. It is proven theoretically that the synchronization criteria based on Lyapunov direct method are more flexible than the criteria based on Gerschgorin disc theorem, and the further more flexible criteria can be obtained by choosing the proper Lyapunov function. Rayleigh-Duffing equation as a numerical example verifies the theoretical results.