Adaptive backstepping synchronization between chaotic systems with unknown Lipschitz constant

The error between two nonlinear terms is a key point of many synchronization problems, however, the Lipschitz constant of the nonlinear term is not always easy to calculate for the stability analysis of the controlled error system, thus the nonlinear systems with unknown parameters and unknown Lipschitz constant is considered in this paper. Their scalar synchronous controller is proposed based on the thought of backstepping design. Without the need to evaluate the invariant set and calculate the Lipschitz constant, an assistant adaptive estimator is designed for the Lipschitz constant. What's more, as a problem solving skill, two different estimators are used on the same unknown parameter. Finally, the synchronization control for both chaotic autonomous Van der Pol-Duffing (ADVP) systems and chaotic Genesio systems with unknown parameters are given as examples to verify the control effect.