Robust synchronization and anti-synchronization of identical Φ6 oscillators via adaptive sliding mode control

The main goal of this paper is to propose the single input robust adaptive sliding mode controllers to accomplish synchronization and anti-synchronization between two identical Φ6 Duffing or Van der Pol oscillators with unmodel dynamics and external disturbances. Unlike directly eliminating the nonlinear dynamics by active control and sliding mode control in the literature, the proposed sliding mode controllers include the equivalent control part, which is only proportional to the synchronized error states, and the switching control part, where the discontinuous control functions have adaptive feedback gains. Sufficient conditions are provided based on the Lyapunov stability theorem and numerical simulations are performed to verify the effectiveness of presented schemes.