Shape synchronization of drive-response for a class of two-dimensional chaotic systems via continuous controllers

This paper investigates the drive-response synchronization in shape for a class of two-dimensional continuous systems of chaos. The shape of the chaotic attractor of the drive chaotic system is considered in this paper. Using the signed curvatures of plane curves to describe the shapes of trajectories for drive and response systems, the continuous controller for shape synchronization is synthesized based on the fundamental theorem on plane curves in classical differential geometry. The continuous controller synthesized can guarantee that the response system is synchronized with the drive chaotic system in shape. The shape synchronization is obtained in spite of different dimensions in drive and response systems. Finally, the Duffing oscillator is utilized as an illustrative example. Simulation results show that the method proposed in this paper is effective for the application of secure communication.