Multivalued synchronization by Poincaré coupling

This work presents multivalued chaotic synchronization via coupling based on the Poincaré plane. The coupling is carried out by an underdamped signal, triggered every crossing event of the trajectory of the master system through a previously defined Poincaré plane. A master–slave system is explored, and the synchronization between the systems is detected via the auxiliary system approach and the maximum conditional Lyapunov exponent. Due to the response to specific conditions two phenomena may be obtained: univalued and multivalued synchronization. Since the Lyapunov exponent is not enough to detect these two phenomena, the distance between the pieces of trajectories of the slave and auxiliary systems with different initial conditions is also used as a tool for the detection of multivalued synchronization. Computer simulations using the benchmark chaotic systems of Lorenz and Rössler are used to exemplify the approach proposed.

► A coupling based on Poincaré planes is proposed and multivalued synchronization of chaos is explored.
► The generalized synchronization in a chaotic master–slave system is explored by the auxiliary system approach.
► Conditional Lyapunov exponents are calculated to detect synchronization of chaos.
► The concept of Poincaré coupling is shown by using the Rossler and Lorenz systems as benchmark to study synchronization.