Synchronization of unstable orbits using adaptive control

Abstract

We propose a method of controlling nonlinear and chaotic systems which is able to synchronize the phase space trajectory to a desired unstable orbit. The desired orbit could be an unstable periodic orbit or a chaotic orbit. The method uses the procedure of adaptive control and introduces time dependent changes in the system parameters. The changes in the parameter values depend on the deviations of the variables of the system from the desired orbit and the deviations of the controlled parameters from their values corresponding to the desired orbit. We illustrate our method using the Lorenz and Rossler systems. We also show that our method may be useful for communication purposes.