Synchronization by feedback and adaptive control

Abstract

A feedback technique which synchronizes the phase space trajectory of a nonlinear dynamical system to a desired chaotic orbit is introduced. The same technique could be used to control the system to a desired unstable periodic orbit. The feedback which depends on the deviations of the variables from the desired orbit is given to a few of the system variables while the other variables evolve freely. This forces all the variables to synchronize with the corresponding variables of the desired orbit. It is further shown that an adaptive control together with the feedback technique could be used to synchronize the system to an unstable orbit. The system can recover from deviations introduced by sudden changes in the system parameter and the system synchronizes to the desired unstable orbit which may be chaotic. We demonstrate these ideas using the Lorenz system, Rossler system, and the resonant three-wave coupling equations from plasma physics.