Controlling the Lorenz system with delay

Abstract

A generalized method for adaptive control, synchronization of chaos and parameter identification in systems governed by ordinary differential equations and delay-differential equations is developed. The method is based on the Lagrangian approach to fluid dynamics. The synchronization error, defined as a norm of the difference between the state variables of two similar and coupled systems, is treated as a scalar fluid property advected by a fluid particle in the vector field of the controlled response system. As this error property is minimized, the two coupled systems synchronize and the time variable parameters of the driving system are identified. The method is applicable to the field of secure communications when the variable parameters of the driver system carry encrypted messages. The synchronization method is demonstrated on two Lorenz systems with variable parameters. We then apply the method to the synchronization of hyperchaos in two modified Lorenz systems with a time delay in one the state variables. © 2008 Springer-Verlag Berlin Heidelberg.