Adaptive control design in chaos synchronization as alternative of Lyapunov´s direct method

As is well known various neuron models show interesting behavior as producing stable, non-sinusoidal limit cycles or even chaotic regimes of motion. Realization of precise tracking of a chaotic trajectory generated by a “master system” by a controlled, quite different “slave” system also having aptitude for chaotic motion is an interesting challenge. Lyapunov´s 2^nd or “Direct” Method is generally used in the literature for such purposes. It is well known that finding appropriate Lyapunov function is rather an “art” that needs special mathematical skills so it needs very talented designers. As an alternative of Lyapunov´s Direct Method the use of “Robust Fixed Point Transformations (RFPT)” for the synchronization of the motion of different chaotic systems is studied in this paper via numerical simulations. It is shown that the RFPT-based method is very efficient. In the present paper the signal generated by a 1^st order Chua-Matsumoto Circuit serves as a reference trajectory to be racked by a 2^nd order chaotic system, the Duffing Oscillator. The interesting finding is reported that while the “brute force” solution achieved by very big simple linear feedback coefficients fails after a short period the adaptive controller operating on the basis of very loose, purely kinematically designed feedback coefficient successfully stabilizes the synchronization.