We consider a class of autonomous, continuous time, chaotic dynamical systems the state equations of which can be represented in so-called Lur’e form. In particular we consider Chua’s circuit which is a paradigmatic chaotic system belonging to this class. It is shown that the dynamic behavior of such a system can be influenced in such a way as to obtain out of chaotic behavior a desired periodic orbit corresponding to an unstable periodic trajectory which exists in the system. This kind of control can be achieved via injection of a single continuous time signal representing the output of the system associated with an unstable periodic orbit embedded in the chaotic attractor. Further, we investigate the case when this signal is sampled, i.e. we supply to the system the control signal at discrete time moments only. We show via extensive numerical simulations that effective control can be achieved with a low number of samples only. Control proves to be very robust. Despite the presence of scaling of system variables and noise of a significant level introduced by signal quantization and offset, we were still able to control the chosen orbit (although with growing noise level the orbit becomes more and more distorted but maintains the same periodicity). We present a variety of simulation results to support this claim. First laboratory confirmations are also included. We claim that this method, proved to be functional for controlling chaos in Chua’s circuit, is also applicable to any chaotic system of the Lur’e type with a single nonlinearity.
Adaptive synchronization of the fractional-order unified chaotic system
