Delay Control And Synchronization Of Several Oscillations: Method, Theory And Application

The thesis investigates oscillation suppression and synchronization in nonlinear systems that feedback controls or couplings with time delays are introduced. As a matter of fact, time delay is an omnipresent phenomenon in various systems due to the unavoidable physical distance of signal transmission. Also, varieties of control techniques with time delays have been fruitfully developed for tracking unstable steady states or unstable periodic orbits in many physical, chemical, and biological systems. A representative example is the technique, called "time delayed autosynchronization"(TDAS) and proposed originally by Pyragas for stabilizing unstable periodic orbits in chaotic systems. Moreover, time delay is always imported for modeling complex systems including gene regulation networks and complex neural networks. Note that not only the model controlled by the TDAS but also some complex networks with time delays can be reduced into some specific classes of normal forms with time delays. Based on this observation, we in this paper study the dynamics of a super-critical Hopf bifurcation with time delays, and establish an accurate relation between the natural frequency and the form of control gain for a successful oscillation suppression. We further employ these established relation to the FitzHugh-Nagumo model and a class of complex networks for achieving oscillation suppression and synchronization. All the theoretical results are attributed to the necessary and sufficient condition that we establish for stability in a transcendental equation with complex coefficients. Indeed, we use the Brouwer fixed point theorem, theory of differential equations, and other analytic methods to establish these results theoretically. In addition, we use the theory of complex analysis and computational techniques to investigate oscillation suppression in some typical discrete dynamical model including the Chialvo neural model. Finally, we use the theory of center manifold, Floquet theory and computational method to study the stabilization of periodic orbit in the well-known Chen’s system, and show how to find out the real periodic orbit possessed by the original system. The thesis is closed by some concluding remarks.