Qualitative Analysis Of Random Shear System And Its Application

Using the theory of stochastic process, this thesis establishes a framework of meth-ods and techniques, by which we theoretically give useful criteria on the stochastic stability and convergence rate in a sense of probability one for some generic dynamical systems with random switch configurations.First, this thesis briefly review some recent progress achieved on dynamical sys-tems with random switches. Second, this thesis establishes practically useful criteria on stochastic stability for discrete, continuous, and time-delayed continuous dynamical systems, respectively, in Chapters 2,3 and 4. Meanwhile, the thesis investigates the dynamical evolution of complex dynamical networks, which indicates the advantages of random switch systems over the deterministic systems. All these results and findings confirm a broad perspective of the applicability of the theoretical criteria established in this thesis. Finally, this thesis is closed with some concluding remarks on the possible directions of future works.It is worthwhile to mention that the models we discuss in this thesis allow to include infinitely many subsystems of random switch, and allow the switch duration to obey some probability density functions with unbounded domains. Compared to the existing criteria which were established by matrix theory in the literature, our criteria is more convenient for analyzing and figuring out what kind of random switch configurations play important roles in synchronization of complex dynamical networks with random switched.More importantly, with a completion of probability space and deeper analysis, we surprisingly find that in every switch duration the connection node number for synchronization in random switch networks is quite smaller than the connection node number in conventional deterministic networks. We believe that this finding will high-light the real applicability of dynamical system with switches.