Study On Chaotic Generalized Synchronization And Synchronization In Dynamic Networks

In the rencent years, the study on chaos and chaos synchronization has attracted increased attention from scientists and engineers. Firstly in this dissertation, a brief introduction to the historical backgrounds of chaos and complex networks, research progress and typical methods for chaotic synchronization are given. Then the dissertation mainly focuses on the generalized synchronization of two chaotic different systems and projective synchronization of complex dynamical networks. The author's main research work and contributions are as follows:(1) The existence of two types generalized synchronization(GS) of chaotic unidirectional coupled systems is studied. Based on the modified system approach, GS is classified into three types: equilibrium GS, periodic GS, and C-GS, when the modified system has an asymptotically stable equilibrium, asymptotically stable limit cycles, and chaotic attractors, respectively. The existence of the first two type GS can be converted to the problem of compression fixed point under certain conditions. Strict theoretical proofs are given to exponential attractive property of generalized synchronization manifold. By using the Schauder Fixed Point Theorem in H?lder continuous function space, the existence of two types H?lder continuous inertial manifolds, equilibrium GS and periodic GS, with certain conditions will be theoretically proved.(2) Furthermore, we studies the existence of two types generalized synchronization of bi-directionally coupled chaotic oscillators. More complex situations appear. When the modified system of Y oscillator is chaotic, while the modified system of X oscillator collapses to a stable equilibrium point or periodic orbit, the existence of generalized synchronization can be converted to the problem of Lipschitz function family's compression fixed point under certain conditions. Proofs are also given to the existence of bi-directionally coupled smooth and H?lder continuous generalized synchronization manifold.(3) Impulsive projective (cluster) synchronization in 1 + N coupled chaotic systems are investigated with the drive-response dynamical network (DRDN) model. Based on impulsive stability theory, some simple but less conservative criteria are achieved for projective synchronization in DRDNs. It is worth noting that any width of impulsive intervalΔ≥0 can be arbitrarily selected to implement projective synchronization(PS) with desired scaling factor under certain conditions. Furthermore, impulsive pinning scheme is also adopted to direct the (cluster) scaling factor onto the desired value.(4) By using simple linear control, some sufficient criteria for projective synchronization with different scaling factors of the response networks of symmetrical and asymmetrical coupling are given based on Lyapunov function method.(5) A new model of N coupled chaotic systems networks without the drive oscillator is established. By designing an adaptive scheme, sufficient criteria for the projective synchronization with different scaling factors PSDF in symmetrical and asymmetrical coupled networks are obtained based on the Lyapunov function method and the left eigenvalue theory.For all the above theoretical results, we proposed numerical simulations which verity the corresponding theoretical results.