Based On The Dynamics Of Small World Network Of Coupled Map Lattices

The synchronization dynamics in the small-world networks of both invertible and discontinuous piece-wise linear maps is investigated. The characteristic quantities, such as synchronous order parameter, mean-field and mean square deviation that described the collective dynamics are calculated for different randomly reconnected probability p and coupling strengthε, and the Lyapunov exponential spectrum is computed. The coupled map lattice model has superiority features itself that the other models like the partial differential equation model, the coupled ordinary equation model and the cellular automata model without comparing, so it becomes a very important model and effective tool to do research on the dynamic behaviors of chaos. Since the small-world model compares to the regular network model and the completely random network model can better describe the real network, and the research objects in the dynamic behaviors of chaos are usually the completely regular networks or completely random networks, the lattices of which are controlled by smooth maps. But in the realistic systems, the discontinuous chops are always emerging, and we can depict these phenomenon by discontinuous maps. So it is worth of doing research on the model which establishes on the base of small-world and the lattices are controlled by discontinuous maps. The text mainly uses the theory of nonlinear dynamics and model to study the dynamics of the complex system, and most of tasks are numerically calculated. On the one hand it is good for us to understand the dynamic phenomenon in this model, on the other hand we can put the results to use in the realistic systems in order to improve the property of them.The text calculated the correlation quantities such as characteristic quantities and the largest lyapunov exponent and gained the main conclusion are as follows:(1) The model which establishes on the base of small-world and the lattices are controlled by discontinuous maps can demonstrate very affluent phenomenon, and we can see them from the characteristic quantities of which depict the group dynamics;(2) We find that in some attraction zones there emerges a very special structure before the system getting to synchronization, and we analyse the structure numerically and theoretically, we redefine the phase and gain the analytic expression of synchronous order parameter;(3) We also find that the system undergoes five dynamic behaviors in some attraction zones:the frozen random pattern mode, the pattern selection mode, the pattern competition intermittency mode, the fully developed turbulence mode, the synchronization mode and the and cycle grafts mode. The first four patterns are the same with K. Kaneko's analytical results about the unimodal model, and proves that some theoretic results derived from the low dimension are universal;(4) With the increase of the ratio of randomly adding edges, in some attraction zones the synchronization region enlarges, and in some other zones we can't observe the synchronization, and the dynamic behaviors in these situations are chaos.