| The collective dynamics is of the locally coupled map lattices consisting of the both discontinuous and non-invertible maps. in the same state of period-5, period-5 coexisting with band-11 chaos, period-5 and band-6 chaos. The order parameters for synchronization and the largest Lyapunov exponents are calculated and analyzed when the states of the single map are in the period-5, the coexistence of period-5 and band-11 chaos, another period-5, and the band-6 chaos states, respectively. The details of the spatiotemporal dynamics of the system are observed through the space-amplitude plots and space-time diagrams.The result reveals that there are three kinds of dynamics when the single map is in its period-5 state. They are the frozen random patterns, the synchronization and the fully developed turbulence. The synchronization appears at the smaller coupling strength when the state of the single map is in the coexistence state of the period-5 and the band-11 chaos. The phase synchronization can be observed as the single map is in the state of band-6 chaos. The mechanisms for these phenomena are analyzed and their characteristics are described.The most important results in this work are the discovery of a peculiar prelude dynamics before synchronization and a periodic dynamics with a similar structure to the prelude dynamics. The analysis on the first type of the phenomena reveals that the spatial sites are decomposed into two synchronized clusters which synchronize to adjacent trajectories of the periodic orbit of the single map. With the time evolution, one of them expands but another one contracts until all the sites approach to the same synchronized state. The further study shows that this process is actually related to the pair-creation of a left-moving and a right-moving soliton-like waves These two soliton-like waves separate the space into two sub-clusters, and the moving of the two soliton-like waves causes the expanding of one cluster and the contracting of the other one. They collide with each other when the contracting cluster disappears, which results in the pair-elimination of the two soliton-like waves and produces a "dissipative breather" simultaneously. The breather decays, and its death induces the complete synchronization. For the second type of the phenomena, the system does not synchronize with time, but the order parameter varies periodically as the time increase. There appear a sub-process that has the same structure to the above-mentioned prelude dynamics and a recovery sub-process that pushes the system back to the beginning of the prelude dynamics in each cycle. In the former sub-process, the two sub-clusters in the system space evolve in the same way to the prelude structure. This sub-process ends up with the disappearing of the contracting cluster where the boudoirs of the two clusters merge into a small region in the space. Then the recovery process begins, and the dynamical variables in the merged region adjust themselves until a site in the middle of the region becomes the seed of a new expanding cluster. Right after this, a new cycle starts, and this process repeats till infinity. This phenomenon can also be explained by the concepts of soliton and breather. Based upon the analysis above, the analytical expressions of the order parameter in the peculiar prelude dynamics and the periodic dynamics are obtained through the introduction of the directory phase, which fits into the numerical results very well. |
