Influence Of Time-Delay On The Dynamical Characteristics In Nonlinear Network

The field of non-linear dynamical systems has been considered as one of the important breakthroughs in science between the end of the last century and the beginning of this century, especially in the study of chaotic systems. In contrast with other disciplines, this area is still relatively young, but there is no question that it is becoming more and more important in a variety of scientific disciplines. This paper mainly introduces the effect on the chaotic dynamics driven by the delay signal of itself, especially chaotic synchronization. It includes anticipated synchronization, complete synchronization and retarded synchronization. Research work of this dissertation has two main parts as follows:One part of the present thesis is devoted to show the control of the single chaotic system with time-delayed feedback and control the chaotic system to different states. By adjusting the delay times, we can control the system to the stable point, single period state, multi-period states or exceeded chaotic state.The other part of the present thesis shows the dynamic research on the coupling nonlinear array, especially four categories of single coupling array: ring arrays, chain arrays, the coupled ring and linear arrays and star arrays. Chaos synchronization has been intensively studied in this paper. (1) For open linear geometries (Chain), the chaotic cells are seen to synchronize consecutively as a synchronization wave spread through the arrays under no time-delay conditions. When we increase the delay time, many different synchronization states have been found, such as anticipated synchronization, complete synchronization, retarded synchronization. (2) For circular loops (Ring), when our system has no delay time, it is found that there is a critical number of cells above which the uniform synchronized state is not stable and exhibits chaotic rotating waves or periodic rotating waves. But when we increase the delay time, the states of chaotic systems can lead to rich dynamical behaviors. (3) For the coupled ring and linear arrays, when our system has no delay time, it is found that the chaotic rotating waves generated from the ring propagate with spatial periodic synchronization along the chain, that is to say, two chaotic oscillators in the chain are synchronization if the number of oscillators (spatial distance) between them is a multiple of oscillator number in the ring. But when increasing the delay time, our system also exhibit spatial periodic anticipated, complete, and retarded chaos synchronization. (4) For star array, when the system has no delay time and parameters of the oscillators are all same, it is found that the oscillators are synchronous, and the layered chaotic synchronization will be earlier appeared than the vertical chaotic synchronization. If the layered oscillators are same and the vertical oscillators are different, there will be only the layered chaotic synchronization occurring. When we increase the delay time in the star array system, even if the parameter of all oscillators are all same and the coupling coefficients are in the area in which all oscillators are synchronization without time-delay, there may be un-synchronous in the all oscillators with delayed time in certain region.