Delay Stability And Bifurcation Of 2-d Discrete Dynamical System

The chaos theory studies the intrinsic randomness discipline in the nonlinear course of the nature. The spatiotemporal chaos system is a chaos system in which the state variable is about both the time and the space. The 2-D (Two Dimensions) discrete dynamic system is a kind of important nonlinear system establishing in the theory of spatiotemporal theory. It has become an academic research area in recent years because of its importance in solving practical problem and its wide-range applications. This paper is about a family of 2-D delayed discrete dynamic system, the content is divided into different categories shown as follows:The stability of the dynamic system solution is a hotspot in the nonlinear dynamic system. Under the base of the 1-D( One Dimension) system, we discuss the stability condition of one family 2-D Logistic map, and we give the examples of the stability case and instability case separately. The study of the undelayed system is the base of the next step of the research.The delayed 2-D discrete dynamic system is one kind of important, widely used nonlinear system. It is the emphasis doing research on delayed 2-D system in this paper. We give the stability condition, then simulate the state diagram of 2-D Logistic system at the stability case and instability case separately. Further more, we give two definition of exponent stability. The stability condition is given based on the definition of exponent stability. Finally, we obtain the effect of the delay through comparing the stability condition of delayed system with undelayed one.The parameters exert important influence to no matter one-dimension or two-dimension system's performance. Regarding the Logistic map system, in whichthe most important parameter is the constantμ. In our delayed 2-D system, there isanother important parameterξ, which stands for the coupled coefficient of thecurrent state and the delayed state. We get the spatiotemporal bifurcation diagram with the change of these two parameters, according to which to analyze the system characteristic. Finally we carry on the analogy with the one-dimension system. We give the conclusion and the forecast at the end of the paper.