The Generation Of Stably Dissipative Matrix And Its Application In Dynamical Systems

This paper considers the Lotka-Volterra system which consists of the following ordinary differential equations: The above system is one of the important equation models in applied mathematics. At present, it is not only widely used in many fields such as physics, chemistry, biol-ogy, evolutionary game theory, economics and other social sciences, but also in many popular subjects as neural networks, biochemical reactions, cell evolution, resource management and epidemiology.As is well-known, there is a close relationship between the dynamics properties of Lotka-Volterra system and the algebraic properties of their interaction matrix A=(aij)n×n.Hence, according to the different properties of the interaction matrix A=(aij)n×n, the Lotka-Volterra system can be summarized into three cases:cooperative(or competitive),conservative and dissipative.In the practical application of the Lotka-Volterra systems, because the date of in-teraction matrix cannot be completely accurate, it’s necessary to consider small pertur-bation on the interaction matrix. Hence, the stably dissipative(SD) concept comes into being. In comparison with the research on other classes of systems, however, studying the stably dissipative ones is relatively rare, and graph theory is used to deal with stably dissipative. Here, maximal SD graph play an important role in the criterion and genera- tion of SD matrix. This paper will study the generation of SD matrix and the dynamical behavior of the corresponding system as following three part:Firstly, according to the mapping process for low-dimensional maximal SD graph, it comes out a method for getting high-dimensional graph. Here, some ideals are pre-sented when considering computer algorithm; Second, proceeding from the combina-tion of graphics, combined with simple algebraic properties, it gives several methods for constructing SD matrix, including the law of superposition, building bridges, adding strong connection and so on; Lastly, based on a class of5-dim dynamical system, it studies the dynamical behaviors in the view of biotic intrusion. And prove the period-icity for a class of6-dim system. Meanwhile, it comes out such two conclusions:near initial values induce near behavior; separated values still can induce near behavior, in the relationship between initial value and its long time behavior.