Stability Analysis Of Lotka-Volterra Neural Network

The human brain has many intelligence behaviors,such as thinking,cognition,learning and memory,which come from the neurons that make up the brain.A large number of simple neurons interconnected form a neural network,which is both a highly nonlinear dynamical system and an adaptive organization system.The neural network has strong mathematical theory,such as nonlinear dynamics,the theory of artificial neural networks,biological neural networks,power system principle,differential equations,differential equations,functional differential equations,computer simulation and so on.The research of neural network research is to study the neural network dynamic behavior,many important theoretical results have been published in the "Nature","Science" "Neural Computation","IEEE Trans.Neural Networks","Neural Networks" and other well-known international academic journals.The application of neural network is more and more widely.For example,it have important applications in the fields of associative memory,pattern recognition,combinatorial optimization,signal processing,signal detection,system optimization,biometric recognition,remote sensing technology.And it can deal with many similar decision-making important neural computing problems.Therefore neural network has the application background and the research value.In this thesis,the object of this thesis is a Lotka-Volterra neural network with two-dimensional,which is also called a bi-ecological model-predation and prey model.It is a biological mathematics model.Biological mathematics is entering the new era,with quantitative method to solve some large-scale complex ecological problems.It has been widely used in agriculture,environmental science,population control,social science and medicine.It is very important to study the stability and permanence of the species in the biosphere.And many scholars have done a great deal of theoretical work.In particular,along with the development of computer technology,it gradually become possible that the artificial neural network is simulated by computer.Therefore,scholars have again raised the upsurge of studying the Lotka-Volterra ecological model.The thesis is mainly concerned with the dynamical properties of equilibria for Lotka-Volterra system with two-dimensional.Mainly include:this thesis not only discusses the type and stability of equilibrium point but also describes the trend of the system solution curve near the equilibrium point when the system parameters change.As follows:(1)In chapter 2,when equilibrium points of the system are elementary singular points,we analyze the nonlinear system by using Hartman linearization,and obtain the trajectory figures of the solution curves.(2)In chapter 3,when equilibrium points of the system are higher-order singular points,according to the change of the eigenvalues,blow-up methods,the center manifold theorem and stretching transformation are utilized to analyze the orbit distribution near the higher-order singular point of the system.Two theorems about the stability of higher-order singular points are obtained,and a judge theorem concerning the system parameters and the type of equilibrium points is developed.When one of the eigenvalues of the system is zero and the other is positive,we use two kinds of blowing up techniques to analyze the system.The results of the two methods are consistent.At the same time,the shortcomings of the two methods are pointed out.Then simulation examples are used in the fifth chapter.And the numerical simulations are carried out by using Maple software,and the correctness of the developed theorem is verified.(3)We want to study the distribution of trajectories for plane autonomous system in the whole plane.We not only know singularity points of the system in finite plane,but also need to understand the trend which the system trajectory extends to the infinite distance,that is,the behavior of the trajectory at infinity.Therefore,in this thesis,we also introduce the infinite singular points of the autonomous system,and how to judge that whether a system has infinite singular points.