Melnikov Method In The Perturbation System Chaos Judgment And Chaos Control

Li and Yorke give the definition of"chaos"using strict mathematical language in 1975, People pay much attention on chaos and began researching it windly, With the developing of research, Many great achievements of chaos control theory have been obtained. The theorys have been widely used in various fields. Therefore, Study of chaos theory is significant; The core problem of researching a system's chaotic properties is studing the proximal property and the topological property of orbits of a system; The sensitive dependence on initial system and chaos control of the system may become a reality. This paper studies the stability of the ecosystem and chaos under Smale map, Further engineering systems is given control of the chaotic state of perturbation.In chapter 1, We introduce the historical development and present of chaos theory briefly, Pointing the necessity of studying chaos. Besides, We also introduce our intention of selecting such a topic and our research contents.In chapter 2, We introduce some basic concepts and theories of chaotic dynamical systems ane a number of background knowledge and existing results in this papers.In chapter 3, We study the chaotic state of equilibrium and chaos in Eco-system model;On the one hand, By solving the model, We access the balance between variable track and use Melnikov function given the threshold, The chaotic state of the system made a cautionary analysis; On the other hand, If the system has a very small perturbation, We get a system(May be different from the original system), We can see the system has the sense of Smale horsehoe chaos state by Melnikov function;The system is a hyperbolic saddle point near the sensitive dependence on initial, This further instructions from the theory: When the number of initial x 0 and y 0 fully close to the hyperbolic saddle point, Regardless of initial point (x1,y1) and (x0,y0) how close, The solution of the new system is large enough in t, Point(φ(t,x1,y1),ψ(t,x1,y1)) and (φ(t,x0,y0),ψ(t,x0,y0))distance is always larger than a certain number of normal.In chapter 4, We study the chaos control method in some applications in engineering, First, Numerical simulation shows the chaotic state and the state in the case of steady state perturbation image; Second, Melnikov method for further analysis in the use of the system in the sense of Smale horseshoe chaos state; If the system parameters in a small range of control make the system state.