Analysis And Study On The Existence Of Chaos In Dynamical Systems

Dynamical system is a hot spot of research in mathematics in 20th century. Since thesecond half of 20th century, chaos theory, a branch of dynamical system, has been developedunprecedentedly. As a very typical behavior of some nonlinear systems, chaos is a state ofmotion, which is deterministic but is not predictable. How to suppress harmful chaos andenliven useful chaos becomes an interesting topic. It is very critical that whether a nonlinearsystem is chaotic, or it will be chaotic if the system satisfies some of conditions. Melnikovmethod is a sufficient condition of the existence of chaos in dynamical systems.This paper constructed three models (the nonlinear terms had degree one and two, degreeone and three, degree three and five, respectively.) We could use the Melnikov method todetermine whether the three models were generating chaos. For the first two models, we gavean exact solution of the homoclinic orbit or heteroclinic orbit, then wrote the Melnikovfunction, and finally found the conditions that would lead to chaos through the relatedtheorems. For the third model, because the nonlinear terms had degree three and degree five,so we could not find the analytical expression of the function. Therefore, based on thenumerical integral method, we got the chaos threshold finally.