Analysis On The Chaotic Motion Of Some Stochastic Nonlinear Dynamic Systems

In this text, we introduce the appearance and development of the nonlinear dynamic system and chaos;give the different definitions of chaos, the routes to chaos and the judgment of chaos. We introduce a famous analytic method——Melnikov method, which can be used to deal with the second order ordinary differential equations with weak period perturbation term and homoclinic orbit or heteroclinic orbit and the three order differential equations with saddle-focus homoclinic orbit. For these two classes of equations, we can establish a two-dimensional Poincare map using some technique. Consider the distance between the stable manifold and the unstable manifold of the saddle of the Poincare map, using an integral called Melnikov function which is direct proportion of the distance to judge if the system will appear transversal homoclinic point and transversal heteroclinic point, and then to judge if the system exist chaotic motion.The stochastic perturbation is unavoidable and cannot be neglected in many practical problems, so the study on stochastic drive is more and more important. Here the stochastic drive we used is bounded noise, then the Melnikov function turns into the Melnikov process, so we need to say if the Melnikov process has simple zero in some probability or statistical sense, here we consider the mean sense and mean square sense only. We also give examples about two classes of Lienard equations and a Duffing equation. Through the examples we can see the difficulty on dealing with the simple zero and the complexity in calculating of the stochastic system using the Melnikov method. So we introduce the rate of phase space flux theory to translate the problem of solving the simple zero of the Melnikov function to solving the zero of the phase space flux function, avoid the problem of solving the simple zero of the Melnikov function and predigest the calculation.