Dynamic Behaviors For A Class Of Two Coupled Nonlinear System

The thesis deals with the dynamic behaviors for a class of two coupled nonlinear system by using the method of dynamical system. Such system has wide application in many physics problems, such as celestial mechanics, plasma physics, nonlinear op-tics. It is important not only in theory but also in practicality to study and understand more about the properties of various solutions. As we all know, it is difficult to ob-tain the explicit solution for this nonlinear dynamical issues. Besides the numerical method and asymptotic analysis method, bifurcation and chaos theory of dynamical system is used to analyze the dynamics of the system according to different parameters.Especially for the equilibrium solution,periodic solution,quasi-periodic solution,homoclinic and heteroclnic solution > chaotic solution, the bifurcation and stabilityanalysis of such solutions have been studied. And some special cases of complete integrability,nonintegrability have been studied with the given parameters.The system we dealt with is given bywhere the parameters A₁,A₂,b_ij(i,j=1,2)are arbitrary real numbers.In this system, the coefficients of the coupled item are b₁₂ and b₂₁, so we can classify the system according to the different cases of these coefficients. i) In the case of b₁₂ = b₂₁ = 0, the two equations of the system are decoupled, and both of them are plane Hamiltonian system, it is easy to analysis the type of equilibrium points and draw it's phase graph by using the theory of plane dynamical system. ii) When there is only one of the coefficients being zero, base on the symmetry of the system, without loss of generality, we can let b₁₂ = 0,b₂₁≠0, so the first equation of the system can be solved alone, then take the solution x₁(t) back to the coupled item x₁²x₂ in the second equation. So the system(2) become to be a non-autonomous system of nonlinear differential equations. In this article, we focus on the periodic solution x₁{t), which means the periodic orbit x₁{t) in the first equation, so the second equation is a plane Hamiltonian system with a periodic perturbation. Melnikov method is used to analysis the existence of homoclinic orbit and subharmonic periodic orbit of the system. iii) When both of the two coefficients are not zero, if b₁₂ and b₂₁ are with the same sign, then by the appropriate scaling transformation, the system is a Hamiltonian system with two-degree-of-freedom. Lyapunov center theorem is used to explain the existence of periodic orbit. When the equilibrium points are in the type of saddle and saddle-center, we analysis the chaos of the system by the Melnikov method.