Constructing A Kind Of Dynamic System Having Silnikov's Saddle-Focus Homoclinic And Heteroclinic Orbit

Chaos which is related to homoclinic and heteroclinic orbit is a complicated dynamical behavior in a system. Rossler dual principle shows that a system may be chaotic, which consists of an ordinary two-variable chemical oscillator and an ordinary single-variable chemical hysteresis system. Furthermore, it provides the systematic way to construct the system having homoclinic and heteroclinic orbit. In this paper, a chaotic system coupled by fast subsystem and slow subsystem is constructed using the Rossler dual principle. Then we connect the orbit of fast subsystems and low subsystems making use of singular perturbation theory. Finally we get the homoclinic and heteoclinic orbit of Silnikov's type. Silnikov homoclinic and heteroclinic theory guarantees that the systems have Silnikov phenomenon and get more abundant results. For example, a three-dimension system having Silnikov heteroclinic orbit connects two saddle-focus points; a three-dimension system having Silnikov heteroclinic orbit connects three saddle-focus points; a three-dimension system having Silnikov heteroclinic orbit connects four saddle-focus points which are symmetrical and nonsymmetrical; Numerical simulation results are also given to demonstrate the theoretical analysis.The main results and methods are as follows 1 Introduce the Silnikov homoclinic and heteroclinic orbit theory; 2 Analyzing dynamic behavior of the fast-subsystem:the differences between perturbations and unperturbations are big in the intersection. The points of intersection will be separated; 3 Analyzing the dynamic behavior of the slow subsystems:the character of the equilibriums in the nullclines make the possibility of the homoclinic and heteroclinic orbit's existence; 4 By using the Rossler dual principle, the fast subsystem and slow subsystem will be coupled. Analyze the existence of homoclinic and heteroclinic orbit and the whole dynamical behavior in the constructing systems, and simulate the systems in order to demonstrate the theoretical analysis; 5 By analyzing the eigenvalue of equilibrium of the constructing systems, and proving the system meeting the theory. So the homoclinic and heteroclinic orbit in constructing system is of Silnikov type.