The Density Of Algebraic Curve Homoclinic Cycle For An Integrable Quadratic System

In the past several decades, the problem of bifurcations of homoclinic or heteroclinic cycles in a plane polynomial system has interested many scholars, especially for quadratic systems, many good results have been achieved . Based on the work before, the present paper is devoted to researching the bifurcations of homoclinic cycles and some relational question in the quadratic system. This paper is divided into three parts.In the first part, the generic quadratic system with a hyperbolic weak saddle is discussed. By transforming the system to a certain normal form which possesses minimum parameters and using Zhu Deming's saddle for-mula([2]), we obtain all the parametric conditons for the system being locally integrable. Further we analysis all the quadratic integrable systems and get the following conclusion: under the nondegenrative linear coordinate transformation, the quadratic integrable system with homoclinic cycle all can be changed to a Hamiltonian system and a symmetric integrable system.In the second part, the above two kind of integrable systems are discussed. Let S is the set of points in the parameter space which correspond to the system possesing homoclinic cycle and within it the singular point being a center, let T is the subset of S, and the homoclinic cycle is algebraic curve, then we get the result: T is density in S. Further, we classify all algebratic cure homoclinic cycles as finite kinds according to the property of curves. Last, by perturbing the symmetric system properly and using Melnikov's method, we get the conditions for the perturbed system arising chaotic behavior.In the third part, disturb the Hamiltonian system which possesses curbic curve homoclinic cycle, such that the disturbed system still keeps the existence of homoclinic cycle, and the curbic curve develops to sextic curve , while the singular point becomes from a center to a focus. Afterwards, disturb the homoclinic bifurcation system again to arise limit cycle, as a re-sult,with the parameter changing the cycle decreases and finally disappears to a fine focus (Holf bifurcation).