R^3中一类四次可逆系统的周期轨与极限环问题

The Periodic Orbits and Limit Cycles of a Family of Reversible Systems of Degree Four in R^3

本文研究了R^3中一类四次可逆系统及其扰动系统的周期轨与极限环问题,利用Poincare紧化理论讨论了相关平面系统的定性性质,证明了所考虑的系统存在无穷多对称周期轨的结论.然后借助一系列技巧性变换,并利用平均方法证明了R^3中这类四次可逆系统的扰动系统在其某周期轨邻域至少存在两个具有不同稳定性的极限环的结论.

This paper＇ is concerned with the periodic orbits and limit cycles of a reversible system and its perturbed system in R^3. By using the theory of Poincare compactification we first study the qualitative properties of the related planar system, and we get the results that the reversible systems have infinitely many symmetric periodic orbits. Then through a series of skillful changes of variables and using the averaging method, we prove that at least two limit cycles with different stability properties exist near the periodic orbit γ of the perturbed system.