离散模型三个不变闭曲线的分支存在性研究

The Existence Research of Three Invariant Closed Curves for Discrete Dynamical Models

主要在离散模型余维2的分支理论的基础上，研究了离散模型三个不变闭曲线的存在性．在模型消掉全部的二次、四次项和部分三次、五次项的基础上，利用余维2的分支理论和复正规形方法进一步化简方程，消掉全部六次项和部分七次项，再化成极坐标形式，得到了三个不变闭曲线的存在性和稳定性．当相关系数L3（0）〉0（L3（0）〈0）时，满足一定条件会出现三个不变闭曲线，其中里边的不变闭曲线不稳定（稳定），中间的不变闭曲线稳定（不稳定），外边的不变闭曲线不稳定（稳定）．最后通过举例展示了方法的应用过程，也通过数值模拟显示了方法和理论的正确性．同时，给出的n个不变闭曲线的存在性讨论、计算公式等在一定程度上推广和丰富了离散模型分支理论的研究．

Based on the bifurcation theory of the codimension two, we study the bifurcation of codimension three generating three invariant closed curves. On the basis of the model with eliminating all quadratic,quaxtic terms and several cubic, quintic terms,we can eliminate all sextic items, and part of septic items. Using the complex normal form method and the bifurcation theory of the codimension two, we can simplify the equations to a polar form. We get the conditions for the existence and stability of three invariant closed curves. WhenL3 （0） 〉 0（L3 （0） 〈 0） and certain conditions hold, there will exist three invaxiant closed curves. The inside invariant closed curves is unstable （stable）, the middle invaxiant closed curves is stable （unstable）, the outer one is unstable（stable）. Finally, examples and numerical simulations axe presented to show our results. Meanwhile, the existence of ninvariant closed curves and the formula to determine its stability generalize and enrich the bifurcation theories of discrete dynamical models.