Shimiza-Morioka系统异宿轨道存在性证明

Existence of heteroclinic orbit in Shimiza-Morioka system

针对Shimiza-Morioka系统用待定系数法证明了其系统中存在异宿轨道的存在性。首先将Shimiza-Mo-rioka系统转换为只含一个变量的非性线微分方程;然后证明该非线性微分方程存在一个指数形式的无穷级数展开式表示的异宿轨道;最后证明了该无穷级数展开式一致收敛性,结合Si’lnikov不等式,证明了该系统中存在Smale马蹄,因而是Si’lnikov意义下的混沌。最终,异宿轨道决定Shimiza-Morioka系统中混沌吸引子的几何结构。

This paper applied the undetermined coefficient method to prove the existence of heteroclinic orbit in Shimiza-Mo-rioka system.First of all,it converted the Shimiza-Morioka system to a nonlinear differential equation with only one variable.Secondly,it verified the nonlinear differential equation to have a heteroclinic orbit expressed by the infinite series expansion with the exponential form.Finally,it proved the uniform convergence of the series expansion of the heteroclicic.Combining the existence of heteroclinic orbit with Si’lnikov inequalities,Smale horseshoses has been found in the Shimiza-Morioka system,and it is chaotic in the sense of Si’lnikov.As a result,it is this heteroclinic orbit that determines the geometric structure of the Shimiza-Morioka chaotic attractor.