具有平方、立方非线性项的耦合动力学系统1∶2内共振分岔

1∶2 Internal Resonance of Coupled Dynamic System With Quadratic and Cubic Nonlinearities

对一类具有平方、立方非线性项的耦合动力学系统 1∶2内共振情形进行了研究· 首先 ,用直接方法求出该系统 1∶2内共振时的NormalForm ,该系统的NormalForm中 ,不仅含有平方非线性项 ,同时还含有立方非线性项· 通过采用适当的变量变换 ,将 4维分岔方程约化成 3维 ,进而得到单变量 4次分岔方程· 最后用奇异性理论 ,研究了一类普适开折的分岔特性· 该方法可用于 4维中心流形上流的强内共振时的分岔行为分析·

The 1∶2 internal resonance of coupled dynamic system with quadratic and cubic nonlinearities is studied. The normal forms of this system in 1∶2 internal resonance were derived by using the direct method of normal form. In the normal forms, quadratic and cubic nonlinearities were remained. Based on a new convenient transformation technique, the 4_dimension bifurcation equations were reduced to 3_dimension. A bifurcation equation with one_dimension was obtained. Then the bifurcation behaviors of a universal unfolding were studied by using the singularity theory. The method of this paper can be applied to analyze the bifurcation behavior in strong internal resonance on 4_dimension center manifolds.