Nonexistence of ultra-subharmonic periodic solutions for a class of nonautonomous dynamic system
一类非自治动力系统超次谐周期解的不存在性

In the study of the nonlinear dynamics, Melnikov function is widely used as a criterion to check whether subharmonic or ultra-subharmonic bifurcation even chaos will occur in a perturbed Hamilton system. However, for the most cases, the classical Melnikov method can merely show the existence of subharmonic periodic orbits. Such a result is attributed to that only first order approximation is adopted in the classical Melnikov method. So higher-order Melnikov method is developed to determine the existence of the ultra subharmonic periodic solution. In this paper, a class of non-autonomous differential dynamic system is studied. It is proved that if there exists a periodic solution in such a system, the solution can only be subharmonic, and the existence of ultra-subharmonic periodic solution is impossible. Moreover, the nonexistence of R-type ultra-subharmonic periodic solution defined for a specified planar system is also confirmed. As an application of above conclusions, some typical examples are investigated. The results demonstrate that second-order Melnikov method used to justify the existence of ultra-subharmonic periodic orbits in a planar perturbation system may lead to a wrong conclusion. A simple geometric explanation is also provided.