In this paper we investigate the necessary condition for the existence of the periodic solution of honeycomb sandwich plate dynamic system of two-degree-of-freedom. We establish the curvilinear coordinates frame on closed orbits of the unperturbed system of the honeycomb sandwich plate dynamic system and construct successor function. Then we get the necessary condition of the existence of periodic solution by judging the existence of the successor functions. The existence of periodic solutions is important for studying the stability of sandwich plates.
In this paper, the peakons and bifurcations in a generalized Camassa-Holm equation are studied by using the bifurcation method and qualitative theory of dynamical systems. First, the averaged equation is obtained by introducing linear transform and traveling wave transform to the generalized Camassa-Holm equation. Then, we applied the bifurcation theory of planar dynamical system and maple software to investigate the averaged equation. The phase portrait of the system under a parameter condition is obtained. Finally, we get the peakons from the limit of general single solitary wave solution.
In this paper, we investigate a class of three dimensional nonlinear dynamical systems whose unperturbed systems have a family of periodic orbits. Firstly, we establish the moving Frenet Frame on these closed orbits. Secondly, the successor functions are defined by the orbits which go through the normal plane. Finally, by judging the existence of solutions of the equations obtained from the Successor functions, we obtain the necessary condition for the existence of periodic solutions of these three dimensional nonlinear dynamical systems. The result has important significance for the basic research of applied mechanics.
In this paper, the sufficient condition for the existence of periodic solution of a class of three dimensional nonlinear dynamical systems is investigated. The moving Frenet frame is established on the closed orbit and the successor functions near the closed orbit are defined. According to the study of the existence of solution of the equation which obtained from the successor functions with the implicit function theorem, the sufficient condition for the existence of periodic solution of these systems is obtained. The results in this paper have important significance to decide the existence of periodic solution of three dimensional nonlinear dynamical systems.
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