The bifurcations of traveling wave solutions for two coupled variant Boussinesq equations introduced as a model for water waves are studied in this paper. Transition boundaries have been presented to divide the parameter space into different regions associated with qualitatively different types of solutions. The conditions for the existence of solitary wave solutions and uncountably infinite, smooth, non-smooth and periodic wave solutions are obtained. The explicit exact traveling wave solutions are presented by using an algebraic method.

By using the theory of planar dynamical systems to a compound KdV-type nonlinear wave equation, the bifurcation boundaries of the system are obtained in this paper. These bifurcation sets divide the parameter space into different regions, which correspond to qualitatively different phase portraits and therefore different types of the solutions may exist in different regions. The parameter conditions for the existence of solitary wave solutions and uncountably infinite, many smooth and non-smooth, periodic wave solutions are therefore obtained.

In this paper, the dynamic behaviour of a double pendulum system in the vicinity of several compound critical points is explored through both analytical and numerical approaches. Four types of critical points are considered, which are characterized by a double zero eigenvalue, a simple zero and a pair of pure imaginary eigenvalues, and two pairs of pure imaginary eigenvalues including resonant and non!resonant cases. With the aid of normal form theory, the explicit expressions for the critical bifurcation lines leading to incipient and secondary bifurcations are obtained. Possible bifurcations leading to 2-D and 3-D tori are also investigated. Closed form stability conditions of the bifurcation solutions are presented. A time integration scheme is used to and the numerical solutions for these bifurcation cases, which agree with the analytic results. Finally, numerical simulation is also applied to obtain double-period cascading bifurcations leading to chaos.

本文通过引入非线性频率，利用Floquet理论及解通过转迁集时的特性，研究了不可通约两周期激励作用下的Duffing方程在一次近似下的各种分岔模式及其转迁集，并指出其通向混沌可能的途径。