The dynamical behavior of two coupled parametrically excited van der pol oscillators is investigated in this paper. Based on the averaged equations, the transition boundaries are sought to divide the parameter space into a set of regions, which correspond to dilerent types of solutions. Two types of periodic solutions may bifurcate from the initial equilibrium. The periodic solutions may lose their stabilities via a generalized static bifurcation, which leads to stable quasi-periodic solutions, or via a generalized Hopf bifurcation, which leads to stable 3D tori. The instabilities of both the quasi-periodic solutions and the 3D tori may directly lead to chaos with the variation of the parameters. Two symmetric chaotic attractors are observed and for certain values of the parameters, the two attractors may interact with each other to form another enlarged chaotic attractor.

分析了耦合vander Pol振子参数共振条件下的复杂动力学行为。基于平均方程，得到了参数平面上的转迁集，这些转迁集将参数平面划分为不同的区域，在各个不同的区域对应于系统不同的解。随着参数的变化，从平衡点分岔出两类不同的周期解，根据不同的分岔特性，这两类周期解失稳后，将产生概周期解或3-D环面解，它们都会随参数的变化进一步导致混沌。发现在系统的混沌区域中，其混沌吸引子随参数的变化会突然发生变化，分解为两个对称的混沌吸引子。值得注意的是，系统首先是由于2-D环面解破裂产生混沌，该混沌吸引子破裂后演变为新的混沌吸引子，却由倒倍周期分岔走向3-D环面解，也即存在两条通向混沌的道路：倍周期分岔和环面破裂，而这两种道路产生的混沌吸引子在一定参数条件下会相互转换。

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.

The dynamics of a 1 + 1 unidirectional non-linear wave equation which combines the linear dispersion of the Korteweg-de Vries (KdV) equation with the non-linear/non-local dispersion of the Camassa–Holm (CH) equation is explored in this Letter. Phase plane analysis is employed to investigate the bounded traveling-wave solutions. By considering the properties of the equilibrium points and the relative position of the singular line, transition boundaries have been derived to divide the parameter space into regions in which different types of phase trajectories can be observed. The explicit expressions of different types of solutions have been presented, which contain both the KdV solitons and the CH peakons as limiting cases.