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.

ln this paper, a double pendulum system is studied for analyzing the dynamic behaviour near a critical point characterized b9 non-semisimple 1:1 resonance. Based on normal from theory, it is shown that two phase-locked periodic solutions may bofircate from an initial equilibrium, one of them is unstable and the other may be stable for certain values of parameters. A secondary bifurcation from the stable periodic solution yields a family of quasi-periodic solutions lying on a two dimensional torus, further cascading bifurcnations from the quasi-periodic motions lead to two chaos via period-doublin9 route. It is shown that all the solutions and chaotic motions are obtained under positive dampin9.

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。

本文研究了已具有静变形的受周期激励作用下浅拱在1:2内共振条件下的分岔特性，进而按系统的运动形式将整个参数平面分成不同的区域，得到了物理参数平面上浅拱的定常运动分布情况，结合数值分析方法详细分析了系统在各个区域内特别是Hopf分岔区域内系统的动力学特性，指出系统模态相互作用的规律及其通向混沌的过程。

通过对非线性Duffing方程解的稳定性进行研究，得到了其周期一解失稳的转迁集的解析表达式，同时应用广义牛顿法，得到了Duffing方程对称破缺分岔转迁集的解析表达式，与Ueda用模拟计算机的方法和A. Y. T. Leung用增量谐波平衡数值方法的结果吻合良好，克服了用模拟计算机或数字计算机确定物理参数平面上的转迁集计算工作量十分大的困难。