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.

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。

研究了受轴向激励屈曲简支梁的复杂动力学行为，分析系统内共振产生的条件，指出在内共振情况下系统的能量会通过二次分叉在其高阶模态和低阶模态之间传递，并通过数值分析证实了上述结论。