Traveling Wave Solutions For A Branch Of The Jm Equation And A Class Of Chaotic Pll Equations And Harmonic Branch

The paper includes two parts. In the first part, J-M equation is considered by using the bifurcation theory of dynamical system. Also, all the exact traveling wave solutions in a family of special curved surfaces are obtained. This part consists of six sections. In section 1, the introduction is stated and two parameter conditions are obtained. In sections 2 and 3 ,under conditions (1) or (2), the bifurcation sets and phase portraits of (1.9) are given, respectively. In sections 4 and 5, corresponding to sections 2 and 3, all exact and explicit formulas of traveling wave solutions under given parameter conditions are showen. In section 6, the main results are given.In the second part, a family of PLL equations are investigated by using Melnikov method. the existence of chaos in the sense of Smale horseshoes is showen and the partitions of the regions of chaos and subharmonic bifurcations are obtained. This part includes four sections. In section 1, the Melnikov method and the system which will be discussed are narrated. In section 2, the qualititative property of the unperturbed system is studied. In section 3 and 4, the existence of chaos and subharmonic bifurcations is proved and their existent regions are given.