Analysis Of Some Non-smooth Factors In Nonlinear Dynamical Systems

The nonlinear phenomenon has been an extensively investigated topic over the past decades. Nonlinear evolution equation and nonlinear electrical circuits are two of the important branches of nonlinear science. Some non-smooth problems in this area are explored in this paper.Firstly, traveling wave solutions to the Generalized KdV equation with singular line and singular curve is investigated. The parameters, especially the singular line, may influence the properties of equilibrium points on the vector field, which, therefore, leads to different types of the phase structures. Bifurcation theory can be employed to derive the transitions boundaries, which divide the parameter space into several regions associated with different types of phase portraits. Combining with the investigation of the Hamilton values, the trajectories can be classified into groups corresponding to different forms of wave solutions. It is pointed out that different wave solutions may coexist with the same parameter conditions. Furthermore, the singular line may cause the singular waves patterns, such as peaked periodic waves, compactons and kink-compactons.Secondly, dynamic behavior of piecewise linear autonomous circuit is investigated. For the autonomous piecewise double-scroll circuit, two symmetric equilibrium points exist, each of which may lead to periodic movements via Hopf bifurcation and further to chaos via sequences of period-doubling bifurcations. When the trajectories of the two chaotic attractors pass across the switching boundaries, interaction between the two chaotic attractors may occur, which leads to an enlarged chaotic attractor with symmetric structure.Finally, dynamic behavior of piecewise linear non-autonomous circuit is investigated. For the circuit with external periodic excitation, the periodic solutions can be approximated by generalized equilibrium points as the trajectories do not pass across the switching boundaries, which may result in two symmetric quasi-periodic solutions (QSs) via the Hopf bifurcations related to the GEPs. As the trajectories expand to pass across both two switching boundaries, the interaction between the two symmetric QSs occurs, leading to a double scroll chaotic attractor with symmetric structure, the trajectory of which may cycle around both the two QSs in turn, while pass across the switching boundaries quickly, because of the large real part of the eigenvalues related to GEPs in the neighbourhood of the switching boundaries. However, when the periodic orbit passes across both the two switch boundaries, two routes to chaos can be observed, the first of which is caused by the sequence of period-doubling bifurcation from an unsymmetric periodic solution, while the second of which is caused by the instability of a quasi-periodic solution from the Hopf bifurcation of a periodic orbit with symmetric structure.