Study of complex behavior in a time-delayed van der Pol’s electromagnetic system (Ⅰ)——The phenomena of bifurcations and chaos

Abstract

In this paper, a simple infinite dimensional system (time-delayed van der Pol’s electromagnetic system) with rigorous solution is developed. Based on Poincaré mapping of the right-traveling voltage wave at the left end of the transmission line (x=0), the phenomena of bifurcations and chaos are investigated with the variation of system parameters E and λ. Numerical results show that there are very complex nonlinear dynamical behaviors in this time-delayed system, such as attractor co-existing, intermittent chaos, quasi border collision bifurcation to chaos and period-adding phenomena. In the meantime of studying the temporal chaotic behaviors, the spatial chaotic behaviors are preliminarily analyzed. Through depicting spatial distribution profile of the voltages, the different spatial patterns are observed in the time-delayed van der Pol’s electromagnetic system with the variation of system parameters E and λ, such as chaos, period and so on.