Analysis Of Dynamic Behavior And Mechanism Of Piecewise Linear System With Three Time Scales

Non-smooth dynamical systems has widely existed and applied in the industry and engineering field. The study of the complex nonlinear phenomena and the mechanism of bifurcation in this system has become one of the hot topics in the research of nonlinear dynamics. Numerous studies have shown that compared with the smooth system, not only the conventional bifurcation might occur in non-smooth system, but also might occur special bifurcation which the former system does not have,such as the sliding bifurcation, the grazing bifurcation and the corner-collision bifurcation, etc. As the typical system of the third class of non-smooth system, the piecewise linear system has been widely studied in flight control, robot control, chemical industry and other fields.However, at the moment, the research related to this type of system was mostly based on single time scale, and multiple time scales were involved in many practical application systems. In consequence, it is actually essential to make a further research on non-smooth system with multiple time scales.Based on the background above, the complex dynamic behavior and its mechanism of the three time scales coupling piecewise linear system are revealed in this paper. First of all, based on Chua’s circuit, a four dimensional non-smooth dynamical system which containing three time scales has been established by introducing periodically alternate current source and selecting the appropriate parameters. Conventional bifurcations of generalized autonomous system for the fastest time scales has been investigated as well as the unconventional bifurcations generated from trajectory crossing the interfaces of non-smooth systems.So a single cross unconventional fold bifurcation and twice cross unconventional Fold-Hopf bifurcation have been discovered under thecondition of different parameters. Meanwhile, with the method of enveloping slow-fast analysis, bifurcation diagram of equilibrium point for the external excitation’s extreme value has been overlapped with phase diagram in the fastest subsystem. The mechanism of different bursting oscillations has been discussed, and the relationship of mutual effects between three dimensions and influence of various irregular bifurcation on system dynamics behaviors have been revealed.Secondly, on the basis of the contents above, the non-smooth boundaries were increase by adjusting the characteristics of the nonlinear resistance, and the dynamic behavior of three time scales system containing two symmetric non-smooth boundaries were analyzed.Through analyzing the stability of equilibrium points and non-smooth bifurcation, it was found that with the change of parameters, the stability of the equilibrium points in the different regions changed, which caused the different form of the quiescent states and spking states from bursting oscillations, and received two periodic bursting phenomena namely the double fold/fold burster and the double fold/Hopf/fold burster. The mechanism of non-smooth bifurcation, which was related to the process of the mutual transformation in quiescent states and spking states under different scale effects, was discussed with the enveloping slow-fast method. Combined with the system phase diagram and time history diagram, the reflects of three dimensions in the process of systematic busting oscillations were explained.