Nonlinear behaviors as well as the bifurcation mechanism in switched dynamical systems

Abstract

Dynamical model of a typical centrifugal flywheel governor system with periodic switches between two forms of external torque has been established, in which the state variables may be governed by two subsystems corresponding to autonomous and non-autonomous systems with periodic excitation, respectively. Two types of bifurcation can be observed in the autonomous subsystem, where the Hopf bifurcation may lead to the occurrence of periodic oscillation, while the fold bifurcation may result in the disappearance of the equilibrium points. The dynamics often behaves in periodic oscillations with the same frequency as the periodic switch, in which the switching points may divide the trajectory of the orbit into two groups of segments governed by the two subsystems, respectively. Cascading of period-doubling bifurcations related to the Floquet multiplier passing across the unit cycle can be observed, which cause the system to evolve to chaotic oscillations. Furthermore, because of the symmetry of the vector field, two attractors symmetric to each other can be obtained, which may interact with each other to form an enlarged attractor with the variation of the parameters. Generalized Hopf bifurcation may lead to the alternation of the behaviors between periodic movements and quasi-periodic oscillations.