On symmetry-breaking bifurcation in the periodic parameter-switching Lorenz oscillator

By introducing the periodic parameter-switching signal to the Lorenz oscillator, a switched dynamic model is established. In order to investigate the mechanism of the behaviors of the whole system, bifurcation sets of the subsystems are derived and the Poincaré map of the switched system is defined by suitable local sections and local maps. Under certain parameter conditions, symmetric periodic oscillations may be observed. With the variation of parameter, the symmetry-breaking bifurcation mechanisms of the symmetric periodic oscillations can be understood by calculating the associated maximal Lyapunov exponent and Floquet multiplies. Meanwhile, the parameter values of the related local bifurcations, such as saddle-node, pitchfork and period-doubling bifurcations are calculated based on the Floquet multiplies.

By introducing the periodic parameter-switching signal to the Lorenz oscillator, a switched dynamic model is established. In order to investigate the mechanism of the behaviors of the whole system, bifurcation sets of the subsystems are derived and the Poincar6 map of the switched system is defined by suitable local sections and local maps. Under certain parameter conditions, symmetric periodic oscillations may be observed. With the variation of parameter, the symmetry-breaking bifurcation mecha- nisms of the symmetric periodic oscillations can be understood by calculating the associated maximal Lyapunov exponent and Floquet multiplies. Meanwhile, the parameter values of the related local bifurcations, such as saddle-node, pitchfork and peri- od-doubling bifurcations are calculated based on the Floquet multiplies.