Chaotic bubbles and phase locking for a shaker system in the vicinity of three coexisting critical points

In this paper, the dynamical model of the shaker system, Poincaré maps, Jacobian matrix and power spectrum are established. Different phase-locking phenomena and chaotic bubbles are investigated in the vicinity of three coexisting critical points including Hopf–Hopf bifurcation point, 1:3 resonance point and 1:4 resonance point. In two strong resonance cases, phase-locking dynamics and associated bifurcations are easily to occur. Coexisting attractors have also been introduced to provide mechanisms for chaotic bubbles with connections between pieces. The occurrence of phase locking on doubling torus to multi-period leads to interruption of torus-doubling bifurcation. Isolated chaotic bubbles are birth via period-doubling bifurcation of such a multi-period. Phase-locking phenomena on T2 torus are also observed in such a neighborhood of critical points. The number of periods on torus by phase locking can be identified by power spectrum methods. The system parameters may be optimized by studying of phase-locking dynamics of this system.