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Abstract: 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. [Copyright &y& Elsevier]

Abstract: We study three critical curves in a quasiperiodically driven system with time delays, where occurrence of symmetry-breaking and symmetry-recovering phenomena can be observed. Typical dynamical tongues involving strange nonchaotic attractors (SNAs) can be distinguished. A striking phenomenon that can be discovered is multistability and coexisting attractors in some tongues surrounding by critical curves. The blowout bifurcation accompanying with on-off intermittency can also be observed. We show that collision of attractors at a symmetric invariant subspace can lead to the appearance of symmetry-breaking. [Copyright &y& Elsevier]

Abstract: We discuss strange nonchaotic attractors (SNAs) in addition to chaotic and regular attractors in a quasiperiodically driven system with time delays. A route and the associated mechanism are described for a special type of attractor called strange-nonchaotic-attractor-like (SNA-like) through torus bifurcation. The type of attractor can be observed in large parameter domains and it is easily mistaken for a true SNA judging merely from the phase portrait, power spectrum and the largest Lyapunov exponent. SNA-like attractor is not strange and has no phase sensitivity. Conditions for Neimark–Sacker bifurcation are obtained by theoretical analysis for the unforced system. Complicated and interesting dynamical transitions are investigated among the different tongues. [Copyright &y& Elsevier]