Complex aperiodic mixed mode oscillations induced by crisis and transient chaos in a nonlinear system with slow parametric excitation

For dynamic systems with multiple time scales, its long-term behaviors usually present as mixed mode oscillations (MMOs). The formation of MMOs is closely related to the fast transitions evoked by the bifurcations of the fast sub-system. Up till now, most of the results reported in deterministic systems focus on periodic MMOs, namely the holistic structure of waveform shows periodicity, no matter whether the slow manifolds that the fast–slow flow follows are periodic or not. Moreover, since the motion of the fast–slow flow is modulated by the critical manifolds, qualitative influences of transient behaviors on the holistic structure of MMOs are less concerned. While in this paper, three patterns of aperiodic MMOs are analyzed based on a three-dimensional nonautonomous system under slow parametric excitation. According to the types of manifolds among which the fast transitions take place, these MMOs can be named as aperiodic "cycle-chaos-cycle" MMOs, aperiodic "cycle-chaos-cycle" MMOs with additional chaotic bursters, and aperiodic "chaos-cycle-point" MMOs. By combining the fast–slow analysis on certain time intervals and the probe into the transient chaos near boundary crisis, generation mechanisms of these MMOs are investigated. Our results show that the relative location between the boundary crisis value and the minimum of slow variable will qualitatively change the pattern of large amplitude oscillations (LAOs); thus, the composition of bursters can be decided by the intensity of transient chaos. On the other hand, the small amplitude oscillations (SAOs) evoked by interior crisis may contain inner structures related to bifurcation delay and sharding phenomenon. Particularly, interior crisis can lead to merging phenomena of chaotic attractors. Thus, when the chaotic LAOs stages are over, the fast–slow flow can transit to upper or lower branch in a irregular way, which results in the uncertain locations of SAOs.