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Purpose <p>The paper displays several interesting phenomena in a special types of chaos, i.e., the chaotic bursting oscillations caused by the coupling of different time scales in the vector field. The trajectory in spiking stage may oscillate according to regular cycles and chaos in turn, which suggest that regular attractors seem to be embedded on the chaotic attractor.</p> Method <p>By regarding the whole exciting term as a slow-varying parameter, the full system can be treated as a generalized autonomous fast subsystem. The dynamics of fast subsystem can employed to reveal the bifurcation mechanism of the bursting oscillation in the full system upon the transformed phase portrait.</p> Results <p>For a chaotic oscillator, when the coupling of slow and fast scale is introduced, chaotic bursting oscillations can also be observed. The trajectory may behave in different forms, because of the effect of periodic windows in the fast subsystem. When large ratio between the two time scales is taken, the trajectory may pass directly across the short windows, leading to the disappearance of influence for the related behaviors on the full system.</p> Conclusion <p>For a chaotic slow-fast dynamical system, when the attractors in the fast subsystem are stable enough, it may attracts the trajectory to settles down once the trajectory passes across the related region, leading to regular oscillations in chaos. Whether the influence of short windows in the fast subsystem on the full dynamics appears or not depends on the ratio between the two scales, because of the delay effect of bifurcations.</p>

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Regular Oscillations Embedded in Chaotic Attractor Induced by Slow-Fast Dynamics

  • 期刊论文

  • 发表日期：

    2025年8月29日

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  • Yeqiang Chen,
  • Miaorong Zhang,
  • Xiaofang Zhang,
  • Qinsheng Bi

摘要

Purpose

The paper displays several interesting phenomena in a special types of chaos, i.e., the chaotic bursting oscillations caused by the coupling of different time scales in the vector field. The trajectory in spiking stage may oscillate according to regular cycles and chaos in turn, which suggest that regular attractors seem to be embedded on the chaotic attractor.

Method

By regarding the whole exciting term as a slow-varying parameter, the full system can be treated as a generalized autonomous fast subsystem. The dynamics of fast subsystem can employed to reveal the bifurcation mechanism of the bursting oscillation in the full system upon the transformed phase portrait.

Results

For a chaotic oscillator, when the coupling of slow and fast scale is introduced, chaotic bursting oscillations can also be observed. The trajectory may behave in different forms, because of the effect of periodic windows in the fast subsystem. When large ratio between the two time scales is taken, the trajectory may pass directly across the short windows, leading to the disappearance of influence for the related behaviors on the full system.

Conclusion

For a chaotic slow-fast dynamical system, when the attractors in the fast subsystem are stable enough, it may attracts the trajectory to settles down once the trajectory passes across the related region, leading to regular oscillations in chaos. Whether the influence of short windows in the fast subsystem on the full dynamics appears or not depends on the ratio between the two scales, because of the delay effect of bifurcations.