Divergences and Regulation of Bursting Solutions in Frequency Switching Systems

Mechanics Mathematics Physics
bursting solutions divergent modes frequency switching scheme divergence modulation dynamical mechanism

Academic Background

In nonlinear dynamical systems, frequency switching has garnered significant attention in recent years due to its ubiquitous presence in the real world and its distinctive fast-slow dynamics. Frequency switching can induce divergence behaviors at certain switching thresholds, destabilizing bursting solutions in slowly excited vector fields related to transcritical bifurcations. Such instabilities are particularly common in engineering applications and can potentially cause fundamental damage to the operational integrity of systems. Therefore, studying the stability of bursting solutions in frequency switching systems and their regulation methods is of great importance for understanding and predicting system dynamics.

Source of the Paper

This paper is co-authored by Jiahao Zhao, Xiujing Han, Jiadong Wang, and Meng Han, all affiliated with the Faculty of Civil Engineering and Mechanics at Jiangsu University in Jiangsu Province, China. The paper was accepted on February 23, 2025, and published in 2025 in the journal Nonlinear Dynamics (DOI: 10.1007/s11071-025-11045-y). The research was supported by the National Natural Science Foundation of China (Grant Nos. 12272150 and 12072132).

Research Process

1. Conditions and Regulation Methods for Frequency Switching-Induced Divergence

The study first analyzed the necessary conditions for frequency switching-induced divergence in bursting solutions. Based on a typical slowly excited vector field, the research identified two necessary conditions leading to divergence. By rationally adjusting the switching scheme, the study successfully blocked the divergence threshold window, generating a set of stable sliding bursting solutions valid for any reasonable threshold. Specifically, by reversing the original switching scheme (from a 2:1 to a 1:2 ratio), the study reconfigured the distribution of dynamic segments on the switching boundary, enabling the system trajectory to achieve smooth frequency switching through stable segments, thereby avoiding divergence.

2. Impact of Reference Excitation Frequency Perturbation on System Stability

Building on the regulation of frequency switching-induced divergence, the study further explored the impact of perturbations in the reference excitation frequency ω on system stability. The research found that ω perturbations could induce four previously undocumented frequency-modulated divergent modes. Through numerical simulations and bifurcation diagram analysis, the study revealed the triggering mechanisms and distribution patterns of these divergent modes. Specifically, the study summarized the frequency-threshold distribution spectra for each divergent mode under different initial conditions and analyzed the dynamical mechanisms underlying the origins and distribution of these modes.

3. Dynamical Mechanisms of Divergent Modes

The study provided a detailed analysis of the dynamical mechanisms of the four divergent modes:

  • Divergent Mode A: The trajectory completes the frequency switching and diverges from x=0 after more than three-quarters of the excitation period. This mode arises from the mismatch between the trajectory and the target attracting basin due to transcritical bifurcation delay.
  • Divergent Mode B: The trajectory diverges approximately one-quarter of the excitation period after completing the frequency switching. This mode results from the trajectory failing to stabilize at the transcritical bifurcation point.
  • Divergent Mode C: The trajectory diverges directly before reaching one-quarter of the excitation period. This mode occurs because the trajectory fails to match the contraction rate of the attracting basin during initial oscillations.
  • Divergent Mode D: The trajectory diverges after several full periods. This mode represents an intermediate state between stable bursting solutions and other divergent modes.

Key Findings

Through numerical simulations and bifurcation diagram analysis, the study yielded the following key results:

  1. Conditions for Frequency Switching-Induced Divergence: The research identified two necessary conditions for divergence and successfully blocked the divergence threshold window by adjusting the switching scheme.
  2. Frequency-Modulated Divergent Modes: The study discovered that ω perturbations could induce four frequency-modulated divergent modes and summarized the frequency-threshold distribution spectra for each mode under different initial conditions.
  3. Dynamical Mechanisms: The study revealed the triggering mechanisms and distribution patterns of these divergent modes, providing a theoretical foundation for understanding and regulating the stability of bursting solutions in frequency switching systems.

Conclusion and Significance

Based on a typical transcritical bifurcation vector field, this paper proposed, for the first time, regulation methods for frequency switching-induced divergence and uncovered novel divergent modes induced by perturbations in the reference excitation frequency. The study not only offers new insights into the stability analysis of frequency switching systems but also provides theoretical support for system design and regulation in practical engineering applications. The results indicate that the triggering of divergent behaviors primarily depends on the boundedness of the attracting basin in phase space, a conclusion applicable not only to transcritical bifurcation vector fields but also to other dynamical systems with bounded attracting basins.

Highlights

  1. Regulation Methods for Divergence: By rationally adjusting the switching scheme, the study successfully blocked the divergence threshold window, generating stable sliding bursting solutions.
  2. Frequency-Modulated Divergent Modes: The study, for the first time, revealed four frequency-modulated divergent modes induced by reference excitation frequency perturbations and summarized their frequency-threshold distribution spectra.
  3. Dynamical Mechanisms: The study provided a detailed analysis of the triggering mechanisms and distribution patterns of these divergent modes, offering new theoretical foundations for understanding and regulating the stability of bursting solutions in frequency switching systems.

Other Valuable Information

The study also noted that the boundedness of the attracting basin is closely related to the manifold analysis of unstable saddle equilibrium solutions. Future research can leverage this analysis to predict the stability of switching systems and circumvent divergent behaviors.