Periodic forces combined with feedback induce quenching in a bistable oscillator

TL;DR

The paper investigates how periodic forcing, especially when combined with nonlinear (quadratic) feedback, can induce oscillation quenching in a bistable oscillator modeled by an extended Stuart–Landau equation. By applying an averaging method to both additive and multiplicative forcing scenarios, the authors transform the non-autonomous system into autonomous reduced systems and perform bifurcation analyses, revealing mechanisms such as SNIC and homoclinic bifurcations that govern the transition from limit cycles to fixed points. They show that additive forcing can produce 1:1 synchronization and amplitude modulation, while multiplicative forcing with quadratic feedback robustly triggers quenching within specific frequency bands; in Case 2, quenching is linked to a monostable origin attractor and a sequence of bifurcations. The work advances understanding of state-transition methods in bistable, non-autonomous oscillators and suggests potential applications for terminating abnormal oscillations in real-world systems, including biomedical contexts like deep brain stimulation or implantable devices.

Abstract

The coexistence of an abnormal rhythm and a normal steady state is often observed in nature (e.g., epilepsy). Such a system is modeled as a bistable oscillator that possesses both a limit cycle and a fixed point. Although bistable oscillators under several perturbations have been addressed in the literature, the mechanism of oscillation quenching, a transition from a limit cycle to a fixed point, has not been fully understood. In this study, we analyze quenching using the extended Stuart-Landau oscillator driven by periodic forces. Numerical simulations suggest that the entrainment to the periodic force induces the amplitude change of a limit cycle. By reducing the system with an averaging method, we investigate the bifurcation structures of the periodically-driven oscillator. We find that oscillation quenching occurs by the homoclinic bifurcation when we use a periodic force combined with quadratic feedback. In conclusion, we develop a state-transition method in a bistable oscillator using periodic forces, which would have the potential for practical applications in controlling and annihilating abnormal oscillations. Moreover, we clarify the rich and diverse bifurcation structures behind periodically-driven bistable oscillators, which we believe would contribute to further understanding the complex behaviors in non-autonomous systems.

Figures (10)

• Figure 1: (a) A bifurcation diagram of Eq. \ref{['esl_r']}. The solid and dotted lines represent the stable and unstable branches, respectively. (b) A typical phase plane of Eq. \ref{['esl']} when $a=b=1.0,\: \mu=-0.1$. The red dot shows a stable fixed point at the origin. The solid and dashed red lines show the stable and unstable limit cycles, respectively.
• Figure 2: The amplitude (a) and frequency (b) response curves of Eq. \ref{['esl_p']}. Panel (a) shows the maxima and minima of the amplitude $r$, whereas panel (b) plots the long-time average of the frequency of the oscillator $\langle \omega \rangle$. In both panels, the horizontal axis represents $\Omega$ (the frequency of periodic force). Note that panel (b) suggests that the entrainment to the periodic force occurs when $\Omega \in [\Omega_{\mathrm l}, \Omega_{\mathrm u}]$ with $\Omega_{\mathrm l} \simeq 1.12$ and $\Omega_{\mathrm u} \simeq 2.23$. For numerical simulation, we set $A = 0.5, \: x(0)=3.0,\: y(0)=0.0$ and $t_{\rm stim} = 500$, where $t_{\rm stim}$ denotes the time when the periodic force is added.
• Figure 3: The comparison between the original model \ref{['esl_p']} and averaged equation \ref{['esl_p_XY']} when $A=0.5$. The blue dots represent the maxima and minima of $r$ averaged over $\frac{\pi}{\Omega}$, obtained by the numerical simulation of the original system \ref{['esl_p']}. The red lines show the maxima and minima of $\bar{r}$, obtained by the numerical simulation of the averaged system \ref{['esl_p_XY']}.
• Figure 4: The bifurcation diagram of Eq. \ref{['esl_p_XY']} when $A = 0.5$. We plot the amplitude $\bar{r}$ of equilibrium states against different values of $\Omega$. The red solid, dotted, and dash-dotted lines represent the stable focus, unstable focus, and saddle points, respectively. The blue solid lines show the maxima and minima of the amplitudes of stable limit cycles, whereas the blue dotted lines show those of unstable limit cycles. Combining with the results of phase plane analysis (Fig. \ref{['esl_freq_pp_xy']}), this figure suggests that 1:1 synchronization starts at $\Omega \simeq 1.11$ and ends at $\Omega \simeq 2.24$. SN: saddle-node bifurcation, supH: supercritical Hopf bifurcation, subH: subcritical Hopf bifurcation.
• Figure 5: Phase planes of Eq. \ref{['esl_p_XY']} for different values of $\Omega$. We fix $A=0.5$ in this figure. The blue lines show the flows. The red dots and red circles represent the stable and unstable fixed points, respectively. The red solid lines and red dashed lines denote the stable and unstable limit cycles, respectively. See main texts for the explanation of each panel [(a)-(h)].