Pace in Concert with Phase: Rate-induced Phase-tipping in Birhythmic Oscillators

TL;DR

This work analyzes rate-induced phase-tipping (RP-tipping) between two coexisting stable limit cycles in birhythmic oscillators, showing that tipping requires both a fast, appropriately scaled input change and the system being in specific tipping phases. By formulating two autonomous birhythmic models—the birhythmic van der Pol oscillator and the Decroly-Goldbeter glycolysis model—it links non-autonomous RP-tipping to properties of the frozen autonomous system, notably the phase along a limit cycle and partial basin instability along the parameter path. The authors construct RP-tipping diagrams for monotone and non-monotone parameter shifts, demonstrating one-way tipping and sequences of alternating tipping between limit cycles, and they reveal how the tipping phase intersects with basin-unstable phases. These insights provide a framework for predicting abrupt, potentially irreversible transitions in systems with birhythmic dynamics subject to time-varying forcing, with broad implications for engineering, biology, and ecology where multi-periodic behavior and non-autonomous inputs are common.

Abstract

We study rate-induced phase-tipping (RP-tipping) between two stable limit cycles of a birhythmic oscillator. We say that such an oscillator RP-tips when a time variation of an input parameter preserves the bistability of the limit cycles but induces transitions from one stable limit cycle to the other, causing abrupt changes in the amplitude and frequency of the oscillations. Crucially, these transitions occur when: the rate of change of the input is in a certain interval bounded by critical rate(s), and the system is in certain phases of the cycle. We focus on two illustrative examples: the birhythmic van der Pol oscillator and the birhythmic Decroly-Goldbeter glycolysis model, each subjected to monotone and non-monotone shifts in their input parameters. We explain RP-tipping in terms of properties of the autonomous frozen system, including the phase of a cycle and partial basin instability along the parameter path traced by the changing input. We show that RP-tipping can occur as an irreversible one-way transition or as a series of transitions between the stable limit cycles. Finally, we present RP-tipping diagrams showing combinations of the rate and magnitude of parameter shifts and the phase of the oscillation that give rise to this genuine non-autonomous instability.

Figures (19)

• Figure 1: Time series of the birhythmic van der Pol oscillator \ref{['vdp_with_xy']} exhibiting limit cycle oscillations with different amplitudes and frequencies. The horizontal dotted lines indicate the maxima and minima of the large-amplitude stable limit cycle $\Gamma_1$ (green), the unstable limit cycle $\theta$ (blue), and the small-amplitude stable limit cycle $\Gamma_2$ (red) along with the amplitudes $A_{\Gamma_1}$, $A_{\theta}$, and $A_{\Gamma_2}$ (marked with double sided arrows), respectively. The dashed black line at $y = 0$ denotes an unstable equilibrium point $e_0$. The parameter values are $d = -0.1, \alpha = 0.114$, $\beta = 0.003$, and $\mu = 0.6$.
• Figure 2: (a) Two-parameter bifurcation diagram of the birhythmic van der Pol oscillator \ref{['vdp_with_xy']} with variations in the parameters $d$ and $\mu$. $F_{l1}$, and $F_{l2}$ represent the fold/saddle-node bifurcation of limit cycles (SNLC) curves where a stable limit cycle and an unstable limit cycle merge and disappear, and $H$ represents the Hopf bifurcation curve. The grey-shaded region-I marks the coexistence of two stable limit cycles $\Gamma_1$ and $\Gamma_2$ (birhythmicity) separated by an unstable limit cycle $\theta$; in region-II, the small amplitude stable limit cycle $\Gamma_2$ exists alone (monorhymicity); region-III marks the existence of a stable steady state (point attractor); and in region-IV, the large amplitude stable limit cycle $\Gamma_1$ exists alone (monorhymicity). The vertical dotted line represents the value of $d=-0.05$ for which the one-parameter bifurcation diagram is plotted. (b) One-parameter bifurcation diagram exhibiting the maxima and minima of each limit cycle with variations in the parameter $\mu$. The stable limit cycles are marked with solid (red and green) curves, and the unstable limit cycle is marked with a dashed (blue) curve. The dashed horizontal line at $x=0$ represents an unstable equilibrium point. The grey-shaded region represents the existence of birhythmicity. Other parameter values are $\alpha = 0.093$, and $\beta = 0.0019$.
• Figure 3: Phase portraits of the birhythmic van der Pol oscillator \ref{['vdp_with_xy']} corresponding to distinct regions shown in Fig. \ref{['fig: Figure_bifn_vdp']}(a): For (a) $d = -0.001$ and $\mu = 1$ (region I: birhythmicity); (b) $d = 0.1$ and $\mu = 1$ (region II: monorhythmicity - small amplitude stable limit cycle $\Gamma_2$); (c) $d = 0.15$ and $\mu = 0.05$ (region III: stable equilibrium point); and (d) $d = -0.15$ and $\mu = 0.05$ (region IV: monorhythmicity - large amplitude stable limit cycle $\Gamma_1$). $e_0$ marked with an open circle stands for an unstable equilibrium point, and a filled circle stands for a stable equilibrium point. The blue dashed curve $\theta$ stands for the unstable limit cycle. The black curves headed with an arrow represent some exemplary trajectories. Other parameter values are same as in Fig. \ref{['fig: Figure_bifn_vdp']}.
• Figure 4: (a) Two-parameter bifurcation diagram of the Decroly-Goldbeter glycolysis model \ref{['gly_model']} with variations in the parameters $v$ and $\sigma_i$. $F_{l1}$, and $F_{l2}$ represent the SNLC curves where a stable limit cycle and an unstable limit cycle merge and disappear, and $H$ represents the Hopf bifurcation curve. The filled circle marked by GH at $(v,\sigma_i) \approx (0.3914,0.5195)$ represents the generalised Hopf (Bautin) bifurcation point where $F_{l1}$ and $H$ merge. The grey-shaded region-I marks the coexistence of two stable limit cycles $\Gamma_1$ and $\Gamma_2$, separated by an unstable limit cycle $\theta$; in region-II, the large amplitude stable limit cycle $\Gamma_1$ coexists with a stable equilibrium point (hard-excitation); region-III marks the existence of a stable equilibrium point; and in region-IV, a single stable limit cycle exists. The vertical dotted line represents $v=0.31$, for which the one-parameter bifurcation diagram is plotted. (b) One-parameter bifurcation diagram with variations in the input parameter $\sigma_i$. The maxima and minima of stable limit cycles are marked with solid (red and green) curves, and those of the unstable limit cycle are marked with dashed (blue) curves. The almost horizontal dashed (black) line represents an unstable equilibrium point, and the solid (black) line represents a stable equilibrium point. The grey-shaded region represents the existence of birhythmicity. Other parameter values are $K = 10$, $L = 3.6 \times 10^6$, $\sigma_M = 10$, $n = 5$, $q = 1$, and $k_s = 0.06$.
• Figure 5: Phase portraits illustrating different dynamics of the Decroly-Goldbeter glycolysis model \ref{['gly_model']}, corresponding to distinct regions as outlined in Fig. \ref{['fig: Figure_bifn_gly']}(a): For (a) $v = 0.29$ and $\sigma_i = 1.15$ (region I: birhythmicity); (b) $v = 0.34$ and $\sigma_i = 1.1$ (region II: hard excitation); (c) $v = 0.34$ and $\sigma_i = 1.6$ (region III: stable equilibrium point); and (d) $v = 0.34$ and $\sigma_i = 0.7$ (region IV: monorhythmicity). $e_0$ marked with an open circle stands for an unstable equilibrium point, and a filled circle stands for a stable equilibrium point. The blue dashed curve $\theta$ stands for the unstable limit cycle. The black curves headed with an arrow represent some exemplary trajectories. Other parameters are same as in Fig. \ref{['fig: Figure_bifn_gly']}.