Effect of filtered feedback on birhythmicity: Suppression of birhythmic oscillation

The birhythmic oscillation, generally known as birhythmicity, arises in a plethora of physical, chemical, and biological systems. In this paper we investigate the effect of filtered feedback on birhythmicity as both are relevant in many living and engineering systems. We show that the presence of a low-pass filter in the feedback path of a birhythmic system suppresses birhythmicity and supports monorhythmic oscillations depending on the filtering parameter. Using harmonic decomposition and energy balance methods we determine the conditions for which birhythmicity is removed. We carry out a detailed bifurcation analysis to unveil the mechanism behind the quenching of birhythmic oscillations. Finally, we demonstrate our theoretical findings in analog simulation with electronic circuit. This study may have practical applications in quenching birhythmicity in several biochemical and physical systems.

Figure 1

(a) Two-parameter bifurcation diagram in $\alpha \text{−}\beta$ space for $\mu =0.1$. Solid line represents saddle-node bifurcation of limit cycles and they meet at a codimension-2 cusp bifurcation point. (b) Monorhythmic oscillation for $\alpha =0.175$ and $\beta =0.0055$ [filled triangle in (a)]. (c) Birhythmic oscillation for $\alpha =0.12$ and $\beta =0.003$ [filled circle in (a)].

Figure 2

(a) Plot of $f\left(A^2\right)\text{−}{A}^2$ for the parameter set $\mu =0.1$, $\alpha =0.114$, $\beta =0.003$, and $d=0.4$ for different values of the cut-off frequency, $\gamma$. Solid (hollow) circle indicates stable (unstable) solutions. ${\mathrm{\Delta }}_L$ denotes the distance between the left-most stable root and the unstable root. (b) Plot of the left-most stable and unstable roots in the $\gamma \text{−}{A}^2$ plane. (c) Variation of ${\mathrm{\Delta }}_L$ with $\gamma$: At $\gamma ={\gamma }_c$ (here ${\gamma }_c=11.09$), ${\mathrm{\Delta }}_L$ becomes zero and saddle node bifurcation of limit cycles occur.

Figure 3

(a) Two parameter bifurcation diagram in $d\text{−}\gamma$ space for $\mu =0.1$, $\alpha =0.114$, and $\beta =0.003$. For $\gamma =1$ and $d=d_c$ monorhythmic zone has maximum spreading. (b) One-parameter bifurcation diagram with $\gamma$ for $d=0.4$ [along the vertical broken line in Fig. 3]; at $\gamma ={\gamma }_c=11.09$ SNLC occurs [in agreement with the analytical result of Fig. 2].

Figure 4

Time series and phase plane plots. [(a) and (b)] $\gamma =14$: Birhythmic oscillations, the trajectory in the middle of two LCs in (b) represents unstable LC. [(c) and (d)] $\gamma =2$: A single monorhythmic LC. The dark gray (red) line is for the initial conditions ${\mathcal{I}}_S\equiv (x_0,{\dot{x}}_0)=(0.1,0)$; the light gray (green) line is for the initial condition ${\mathcal{I}}_L\equiv (x_0,{\dot{x}}_0)=(7,0)$. Note that in (d) both initial conditions result in a single LC confirming a monorhythmic oscillation. Other parameters are $\mu =0.1$, $\alpha =0.114$, $\beta =0.003$, and $d=0.4$.

Figure 5

(a) Two-parameter bifurcation diagram in $\mu \text{−}\gamma$ space for $\alpha =0.114$, $\beta =0.003$, and $d=0.4$; (b) two-parameter bifurcation diagram in $\alpha \text{−}\gamma$ space (for $\mu =0.1$, $\beta =0.003$, and $d=0.4$). (c) Two-parameter bifurcation diagram in $\beta \text{−}\gamma$ space (for $\mu =0.1$, $\alpha =0.114$, and $d=0.4$). (d) Two-parameter bifurcation diagram in $\alpha \text{−}\beta$ space with (solid line) and without filtering ($\mu =0.1$ and $d=0.4$): A decreasing $\gamma$ suppresses the birhythmic zone.

Figure 6

The experimental circuit diagram. It is made compatible to be controlled and acquired by DAQ and Labview. For description and parameter values see text.

Figure 7

The experimental time series and phase plane diagrams. [(a) and (b)] $R_F=18.25$ $\mathrm{k}\mathrm{\Omega }$: Birhythmic oscillations. [(c) and (d)] $R_F=52.6$ $\mathrm{k}\mathrm{\Omega }$: Monorhythmic oscillations with large-amplitude single LC. The initial conditions are ${\mathcal{I}}_L\equiv (V_1,V_2)=(4.2V,0.2V)$ for large-amplitude LC and ${\mathcal{I}}_S\equiv (V_1,V_2)=(0.2V,0.2V)$ for small-amplitude LC. For other parameters see text.