Oscillation Quenching Induced By Time-Varying Coupling Functions

Abstract

The oscillatory dynamics of natural and man-made systems can be disrupted by their time-varying interactions, leading to oscillation quenching phenomena in which the oscillations are suppressed. We introduce a framework for analyzing, assessing, and controlling oscillation quenching using coupling functions. Specifically, by observing limit-cycle oscillators we investigate the bifurcations and dynamical transitions induced by time-varying diffusive and periodic coupling functions. We studied the transitions between oscillation quenching states induced by the time-varying form of the coupling function while the coupling strength is kept invariant. The time-varying periodic coupling function allowed us to identify novel, non-trivial inhomogeneous states that have not been reported previously. Furthermore, by using dynamical Bayesian inference we have also developed a Proportional Integral (PI) controller that maintains the oscillations and \red{prevents oscillation quenching from occurring}. In addition to the present implementation and its generalizations, the framework carries broader implications for identification and control of oscillation quenching in a wide range of systems subjected to time-varying interactions.

Figures (5)

• Figure 1: Bifurcation analysis of the oscillation quenching induced by time-varying diffusive coupling function. (a) the temporal evolution of the coupling coefficients $c_1(t)$ and $c_2(t)$. (b) and (c) are the state responses $x_1(t)$ and $x_2(t)$ of the two oscillators, illustrating the transitions from Limit Cycle (LC) to Amplitude Death (AD) and Oscillation Death (OD). The enlarged inset shows the LC oscillations. (d) bi-parameter ($c_1,c_2$) bifurcation diagram, showing the AD, OD, and LC regions. (e)–(f) showing the specific bifurcations, including Saddle-Node (SN), Hopf bifurcation (HB), and Pitchfork bifurcation (PB).
• Figure 2: Time-varying coupling functions for time-invariant coupling strength. (a) shows the time-variability of the coupling coefficients $c^{1}_{1}(t),c^{1}_{2}(t)$and the time-invariant net coupling strength $\varepsilon^{1} (t)$. (b)-(f) present the variability of the form of coupling function with time -- the arrows indicate the time instance of the coupling function. For comparison the amplitude on z-axis is the same for all $q_{1}(x_1,x_2)$.
• Figure 3: Bifurcation analysis of the oscillation quenching induced by time-varying coupling functions. (a) bi-parametric ($c^{1}_1,c^{1}_2$) bifurcation diagram, showing the Amplitude Death (AD), Oscillation Death (OD), Limit Cycle (LC) and the Inhomogeneous LC (IHLC) regions. The quarter circle (thick red line) indicates the time-variability of $c^{1}_1(t),c^{1}_2(t)$ along the parameter space. (b) an example of the dynamical time-evolution of $x_2(t)$ showing transition between AD, OD and LC. The enlarged inset shows the LC oscillations. The parameters $c^{1}_1(t),c^{1}_2(t)$ which induce the transition in (b) are varied along the quarter-circle line in the left-bottom corner of (a). Specific Hopf Bifurcation (HB) and Supercritical Pitchfork Bifurcation (PB) for the $x_1$ state (c) and the $x_2$ state (d) in respect of the $c^{1}_2$ coupling parameter.
• Figure 4: Analysis of the periodic OD solutions. Bifurcation diagram (a) showing the bifurcations including the Saddle Node (SN), PB and Subcritical Pitchfork Bifurcation (PBS). The left vertical dashed line delimits the space into oscillating (OS) and non-oscillating (AD/OD) regions. For better presentation the next bifurcation branch around $x_{1,2}=-7$ is shown on the inset plot. (b) shows the dynamical time-evolution of $x_1(t)$ and $x_2(t)$ on the periodic OD steady states. The solution function $f_1$ of Eq. (4) is shown in 3D (c) and 2D (d). The coupling function form dominates the specific solution function $f_1$. Note and compare the markers on (a), (c) and (d) which show the same solution-points for the different OD steady states, as they are shown with same color in (b).
• Figure 5: Application of a PI system control for avoiding quenching and maintaining active oscillations. (a) bi-parametric ($c_1,c_2$) bifurcation diagram, showing the AD, OD, LC and IHLC regions. The horizontal line indicates how the coupling parameter $c_1$ varies from LC into AD, while the black part of the line is where the PI controls the variations to avoid AD. (b) shows the evolution of the coupling parameters $c_1$ and $c_2$ and the control parameters: $c_{1,ref}$, $c_{1,agg}$ and $\Delta c_1$. Note the correspondence of $c_1$ and $c_{1,ref}$ between (a) and (b), as indicated with the two arrows. Compare $x_2$ evolution in (c) when there is no control with (d) when the controller is on preventing quenching and maintaining active oscillations.