Time-delay effects on the aging transition in a population of coupled oscillators

We investigate the influence of time-delayed coupling on the nature of the aging transition in a system of coupled oscillators that have a mix of active and inactive oscillators, where the aging transition is defined as the gradual loss of collective synchrony as the proportion of inactive oscillators is increased. We start from a simple model of two time-delay coupled Stuart-Landau oscillators that have identical frequencies but are located at different distances from the Hopf bifurcation point. A systematic numerical and analytic study delineates the dependence of the critical coupling strength (at which the system experiences total loss of synchrony) on time delay and the average distance of the system from the Hopf bifurcation point. We find that time delay can act to facilitate the aging transition by lowering the threshold coupling strength for amplitude death in the system. We then extend our study to larger systems of globally coupled active and inactive oscillators including an infinite system in the thermodynamic limit. Our model system and results can provide a useful paradigm for understanding the functional robustness of diverse physical and biological systems that are prone to aging transitions.

Figure 1

(a) Variation of $\overline{\alpha }$ with $\kappa$ at a fixed value of $\Delta \alpha =1$ obtained from Eq. (6) on setting ${\lambda }_R=0$. Region I corresponds to the phase-locked region and region II corresponds to the amplitude-death region. (b) Marginal stability curves giving the amplitude-death region in $\kappa -\overline{\alpha }$ parametric space for some finite values of delay $\tau$ and $\omega =10$. With an increase in delay $\tau$, the amplitude-death region is seen to expand and move into the space of positive $\overline{\alpha }$.

Figure 2

(a) Death island obtained from Eqs. (18) and (19) for two delay coupled oscillators having different bifurcation parameters ${\alpha }_1=-0.5$ and ${\alpha }_2=1.0$ but identical frequencies $\omega =10$. (b) Death island for two delay coupled oscillators at different values of the average bifurcation parameter $\overline{\alpha }$ for fixed $\Delta \alpha =0.2$ and $\omega =10$. It can be seen that as $\overline{\alpha }$ decreases, the area of the death-island region increases.

Figure 3

Variation of the normalized order parameter $R=|\overline{Z}\left(p\right)|/|\overline{Z}\left(0\right)|$ with the fraction of inactive oscillators $p$ for various values of delay $\tau$ for $N=500$ oscillators considering $a=2$, $b=1$, and $\omega =10$ at ${\kappa }^{\prime }=5$. The critical value of $p$ decreases with an increase in time delay.

Figure 4

Amplitude-death islands in $\kappa -\tau$ parameter space at different values of $p$ (fraction of inactive oscillators) for $N=500$ oscillators with $a=2$, $b=1$, and $\omega =10$. We can see that the death island increases with an increase in the fraction of inactive oscillators, i.e., with decrease in the average bifurcation parameter of the coupled oscillators.

Figure 5

(a) Marginal stability curves at various values of delay $\tau$ for $\Delta =1$ and $\omega =10$. The regions below the curve represent oscillatory states, while the death region lies above the curves. As the delay parameter $\tau$ is increased, the amplitude-death region increases and at a certain value of $\tau$ it touches the $\sigma =0$ axis, implying the amplitude-death state for the system consisting of all active oscillators. (b) Normalized order parameter given by $R\equiv |\overline{Z}\left(\sigma \right)|/|\overline{Z}\left(0\right)|$ (for $N=1000$ and $\omega =10)$ plotted against $\sigma$ at ${\kappa }^{\prime }=2$ for various values of $\tau$. Finite $\tau$ is seen to lower the threshold for amplitude death and thereby hasten aging.