Amplitude death in globally coupled oscillators with time-scale diversity

We analyze amplitude death (AD) in large systems of globally coupled oscillators with randomly distributed time scales. We show that the distribution of characteristic eigenvalues of a large but finite system is well approximated by the continuous and discrete spectra of the system in the thermodynamic limit. The stability analysis from the continuous and discrete spectra of the infinite system provides a fairly accurate prediction for the onset of AD in the large finite system. We prove the argument by examining the stability of AD in a paradigmatic system of coupled Stuart-Landau limit cycles with mismatched time scales. The proposed technique is extended to systems of globally coupled nonlinear oscillators of a general form with time-scale diversity, which is verified in coupled chaotic Rössler oscillators. Our study provides analytical insight into the understanding of the emergence of AD in populations of globally coupled dynamical systems.

Figure 1

Distribution of eigenvalues of the coupled system (2) with $N=30$, $K=5$, $w_j=10$, and $\sigma =0.4$. The time scales ${\tau }_j$'s are uniformly distributed on $[1-\sigma ,1+\sigma ]$. The black squares denote the eigenvalues numerically obtained from the characteristic equation (4), which can be nicely predicted by the discrete and continuous spectra in Eqs. (8) and (9) represented by the red open circle and the red line, respectively. All eigenvalues have a negative real part implying that AD is stable.

Figure 2

Stability regions for AD of the coupled system (1) with $N=30$ in the parameter space of $(\sigma ,K)$ for (a) $w=10$ and (b) $w=5$. The black squares are numerical results for the AD boundary obtained from directly integrating the coupled system (1), where the lower and upper black lines are the theoretical predictions by Eqs. (10) and (11), respectively. The time scales ${\tau }_j$'s are sampled from a uniform distribution over $[1-\sigma ,1+\sigma ]$.

Figure 3

(a) Stability region for AD of the coupled system (1) with $N=30$ in the parameter space of $(K,w)$ for a uniform distribution of $g\left(\tau \right)$ with $\sigma =0.4$. The coupling interval for AD decreases monotonically as $w$ decreases, and completely disappears if $w<w_{\text{min}}=4.95$. The black squares and the black solid line, respectively, denote the numerical simulations and the theoretical predictions as in Fig. 2. (b) The dependence of $w_{\text{min}}$ on $\sigma$. The black square points of $w_{\text{min}}$ are obtained as in (a). The red solid line is a power-law fit $w_{\text{min}}\propto {\sigma }^{\gamma }$ with $\gamma =-0.76$, which is clearly demonstrated in the upper-right inset with the log-log plot.

Figure 4

Effect of the distribution of $w_j$ on AD induced by time-scale diversity in the coupled system (1). The stability regions for AD are depicted in the parameter space of $(\sigma ,K)$ with $w_j$ and ${\tau }_j$ independently and uniformly distributed over $[1-\sigma ,1+\sigma ]$ and $[w-\alpha ,w+\alpha ]$, respectively. $N=100$ and $w=5$ are fixed. For $\alpha >0$, a lot of different combinations for ${\tau }_j$ and $w_j$ are possible, where the two extreme cases are (i) ${\tau }_j=1+\sigma (2j-N-1)/(N-1)$ and $w_j=w+\alpha (2j-N-1)/(N-1)$, and (ii) ${\tau }_j=1+\sigma (2j-N-1)/(N-1)$ and $w_{(N-j+1)}=w+\alpha (2j-N-1)/(N-1)$. The AD region expands for $\alpha =0.5$ with the case (i) (enclosed by the black lines) and shrinks for $\alpha =0.5$ with the case (ii) (enclosed by the blue lines) compared with that for $\alpha =0$ (enclosed by the red lines), as the level of diversity for the effective time scales is enhanced in the case (i) and is weakened in the case (ii) by the distribution of $w_j$.

Figure 5

(a) Bifurcation diagram by plotting the local maxima of the mean of the $x_j(j=1,\cdots ,N)$ for the globally coupled chaotic Rössler oscillators with $N=100$ and a uniform distribution of $g\left(\tau \right)$ with $\sigma =0.4$. With increasing the value of $K$, AD is stabilized within a certain coupling interval of $0.1<K<0.72$. (b) Stability region for AD in the parameter space of $(\sigma ,K)$. The black squares are numerical results for the AD boundary from direct integrations of the $N=100$ coupled chaotic Rössler oscillators as in (a), where the lower and upper solid lines are the theoretical predictions from Eqs. (15) and (16) with the chaotic Rössler oscillator model.

Figure 6

Comparison of the bifurcation boundaries of the AD region by the method of order-parameter expansion in [40] with our results for (a) coupled Stuart-Landau oscillators and (b) coupled chaotic Rössler oscillators. The time scales ${\tau }_j$ are uniformly distributed over $[1-\sigma ,1+\sigma ]$. The prediction of AD from the method of order-parameter expansion in [40] is indicated by the red solid lines. Our theoretical prediction is marked by the black solid lines, and the black squares denote the numerical observation of AD, which are the same as in Figs. 2 and 5, respectively. Our theory gives rise to a better prediction of the boundaries of the AD region compared with the method of order-parameter reduction.