Delay-enhanced coherent chaotic oscillations in networks with large disorders

We study the effect of coupling delay in a regular network with a ring topology and in a more complex network with an all-to-all (global) topology in the presence of impurities (disorder). We find that the coupling delay is capable of inducing phase-coherent chaotic oscillations in both types of networks, thereby enhancing the spatiotemporal complexity even in the presence of $50\%$ of symmetric disorders of both fixed and random types. Furthermore, the coupling delay increases the robustness of the networks up to $70\%$ of disorders, thereby preventing the network from acquiring periodic oscillations to foster disorder-induced spatiotemporal order. We also confirm the enhancement of coherent chaotic oscillations using snapshots of the phases and values of the associated Kuramoto order parameter. We also explain a possible mechanism for the phenomenon of delay-induced coherent chaotic oscillations despite the presence of large disorders and discuss its applications.

Figure 1

The schematic diagram of the array of pendula with periodic boundary conditions.

Figure 2

Bifurcation diagrams of a single pendulum in a ring of three coupled pendula and the largest Lyapunov exponent (of a single pendulum for $C=0.0$ and that of the entire network for $C>0.0$) for different values of the coupling strength $C$ and the coupling delay $\tau$. (a) $C=0.0$ and $\tau =0.0$, (b) $C=0.5$ and $\tau =0.0$, (c) $C=0.5$ and $\tau =1.5$, and (d) $C=0.6$ and $\tau =2.0$ (inset shows that the pendulum exhibits chaotic oscillations for $f_d^{\prime }=1.6$). Red (dark gray) line corresponds to the largest Lyapunov exponent and light blue (light gray) dots correspond to the bifurcation diagram. Note that in all the cases (b)–(d) the three pendula are in a completely synchronized state and, hence, the largest Lyapunov exponent corresponds to the synchronization manifold.

Figure 3

Poincaré points corresponding to the network of pendula in a ring configuration with $N=50$ for the coupling strength $C=0.5$ in the absence of coupling delay. (a) Chaotically oscillating pendula for $f^{\prime }=1.5$ when no disorder is present, (b) periodically oscillating pendula for $20\%$ disorders with fixed $f^{\prime }=f_d^{\prime }=0.5$, and (c) periodically oscillating pendula for $20\%$ disorders with random $f_d^{\prime }\in (0.1,0.5)$.

Figure 4

Spatiotemporal representation of Fig. 3. Here the horizontal axis corresponds to time $t$ and the vertical axis to the oscillator index $N$.

Figure 5

Same as in Fig. 3 but now in the presence of coupling delay $\tau =1.5$ and for different densities of disorders. First row is with fixed value of the disorders $f_d^{\prime }=0.5$ and the second row with random values of $f_d^{\prime }\in (0.2,0.9)$. (a), (d) $20\%$, (b), (e) $50\%$, and (c), (f) $70\%$ of disorders.

Figure 6

Spatiotemporal representation of Fig. 5. Color bar is the same as in Fig. 4.

Figure 7

Snapshots of the phase portraits on the $(X_i,Y_i)$ plane of the ring network with $X_i=\cos {\theta }_i$ and $Y_i=\sin {\theta }_i$. Here (a) corresponds to $C=0.05$ and (b)–(d) correspond to Figs. 5d–5(f) with $C=0.5$ and the strength of the impurities being $20\%,50\%$, and $70\%$, respectively.

Figure 8

Time evolution of the Kuramoto order parameter $r$ for (a) $C=0.05$ [Figs. 7a and 7b] $C=0.5$ with $20\%$ impurity [Fig. 7b].

Figure 9

Poincaré points corresponding to the network of pendula, with $f^{\prime }=1.6$ for chaotic pendula, in a ring configuration with $N=50$ for the coupling strength $C=0.6$ with coupling delay $\tau =2.0$ and for different values of density of disorders. First row is with fixed value of the disorders $f_d^{\prime }=1.1$ and the second row with random values of $f_d^{\prime }\in (0.5,1.2)$. (a), (d) $20\%$, (b), (e) $50\%$, and (c), (f) $70\%$ of disorders.

Figure 10

Spatiotemporal representation of Fig. 9. Color bar is the same as in Fig. 4.

Figure 11

Schematic diagram of a network of five oscillators with all-to-all (global) coupling. In this figure each oscillator gets four delayed input from the remaining oscillators in the network, and also a delayed feedback from itself when $j=i$.

Figure 12

The largest Lyapunov exponents of the network of $N=20$ globally coupled pendula and the bifurcation diagram of a single pendulum in the network for different values of the coupling strength $C$ and the coupling delay $\tau$. The inset shows that the network exhibits chaotic oscillations for the specific values of $f^{\prime }$ chosen in the text. (a) $C=0.2$ and $\tau =0.0$, (b) $C=0.2$ and $\tau =1.5$, and (c) $C=0.3$ and $\tau =2.0$. Red and dark blue (dark gray) lines correspond to the largest Lyapunov exponents and light blue (light gray) dots correspond to the bifurcation diagram.

Figure 13

Poincaré points of globally coupled network of $N=20$ pendula for the coupling strength $C=0.2$ in the absence of coupling delay $\tau =0.0$. (a) Chaotically oscillating pendula for $f^{\prime }=1.4$, (b) periodically oscillating pendula for $20\%$ of disorders with fixed $f_d^{\prime }=0.3$, and (c) periodically oscillating pendula for $20\%$ of disorders with random $f_d^{\prime }\in (0,0.3)$.

Figure 14

Spatiotemporal representation of Fig. 13. Color bar is the same as in Fig. 4.

Figure 15

Poincaré points of globally coupled network of $N=20$ pendula, chaotic for $f^{\prime }=0.92$, for the coupling strength $C=0.2$ and the coupling delay $\tau =1.5$. First row with fixed value of the disorders $f_d^{\prime }=1.5$ and the second row with random values of $f_d^{\prime }\in (1.0,1.7)$. (a), (d) $20\%$, (b), (e) $50\%$, and (c), (f) $65\%$ of disorders.

Figure 16

Spatiotemporal representation of Fig. 15. Color bar is the same as in Fig. 4.

Figure 17

Poincaré points of globally coupled network of $N=20$ pendula, chaotic for $f^{\prime }=0.96$, for the coupling strength $C=0.3$ and the coupling delay $\tau =2.0$. First row with fixed value of the disorders $f_d^{\prime }=1.5$ and the second row with random values of $f_d^{\prime }\in (1.2,2.0)$. (a), (d) $20\%$, (b), (e) $50\%$, and (c), (f) $65\%$ of disorders.

Figure 18

Spatiotemporal representation of Fig. 17. Color bar is the same as in Fig. 4.

Figure 19

Snapshots of the phase portraits on the $(X_i,Y_i)$ plane for a globally coupled network of pendula. Here (a) corresponds to $C=0.05$, and (b)–(d) corresponds to Figs. 15d, 15e, 15f with $C=0.2$ and the strength of the impurities being $20\%,50\%$, and $65\%$, respectively.

Figure 20

Time evolution of the Kuramoto order parameter $r$ for (a) $C=0.05$ and (b) $C=0.2$ with $20\%$ impurity.