Networks on the edge of chaos: Global feedback control of turbulence in oscillator networks

Random networks of coupled phase oscillators with phase shifts in the interaction functions are considered. In such systems, extensive chaos (turbulence) is observed in a wide range of parameters. We show that, by introducing global feedback, the turbulence can be suppressed and a transition to synchronous oscillations can be induced. Our attention is focused on the transition scenario and the properties of patterns, including intermittent turbulence, which are found at the edge of chaos. The emerging coherent patterns represent various self-organized active (sub)networks whose size and behavior can be controlled.

Figure 1

(Color online) Dependence of the maximal Lyapunov exponent on the control parameter $\mu$.

Figure 2

(Color online) Dependence of the Kaplan-Yorke embedding dimension on the control parameter $\mu$.

Figure 3

(Color online) Histograms of Lyapunov exponents for different values of the control parameter $\mu$.

Figure 4

(Color online) Mean time-averaged velocity of the network (dots) as function of the feedback intensity $\mu$. Thin lines show the maximum and minimum velocity values.

Figure 5

Distributions of time-averaged velocities of all network elements for three selected values of feedback intensity $\mu$.

Figure 6

(Color online) Instantaneous velocities of several randomly chosen elements $\mu =0.87$.

Figure 7

(Color online) The sequence of active subnetworks observed under gradual decrease of the feedback intensity $\mu$.

Figure 8

(Color online) Dependence of the number of active elements (line) and the size of the largest connected active subnetwork (dots) on the feedback intensity $\mu$.

Figure 9

The largest diameter (dots) of active subnetworks as function of the feedback intensity $\mu$.

Figure 10

Space-time diagram showing the dynamical activity pattern in the chain of 29 elements belonging to the active subnetwork.

Figure 11

(Color online) Time-averaged magnitude of the global signal (solid line) and mean time-averaged magnitude of local signals (dash line) as functions of the feedback intensity $\mu$.

Figure 12

Time-dependent spectral densities of the global signal at different feedback intensities $\mu$. The absolute magnitude of the signal is much larger at $\mu =0.266$, than in other plots shown.

Figure 13

Time-dependent spectral density of a single oscillator above the feedback breakdown transition $(\mu =0.266)$.

Figure 14

(Color online) Time-dependent spectrum plot of (above) the global signal and (below) collective signals of two specially selected groups of oscillators. $\mu =0.265$.