Chimeralike states in networks of bistable time-delayed feedback oscillators coupled via the mean field

We study the collective dynamics of oscillators in a network of identical bistable time-delayed feedback systems globally coupled via the mean field. The influence of delay and inertial properties of the mean field on the collective behavior of globally coupled oscillators is investigated. A variety of oscillation regimes in the network results from the presence of bistable states with substantially different frequencies in coupled oscillators. In the physical experiment and numerical simulation we demonstrate the existence of chimeralike states, in which some of the oscillators in the network exhibit synchronous oscillations, while all other oscillators remain asynchronous.

Figure 1

Block diagram of a ring system with time-delayed feedback. Numerals 1, 2, and 3 designate points where an external signal can be fed into the system.

Figure 2

Plot of function (2) for $a=1.5$, $b=2.3$, $c=1.78$, and $d=1.57$. A, B, and C are unstable fixed points.

Figure 3

Dependencies $\mathrm{\Delta }{\phi }_1\left(\gamma \right)$ and $\mathrm{\Delta }{\phi }_2\left(\gamma \right)$ described by Eqs. (11) and (12) (a), Eqs. (13) and (14) (b), and Eqs. (15) and (16) (c) for ${\nu }_1=1/200$ and ${\nu }_2=3/200$. Regions with synchronous behavior of oscillators in both clusters are denoted by SS. Regions with asynchronous behavior of oscillators in both clusters are denoted by AS. Regions in which chimeralike states take place are denoted by CS1 and CS2.

Figure 4

Dependencies $\mathrm{\Delta }{\phi }_1\left({\tau }_m\right)$ and $\mathrm{\Delta }{\phi }_2\left({\tau }_m\right)$ for ${\nu }_1=1/200$ and ${\nu }_2=3/200$. Regions with synchronous behavior of oscillators in both clusters are denoted by SS. Regions with asynchronous behavior of oscillators in both clusters are denoted by AS. Regions in which chimeralike states take place are denoted by CS1 and CS2.

Figure 5

Block diagram of a network of time-delayed feedback oscillators globally coupled via the mean field. The first and Nth oscillators are depicted. The delay lines, nonlinear elements, and filters are denoted by DL, NE, and F, respectively. The summary amplifier is denoted by Σ.

Figure 6

Experimental time series of voltage in eight coupled electronic oscillators for $|\mathrm{\Delta }{\phi }_1|=0.002\pi$ and $|\mathrm{\Delta }{\phi }_2|=0.006\pi$ (a), $|\mathrm{\Delta }{\phi }_1|=0.47\pi$ and $|\mathrm{\Delta }{\phi }_2|=0.51\pi$ (b), and $|\mathrm{\Delta }{\phi }_1|=0.8\pi$ and $|\mathrm{\Delta }{\phi }_2|=0.99\pi$ (c). The time series of periodic and chaotic oscillators are shown at the top and at the bottom of the figures, respectively. The same set of colors is used for both periodic and chaotic time series.

Figure 7

Space-time plots of the network of eight coupled experimental oscillators for $|\mathrm{\Delta }{\phi }_1|=0.002\pi$ and $|\mathrm{\Delta }{\phi }_2|=0.006\pi$ (a), $|\mathrm{\Delta }{\phi }_1|=0.47\pi$ and $|\mathrm{\Delta }{\phi }_2|=0.51\pi$ (b), and $|\mathrm{\Delta }{\phi }_1|=0.8\pi$ and $|\mathrm{\Delta }{\phi }_2|=0.99\pi$ (c).

Figure 8

Time series of oscillations in the coupled model time-delay systems (3) for $|\mathrm{\Delta }{\phi }_1|=0.04\pi$ and $|\mathrm{\Delta }{\phi }_2|=0.12\pi$ (a), $|\mathrm{\Delta }{\phi }_1|=0.36\pi$ and $|\mathrm{\Delta }{\phi }_2|=0.69\pi$ (b), and $|\mathrm{\Delta }{\phi }_1|=0.64\pi$ and $|\mathrm{\Delta }{\phi }_2|=0.87\pi$ (c).

Figure 9

Space-time plots of the network of eight coupled model oscillators (3) for $|\mathrm{\Delta }{\phi }_1|=0.04\pi$ and $|\mathrm{\Delta }{\phi }_2|=0.12\pi$ (a), $|\mathrm{\Delta }{\phi }_1|=0.36\pi$ and $|\mathrm{\Delta }{\phi }_2|=0.69\pi$ (b), and $|\mathrm{\Delta }{\phi }_1|=0.64\pi$ and $|\mathrm{\Delta }{\phi }_2|=0.87\pi$ (c).

Figure 10

Time series of oscillations in the coupled model time-delay systems (4) for a chimeralike state for $|\mathrm{\Delta }{\phi }_1|=0.72\pi$ and $|\mathrm{\Delta }{\phi }_2|=0.40\pi$.

Figure 11

Space-time plot of the network of eight coupled model oscillators (4) illustrating a chimeralike state for $|\mathrm{\Delta }{\phi }_1|=0.72\pi$ and $|\mathrm{\Delta }{\phi }_2|=0.40\pi$.

Figure 12

Snapshots of variables $x_i\left(t\right)$ in eight coupled model oscillators (3) in the presence of delay in the mean field for $|\mathrm{\Delta }{\phi }_1|=0.05\pi$ and $|\mathrm{\Delta }{\phi }_2|=0.15\pi$ (a), $|\mathrm{\Delta }{\phi }_1|=0.24\pi$ and $|\mathrm{\Delta }{\phi }_2|=0.72\pi$ (b), $|\mathrm{\Delta }{\phi }_1|=0.63\pi$ and $|\mathrm{\Delta }{\phi }_2|=1.89\pi$ (c), and $|\mathrm{\Delta }{\phi }_1|=0.91\pi$ and $|\mathrm{\Delta }{\phi }_2|=2.73\pi$ (d).