Chimeralike states in a network of oscillators under attractive and repulsive global coupling

Arindam Mishra, Chittaranjan Hens, Mridul Bose, Prodyot K. Roy, and Syamal K. Dana

Phys. Rev. E 92, 062920 – Published 22 December 2015

Abstract

We report chimeralike states in an ensemble of oscillators using a type of global coupling consisting of two components: attractive and repulsive mean-field feedback. We identify the existence of two types of chimeralike states in a bistable Liénard system; in one type, both the coherent and the incoherent populations are in chaotic states (which we refer to as chaos-chaos chimeralike states) and, in another type, the incoherent population is in periodic state while the coherent population has irregular small oscillation. We find a metastable state in a parameter regime of the Liénard system where the coherent and noncoherent states migrate in time from one to another subpopulation. The relative size of the incoherent subpopulation, in the chimeralike states, remains almost stable with increasing size of the network. The generality of the coupling configuration in the origin of the chimeralike states is tested, using a second example of bistable system, the van der Pol-Duffing oscillator where the chimeralike states emerge as weakly chaotic in the coherent subpopulation and chaotic in the incoherent subpopulation. Furthermore, we apply the coupling, in a simplified form, to form a network of the chaotic Rössler system where both the noncoherent and the coherent subpopulations show chaotic dynamics.

Received 1 June 2015

DOI:https://doi.org/10.1103/PhysRevE.92.062920

Authors & Affiliations

Arindam Mishra^1,2, Chittaranjan Hens³, Mridul Bose¹, Prodyot K. Roy⁴, and Syamal K. Dana²

¹Department of Physics, Jadavpur University, Kolkata 700032, India
²CSIR-Indian Institute of Chemical Biology, Kolkata 700032, India
³Department of Mathematics, Bar-Ilan University, Ramat Gan 529002, Israel
⁴Department of Mathematics, Presidency University, Kolkata 700073, India

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Vol. 92, Iss. 6 — December 2015

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Images

Figure 1
Network of Liénard system. Phase diagram in the $K − ε$ plane. Different dynamical states are noted as one-cluster (striped region), two-cluster in light gray (yellow), three-cluster (hashed region), four-cluster in black (brown), chimeralike states dark gray (blue), metastable states (white). Dynamics of all the oscillators are shown in different parameter regimes: attractors of (a) Chimera-I ( $K = 1.6, ε = - 1.8$ ); (b) three-cluster ( $K = 1.7, ε = - 0.47$ ); (c) metastable state ( $K = 0.91, ε = - 1.95$ ); (d) two-cluster ( $K = 1.7, ε = - 0.09$ ); (e) two-cluster ( $K = 0.7, ε = - 1.8$ ); (f) four-cluster ( $K = 0.7, ε = - 0.59$ ); (g) Chimera-II ( $K = 0.3, ε = - 1.8$ ). Time series of all the oscillators in (h) one-cluster state ( $K = 0.2, ε = - 0.2$ ).
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Figure 2
Chimera-I state in the network of Liénard system for $K = 1.6$ and $ε = - 1.8$ . (a) Snapshot in polar coordinate of all the oscillators, synchronized subpopulation in square (red) and nonsynchronous subpopulation in circles (blue). (b) Temporal dynamics of all the nodes in the network. (c) Amplitude of the mean field component contributed by the synchronous (red horizontal line) and by the asynchronous (blue line) subpopulations. Black dashed line shows the amplitude of the total mean field. (d) Variation of the fraction of the noncoherent subpopulation ( $\frac{n}{N}$ ) with $N$ in Chimera-I state for $K = 1.6$ and $ε = - 1.8$ .
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Figure 3
Chimera-II state in the network of Liénard system for $K = 0.3$ and $ε = - 1.8$ . (a) Snapshot in polar coordinate of all the oscillators, synchronized population in square (red) and nonsynchronous population in circle (blue), (b) Temporal dynamics of all the nodes in the network.
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Figure 4
Temporal dynamics of $x_{i}$ for all the Liénard oscillators for $K = 0.91, ε = - 1.95$ . Clustering is seen in the top left panel ( $N = 10$ ) after a long time. Right panels ( $N = 100$ ) show no clustering. Top and bottom panels are split in time to show the dynamics for a long time.
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Figure 5
Network of forced van der Pol-Duffing oscillators. (a) Strength of incoherence as a function of $ε$ . (b) Polar plot showing amplitude and phase distribution of all oscillators. (c) Temporal dynamics of all nodes for $K = 0.3$ and $ε = - 3$ . (d) Poincaré plot of synchronized (green) and desynchronized (black) oscillators in the chimera state. (e) Corresponding attractors. (f) Polar plot showing three-cluster state (circle, square, and triangle) for $K = 0.3$ and $ε = - 4.7$ . (g) Trajectory of a three-cluster state with the Poincaré plot in the inset.
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Figure 6
Chimera state for Rössler oscillators for $K = 0.151$ and $ε = - 0.47$ . Snapshot of $x_{i} (t)$ at left panel and phase $ϕ_{i}$ at right panel in the top row for all the oscillators. Middle row (left panel) shows mean frequency of all oscillators, right panel shows spatio-temporal evolution of all nodes. Left panel of the last row shows the Poincaré plot for synchronized (red) and desynchronized (black) oscillators. Corresponding attractors from the coherent and incoherent regions are in the right panel.
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