Optimal design of tweezer control for chimera states

Chimera states are complex spatio-temporal patterns which consist of coexisting domains of spatially coherent and incoherent dynamics in systems of coupled oscillators. In small networks, chimera states usually exhibit short lifetimes and erratic drifting of the spatial position of the incoherent domain. A tweezer feedback control scheme can stabilize and fix the position of chimera states. We analyze the action of the tweezer control in small nonlocally coupled networks of Van der Pol and FitzHugh-Nagumo oscillators, and determine the ranges of optimal control parameters. We demonstrate that the tweezer control scheme allows for stabilization of chimera states with different shapes, and can be used as an instrument for controlling the coherent domains size, as well as the maximum average frequency difference of the oscillators.

Figure 1

Schematic representation of the tweezer control. $Z_1$ and $Z_2$ denote the complex order parameters for one-half of the oscillator population, respectively.

Figure 2

(a) Symmetric and asymmetric control inputs $|Z_{\mathrm{s}}|$ (blue line) and $Z_{\mathrm{a}}$ (red line), respectively, for $N=24$ Van der Pol oscillators, $R=8,\varepsilon =0.2,K_s=1,K_a=2$, after transient time $T=500000$. Solid black lines denote time averaged values of $Z_{\mathrm{s}}$ and $Z_{\mathrm{a}}$. (b), (c) Instantaneous invasiveness of symmetric control $K_{\mathrm{s}}\mathrm{\Delta }{Z}_{\mathrm{s}}$ for $K_{\mathrm{a}}=2$ (b) and of asymmetric control $K_{\mathrm{a}}\mathrm{\Delta }{Z}_{\mathrm{a}}$ for $K_{\mathrm{s}}=1$ (c) (lines with circles). The black (dotted) and purple (full) lines in panel (b) show the standard deviation of the mean phase velocity profile ${\mathrm{\Delta }}_{\omega }$ and the size of coherent domain for chimera states stabilized by tweezer control, respectively.

Figure 3

Control regimes of Van der Pol oscillator: Standard deviation ${\mathrm{\Delta }}_{\omega }$ of the mean phase velocity profiles for controlled chimera states in system Eqs. (1, 2, 3, 4, 5, 6) in dependence on the control gains $K_{\mathrm{s}},K_{\mathrm{a}}$ for $N=24$ oscillators, $R=8,a=0.02,\mathrm{\Delta }T=500000$: (a) $\varepsilon =0.2$, (b) $\varepsilon =1$, (c) $\varepsilon =5.$ Corresponding sizes of coherent domains for chimera states: (d) $\varepsilon =0.2$, (e) $\varepsilon =1$, (f) $\varepsilon =5.$ White dots A, B, C, and D in (a),(d) correspond to the examples shown in Fig. 5.

Figure 4

Time-averaged order parameter $\langle \left|Z_{\mathrm{s}}\right|\rangle$ as a function of $K_{\mathrm{s}}$ (for fixed $K_{\mathrm{a}}=2)$ and as a function of $K_{\mathrm{a}}$ (for fixed $K_{\mathrm{s}}=1)$. Parameters: $N=24$ Van der Pol oscillators, $R=8,\varepsilon =0.2,K_{\mathrm{s}}=1,K_{\mathrm{a}}=2$. Trajectory length $\mathrm{\Delta }T=500000$ and the same length of the discarded transient.

Figure 5

Mean phase velocities for a system of $N=24$ Van der Pol oscillators, $R=8,a=0.02,\varepsilon =0.2$ after a transient time of $T=495000$: (a) $K_{\mathrm{s}}=0.5,K_{\mathrm{a}}=2$; (b) $K_{\mathrm{s}}=1,K_{\mathrm{a}}=2$; (c) $K_{\mathrm{s}}=1.5,K_{\mathrm{a}}=2$; (d) $K_{\mathrm{s}}=2,K_{\mathrm{a}}=2$. (e) Average frequency profiles corresponding to the chimera states shown in (a)–(d). Trajectory length $\mathrm{\Delta }T=500000$. Values of control gains $K_s,K_a$ are denoted by white dots in Figs. 3 and 3.

Figure 6

Control regimes of FitzHugh-Nagumo oscillator: Standard deviation ${\mathrm{\Delta }}_{\omega }$ of the mean phase velocity profiles for controlled chimera states in system Eqs. (9)–(13) depending on the control gains $K_{\mathrm{s}},K_{\mathrm{a}}$ for $N=24$ oscillators, $R=8,\varepsilon =0.15,b=0.15,\mathrm{\Delta }T=500000$: (a) $a=0.1$, (b) $a=0.3$, (c) $a=0.5$ Corresponding sizes of coherent domains for chimera states: (d) $a=0.1$, (e) $a=0.3$, (f) $a=0.5$. White dots E, F, G, and H in (c),(f) correspond to the examples shown in Fig. 7.

Figure 7

(a) Mean phase velocities for a system of $N=24$ FitzHugh-Nagumo oscillators, and $R=8,\varepsilon =0.15,b=0.15,a=0.5$, after a transient time $T=495000$; (b) phase velocity profile averaged over $\mathrm{\Delta }T=500000$ (top panel), snapshot of variables $u_k$ (middle panel), and snapshot in the $(u_k,v_k)$ phase space at time $t=500000$ (bottom panel, cubic nullcline of the uncoupled FitzHugh-Nagumo unit shown in black), control gains $K_{\mathrm{s}}=1,K_{\mathrm{a}}=1$, denoted by point E in Figs. 6 and 6. Same for other values of control gains: (c),(d) $K_{\mathrm{s}}=1.2,K_{\mathrm{a}}=1.2$; (e),(f) $K_{\mathrm{s}}=2,K_{\mathrm{a}}=2$; (g),(h) $K_{\mathrm{s}}=3,K_{\mathrm{a}}=3$, denoted by F, G, H, correspondingly, in Figs. 6 and 6.