Robustness of chimera states for coupled FitzHugh-Nagumo oscillators

Chimera states are complex spatio-temporal patterns that consist of coexisting domains of spatially coherent and incoherent dynamics. This counterintuitive phenomenon was first observed in systems of identical oscillators with symmetric coupling topology. Can one overcome these limitations? To address this question, we discuss the robustness of chimera states in networks of FitzHugh-Nagumo oscillators. Considering networks of inhomogeneous elements with regular coupling topology, and networks of identical elements with irregular coupling topologies, we demonstrate that chimera states are robust with respect to these perturbations and analyze their properties as the inhomogeneities increase. We find that modifications of coupling topologies cause qualitative changes of chimera states: additional random links induce a shift of the stability regions in the system parameter plane, gaps in the connectivity matrix result in a change of the multiplicity of incoherent regions of the chimera state, and hierarchical geometry in the connectivity matrix induces nested coherent and incoherent regions.

Figure 1

Snapshots of the variables $u_k$ and mean phase velocities ${\omega }_k$ for inhomogeneous oscillators. [(a)–(c)] $r=0.35,\sigma =0.1$; [(d)–(f)] $r=0.33,\sigma =0.28$; [(g)–(i)] $r=0.25,\sigma =0.25$; values of ${\delta }_a$ shown for each panel above the mean phase velocities plots. System parameters: $N=1000,\varphi =\pi /2-0.1,a_{\text{mean}}=0.5$, and $\varepsilon =0.05$.

Figure 2

Averaged mean phase velocities for chimera states with one and multiple incoherent domains with increasing system inhomogeneity. Black denotes phase velocities for the system of identical oscillators $a_k=0.5,k=1,\cdots ,N$, red (gray) denotes mean phase velocities averaged over 100 realizations, gray (light gray) denotes variations of phase velocities $({\omega }_{\text{min}},{\omega }_{\text{max}})$ in 100 realizations. [(a)–(c)] $r=0.35,\sigma =0.1$; [(d)–(f)] $r=0.33,\sigma =0.28$; [(g)–(i)] $r=0.25,\sigma =0.25$; values of ${\delta }_a$ are shown inside each panel. Other system parameters as in Fig. 1.

Figure 3

Snapshots of the variables $u_k$ (left panels) and mean phase velocities (right panels) for the system (3). Probability of adding new links: (a) $p=0.05$, (b) $p=0.1$, (c) $p=0.2$, and (d) $p=0.3$. Parameters: coupling radius $r=0.35$, coupling strength $\sigma =0.1,a_k=a=0.5$, and other parameters as in Fig. 1.

Figure 4

(a) Stability regions for chimera states with one incoherent domain in the plane of coupling radius $r$ and coupling strength $\sigma$. The gray region corresponds to the system with regular nonlocal coupling topology $(p=0)$, and lines denote the borders of stability region for $p=0.05,0.1,0.2,0.3$. The inset shows the same plots rescaled to the effective coupling radius $\tilde{r}$. (b) Fraction $f_{\text{Chimera}}$ of successful realizations of chimera state starting from random initial conditions in 500 numerical experiments for each parameter set. The parameter values denoted by black dots in panel (a) are $A(r=0.3,\sigma =0.1),B(r=0.35,\sigma =0.1)$, and $C(r=0.35,\sigma =0.2)$. Other parameters as in Fig. 1 with $a_k=a=0.5$. (c) Fraction $f_{\text{Chimera}}$ of successful realizations of chimera state in the network with rewired links; all parameters as in panel (b).

Figure 5

Snapshots and corresponding mean phase velocity profiles for different connectivity distributions. (a) Symmetric connectivity distribution $R_L=330,R_R=330$, (b) asymmetric connectivity distribution $R_L=0,R_R=660$, and (c) displaced asymmetric connectivity distribution, $R_L=0,G=200,R_R=660$. Other parameters are $N=1000,\varepsilon =0.05,a=0.5,\varphi =\pi /2-0.1,\sigma =0.28$. All simulations start from the same initial conditions.

Figure 6

Schematic distribution of gaps around a reference node in a ring geometry: (a) symmetrically distributed gaps with $G_L=G_R=G$; (b) asymmetrically distributed gaps.

Figure 7

Snapshots of the variables $u_k$ and corresponding mean phase velocities for different gap positions. Each element is coupled with $2R=500$ others elements symmetrically distributed around each node. The connectivity matrix contains two gaps of size $G_L=G_R=G=100$ symmetrically placed around each node while the values of $R_1$ are indicated on each panel, and $R_2=R-R_1$. Other parameters: $N=1000,\sigma =0.3,a_k\equiv a=0.5$, and $\varepsilon =0.05$. All simulations start from the same initial conditions.

Figure 8

Measures of coherence-incoherence of a chimera state as a function of the size of the inner coupling regions $R_1$. Each element is coupled with $2(R_1+R_2)=500$ others elements symmetrically distributed around each node. The connectivity matrix contains two gaps of size $G_L=G_R=G=100$ symmetrically placed around each node. Other parameters as in Fig. 7. Averages are taken over 10 runs.

Figure 9

Snapshots and mean phase velocity of the variable $u_k$ for one-sided connectivity. Each element is coupled with $R=500$ elements to its right, i.e., $R_3=R_4=0$. The connectivity matrix is asymmetric and the coupling pattern contains one gap of size $G=100$. The values of $R_1$ and $R_2=R-R_1$ are indicated in the panels. Other parameters as in Fig. 7. All realizations start from the same initial conditions.

Figure 10

Measures of coherence-incoherence of a chimera state. Each element is coupled with $R=R_1+R_2=500$ elements to its right. The connectivity matrix is asymmetric and contains one gap of size $G=100$. All other parameters as in Fig. 9. Averages are taken over 10 runs.

Figure 11

(a) Schematic distributions of hierarchical gaps in a ring geometry based on the triadic (101) Cantor set. (b) Iteration process for construction of the corresponding hierarchical connectivity scheme.

Figure 12

Snapshots of the variable $u_k$ and corresponding mean phase velocities ${\omega }_k$ for different hierarchical connectivity matrices. The fractal dimensions of the hierarchical structures are (a) $d_f=ln3/ln6=0.6132$ (base pattern: $100101)$, (b) $d_f=ln4/ln6=0.7737$ (base pattern: $101110)$, and (c) $d_f=ln5/ln6=0.8982$ (base pattern: $110111)$. Parameters: $N=6^4=1296$ and other parameters as in Fig. 7.

Figure 13

Snapshots of the the variable $u_k$ (on the left) corresponding mean phase velocities ${\omega }_k$ (middle) and details of mean phase velocities (on the right) for different Cantor patterns. The fractal dimensions in all hierarchical structures is constant $d_f=ln4/ln6$. The initiation strings are (a) 110011 and (b) 111010. The left panels are zooms of the corresponding mean phase velocities (middle panels) to demonstrate structural ramifications. Parameters as in Fig. 12.

Figure 14

Snapshots of the the variable $u_k$ and the corresponding mean phase velocities ${\omega }_k$ for different coupling strengths. The fractal dimensions of the hierarchical structure is constant $d_f=ln4/ln6$. The panels correspond to: (a) $\sigma =0.24$, (b) $\sigma =0.27$, (c) $\sigma =0.28$ (d) $\sigma =0.30$ [same as in Fig. 12], (e) $\sigma =0.32$, (f) $\sigma =0.33$. Initiation string is 101110 and other parameters are as in Fig. 12.