Robustness of chimera states for coupled FitzHugh-Nagumo oscillators

Abstract

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.

Chimera states in population dynamics: Networks with fragmented and hierarchical connectivities

“…7 also reveal an additional internal nested structure in both cases which are the imprints of the hierarchical connectivity. Similar nested chimera states have been previously demonstrated by the FitzHugh-Nagumo system without the additional feature of traveling (in)coherent parts [64]. The inset in b) shows in detail the lowest part of the two spectra where the frequencies associated with the traveling motion around the ring are discerned.…”

“…Complex nested chimera structures, when regarded from the viewpoint of population dynamics, show the rich organization which can emerge in communities of nonlinearly interacting populations. Previous studies have demonstrated that the introduction of gaps in a system of coupled elements modifies the chimera states, producing different coherenceincoherence patterns [64]. We test this hypothesis in the case of the LLC model and investigate whether this effect is generic or exclusive to specific models.…”

“…Earlier studies for connectivity matrices with gaps in the FitzHugh-Nagumo system have demonstrated that the position of the gap is important and that it affects the chimera properties modifying all four measures of coherence [64]. To test this for the LLC model we produce the snapshots and corresponding mean phase velocity profiles varying the sizes of the connectivity regions R 1 and R 2 but keeping the total number of links R 1 +R 2 = 400 fixed, while introducing a gap of constant size G R = G = 100 at various positions between the links.…”

“…10 we observe changes in the chimera multiplicity, ranging from one to three (in)coherent regions, merging and splitting of coherent and incoherent regions and shifting of their position in space. The change of the chimera profiles can be quantified statistically by calculating the following measures of coherence [64]: a) the av- (Color online) Chimera states (left) and corresponding mean phase velocity profiles (right) for gap size variation. The gap sizes are indicated in the right panels, while R1 = 100 and R2 = 300.…”

“…4(c)-(f). The local order parameter is defined as follows [5,64]: where Θ j = arctan[(y j − y Q4 )/(x j − x Q4 )] denotes the geometric phase of the jth LLC unit and (x Q4 , y Q4 ) are the coordinates of the nontrivial fixed point Q 4 of the uncoupled system. We use a spatial average with a window size of δ = 25 elements.…”