Multiple scroll wave chimera states

Abstract

We report the appearance of three-dimensional (3D) multiheaded chimera states that display cascades of self-organized spatiotemporal patterns of coexisting coherence and incoherence. We demonstrate that the number of incoherent chimera domains can grow additively under appropriate variations of the system parameters generating thereby head-adding cascades of the scroll wave chimeras. The phenomenon is derived for the Kuramoto model of N identical phase oscillators placed in the unit 3D cube with periodic boundary conditions, parameters being the coupling radius r and phase lag α. To obtain the multiheaded chimeras, we perform the so-called 'cloning procedure' as follows: choose a sample single-headed 3D chimera state, make appropriate scale transformation, and put some number of copies of them into the unit cube. After, start numerical simulations with slightly perturbed initial conditions and continue them for a sufficiently long time to confirm or reject the state existence and stability. It is found, by this a way, that multiple scroll wave chimeras including those with incoherent rolls, Hopf links and trefoil knots admit this sort of multiheaded regeneration. On the other hand, multiple 3D chimeras without spiral rotations, like coherent and incoherent balls, tubes, crosses, and layers appear to be unstable and are destroyed rather fast even for arbitrary small initial perturbations.

Chimerapedia: coherence–incoherence patterns in one, two and three dimensions

“…Top hat kernels. A large number of numerical results [32,33], including sketches of stability diagrams, were obtained for model (5) with the so-called piecewise constant, or top-hat kernel  p s ps ps where σä(0, 1) is the coupling radius. Along with the chimera patterns from tables 1 and 2 and their multiple analogs, remarkable new states, called knotted and linked chimeras, were found in this model [31,33].…”

“…A large number of numerical results [32,33], including sketches of stability diagrams, were obtained for model (5) with the so-called piecewise constant, or top-hat kernel  p s ps ps where σä(0, 1) is the coupling radius. Along with the chimera patterns from tables 1 and 2 and their multiple analogs, remarkable new states, called knotted and linked chimeras, were found in this model [31,33]. Note, however, that the top-hat kernel has a Fourier series with infinitely many terms, so that the analysis of the corresponding self-consistency equation (14) and especially of the corresponding eigenvalue problem(24) requires a significant computational effort that is beyond the scope of the present work.…”

“…The most complicated model considered in this paper is a cubic lattice of coupled phase oscillators Y To the best of our knowledge the model(5)- (6) has not been studied and all our results concerning its symmetric chimera states are therefore new. Moreover, these appear to be the first rigorous mathematical results available for three-dimensional chimera states which have up to now only been studied numerically [29][30][31][32][33].…”

Synchronization patterns in LIF neuron networks: merging nonlocal and diagonal connectivity

“…Chi-mera states have been realized using a number of neuron models, such as the FitzHugh-Nagumo model [19,24,[32][33][34][35], the Hindmarsh-Rose model of bursting neurons [20-22, 29, 36-39], the LIF model [34,[40][41][42][43][44] and others. More recent studies include simulations of chime-ra arXiv:1807.11843v2 [nlin.PS] 13 Oct 2018 states in 2D lattices [34,38] and 3D spatial dimensions [34,39,[44][45][46]. Another striking synchronization example is the subthreshold oscillations: the network splits into two coexisting domains, one domain where all elements perform small-amplitude oscillations at subthreshold values and one domain where the incoherent elements develop a multileveled mean phase velocity distribution [43].…”

Festschrift on the occasion of Ulrike Feudel’s 60th birthday

“…To solve this question the authors study coherence-incoherence transitions in ensembles of non-locally coupled Hénon and Lozi maps, describe their similarities and fundamental differences and find that completely synchronized chaos exists in both systems for dominant coupling, as does desynchronized spatio-temporal chaos for weak coupling. In [16] Maistrenko et al study the effects of non-local coupling in three-dimensional networks of Kuramoto phase oscillators and investigate conditions for the emergence and stability of scroll wave chimeras. As a result of their demanding numerical simulations they find complex chimera states in three dimensions (Hopf links and trefoils) that exist only for nonlocal coupling and sufficiently large phase shift between oscillators.…”