Linked and knotted chimera filaments in oscillatory systems

Breathing chimera states in nonlocally coupled type-I excitable phase oscillators

We explore chimera states in a ring of nonlocally coupled type-I excitable phase oscillators, with each isolated oscillator being restricted to a homogeneous equilibrium state. Our study identifies the presence of breathing chimera states, characterized by their oscillatory dynamics and periodic fluctuations in the global order parameter. Beyond the breathing chimera states with a single coherent cluster, we find the 2n-cluster breathing chimera states, where 2n represents an even number of coherent clusters. These states exhibit the varying phase difference between adjacent clusters and a consistent phase among clusters separated by one intermediate cluster. The number of clusters is found to be modulated by the relative coupling radius. These dynamics for the finite number of oscillators are well confirmed by the Ott-Antonsen ansatz.

Introduction to Focus Issue: Chimera states: From theory and experiments to technology and living systems

One of the pillars of modern science is the concept of symmetries. Spontaneously breaking such symmetries gives rise to non-trivial states, which can explain a variety of phenomena around us. Chimera states, characterized by the coexistence of localized synchronized and unsynchronized dynamics, are a novel example. This Focus Issue covers recent developments in the study of chimera states, from both theoretical and experimental points of view, including an emphasis on prospective practical realization for application in technology and living systems.

Chimera patterns in conservative Hamiltonian systems and Bose-Einstein condensates of ultracold atoms

Experimental realizations of chimera patterns, characterized by coexisting regions of phase coherence and incoherence, have so far been achieved for non-conservative systems with dissipation and exclusively in classical settings. The possibility of observing chimera patterns in quantum systems has rarely been studied and it remains an open question if chimera patterns can exist in closed, or conservative quantum systems. Here, we tackle these challenges by first proposing a conservative Hamiltonian system with nonlocal hopping, where the energy is well-defined and conserved. We show explicitly that such a system can exhibit chimera patterns. Then we propose a physical mechanism for the nonlocal hopping by using an additional mediating channel. This leads us to propose a possible experimentally realizable quantum system based on a two-component Bose-Einstein condensate (BEC) with a spin-dependent optical lattice, where an untrapped component serves as the matter-wave mediating field. In this BEC system, nonlocal spatial hopping over tens of lattice sites can be achieved and simulations suggest that chimera patterns should be observable in certain parameter regimes.

Topological charge-density-vector method of identifying filaments of scroll waves

Scroll waves have been found in a variety of three-dimensional excitable media, including physical, chemical, and biological origins. Scroll waves in cardiac tissue are of particular significance as they underlie ventricular fibrillation that can cause sudden death. The behavior of a scroll wave is characterized by a line of phase singularity at its organizing center, known as a filament. A thorough investigation into the filament dynamics is the key to further exploration of the general theory of scroll waves in excitable media and the mechanisms of ventricular fibrillation. In this paper, we propose a method to identify filaments of scroll waves in excitable media. From the definition of the topological charge of filaments, we obtain the discrete expression of the topological charge-density vector, which is useful in calculating the topological charge vectors at each grid in the space directly. The set of starting points of these topological charge vectors represents a set of phase singularities, thereby forming a line of phase singularity, that is, a filament of a scroll wave.

Chimeras on annuli

Chimeras occur in networks of coupled oscillators and are characterized by the coexistence of synchronous and asynchronous groups of oscillators in different parts of the network. We consider a network of nonlocally coupled phase oscillators on an annular domain. The Ott/Antonsen ansatz is used to derive a continuum level description of the oscillators' expected dynamics in terms of a complex-valued order parameter. The equations for this order parameter are numerically analyzed in order to investigate solutions with the same symmetry as the domain and chimeras which are analogous to the "multi-headed" chimeras observed on one-dimensional domains. Such solutions are stable only for domains with widths that are neither too large nor too small. We also study rotating waves with different winding numbers, which are similar to spiral wave chimeras seen in two-dimensional domains. We determine ranges of parameters, such as the size of the domain for which such solutions exist and are stable, and the bifurcations by which they lose stability. All of these bifurcations appear subcritical.

Chimeras and solitary states in 3D oscillator networks with inertia

We report the diversity of scroll wave chimeras in the three-dimensional (3D) Kuramoto model with inertia for N³ identical phase oscillators placed in a unit 3D cube with periodic boundary conditions. In the considered model with inertia, we have found patterns that do not exist in a pure system without inertia. In particular, a scroll ring chimera is obtained from random initial conditions. In contrast to this system without inertia, where all chimera states have incoherent inner parts, these states can have partially coherent or fully coherent inner parts as exemplified by a scroll ring chimera. Solitary states exist in the considered model as separate states or can coexist with scroll wave chimeras in the oscillatory space. We also propose a method of construction of 3D images using solitary states as solutions of the 3D Kuramoto model with inertia.

Loss of coherence among coupled oscillators: From defect states to phase turbulence

Synchronization of a large ensemble of identical phase oscillators with a nonlocal kernel and a phase lag parameter α is investigated for the classical Kuramoto-Sakaguchi model on a ring. We demonstrate, for low enough coupling radius r and α below π/2, a phase transition between coherence and phase turbulence via so-called defect states, which arise at the early stage of the transition. The defect states are a novel object resulting from the concatenation of two or more uniformly twisted waves with different wavenumbers. Upon further increase of α, defects lose their stability and give rise to spatiotemporal intermittency, resulting eventually in developed phase turbulence. Simulations close to the thermodynamic limit indicate that this phase transition is characterized by nonuniversal critical exponents.

Three-dimensional chimera patterns in networks of spiking neuron oscillators

We study the stable spatiotemporal patterns that arise in a three-dimensional (3D) network of neuron oscillators, whose dynamics is described by the leaky integrate-and-fire (LIF) model. More specifically, we investigate the form of the chimera states induced by a 3D coupling matrix with nonlocal topology. The observed patterns are in many cases direct generalizations of the corresponding two-dimensional (2D) patterns, e.g., spheres, layers, and cylinder grids. We also find cylindrical and "cross-layered" chimeras that do not have an equivalent in 2D systems. Quantitative measures are calculated, such as the ratio of synchronized and unsynchronized neurons as a function of the coupling range, the mean phase velocities, and the distribution of neurons in mean phase velocities. Based on these measures, the chimeras are categorized in two families. The first family of patterns is observed for weaker coupling and exhibits higher mean phase velocities for the unsynchronized areas of the network. The opposite holds for the second family, where the unsynchronized areas have lower mean phase velocities. The various measures demonstrate discontinuities, indicating criticality as the parameters cross from the first family of patterns to the second.