Chimera States: From Theory and Experiments to Technology and Living Systems

We study the synchronization of small-world networks of identical coupled phase oscillators through the Kuramoto interaction and uniform time delay. For a given intrinsic frequency and coupling constant, we observe synchronization enhancement in a range of time delays and discontinuous transition from the partially synchronized state with defect patterns to a glassy phase, characterized by a distribution of randomly frozen phase-locked oscillators. By further increasing the time delay, this phase undergoes a discontinuous transition to another partially synchronized state. We found the bimodal frequency distributions and hysteresis loops as indicators of the discontinuous nature of these transitions. Moreover, we found the existence of Chimera states at the onset of transitions.

Inverse stochastic resonance is a nonlinear response of an oscillatory system to noise, where the frequency of noise-perturbed oscillations becomes minimal at an intermediate noise level. We demonstrate two generic scenarios for inverse stochastic resonance. We consider a paradigmatic model of two adaptively coupled stochastic active rotators whose local dynamics is near a bifurcation threshold. In the first scenario, inverse stochastic resonance emerges due to biased switching between oscillatory and quasi-stationary metastable states derived from attractors of the noiseless system. In the second scenario, inverse stochastic resonance arises due to a trapping effect associated with a noise-enhanced stability of an unstable fixed point. The details of the mechanisms behind the resonant effect are explained in terms of slow-fast analysis of the corresponding noiseless systems.

Amplitude chimera (AC) is distinct from other chimera patterns, like phase chimeras and amplitude mediated phase chimeras. Unlike other chimeras, in the AC pattern all the oscillators have the same phase velocity, however, the oscillators in the incoherent domain show periodic oscillations with randomly shifted origin. In this paper we investigate how the effect of local low pass filter (LPF) affects the occurrence of AC patterns. As dispersion and dissipation is present in every practical coupling path and these two effects can be mathematically modeled by LPF, we are motivated to investigate the effect of LPF on the occurrence of AC. We show that a low-pass or all-pass filtering is actually detrimental to the occurrence of AC. As we increase the effect of LPF an AC transforms into a synchronized pattern. We also show the effect of LPF can revoke the symmetry-breaking steady state (oscillation death state). Our study will shed light on the understanding of many biological systems where spontaneous symmetry-breaking and local filtering occur simultaneously.

Networks are suitable mathematical tools for representing the complexity of biosystems and measure the impact of perturbing conditions, such as disease. In the cancer context, the concept of network entropy has guided many studies focused on comparing equilibrium to disequilibrium (i.e., perturbed) conditions. Since these conditions reflect both structural and dynamic properties of network interaction maps, the derived topological characterizations offer precious support to conduct cancer inference. The focus is on specific properties of networks only partially utilized or studied in the oncological domain: controllability, synchronization and symmetry. The examples consider the complexity of metastatic processes from a computational viewpoint. The network states showing specific modular configurations can be useful to interpret cancer dysregulation dynamics and the emergence of disease phenotypes.

We consider an ensemble of phase oscillators in the thermodynamic limit, where it is described by a kinetic equation for the phase distribution density. We propose an ansatz for the circular moments of the distribution (Kuramoto-Daido order parameters) that allows for an exact truncation at an arbitrary number of modes. In the simplest case of one mode the ansatz coincides with that of Ott and Antonsen [Chaos: v. 18, 037113 (2008)], which is commonly utilised for modelling and analysis of chimera states. Our ansatz generalises the finite-dimensional description to arbitrary dimensions and is thus capable of representing higher-order dynamics such as quasiperiodicity and chaos, two examples of dynamics commonly associated with chimeras. The extensions also generalise for populations with a Cauchy-Lorentzian distribution of natural frequencies. In the case of oscillators driven by independent Gaussian white noise forces, the reductions break but may be used for an approximate description of the dynamics.

We obtain exact results on the autocorrelation of the order parameter in the nonequilibrium stationary state of a paradigmatic model of spontaneous collective synchronization, the Kuramoto model of coupled oscillators, evolving in presence of Gaussian, white noise. The method relies on an exact mapping of the stationary-state dynamics of the model in the thermodynamic limit to the noisy dynamics of a single, non-uniform oscillator, and allows to obtain besides the Kuramoto model the autocorrelation in the equilibrium stationary state of a related model of long-range interactions, the Brownian mean-field model. Both models show a phase transition between a synchronized and an incoherent phase at a critical value of the noise strength. Our results indicate that in the two phases as well as at the critical point, the autocorrelation for both the model decays as an exponential with a rate that increases continuously with the noise strength.

Chimera states, dynamical states composed of coherent and incoherent parts in an ensemble of coupled identical oscillators, have been proven to be important symmetry-breaking phenomena in various natural and man-made systems. A suitable network architecture that facilitates a deeper understanding of their dynamics is a two-population network, for, unlike a system of spatially extended oscillators along a one-dimensional ring, it inherently possesses an invariant synchronized manifold. In this talk we discuss the dynamical and spectral properties of finite-sized chimeras in two-population networks. We address the dependence of these chimera states on initial conditions and define a Poisson chimera and a non-Poisson chimera according to the observable chimera dynamics. We also explain how the finite-size Poisson chimera is related to that in the continuum limit. We perform a Lyapunov analysis of the chimera states and elucidate their spectral properties using both the macroscopic dynamics and the symmetry-induced cluster pattern dynamics. Finally, we consider a nonlocal intra-population network and an ensemble of Stuart-Landau limit cycle oscillators that possess amplitude degrees of freedom. We demonstrate that both variations may act as small heterogeneities that render Poisson chimeras attracting.

R.Levchenko, P.Jaros, T.Kapitaniak, and Y.Maistrenko We demonstrate that solitary states behavior can be observed in ensembles of phase oscillators with unidirectional coupling. For a small network consisting of only three identical oscillators (cyclic triple), tiny solitary islands arise in the parameter space. They are surrounded by "stormy" switching behavior caused by a collision of rotating waves propagating in opposite directions. For larger networks, N=100 (cyclic century), the islands merge into a single continent, which incorporates the world of chimeras/solitary states of different configurations. The phenomenon inherits from networks with intermediate ranges of the unidirectional coupling, and it diminishes as the coupling range decreases[1]. [1]. P. Jaros, R. Levchenko, T. Kapitaniak, and Yu. Maistrenko. Chimera states for directed networks. Chaos 31, 103111 (2021).

The phenomenon of the chimera state symbolizes the coexistence of coherent and incoherent sections of a given population. This phenomenon identified in several physical and biological systems presents several variants, including the multichimera states and the traveling chimera state. Here, we numerically study the influence of a weak external electric field on the dynamics of a network of Hindmarsh-Rose (HR) neurons coupled locally by an electrical interaction and nonlocally by a chemical one. We first focus on the phenomena of traveling chimera states and multicluster oscillating breathers that appear in the electric field’s absence. Then in the field’s presence, we highlight the presence of a chimera state, a multichimera state, an alternating chimera state, and a multicluster traveling chimera.

In ecology, species ability of colonizing new patches in search for better conditions of resources is known to increase the structural stability of population dynamics in ecosystems. By contrast, such spatial dispersal constitutes interconnections among patches that facilitates synchrony of their oscillations. This synchronizing aspect may elevate the risk of local extinctions to become global, causing the collapse of the whole ecosystem. In this context, we offer a mechanism to avoid this downside in which unstable, and transient, dynamics, at the level of single patches emerge as stable, and perpetual, in the connected system. Specifically, unstable chaotic sets in the local population dynamics of each patch behave as stable in the spatially distributed system and generate alternative desynchronized states. Sets of initial conditions leading to undesired synchronized solutions and the ones leading to desynchronization are shown and analysed. Despite of the generality of the reported phenomenon, we demonstrate our findings, and explain the mechanism, in a food web containing three trophic levels and a diffusive coupling to represent the connections due to dispersal.

We are interested in the possibility of chimera states in networks of coupled quantum oscillators. Starting with small systems, we investigate how such states can be characterized. Additional insights are gained by comparing the quantum systems with their analogues of classical oscillators under the addition of noise. This way, we want to identify quantum chimera states and show their qualitative similarities with classical chimeras as well as possibly their special quantum features.

We study a two-layer network of identical Kuramoto oscillators, where each layer is a non-locally coupled ring. We demonstrate how transitions can be induced between its different synchronised and partially synchronised states, and the phenomenon of a floating breathing chimera. Significantly we show that in such a system modulating the interlayer coupling by noise can play a similar role to increasing the interlayer coupling strength. In particular, noise can counter-intuitively cause the two layers to synchronise with each other, and in some cases cause the layers to completely synchronise within themselves. We also show how a different kind of disturbance caused by a slight mismatch in the coupling in both layers can play a similar synchronising role, suggesting how in such a system small disturbances surprisingly generally improve synchronisation. We discuss how such disturbances could be beneficial to cortical function and show how 1/f noise can play an especially efficient role in synchronising the system.

Phase synchronization in networks of oscillators is an ubiquitous phenomenon characterized by a large variability of dynamics in the neighborhood of the transition to phase synchronization. We study networks of Kuramoto oscillators with Watts-Strogatz and distance-dependent topologies to show that synchronization malleability -- the coexistence of various spatio-temporal synchronization patterns with different synchronization degrees -- results from an intricate interplay between the dynamics of the oscillators and the topology of the network. In the transition region from desynchronized behavior to complete phase synchronization, the dynamics is determined by a large number of coexisting attractors representing different synchronization patterns. Networks in this region are very sensitive to changes in parameters: even changes to the frequency of single units can alter the whole network's behavior in complex ways. In addition, we find that this malleability is most pronounced when short-range and long-range connections coexist in the network. This phenomenon occurs for several types of parameter distributions and dynamical models besides phase oscillators, and we argue it is a manifestation of the sample-to-sample fluctuations that occur in finite-size systems near phase transitions. By describing this phenomenon here, we hope to call attention to the significant consequences of these fluctuations to the dynamics of finite complex networks. This can then be used to understand the sensitivity of networks and to design protocols to control their behavior.

When systems are coupled to external reservoirs a variety of phenomena with no counterpart in closed systems may be observed. One of these is synchronization, which is a hallmark of collective behavior in nonequilibrium systems. However, unavoidable defects, local deformations caused by ambient conditions as well as long-term degradation can have a large impact on the collective be- havior or even destroy the synchronicity altogether. Especially at the nanoscale, it is desirable to investigate universal principles to enhance the robustness of synchronization. On the other hand, the application of topology has become an integral part in condensed matter physics leading to the discovery of phases characterized by global invariants rather than by local order parameters. These new phases of matter exhibit an unusual robustness to the adverse e↵ects of impurities and defects. In this talk, we will show a possible route towards integrating topological concepts with nonlinear, open system dynamics. By adjusting the coupling in a network of Van der Pol oscillators, we realize topological models like the Sue-Schrie↵er-Heeger chain or a kagome lattice. In the resulting systems, signatures of the underlying topology emerge as di↵erent synchronization mechanism between the bulk and the edges. Moreover, the synchronized edge states are topologically protected against local disorder demonstrating the availability of topological features even under open system conditions far away from equilibrium.