Lyapunov spectra and collective modes of chimera states in globally coupled Stuart-Landau oscillators

Chimeras in globally coupled oscillators: A review

The surprising phenomenon of chimera in an ensemble of identical oscillators is no more strange behavior of network dynamics and reality. By this time, this symmetry breaking self-organized collective dynamics has been established in many networks, a ring of non-locally coupled oscillators, globally coupled networks, a three-dimensional network, and multi-layer networks. A variety of coupling and dynamical models in addition to the phase oscillators has been used for a successful observation of chimera patterns. Experimental verification has also been done using metronomes, pendula, chemical, and opto-electronic systems. The phenomenon has also been shown to appear in small networks, and hence, it is not size-dependent. We present here a brief review of the origin of chimera patterns restricting our discussions to networks of globally coupled identical oscillators only. The history of chimeras in globally coupled oscillators is older than what has been reported in nonlocally coupled phase oscillators much later. We elaborate the story of the origin of chimeras in globally coupled oscillators in a chronological order, within our limitations, and with brief descriptions of the significant contributions, including our personal experiences. We first introduce chimeras in non-locally coupled and other network configurations, in general, and then discuss about globally coupled networks in more detail.

Nontrivial twisted states in nonlocally coupled Stuart-Landau oscillators

A twisted state is an important yet simple form of collective dynamics in an oscillatory medium. Here we describe a nontrivial type of twisted state in a system of nonlocally coupled Stuart-Landau oscillators. The nontrivial twisted state (NTS) is a coherent traveling wave characterized by inhomogeneous profiles of amplitudes and phase gradients, which can be assigned a winding number. To further investigate its properties, several methods are employed. We perform a linear stability analysis in the continuum limit and compare the results with Lyapunov exponents obtained in a finite-size system. The determination of covariant Lyapunov vectors allows us to identify collective modes. Furthermore, we show that the NTS is robust to small heterogeneities in the natural frequencies and present a bifurcation analysis revealing that NTSs are born or annihilated in a saddle-node bifurcation and change their stability in Hopf bifurcations. We observe stable NTSs with winding number 1 and 2. The latter can lose stability in a supercritical Hopf bifurcation, leading to a modulated 2-NTS.

Amplitude-mediated chimera states in nonlocally coupled Stuart-Landau oscillators

Chimera states achieve the coexistence of coherent and incoherent subgroups through symmetry breaking and emerge in physical, chemical, and biological systems. We show the presence of amplitude-mediated multicluster chimera states in nonlocally coupled Stuart-Landau oscillators. We clarify the prerequisites for having different types of chimera states by analytically and numerically studying how phase transitions occur between these states. Our results demonstrate how the oscillation amplitudes interact with the phase degrees of freedom in chimera states and significantly advance our understanding of the generation mechanisms of such states in coupled oscillator systems.

Bumps, chimera states, and Turing patterns in systems of coupled active rotators

Self-organized coherence-incoherence patterns, called chimera states, have first been reported in systems of Kuramoto oscillators. For coupled excitable units, similar patterns where coherent units are at rest are called bump states. Here, we study bumps in an array of active rotators coupled by nonlocal attraction and global repulsion. We demonstrate how they can emerge in a supercritical scenario from completely coherent Turing patterns: a single incoherent unit appears in a homoclinic bifurcation, undergoing subsequent transitions to quasiperiodic and chaotic behavior, which eventually transforms into extensive chaos with many incoherent units. We present different types of transitions and explain the formation of coherence-incoherence patterns according to the classical paradigm of short-range activation and long-range inhibition.

Attracting Poisson chimeras in two-population networks

Chimera states, i.e., dynamical states composed of coexisting synchronous and asynchronous oscillations, have been reported to exist in diverse topologies of oscillators in simulations and experiments. Two-population networks with distinct intra- and inter-population coupling have served as simple model systems for chimera states since they possess an invariant synchronized manifold in contrast to networks on a spatial structure. Here, we study dynamical and spectral properties of finite-sized chimeras on two-population networks. First, we elucidate how the Kuramoto order parameter of the finite-sized globally coupled two-population network of phase oscillators is connected to that of the continuum limit. These findings suggest that it is suitable to classify the chimera states according to their order parameter dynamics, and therefore, we define Poisson and non-Poisson chimera states. We then perform a Lyapunov analysis of these two types of chimera states, which yields insight into the full stability properties of the chimera trajectories as well as of collective modes. In particular, our analysis also confirms that Poisson chimeras are neutrally stable. We then introduce two types of "perturbation" that act as small heterogeneities and render Poisson chimeras attracting: A topological variation via the simplest nonlocal intra-population coupling that keeps the network symmetries and the allowance of amplitude variations in the globally coupled two-population network; i.e., we replace the phase oscillators by Stuart-Landau oscillators. The Lyapunov spectral properties of chimera states in the two modified networks are investigated, exploiting an approach based on network symmetry-induced cluster pattern dynamics of the finite-size network.