Between synchrony and turbulence: intricate hierarchies of coexistence patterns

ARTICLE https://doi.org/10.1038/s41467-021-25907-7 OPEN Between synchrony and turbulence: intricate hierarchies of coexistence patterns 1234567890():,; Sindre W. Haugland 1, Anton Tosolini1 & Katharina Krischer 1✉ Coupled oscillators, even identical ones, display a wide range of behaviours, among them synchrony and incoherence. The 2002 discovery of so-called chimera states, states of coexisting synchronized and unsynchronized oscillators, provided a possible link between the two and definitely showed that different parts of the same ensemble can sustain qualitatively different forms of motion. Here, we demonstrate that globally coupled identical oscillators can express a range of coexistence patterns more comprehensive than chimeras. A hierarchy of such states evolves from the fully synchronized solution in a series of cluster-splittings. At the far end of this hierarchy, the states further collide with their own mirror-images in phase space – rendering the motion chaotic, destroying some of the clusters and thereby producing even more intricate coexistence patterns. A sequence of such attractor collisions can ultimately lead to full incoherence of only single asynchronous oscillators. Chimera states, with one large synchronized cluster and else only single oscillators, are found to be just one step in this transition from low- to high-dimensional dynamics. 1 Physics Department, Nonequilibrium Chemical Physics, Technical University of Munich, James-Franck-Str. 1, D-85748 Garching, Germany. ✉email: NATURE COMMUNICATIONS | (2021)12:5634 | https://doi.org/10.1038/s41467-021-25907-7 | www.nature.com/naturecommunications 1 ARTICLE NATURE COMMUNICATIONS | https://doi.org/10.1038/s41467-021-25907-7 O ne of the big problems in physics is how highdimensional disorder in space and time may emerge from a spatially ordered, in the simplest case uniform, state with low-dimensional dynamics1. Exploring different paths from order to spatiotemporal disorder and their universal character is central for a deeper understanding of complex emergent behaviour such as spatiotemporal chaos in reaction-diffusion systems2,3 or turbulence in hydrodynamic flows4,5. Ensembles of coupled oscillators are one class of apparently simple dynamical systems that yet may adopt states ranging from full synchrony to complete incoherence, and which has provided insights in virtually any discipline, ranging from the natural sciences to sociology6,7. During the last two decades, a kind of hybrid phenomenon, in which synchronized and incoherent oscillators coexist in an ensemble of identical oscillators8, coined a chimera state9, has received considerable attention (see reviews10–12 and the references therein), not least since it can be considered a “natural link between coherence and incoherence”13. In an earlier study employing globally coupled logistic maps14, four different classes of behaviour were found, including a large variety of partially ordered states, some of which were later classified as chimeras15. Yet, the bifurcation structure between the different classes was not resolved. In this article, we study the bifurcations from synchrony, via clustered and partially clustered states to full incoherence in a system of globally coupled oscillators with nonlinear coupling, with simulations and bifurcation analysis for an increasing number of oscillators. Here, chimera states are just one of a multitude of coexistence patterns, all consisting of clusters, that is, internally synchronized groups of oscillators, of widely different sizes and dynamics, and possibly including one or several single oscillators. The path towards complete incoherence begins with a symmetry-breaking cascade of cluster-splitting period-doubling bifurcations, wherein the currently smallest cluster is repeatedly split into two, leading to hierarchical clustering. Due to the high symmetry of the system, each symmetry-breaking produces many equivalent mirror-image variants of each outcome state, multiplying the number of attractors and leading to an ever more crowded phase space16. At some point, each variant collides with some of its mirror-images, creating larger attractors with higher symmetry. Usually, this blows up some of the clusters, the resulting single oscillators henceforth moving similarly on average. A succession of such symmetry-increasing bifurcations destroys first the smallest clusters, and then the larger ones, partially mirroring the former cluster-splitting cascade and ultimately creating a completely incoherent state. A chimera state, consisting of one synchronized cluster and otherwise only single, incoherent oscillators is often the second to last state of the sequence. The model we employ is an ensemble of N Stuart-Landau oscillators W k 2 C, k = 1, …, N, with nonlinear global coupling17: dW k ¼ W k  ð1 þ ic2 ÞjW k j2 W k dt  ð1 þ iνÞhWi þ ð1 þ ic2 ÞhjWj2 Wi; ð1Þ where c2 and ν are real parameters and h ¼ i ¼ 1=N ∑Nk¼1 ¼ denotes ensemble averages. The Stuart-Landau oscillator itself is a generic model for a system close to a Hopf bifurcation, that is, to the onset of self-sustained oscillations18. Networks of such oscillators have previously been found to exhibit a wide range of dynamics, many of them occurring for linear global coupling19–23. The nonlinear global coupling in Eq. (1) stands out by featuring two qualitatively different chimera states, each of them deduced to somehow emerge from a corresponding type of two-cluster solution24. Originally, this coupling was inspired by 2 electrochemical experiments, wherein the oxide layer on a silicon electrode displays a wide range of spatiotemporal patterns17. A few experimental measurements reminiscent of new results in Eq. (1) will be discussed later in this article. Because the oscillators are identical and the coupling is global, the system is SN -equivariant: If WðtÞ 2 CN is a solution, then so is γWðtÞ 8 γ 2 SN , where SN is the symmetric group of all permutations of the N oscillators25. Or in less mathematical terms: If we start at a solution to Eq. (1) and interchange the trajectories of any two oscillators, the result is still a solution. Further, the average 〈W〉 is confined to simple harmonic motion with frequency ν, as shown by taking the ensemble average of the whole equation:
 dW k d ¼ hWi ¼ iνhWi ) hWi ¼ η eiνt ; ð2Þ dt dt where η 2 R is an additional parameter, implicitly set by choosing the initial condition. This constraint also implies that for a Poincaré map26 defined by sampling the system with frequency ν, the average of the N components of the map will always be constant. Thus the nonlinear constraint in the timecontinuous Eq. (1) becomes a linear constraint in the timediscrete map. Results The fully synchronized solution Wk = ηe−iνt ∀k always exists and is stable for sufficiently large values of η. It loses stability in either an equivariant pitchfork bifurcation, producing separate clusters that continue to orbit th (...truncated)