Chaotic Chimera Attractors
in a Triangular Network of Identical Oscillators

Comprehending the dynamics of coupled oscillator ensembles is crucial for various applications, including laser physics [1, 2] or Josephson junctions [3, 4] to biology [5, 6, 7] and neural science [8, 9]. Among the fascinating collective behaviors exhibited by identical oscillators is the coexistence of synchrony and asynchrony. This intriguing state was first identified by Battogtokh and Kuramoto [10] 20 years ago, and was subsequently named a ‘chimera state’ [11] to underscore its peculiarity.

While originally chimera states were investigated in a spatially extended ring geometry [12, 13, 14], simpler settings, in particular two-population networks revealed essential properties of chimera states [15, 16]. Here, a chimera state consists of one fully synchronized and one incoherent populations. The success of this model relied on two pillars: First, each oscillator is sinusoidally influenced by an effective force that depends only on the macroscopic dynamics and measures the degree of synchrony of the individual populations [17]. Second, dimension reduction methods allow for the investigation of the dynamics of each population in terms of a few macroscopic variables rather than directly studying the microscopic individual oscillators. In the thermodynamic limit, an invariant manifold, called Ott-Antonsen (OA) manifold [18, 19], exists on which the evolution equation for the complex order parameter can be written in a closed form. For finite-sized systems, a so-called Watanabe-Strogatz (WS) transformation [20, 21] maps the microscopic dynamics of each population to three macroscopic variables and $N-3$ independent constants of motion [22, 23, 24, 25].

Much about chimera states was learned from two-population networks of identical Kuramoto-Sakaguchi (KS) phase oscillators, arguably the best-studied oscillator model and the simplest network configuration. Three types of stable chimera states were identified: Stationary chimera states where the incoherent population possesses a constant degree of partial coherence, i.e., a stationary magnitude of the Kuramoto order parametere of the incoherent population and breathing chimeras where the degree of partial coherence, respectively the Kuramoto order parameter oscillates periodically [16, 24, 26], and quasiperiodic chimeras, exhibiting quasiperiodic motion of the order parameter [22, 23]. In contrast, chaotic motion of the macroscopic dynamics seems to require more complex settings, such as the presence of heterogeneities [27, 28, 29], higher order interactions [30, 31] or higher-dimensional individual oscillators [32, 33].

Chaotic chimeras, however, are also far less studied, and the settings under which macroscopic chaotic chimeras exist in ensembles of identical KS phase oscillators are not well understood. Note that when we talk here about chaotic chimera states we refer to (macroscopic) chaotic order parameter dynamics in large ensembles of oscillators, not to chaotic (weak) chimeras in systems of just a few, e.g. three [34] or four oscillators [27], where it is not clear to which order parameter dynamics the chaotic motion translates in the thermodynamic limit [35]. In this paper, we study what seems to be the simplest topology of a network of identical KS phase oscillators that exhibits, as we will show below, chaotic chimera states: a three-population network of identical KS phase oscillators globally coupled within each population and arranged in a triangle with equal inter-population coupling strengths.

Three-, or more generally, multi-population networks can be seen as toy models for networks of networks and have been investigated in different contexts. They may exhibit a variety of dynamical states; besides chimera states [36, 37, 38, 39] and heteroclinic switching between saddle chimeras [40, 41], effects of nonresonant natural frequencies [42], comparison with three-modal natural frequency distribution [43], and the impact of repulsive coupling [44, 45] have been studied.

The macroscopic dynamics of chimera states in three-population networks of identical KS phase oscillators was studied for symmetric solutions where two populations behave identically [36, 37]. In these studies the thermodynamic limit was considered using the OA approach. Two kinds of chimera states were identified, so called DSD- and SDS-type chimeras, respectively. Here, ‘S’ stands for a completely synchronized population while ‘D’ denotes a desynchronized population. On the symmetry-reduced manifold both of these types of chimeras are stable in some range of the parameters and the degree of coherence of the D-populations can be either stationary or breathing. As the author noted [36], nothing can be said about the stability of these symmetric chimeras off the symmetry reduced manifold or about the existence of non-symmetric solutions, in particular of $\text{D}\text{S}\text{D}^{\prime}$-chimeras, where the $\text{D}^{\prime}$ indicates that the two desynchronized populations display a different order parameter dynamics. This holds for solutions on the full OA manifold as well as outside it. In this context, macroscopic chaotic chimera states showing aperiodic motion of the order parameter of incoherent populations are of particular interest, as this would be an example of macroscopic chaotic chimeras of identical KS phase oscillators in the thermodynamic limit.

This paper addresses these questions and provides detailed dynamical and spectral properties of observable chimera states. Starting with the microscopic equations, we introduce the order parameter dynamics in both the thermodynamic limit on the full OA manifold and finite-sized ensembles using WS reduction in Sec. II. In Sec. III, we elucidate the coexistence of periodic and chaotic antiphase $\text{D}\text{S}\text{D}^{\prime}$ chimera attractors on and off the Ott-Antonsen manifold, respectively and demonstrate that these two states also coexist with a symmetric SDS stationary chimera state on the OA manifold. The results are summarized in Sec. IV.

Consider a system of identical Kuramoto-Sakaguchi phase oscillators, each oscillator $j$ being described by a phase variable $\phi^{(a)}_{j}(t)\in[-\pi,\pi)=:\mathbb{T}$ where $a=1,2,3$ denotes the population of a three-population network and each population consists of $N$ oscillators. The 3$N$ microscopic governing equations are given by

with $j=1,...,N$ and $a,b=1,2,3$. $H_{a}(t)$ represents a mean-field forcing defined by $H_{a}(t):=e^{-i\alpha}\big{(}\mu\Gamma_{a}(t)+\nu\Gamma_{b}(t)+\nu\Gamma_{c}(t)\big{)}$ where $(a,b,c)$ is a permutation of $(1,2,3)$. $\Gamma_{a}(t)$ denotes the complex Kuramoto order parameter of each population, which is defined as

The phase-lag parameter $\alpha$ is conveniently written as $\alpha=\frac{\pi}{2}-\beta$. In this work, we fix $\beta=0.025$. Since the oscillators are identical, we can set $\omega=0$ and the intra-population coupling strength $\mu=1$. The inter-population coupling strength $\nu$, which we assume to be always weaker than the intra-population coupling strength, is expressed through $\nu=1-A$ where $A\in[0,1]$. Note that the larger the parameter $A$, the weaker the coupling between populations [36]. In matrix form, the coupling strength of this triangular, symmetric network [37] is thus given by

for $a,a^{\prime}=1,2,3$.

Below we analyze the dynamics of Eq. (1) on the levels of the OA and WS reductions. To derive the global dynamics of the finite-sized system, we exploit the WS transformation [23, 21, 17]:

for $j=1,...,N$ and $a=1,2,3$. For each population, the radial variable $\rho_{a}(t)$ measures the degree of coherence, and $\Phi_{a}(t)$ the mean phase; note, though, that these quantities are not exactly identical to $r_{a}(t)$ and $\Theta_{a}(t)$ of the Kuramoto order parameter. The second angular variable, $\Psi_{a}(t)$, specifies the motion of the oscillators with respect to the mean phase and $\{\psi_{j}^{(a)}\}_{j=1}^{N}$ are $N-3$ independent constants of motion that satisfy three constraints: $\sum_{j=1}^{N}\cos\psi_{j}^{(a)}=\sum_{j=1}^{N}\sin\psi_{j}^{(a)}=0$ and $\sum_{j=1}^{N}\psi_{j}^{(a)}=0$ for $a=1,2,3$ [21, 22, 23, 17]. The WS macroscopic variables are related to the complex Kuramoto order parameter via [22, 23]:

where $\gamma_{a}$ is defined by

for $a=1,2,3$. With these definitions, the governing equations of the 9D WS variables are given as [22, 23, 24]

for $a=1,2,3$.

The OA dynamics can be obtained from the WS dynamics for uniform constants of motion $\psi_{j}^{(a)}=-\pi+\frac{2\pi(j-1)}{N}$ for $j=1,...,N$, taking the thermodynamic limit $N\rightarrow\infty$. Under this condition, the Kuramoto order parameter is exactly described by $\Gamma_{a}(t)=\rho_{a}(t)e^{i\Phi_{a}(t)}$ since $\gamma_{a}=1$ for $a=1,2,3$ and the governing equations read [36, 22, 23]

for $a=1,2,3$.

We first focus our attention on observable, i.e., stable chimera states in the thermodynamic limit which live on the Ott-Antonsen manifold. As mentioned above, there are two basic types of chimera states, SDS- and DSD-types chimeras. The detailed analysis of the symmetric SDS and DSD chimera states, which do not lead to a chaotic chimera state, are given in Appendix. A. Here, we concentrate on the symmetry-broken $\text{D}\text{S}\text{D}^{\prime}$ chimera states, which are connected to the chaotic chimeras.

A bifurcation diagram of DSD-type chimera states is depicted in Fig. 1. The black curves correspond to the symmetric DSD chimeras with $\rho_{1}(t)=\rho_{3}(t)<1$ while $\rho_{2}(t)=1$ and $\Phi_{1}(t)=\Phi_{3}(t)\in\mathbb{T}$ that live in the symmetry-reduced manifold. Their dynamics within this manifold was studied in Ref. [36]. Our results are in agreement with these data, but reveal that all symmetric DSD chimera states, i.e. also those that were found to be stable within the reduced manifold, have at least one transversally unstable direction, which drives the two symmetric D-populations in opposite directions, and are thus unstable (see Appendix. A.2 for details). Not reported before are the asymmetric $\text{D}\text{S}\text{D}^{\prime}$ chimeras with $\rho_{1}(t)\neq\rho_{3}(t)<1,\Phi_{1}(t)\neq\Phi_{3}(t)$ that live off the reduced manifold. Among these states are stable asymmetric $\text{D}\text{S}\text{D}^{\prime}$ chimera states (blue) in which the two incoherent populations exhibit antiphase character: $\rho_{1}(t)=\rho_{3}(t-\frac{T}{2})$ where $T$ is the period of the oscillation, and thus, $\rho_{1}(t)\neq\rho_{3}(t)$ for $t\in\mathbb{R}_{+}$ almost everywhere. Time series of the three moduli of the order parameters are shown in Fig. 2 (a). These antiphase $\text{D}\text{S}\text{D}^{\prime}$ chimeras are born in a double-homoclinic cycle of the stationary symmetric DSD chimeras, which is point-symmetric in the projection on the $\rho_{1}\rho_{3}$-plane with respect to the diagonal line ($\rho_{1}=\rho_{3}$). The homoclinic bifurcation (Hom) lies very close to a pitchfork bifurcation (PF) at which a pair of stationary $\text{D}\text{S}\text{D}^{\prime}$ chimeras (red) bifurcates off the stationary DSD chimera state (black). The homoclinic bifurcation throws off unstable antiphase $\text{D}\text{S}\text{D}^{\prime}$ chimeras (dashed blue) that are at somewhat larger values of $A$ stabilized in a subcritical torus bifurcation (TR). Further increasing $A$, the antiphase chimera state eventually becomes unstable again via a subcritical torus bifurcation.

Though the antiphase $\text{D}\text{S}\text{D}^{\prime}$ chimera state results from the broken symmetry due to the transversal instability, the dynamics of them inherits another symmetry of the solution: $\dot{\Phi}_{a}(t)=\dot{\Phi}_{a}(t-T),\dot{\rho}_{a}(t)=\dot{\rho}_{a}(t-T)$ for $a=1,3$ and $\dot{\Phi}_{2}(t)=\dot{\Phi}_{2}(t-\frac{T}{2}),\dot{\Phi}_{1}(t)=\dot{\Phi}_{3}(t-\frac{T}{2})$ which leads to $\rho_{1}(t)=\rho_{3}(t-\frac{T}{2})$. Thus, antiphase chimera states possess just 2 effective degrees of freedom. This fact is also reflected in the spectrum of the Lyapunov exponents (LEs) (Fig. (2) (b)) [46, 47, 48]. There are two zero Lyapunov exponents due to continuous symmetries, i.e., time and phase shift invariance, with two slightly negative and two prominently negative Lyapunov exponents. Hence, the antiphase $\text{D}\text{S}\text{D}^{\prime}$ chimeras in the OA manifold are a stable periodic motion, i.e., a periodic antiphase chimera.

In Fig. 2 (c), the antiphase chimera state in a finite-sized system obtained from the WS dynamics (7) with uniform constants of motion is shown. Although it displays oscillating antiphase motion of the two incoherent populations similar to the one of the OA dynamics, its motion is not periodic but rather aperiodic, and thus the state does not possess the above mentioned spatio-temporal symmetry, i.e., $\rho_{1}(t)\neq\rho_{3}(t-\frac{T}{2})$. In fact, the Lyapunov analysis (Fig. 2 (d)) clearly shows two positive Lyapunov exponents. In other words, the antiphase chimera state of the small-sized system in the Poisson submanifold (defined as a manifold of finite size $N$ as close as possible to the OA manifold, and obtained with uniform constants of motion) is chaotic. Furthermore, besides the evidence of the two positive LEs, the chaotic motion of the WS dynamics in small-$N$ systems can be detected in a Poincaré section defined by $\Psi_{1}\equiv 2\pi$ (Fig. 3 (a)) [24]. For $N=30$, the dynamics in the section follows a scattered motion on a band-like region, as expected for motion on a chaotic attractor.

Opposed to this, the dynamics of the large system with $N=100$ oscillators resides on a one-dimensional curve in the Poincaré section (Fig. 3 (b)) suggesting that the chaotic motion is restricted to small system sizes. This conjecture is further supported by calculations of the two largest Lyapunov exponents as a function of system size $N$. In Fig. 3 (c) we can see that the two positive Lyapunov exponents decrease until $N\approx 60$ where they become numerically indistinguishable from zero. The deterministic small-size effect that gives rise to a chaotic motion of antiphase chimeras arises from the influence of $\gamma_{a}\in\mathbb{C}$ on the WS variables. For small $N$, $\gamma_{a}$ significantly affects the dynamics given by Eq. (7) and renders the WS dynamics different from the OA dynamics. The irregular motion of the real and the imaginary part of $\gamma_{a}$ as a function of time of the chaotic chimera state depicted in Fig. 3 (a) is displayed in Fig. 3 (d). As $N$ approaches infinity, $\text{Re}(\gamma_{a})\rightarrow 1$ and $\text{Im}(\gamma_{a})\rightarrow 0$, so that Eq. (7) becomes identical to Eq. (8) and the aperiodic WS dynamics coincides with the periodic OA motion.

Off the OA manifold, the picture is different. First, we observe that starting from random initial conditions picked from $\mathbb{T}^{3N}$ for simulations of the microscopic dynamics (1), chaotic antiphase chimera states are observed as well. An example is shown in Fig. 4 with $N=40$ where the evolution of the moduli of the Kuramoto order parameters for the microscopic dynamics (Fig. 4 (a)) is depicted together with the radial variables of the WS dynamics (Fig. 4 (b)). The latter is obtained from nonuniform constants of motion that are generated using $\psi_{j}^{(a)}=(1-q)\frac{\pi}{2}+\frac{\pi q(j-1)}{N/2}$ and $\psi_{j+N/2}^{(a)}=-(1+q)\frac{\pi}{2}+\frac{\pi q(j-1)}{N/2}$ with $q=0.85$ for $j=1,...,\frac{N}{2}$ and for $a=1,2,3$ [22, 23]. The moduli of the Kuramoto order parameters exhibit a qualitatively similar envelope to the radial variables, but with further fluctuations superimposed, as expected from the impact of $\gamma_{a}$ in Eq. (5). However, different from the dynamics in the OA manifold, outside of it, the chaotic motion persists for systems as large as $N=4,000$. The two largest Lyapunov exponents of simulations with nonuniformly distributed constants of motion are shown in Fig. 4 (c). Up to about $N=100$ they decrease with system size, but then saturate at a positive, non-zero value, indicating that outside the OA manifold the chaotic attractors exist even in the thermodynamic limit. Likewise, the Poincaré section elucidates a similar scattered characteristics to Fig. 3 (a) even for $N=200$ with nonuniform constants of motion (not shown here). Note that this chaotic motion is not microscopic chaos but rather it is a macroscopic chaos of the order parameter dynamics [49, 50]. Consequently, we show that in the thermodynamic limit, a system of identical Kuramoto-Sakaguchi phase oscillators in three-population networks supports the coexistence of periodic antiphase chimeras within the OA manifold and chaotic antiphase chimeras outside of it. These symmetry-broken chimeras arise due to the transversal instability of symmetric DSD chimeras between two symmetric populations. Above we discussed that in the thermodynamic limit, three populations support the coexistence of periodic antiphase chimeras within the OA manifold and chaotic antiphase chimeras outside it. In fact, in addition to these two, the stationary SDS chimera state coexists with these two states in a wide parameter range (see Appendix. A.1 for details of stability of the symmetric SDS chimeras). Therefore, we evidence the coexistence of three observable chimera states. In addition, the fully synchronized solution is stable in the entire parameter range in which chimera states exist. Yet, from 200 simulations from random initial condition within and off the OA manifold each, at e.g. $A=0.3$ about half of them lead to antiphase chimeras (periodic in the OA and chaotic off the OA) and the other half of them to symmetric SDS chimeras (see Appendix. A.1), evidencing that the basins of the three coexisting chimera states occupy most of the phase space.

Our studies revealed that in three-population networks of identical Kuramoto-Sakaguchi phase oscillators, a variety of qualitatively different chimera states can not only exist, but also coexists and jointly attract most of the initial conditions in phase space.

Previously, symmetric chimera states, namely SDS and DSD-chimeras with two populations behaving alike, were reported to exist in the symmetry-reduced manifold [36]. In this work, we have lifted the symmetry constraint and studied the full dynamics at the level of both microscopic dynamics and macroscopic dynamics, using the Watanabe-Strogatz and the Ott-Antonsen approaches. In full phase space, the symmetric DSD chimeras are unstable to transversal perturbations, i.e. perturbations in which the two D-populations move in opposite directions, resulting in $\text{D}\text{S}\text{D}^{\prime}$ states. These asymmetric $\text{D}\text{S}\text{D}^{\prime}$ chimeras are stable in a wide parameter range. They form chaotic antiphase chimera attractors in both finite-sized systems and in the thermodynamic limit outside the Ott-Antonsen manifold. In the OA manifold, such an antiphase chaotic chimera is rendered periodic. Hence, these two types of antiphase $\text{D}\text{S}\text{D}^{\prime}$ chimeras coexist in the thermodynamic limit. Furthermore, these two types of chimeras coexist with a symmetric stationary SDS chimera state in apparently the entire parameter range of their existence. The simple transition from two- to three-population networks results thus in a quantum jump in the richness of dynamics, and in particular supporting tri-stability of chimera attractors, whereby the three coexisting states encompass stationary, oscillating and chaotic chimeras.

This work illustrates in detail how rich the order parameter dynamics of oscillators with identical natural frequencies in more complex network topologies can be. It thus complements and supports recent results on M-population networks of coupled heterogeneous oscillators [38] and will certainly lead to further investigations in the future.