Abstract
Coupled oscillators, even identical ones, display a wide range of behaviours, among them synchrony and incoherence. The 2002 discovery of socalled 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 clustersplittings. At the far end of this hierarchy, the states further collide with their own mirrorimages 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 highdimensional dynamics.
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Introduction
One 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 lowdimensional dynamics^{1}. 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 reactiondiffusion systems^{2,3} or turbulence in hydrodynamic flows^{4,5}.
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 sociology^{6,7}. During the last two decades, a kind of hybrid phenomenon, in which synchronized and incoherent oscillators coexist in an ensemble of identical oscillators^{8}, coined a chimera state^{9}, has received considerable attention (see reviews^{10,11,12} 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 maps^{14}, four different classes of behaviour were found, including a large variety of partially ordered states, some of which were later classified as chimeras^{15}. 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 symmetrybreaking cascade of clustersplitting perioddoubling bifurcations, wherein the currently smallest cluster is repeatedly split into two, leading to hierarchical clustering. Due to the high symmetry of the system, each symmetrybreaking produces many equivalent mirrorimage variants of each outcome state, multiplying the number of attractors and leading to an ever more crowded phase space^{16}. At some point, each variant collides with some of its mirrorimages, 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 symmetryincreasing bifurcations destroys first the smallest clusters, and then the larger ones, partially mirroring the former clustersplitting 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 StuartLandau oscillators \({W}_{k}\in {\mathbb{C}}\), k = 1, …, N, with nonlinear global coupling^{17}:
where c_{2} and ν are real parameters and \(\langle \ldots \rangle =1/N\mathop{\sum }\nolimits_{k = 1}^{N}\ldots\) denotes ensemble averages. The StuartLandau oscillator itself is a generic model for a system close to a Hopf bifurcation, that is, to the onset of selfsustained oscillations^{18}. Networks of such oscillators have previously been found to exhibit a wide range of dynamics, many of them occurring for linear global coupling^{19,20,21,22,23}. 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 twocluster solution^{24}. Originally, this coupling was inspired by electrochemical experiments, wherein the oxide layer on a silicon electrode displays a wide range of spatiotemporal patterns^{17}. 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 \({{\mathbb{S}}}_{N}\)equivariant: If \({{{{{{{\bf{W}}}}}}}}(t)\in {{\mathbb{C}}}^{N}\) is a solution, then so is \(\gamma {{{{{{{\bf{W}}}}}}}}(t)\,\forall \,\gamma \in {{\mathbb{S}}}_{N}\), where \({{\mathbb{S}}}_{N}\) is the symmetric group of all permutations of the N oscillators^{25}. 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:
where \(\eta \in {\mathbb{R}}\) is an additional parameter, implicitly set by choosing the initial condition. This constraint also implies that for a Poincaré map^{26} 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 W_{k} = 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 the origin with frequency ν at different fixed amplitudes, or an equivariant Hopf bifurcation to a T^{2} torus, producing separate modulatedamplitude clusters that henceforth oscillate with two superposed frequencies ν and ω_{H}^{27}. We will focus on the latter and the dynamics arising from these.
The equivariant Hopf bifurcation occurs at \({\eta }_{{{{{{{{\rm{H}}}}}}}}}=1/\sqrt{2}\) for suitable values of c_{2} and ν. For ν = 0.1, which we keep fixed throughout, it does for c_{2} < −0.448^{17}. In this Hopf bifurcation, differently balanced twocluster solutions ranging from (N − 1) − 1 (with all but one oscillator in the largest cluster) to N/2 − N/2 (with half the oscillators in each cluster) emerge from the synchronized solution. Some of these emerge as stable and others as unstable, depending on the value of c_{2}. The balanced N/2 − N/2 solution, with an equal number of oscillators in each cluster, is shown in Fig. 1a, b. The dashed circle marks the enforced path of the ensemble average 〈W〉 = ηe^{−iνt}, which the two clusters orbit on opposite sides as it circles the origin. An unbalanced 3N/4 − N/4 solution, with N_{1} = 3N/4 of the oscillators in one cluster and N_{2} = N/4 in the other, looks as in Fig. 1c, d.
Because 〈W〉 is independent of the individual oscillator dynamics, the value of any oscillator in the frame of reference of the ensemble average is always given by the simple transformation
where w_{k} is the value of W_{k} in the corotating frame. There, the N/2 − N/2 solution from Fig. 1a, b is simply periodic with frequency ω_{H} and looks as in Fig. 1e, f. An unbalanced modulatedamplitude 3N/4 − N/4 solution like that in Fig. 1c, d appears as in Fig. 1g, h. The average of all oscillators in the corotating frame of 〈W〉 is of course always zero. Notably, the global coupling ensures that all solutions for an ensemble size N are also solutions for \(N^{\prime} =nN,\,n\in {\mathbb{N}}\), with every cluster scaled up by a factor of n. For solutions that contain only clusters N_{i} ≥ 2, the stability properties will also be the same for different n^{22,28}.
If we initialize the N/2 − N/2 solution at a point in the c_{2} − η parameter plane were it is stable and from there on gradually change c_{2} and/or η appropriately, one of the two clusters will break up into two smaller clusters. A possible outcome is shown in Fig. 1i, j. The trajectory of the two new clusters is no longer simply periodic, but period2, with a small and a large loop. The N/2 − N/2 solution has thus become unstable in a symmetrybreaking perioddoubling bifurcation, giving rise to a stable N/2 − N/4 − N/4 threecluster solution. This bifurcation also destabilizes less balanced twocluster solutions, such as the 3N/4 − N/4 solution in Fig. 1g, h. In these cases, the smaller of the two clusters is split. The position of of the perioddoubling bifurcation in parameter space depends on the relative sizes of the clusters, as shown by the blue line in Fig. 1k, which tracks the value of c_{2} at which this bifurcation occurs as a function of N_{1}/N for η = 0.67.
For very unbalanced solutions N_{1}/N > 0.8, the smallest cluster is destroyed in a subcritical pitchfork bifurcation (green line). This results in several smaller clusters and/or single oscillators, depending on the relative sizes of the initial two clusters. In some cases, a few oscillators originally in the smaller cluster are also absorbed by the larger one. As the transition is subcritical, these outcome states are not directly related to the initial twocluster solution, but rather belong to a different, coexisting solution branch. They will not concern us further here.
Hierarchical clustering through pervasive stepwise symmetry breaking
If we concentrate on the N/2 − N/2 solution, that is, keep N_{1}/N = 0.5 fixed, we can track the clustersplitting perioddoubling bifurcation in both c_{2} and η simultaneously. A part of the resultant bifurcation line in the c_{2} − η parameter plane is delineated by the leftmost line in Fig. 2c. Beyond this bifurcation, we find a mesh of additional clustersplitting bifurcation curves, creating a hierarchy of successively less symmetric multicluster solutions with various periodicities. Each bifurcation involves the breakup of either one cluster or two similarly behaving clusters and produces several qualitatively different solutions, differing by how the oscillators of the splitting cluster(s) distribute. (For example, the 4 − 4 solution for N = 8 can split into either 4 − 2 − 2, 4 − 3 − 1, 2 − 2 − 2 − 2, 2 − 2 − 3 − 1 or 3 − 1 − 3 − 1.) However, all these solutions will usually not be costable.
Figure 2c shows the stability boundaries of several solutions for N = 16. The N/2 − N/2 = 8 − 8 solution is stable in the upper left. This solution is destabilized at the leftmost blue perioddoubling line. When increasing c_{2} past this line for η > 0.635 (that is, in the upper half of the figure), it gives rise to stable 8 − 4 − 4 and 8 − 5 − 3 solutions (shown in Figs. 1i, j and 2a, respectively). The 8 − 5 − 3 solution is stable within the two dashed lines. Below the dashed green line, this solution in turn produces a stable 5 − 3 − 5 − 3 and unstable 5 − 3 − 6 − 2 and 5 − 3 − 7 − 1 solutions. At the dashed blue line, it undergoes another perioddoubling cluster split to an 8 − 5 − 2 − 1 period4 solution.
Between η = 0.62 and η = 0.635, only the 8 − 4 − 4 solution emerges as stable when crossing the leftmost perioddoubling line. The remaining solid bifurcation lines all affect this solution and its descendants. At the solid green line from c_{2} ≈ − 0.755 to c_{2} ≈ − 0.725 in the lower left, it produces stable 4 − 4 − 4 − 4 and 4 − 4 − 5 − 3 (Fig. 2b) solutions, as well as unstable 4 − 4 − 6 − 2 and 4 − 4 − 7 − 1 solutions. Like the dashed green line, this is an equivariant pitchfork bifurcation, splitting clusters, but not altering the overall periodicity of the ensemble. Below this pitchfork line, the abovementioned 4 − 4 − 4 − 4, 4 − 4 − 5 − 3, 4 − 4 − 6 − 2 and 4 − 4 − 7 − 1 fourcluster solutions also emerge directly from the 8 − 8 solution at the leftmost perioddoubling line.
At the solid blue line directly to the right of the dashed blue one, the 8 − 4 − 4 solution undergoes a perioddoubling bifurcation analogous to that of the 8 − 5 − 3 solution, producing a stable 8 − 4 − 2 − 2 (Fig. 2d) and an unstable 8 − 4 − 3 − 1 period4 solution. The former becomes unstable either at the bottom diagonal green pitchfork line at c_{2} ≈ − 0.72 or at the rightmost blue perioddoubling line. In the latter case (see inset), the 8 − 4 − 2 − 2 solution produces an unstable 8 − 4 − 2 − 1 − 1 and a stable 8 − 4 − 1 − 1 − 1 − 1 period8 solution (Fig. 2e, f).
At the red line in Fig. 2c, the 8 − 4 − 1 − 1 − 1 − 1 solution undergoes a torus bifurcation, whereby a third frequency is added to the dynamics, while all clusters stay intact. The resultant threefrequency motion is resistant to the addition of small random numbers over a nonzero c_{2} interval. This is notable as stable quasiperiodic dynamics with more that two frequencies is usually not observed. It has even been proven that quasiperiodic dynamics with three or more frequencies are in general structurally unstable^{1,29}. However, such stable quasiperiodic motion on T^{3} has also been observed in StuartLandau oscillators with linear global coupling^{20} and could be due to the rotational invariance of the differential equations.
If we initialize the 8 − 4 − 4 solution at c_{2} = − 0.71 and η = 0.63 and slowly increase c_{2} along the horizontal black line in Fig. 2c, the maxima of \({{{{{{{\rm{Re}}}}}}}}({w}_{k})\) for k = 1, …, 16 develop as in Fig. 3a: Initially, there are one maximum of the oscillators in the cluster of eight (blue) and two shared maxima of the two period2 clusters of four (red). When one of these clusters splits up into two smaller clusters of two at the perioddoubling bifurcation PD_{2}, the maxima of these smaller clusters henceforth appear as four distinct yellow lines. In the next perioddoubling bifurcation (PD_{3}), these lines split up into eight.
From the fully synchronized solution to the 8 − 4 − (4 × 1) solution, four discrete steps of symmetry breaking have taken place: one initial equivariant Hopf bifurcation, as well as three equivariant perioddoubling bifurcations. The three last of these steps are shown schematically in Fig. 3b. Similar stepwise symmetry breaking is observed both for larger N and when the smallest cluster does not break up into equalsized parts (Fig. 3c). The larger N is, the more steps occur, at ever closer parameter values, and for N = 256, as many as seven steps can be observed (see Fig. 3d). The N/2 − N/2 twocluster solution thus gives rise to a clustersplitting cascade, producing a multitude of coexisting multicluster states and, most notably, hierarchical clustering.
Symmetryincreasing bifurcation and temporary clusters
At the end of a cascade of clustersplitting perioddoubling bifurcations, a torus bifurcation usually occurs (see e.g. the red bifurcation line in Fig. 2c). The resultant T^{3} motion is usually stable for a nonzero parameter interval, before being superseded by less regular dynamics in a symmetryincreasing bifurcation^{30}, wherein several distinct, but equivalent variants of the same solution collide. These variants exist because Eq. (1) is \({{\mathbb{S}}}_{N}\)equivariant. Thus, any solution remains a solution when any of the oscillators are interchanged, and each solution (except the fully synchronized one) exists in the form of several distinct symmetric variants in phase space. (For example, if we interchange an oscillator from the blue cluster in Fig. 1a with one from the red, the outcome is such a different, but equivalent variant.)
All solutions investigated here are at least periodic in the corotating frame. The attractor corresponding to a stable solution thus occupies more than a single point in the phase space spanned by w_{k}, k = 1, …, N. As these attractors become more complex, and especially as the aforementioned torus bifurcation renders the motion quasiperiodic, the part of phase space they occupy increases in extent. This of course applies equally to all the symmetrized variants of each solution.
At some point, two or more variants might grow to touch each other in phase space. When this happens, the variants involved in the collision merge to become a single instance of a new solution, of which there are fewer distinct mirrorimage variants in total. The attractor on which the new solution lives is correspondingly more symmetric than the attractors of the colliding variants. One symmetryincreasing bifurcation can in general be followed by another, further increasing the attractor symmetry.
In the N = 16 case in Fig. 3a, the first symmetryincreasing bifurcation only disrupts the former rigid cyclic order of the four single oscillators, inherited from the solution in Fig. 2e, f (i.e. that the purple oscillator trails the yellow one, which trails the pink, and so on). In other cases, some of the intact clusters of a certain colliding variant contain oscillators that are in a different cluster in some of the other variants this variant is colliding with. Then, the symmetryincreasing bifurcation destroys these clusters. Such a scenario is schematically shown in Fig. 4: In this N = 8 example of two colliding 4 − 2 − 1 − 1 variants, the cluster of two in one of the variants contains oscillators 5 and 6, while in the other, it contains oscillators 5 and 7. Because the two variants are identical mirrorimages of each other, they must both be treated equally by the collision. Thus, the oscillators 5 and 6, which are clustered in only one of the variants, cannot remain together after the collision, nor can the oscillators 5 and 7. The result is thus a 4 − 1 − 1 − 1 − 1 state in which all four single oscillators behave identically. Similarly for larger ensembles, as the attractor symmetry is increased, the number of single oscillators, in general, grows, in a sense also decreasing the overall order of the ensemble.
The N = 32 clustersplitting cascade in Fig. 3c is also followed by symmetryincreasing bifurcations, and at some point, the longterm clustersize distribution becomes 16 − 9 − (7 × 1). A time series of the resulting solution is shown in Fig. 5a: Here, the seven single oscillators in yellow move similarly to the clusters of four, two and one in the former 16 − 9 − 4 − 2 − 1 solution, being close to deep minima when the red cluster of nine is at a shallow minimum and vice versa. They also repeatedly congregate into loose temporary agglomerations of four, three and two oscillators, respectively. This is further illustrated by Fig. 5b–d, where the crosscorrelation between all oscillator trajectories is calculated every 10^{4} time steps. Two oscillators are said to be in the same cluster if their crosscorrelation is greater than 1 − ε for ε = 10^{−8} (b), ε = 10^{−4} (c) or ε = 10^{−2} (d). Sometimes, a temporary cluster of three detected for a certain ε becomes a cluster of four for larger values of ε, such as the blue cluster at t = 7 ⋅ 10^{4} and the red cluster at t = 1.2 ⋅ 10^{5}. This means that four oscillators are loosely congregating here, but that three of the oscillators are more strongly clustering than the fourth. The ensemble is thus less closely approaching the remains of a formerly stable 16 − 9 − 4 − 2 − 1 attractor in phase space.
Dynamics like those in Fig. 5 have previously been observed by Kaneko in globally coupled logistic maps when the phase space becomes so full of mirrorimage attractors that they inevitably intrude upon each other^{31}. The outcome is a form of chaotic itinerancy^{32}, wherein the system meanders between the attractor ruins of previous attractors, each of them relatively lowdimensional, but connected by higherdimensional transitional motion^{15}.
Also found in globally coupled maps is precisiondependent clustering, wherein trajectories of individual maps that are unclustered when distinguished with high precision appear to repeatedly merge into the ever thicker branches of a clustering tree when the precision is decreased^{14}. In our ensemble, this occurs as a consequence of the symmetrybreaking perioddoubling cascade. For example, past the N = 256 cascade in Fig. 3 (at c_{2} ≈ −0.71162), we encounter a 128 − 64 − 33 − (31 × 1) itinerant solution that for small ε ≤ 10^{−5} is found to have an additional cluster of usually 16, sometimes 18 or 19 oscillators, while the remaining oscillators repeatedly form ephemeral smaller clusters of strongly fluctuating sizes. For ε = 10^{−4}, a cluster of size 15 is also sometimes detected (along with that of 16), and for ε = 10^{−3} the sizes are always 128 − 64 − 33 − 16 − 15, 128 − 64 − 33 − 18 − 13 or 128 − 64 − 33 − 19 − 12. For ε = 10^{−2}, they are 128 − 64 − 33 − 31 throughout, and for ε = 10^{−1}, 128 − 64 − 64. The same pattern to some extent already applies in the quasiperiodic domain of Fig. 3b–d, where clusters are most strongly correlated with those other clusters from which they most recently split.
If we initialize the ensemble in the itinerant state beyond a symmetryincreasing bifurcation and gradually change the parameters back towards more regular motion, the transition to the relevant twocluster solution will simply be the reverse of the one that created the itinerant state. For example, if we initialized the N = 16 ensemble in the state at the right edge of Fig. 3a and slowly decreased c_{2}, this would produce the same sequence of bifurcations. See Supplementary Fig. 4 and Supplementary Note 2.
When the equation parameters are incremented too far into the regime of chaotic itinerancy, the ensemble will often jump to an entirely different solution. Beyond the 16 − 9 − 7derived state in Fig. 5, it e.g. jumps to the blue hitherto costable 16 − 8 − 8derived branch. However, the end result can also be the destruction of all permanent clusters and the motion of only single oscillators on a fully symmetric chaotic attractor. See Supplementary Figs. 5–7 and Supplementary Note 3.
Emergence of a chimera state
In our context, a chimera state is an N_{1} − ((N − N_{1}) × 1)) solution. The modulatedamplitude chimeras previously found in Eq. (1) have significantly more synchronized (N_{1}) than unsynchronized (N − N_{1}) oscillators^{33}. This suggests they have not evolved from balanced twocluster solutions like the ones studied above. Yet, our above results can be used to explain how they are created. If we e.g. initialize an N = 20 ensemble as an 3N/4 − N/4 = 15 − 5 solution (Fig. 6a) for c_{2} = − 0.87 and η = 0.67, the bifurcation diagram in Fig. 1k tells us it will undergo a clustersplitting perioddoubling bifurcation if c_{2} is increased. The resulting 15 − 3 − 2 period2 solution is shown in Fig. 6b. Further up in c_{2}, the cluster of two is split into single oscillators (Fig. 6c). Then, a torus bifurcation smears the previously closed trajectories into continuous bands (Fig. 6d).
Finally, the current 15 − 3 − 1 − 1 variant collides with nine others in a symmetryincreasing bifurcation. This destroys the cluster of three, resulting in a 15 − (5 × 1) chimera state (Fig. 6e). Note how the three oscillators that are temporarily close to each other in Fig. 6e (red, yellow, grey, in the lower left) are not all the same three that were clustered in Fig. 6b–d (red, purple, grey). The ensemble is currently close to the ruin of a different 15 − 3 − 1 − 1 solution variant, and the chimera state is thus also an example of chaotic itinerancy. For N = 200, the transition from a 150 − 50 to a 150 − (50 × 1) solution proceeds along a much more involved, but essentially similar path. See Supplementary Fig. 8 and Supplementary Note 4.
Generality of results I—pitchfork maps
Other theoretical \({{\mathbb{S}}}_{N}\)symmetric systems can also develop as discussed in the previous sections. One such system is the following ensemble of N globally coupled timediscrete maps:
where \({y}_{k}(n)\in {\mathbb{R}}\) denotes the nth iteration of the kth map, k = 1, …, N, and α is a realvalued parameter. Each map y_{k}(n + 1) = (1 + α − ∣y_{k}(n)∣^{2}) ⋅ y_{k}(n) (without the coupling) is modeled on the normal form of the supercritical pitchfork bifurcation ^{26}, \({x}_{n+1}={x}_{n}+\mu {x}_{n}{x}_{n}^{3}\) Altogether, the system (4) is subject to a conservation law:
that is, the ensemble average \(\langle y(n)\rangle =\langle y(0)\rangle \,\forall \ n\in {\mathbb{N}}\) remains constant independent of the individual map behaviour and thus effectively constitutes an additional parameter β = N^{−1}∑_{k}y(0), implicitly set by choosing the initial value of each map y_{k}(0).
For suitable values of α and β, Eq. (4) has stable period1 (i.e. constant) twocluster solutions, among them a balanced N/2 − N/2 solution for any even N. As an example, for α = 0.7 and β = 0.15 this solution is given by y_{k}(n) ≈ − 0.645 for k = 1, …, N/2 and y_{k}(n) ≈ 0.945 for k = N/2 + 1, …, N. (Of course, any other N/2 of the N maps could also be in the first cluster; that would simply constitute a different equivalent variant of the same solution.) Clearly, the ensemble average remains constantly equal to β = 0.15.
Let us now again consider the concrete case N = 16. If α is slowly increased, the N/2 − N/2 = 8 − 8 solution undergoes an equivariant perioddoubling bifurcation at α = 0.708. Like in the StuartLandau ensemble, several threecluster period2 solutions emerge, one of which is the N/2 − N/4 − N/4 = 8 − 4 − 4 solution in Fig. 7a. For a further increase of α, this solution also undergoes a perioddoubling bifurcation, wherein both clusters of four are split into a total of four period4 clusters of two, as seen in Fig. 7b. The cluster of eight (at y_{k}(n) ≈ − 0.7) still remains period1, but has been left out of the figure for a better view. This sequence of twocluster splittings, summarized in Fig. 8a, is strongly reminiscent of the two last steps in Fig. 3b.
For sufficiently large ensemble sizes N, when the N/2 − N/2 solution is destabilized in its perioddoubling bifurcation, several of the resultant threecluster solutions emerge as costable. For N = 128 and β = 0.15, one of the stable solutions is a 64 − 33 − 31 period2 solution whose trajectory looks more or less like that of the N/2 − N/4 − N/4 solution in Fig. 7a. (The maxima of the period2 trajectory of the cluster of 33 are only slightly smaller than those of the cluster of 31, and its minima slightly less deep, in order to fulfill the condition 〈y(n)〉 = β ∀n.) When α is gradually increased for this 64 − 33 − 31 solution, an equivariant perioddoubling also occurs, but here, only the cluster of 31 is split. The result is the 64 − 33 − 16 − 15 period4 solution shown in Fig. 7c. (Here, the cluster of 64 also moves with a small period4 component, due to the asymmetry in the smaller clusters.) If α is increased somewhat further, it results in the 64 − 33 − 16 − 8 − 7 period8 solution in Fig. 7d. Even further upward in α, two more clustersplitting perioddoubling bifurcations occur, resulting in the overall sequence of cluster sizes shown in Fig. 8b. Thus, Eq. (4) undergoes a clustersplitting cascade remarkably similar to that of Eq. (1).
The coupled maps of Eq. (4) also exhibit transitions likely to be symmetryincreasing bifurcations. Past the α interval covered in Fig. 8b, the N = 128 ensemble namely also enters several consecutive α intervals wherein the sizes of only some clusters remain stable for several α increments. Other clusters seemingly appear and disappear erratically, as already observed for the StuartLandau ensemble in Fig. 5. We even encounter 64 − (64 × 1) chimera states, as shown in Supplementary Fig. 10.
The difference between the two systems (1) and (4) lies in the details. For example, we have already seen that in the StuartLandau ensemble, the perioddoubling bifurcation of the N/2 − N/4 − N/4 solution produces a stable N/2 − N/4 − N/8 − N/8 solution, whereas the analogous bifurcation in the coupled maps gives rise to a stable N/2 − (4 × N/8) solution (comparing Figs. 3b and 8a). Another difference can be observed if we track the 128 − 65 − 63 period2 solution to Eq. (4) for N = 256 upward in α for β = 0.15. At first, it will give rise to a stable 128 − 65 − 32 − 31 period4 solution. However, the next bifurcation encountered will not be another perioddoubling, but an equivariant pitchfork splitting the cluster of 65. Thus, the pattern of stepwise cluster splitting, whereby always the smallest cluster is the next one to be split, as observed in both Figs. 3c, d and 8b, ends prematurely. In this case, no more discrete cluster splittings occur and the next qualitative change of the dynamics is a symmetryincreasing bifurcation, as seen in Supplementary Fig. 11 and described in Supplementary Note 6.
Generality of results II—electrochemical experiments
As stated in the introduction, Eq. (1) is inspired by electrochemical experiments. In fact, the theoretical model was originally more complicated, consisting not of discrete identical oscillators, but of a continuous oscillatory medium coupled via both global and local (diffusional) coupling^{17,34}. Later results showed that most of the qualitative dynamics obtained in this extended model could still be reproduced if the diffusion was set to zero^{33}, thus paving the way for our purely globally coupled ensemble. Meanwhile, the experimental system itself has been found to exhibit a vast amount of dynamical phenomena^{17,34,35,36,37,38,39,40,41}. Below, we revisit four different spatiotemporal states representative of solutions in the transition scenario outlined above.
The central component of the experiment is an ntype silicon (Si) electrode, immersed in a fluriodecontaining electrolyte. A voltage is applied across the electrode, which is also illuminated with a laser. Thus, an oxide layer is grown photoelectrochemically on the Si surface. Simultaneously, the fluoride species in the electrolyte etches away the silicon oxide in a purely chemical process^{42}. An ellipsometric setup is used to measure the spatiotemporal changes in the optical pathway through the Si∣SiO_{2}∣electrolyte interface^{34,35,43}.
For a wide range of experimental parameters, the ellipsometric signal can be made to oscillate homogeneously with a simple period^{37}. If the parameters are suitably changed, the electrode undergoes a perioddoubling bifurcation, resulting in two antiphase clusters connected by a mediating region with rather low amplitude. An exemplary snapshot of the electrode in this state is shown in Fig. 9a, together with the time series of a section. The location of the section is indicated by the blue line on the image of the electrode. In the depicted snapshot, a rather high ellipsometric signal, displayed by the red color in the upper part of the electrode, coexists with a rather low signal in the lower right, displayed by the blue color. In the time series below, we recognize the oscillation of the ellipsometric signal; the two regions, connected by the cut, oscillate with the same frequency, but in antiphase to each other. Note that the global time series exhibits a simple periodic oscillation, which, as demonstrated in Supplementary Fig. 12 and described in Supplementary Note 7, defines a rotating frame. Thus, as in the above simulation results, the experimental results are depicted in a rotating frame, that is, the spatial mean of each frame has been subtracted from every point in the same frame.
Figure 9b shows the same electrode after a further parameter variation (see appendix). This time, in order to properly view the spatiotemporal development, the spatial coordinate of the time series is composed of two lines forming an angle. Clearly, the variation of the parameters has resulted in a perioddoubling bifurcation affecting the right and left side of the electrode, corresponding to the upper and lower part of the timeseries spatial coordinate. These regions now oscillate with double the period of the oscillations in the upper part of the electrode and are in antiphase with respect to each other.
In Fig. 9c, a deep blue region can be seen on the right of the electrode snapshot. This region appears rather regular throughout whereas the rest of the electrode is irregularly patterned. In the time series of the spatial cut, the deep blue area appears in the lower quarter. It is indeed found to exhibit simply periodic oscillations, whereas most of the electrode is turbulent. This solution is a chimera state.
Finally, Fig. 9d depicts a state that is turbulent throughout. Here, irregular patterns arise over the entire electrode. The time series shows that the spatial incoherence is accompanied by aperiodic behaviour.
Note that the measurements in Fig. 9a, b were carried out on a different day and with a different electrolyte than the ones in Fig. 9c, d. The electrolyte composition seems to be a crucial parameter for some of the presented states, yet it is a parameter which cannot be easily varied during a measurement day. Spatially homogeneous oscillations can, however, be found with both of the electrolytes used here.
Discussion
In this article, we have shown how a globally coupled system can transition from full symmetry to ever more complex coexistence patterns through a sequence of discrete symmetrybreaking steps. The transition begins with a cascade of clustersplitting bifurcations, and at each step of this cascade, either one cluster or two similarly behaving clusters are split into smaller clusters. In an ideal form of the cascade, the next cluster to split is always the smallest one, ultimately creating a multicluster state with very different cluster sizes, wherein the smallest “cluster” is just a single oscillator. This ideal cascade is schematically depicted in the left part of Fig. 10.
The clustersplitting cascacde is followed by one or more symmetryincreasing bifurcations, breaking ever more clusters up into single oscillators. Even though they destroy clusters, these bifurcations are symmetryincreasing because the single oscillators they produce all behave equally for t → ∞. Thus, the attractor is symmetric with regards to the interchange of any two of these oscillators.
In practice, the symmetryincreasing transition is often cut short by interactions with other solution branches and the ensemble at some point is thrown onto a different, hitherto costable solution. In the ideal case when it is allowed to continue sufficiently long, the end result is ultimately a turbulent state of only single oscillators, all behaving equally in the long run. A chimera state is then the last step but one in the cascade.
Our primary model has been one of \(N={2}^{n},\ n\in {\mathbb{N}}\) StuartLandau oscillators with nonlinear global coupling and our main focus on the case where the fully synchronized solution is initially split into two equal clusters N/2 − N/2. However, the clustersplitting cascade is also observed for less balanced initial states, such as the 15 − 5 solution which transitions via 15 − 3 − 2 to 15 − 3 − 1 − 1 in Fig. 6. The cascade is also not dependent on the choice of N = 2^{n}, but similarly occurs for odd N as well, as shown in Supplementary Fig. 13 and described in Supplementary Note 8. Nor is it conditional upon the particular chosen model, but can be similarly observed in globally coupled timediscrete maps.
Symmetryincreasing bifurcations also occur whether the clustersize distribution of the initial twocluster state emerging from synchrony is N_{1}/N_{2} = 1 (Fig. 3a) or N_{1}/N_{2} = 3 (Fig. 6). Moreover, it is observed in both the StuartLandau oscillators (1) and the pitchfork maps (4). The general outcome of a symmetryincreasing bifurcation, chaotic itinerancy, has previously been found in globally coupled logistic maps, along with both multicluster states, chimeras and precisiondependent clustering, but without an overall explanation of how these phenomena might be bifurcationtheoretically related to each other^{14,15,31,32}. This suggests that the bifurcation scenario uncovered here occurs in those logistic maps as well.
What are the prerequisites for the observed bifurcation scenario? The high permutation symmetry of globally coupled equations is probably a central factor shared by the StuartLandau oscillators and the pitchfork maps (and Kaneko’s logistic maps). \({{\mathbb{S}}}_{N}\) not only has many subgroups, but most of these subgroups have many subgroups as well, and so on. This intricate subgroup structure is mirrored in the hierarchy of successively less symmetric quasiperiodic solutions. Moreover, because \({{\mathbb{S}}}_{N}\) is much larger than those of its subgroups corresponding to the more intricate solutions, there are many mirrorimage variants of the latter, causing the symmetryincreasing bifurcations and the itinerant coexistence patterns that these produce.
The nonlinear nature of the global coupling could be another relevant system property, and all three systems studied here are coupled nonlinearly (as are Kaneko’s logistic maps). However, symmetryincreasing bifurcations have also been observed in StuartLandau oscillators with linear global coupling^{44}, for an ensemble size as small as N = 4. Such an ensemble is of course too small to exhibit an evident clusterhalving cascade, and instead, the symmetryincreasing bifurcation was preceded by a nonequivariant Feigenbaum perioddoubling cascade to chaos. To test whether our full bifurcation scenario can occur for linear global coupling as well, is thus an exciting task for future studies.
In the case of the StuartLandau oscillators, an additional factor required by the clustersplitting cascade is the amplitude variation of the cluster orbits. If the clusters were to have fixed amplitudes, i.e., d∣w_{k}∣/dt = 0 for all oscillators w_{k}, then there could namely only be three different clusters, due to the StuartLandau oscillator being a thirdorder polynomial^{45}. This would render any extended clustersplitting impossible. On the other hand, the fact that clustersplitting takes place by means of perioddoubling seems to imply that the amplitudes must vary over the course of a full oscillation period.
What all discussed symmetryincreasing bifurcations seem to require, is a certain dimensionality of the dynamics before the bifurcation. (That is, trajectories can for example not be zerodimensional fixed points or onedimensional periodic orbits.) In the StuartLandau ensemble with nonlinear global coupling and in the pitchfork maps, this extra dimensionality arises in the torus bifurcation at the end of the clustersplitting cascade. In the aforementioned StuartLandau oscillators with linear global coupling, it is provided by a perioddoubling cascade to chaos, and in the globally coupled logistic maps, the maps are also in the chaotic regime.
Finally, we again consider the experiments. These exhibit a series of patterns which are similar to the ones in the simulations. The antiphase clusters in Fig. 9a emerge from the homogeneous oscillation and bring about a second frequency. Figure 9b, c exhibits coexistence patterns, consisting either of clusters of different frequencies (b) or of the coexistence of a regularly oscillating region and irregular motion (c). Figure 9d is a completely turbulent state. These states are clearly reminiscent of the ones found in the simulations, and we are not aware of other bifurcation scenarios that include these states.
Nevertheless, the connection between the experiments and the discussed bifurcation scenario has yet to be clearly demonstrated. As mentioned above, one difficulty is that the electrolyte composition, which is difficult to change within the same experiment, seems to be a crucial bifurcation parameter. Despite the prior equivalence of many results with and without diffusion, the diffusive coupling on the electrode could also potentially influence the dynamical transition. This could for example be investigated in experiments with amorphous instead of crystalline silicon. Altogether, many effects of different parameters on the dynamics are only poorly understood and the detailed oscillation mechanism is still unknown. There is thus great potential for further studies in this direction. The same applies to the search for other experimental systems that exhibit the discussed transition from synchrony to turbulence.
Methods
StuartLandau oscillators
The differential equations (1) were solved numerically using the Python programming language^{46} (version 2.7 and later 3.8) and the implicit Adams method of the scipy.integrate.ode class of the SciPy library^{47} (version 1.6) with a time step of Δt = 0.01. The data were held and processed in the form of NumPy (version 1.19) arrays^{48} with complexvalued floatingpoint elements and visualized using the Matplotlib library and graphics environment (version 3.3) ^{49}. The numerical results were evaluated using custombuilt functions drawing on the resources of these standard Python libraries, written by S.W.H. Simulations were carried out in the nonrotating frame of Eq. (1), and results in the corotating frame of 〈W〉 were visualized applying Eq. (3) to the data after simulations had been carried out. When not otherwise stated, initial conditions of numerical solutions were random numbers on the real line, fulfilling the global constraint that 〈W〉 = ηe^{−iνt}. This choice was inspired by earlier work^{17}.
Figures 1e and 2c were created using the dynamicalsystems continuation software Auto07p^{50,51} to continue solutions in parameter space. As Auto can only continue fixedpoint and limitcycle solutions, Eq. (1) had to be formulated in the corotating frame of the ensemble average in order to carry out these continuations, yielding
where w_{k} = a_{k} + ib_{k} with \({a}_{k},{b}_{k}\in {\mathbb{R}}\), k = 1, …, N, and
To obtain Fig. 2c, the relevant N = 16 quasiperiodic solutions where first generated using Python simulations. The output data were transferred to the rotating frame, and a time series corresponding to one full period in that frame was used as input for a c_{2} or η oneparameter continuation of each periodic solution, in order to detect the location of the depicted bifurcations. Then, the detected bifurcations were twoparameter continued in c_{2} and η to obtain the depicted bifurcation lines.
To obtain Fig. 1e, Eq. (6) was reduced to a twocluster model by setting a_{k} = a_{c1} and b_{k} = b_{c1} for all k = 1, …, N_{1}, where w_{c1} = a_{c1} + ib_{c1} is the value of the first cluster. All other oscillators k = N_{1} + 1, …, N are in the other cluster w_{c2} = a_{c2} + ib_{c2}. This yields the following equation for the first cluster
with A_{c1} and B_{c1} analogous to Eq. (7), while \({w}_{{{{{{{{\rm{c2}}}}}}}}}=\frac{{N}_{1}}{N{N}_{1}}{w}_{{{{{{{{\rm{c1}}}}}}}}}\), because of the constraint that ∑_{k}w_{k} = 0. Thus, the reduced twocluster model is only twodimensional. The relative size of the first cluster, N_{1}/N, becomes an effective fourth parameter, in addition to c_{2}, ν and η.
Whereas (8) describes the motion of two clusters of sizes N_{1} and N_{2} = N − N_{1}, respectively, it says nothing about intracluster stability and cannot model the breakup of either cluster. To be able to evaluate the internal stability of the clusters, we followed Ku et al.^{22} and added two effectively infinitesimal extra oscillators to the model, which only feel the presence of the two macroscopic clusters, but themselves neither affect the movement of each other, nor that of the macroscopic clusters. Their motion is given by
where P_{1,2} and Q_{1,2} denote composite expressions for the first and second infinitesimal oscillator, of the same form as A_{c1,c2} and B_{c1,c2}:
In the initial state of the continuation, one of these two infinitesimal oscillators is set to follow the same periodic orbit as either of the two clusters. If any bifurcations are detected to make either infinitesimal oscillator leave the macroscopic cluster it started at, this means that cluster has become unstable.
Figure 3a was created by initializing the N = 16 ensemble in the 8 − 4 − 4 configuration at c_{2} = − 0.715 and incrementing c_{2} by Δc_{2} = 10^{−5} every ΔT = 4 ⋅ 10^{4} time steps until c_{2} = − 0.7095 for ν = 0.1 and η = 0.63. At the beginning of each c_{2} step, random numbers ≤10^{−6} were added to the real and imaginary part of each oscillator to provoke the breakup of potential unstable clusters. Maxima of \({{{{{{{\rm{Re}}}}}}}}({w}_{k})\) were plotted for the last 2000 time steps of simulation at each c_{2} steps.
The schematic in Fig. 3b was drawn based on automatically detected cluster sizes at each c_{2} step in the aforementioned c_{2}incremented simulation. These cluster sizes were determined by calculating the pairwise crosscorrelations of the trajectories of all oscillators over the last 2000 time steps at each c_{2} step, respectively. If the crosscorrelation differed from 1 by less than ϵ = 10^{−8}, the two oscillators were deemed to belong to the same cluster. To calculate the crosscorrelations and obtain the clusters, we used SciPy’s builtin scipy.cluster.hierarchy.linkage function.
The schematic in Fig. 3c was determined based on an analogous c_{2}incremented simulations for N = 32, ν = 0.1 and η = 0.63, initialized in the 16 − 16 configuration at c_{2} = − 0.74. Here, Δc_{2} = 2 ⋅ 10^{−5} and ΔT = 10^{4}. The simulation was performed until c_{2} = −0.712, producing the result in Supplementary Fig. 1a. Clusters were calculated as in the N = 16 case based on the last 800 time steps of simulation at each c_{2} step, producing Supplementary Fig. 2b.
The schematic in Fig. 3d was determined based on two analogous c_{2}incremented simulations for N = 256, ν = 0.1 and η = 0.63. In the first of these, the ensemble was initialized in the 128 − 64 − 64 configuration at c_{2} = − 0.7145, from where c_{2} was incremented by Δc_{2} = 10^{−5} every ΔT = 2 ⋅ 10^{4} time steps until c_{2} = − 0.7117, producing the result in Supplementary Fig. 2a. In a second c_{2}incremented simulation for N = 256, the ensemble was initialized at c_{2} = − 0.71172 in the 128 − 64 − 33 − 16 − 15 configuration found there in the prior N = 256 simulation with Δc_{2} = 10^{−5}. From there on, c_{2} was incremented by Δc_{2} = 2 ⋅ 10^{−7} every ΔT = 2 ⋅ 10^{4} time steps until c_{2} = − 0.7116, producing the result in Supplementary Fig. 3a, b. For either simulation, clusters were calculated based on the last 800 time steps of simulation at each c_{2} step, producing Supplementary Figs. 2b and 3b, c, respectively.
Figure 5 b–d were created by simulating the 16 − 9 − (7 × 1) solution in Fig. 5a for T = 10^{6} time steps. Every 10^{4} time steps, the pairwise crosscorrelation between all oscillators was calculated over an interval of 800 time steps, and if the crosscorrelation of two oscillators was found to be greater than 1 − ε for ε = 10^{−8} (Fig. 5b), ε = 10^{−4} (Fig. 5c) or ε = 10^{−2} (Fig. 5d), respectively, they were counted as being in the same cluster.
The solutions in Fig. 6 were obtained by initializing the N = 20 ensemble in a 15 − 5 solution at c_{2} = − 0.88, ν = 0.1 and η = 0.67, and incrementing c_{2} by Δc_{2} = 2 ⋅ 10^{−4} every ΔT = 5000 time steps until c_{2} = − 0.7. Supplementary Figures 4–7 were created based on data obtained analogously to that in Figs. 3 and 6 with parameters as given in their respective captions.
Pitchfork maps
The differential equations (1) were solved numerically using the Python programming language^{46} (version 3.8). The data were held and processed in the form of NumPy (version 1.19) arrays^{48} with complexvalued floatingpoint elements and visualized using the Matplotlib library and graphics environment (version 3.3) ^{49}. The numerical results were evaluated using custombuilt functions drawing on the resources of these standard Python libraries, written by S.W.H.
The αincremented simulations behind Fig. 8 were initialized in the N/2 − N/2 configuration and α was then gradually increased as specified in the captions of Supplementary Figs. 9 and 10. The same applies to Supplementary Fig. 11. At the beginning of each α step, small random numbers ≤10^{−6} were added to the maps to provoke the breakup of potential unstable clusters.
The cluster sizes at each α step of the aforementioned simulations were calculated automatically by comparing the trajectories of all maps during the last 2000 steps at each α value. If the Euclidean distance between the vectors made up by two such map trajectories was found to be less than ε = 10^{−4}, the two maps were said to be in the same cluster.
Electrochemical experiments
For the experiments a custom made three electrode electrochemical PTFE cell is used, with a circular shaped platinum wire as counter electrode and a commercial mercurymercurous sulfate reference electrode^{34}. As working electrode a sample from an ntype Silicon wafer with a (111) crystal orientation and a resistivity of 1–10 Ωcm is used. A 200 nm aluminium back contact is evaporated onto the wafer and annealed at 250 °C for 30 min. To passivate the silicon surface and get rid of organic contamination, the samples are plasmaoxidized.
Before the experiment the sample is brought into contact with the wire in the custom made PTFE WE holder using silver paste. It is subsequently sealed using silicone rubber (Scrintex 901, Ralicks GmbH, Rees Haldern, Germany), leaving free only the active electrode area. After the silicone has dried, the sample holder with the sample is immersed in acetone for 5 min, subsequently in ethanol for 5 min, then methanol for 5 min, then in ultra pure water (R = 18.2 MΩcm) for 10 min and finally it is abundantly rinsed with ultra pure water.
The organic cleaning solvents are AnalaR NORMAPUR grade (VWR Chemicals). The electrolyte components are Suprapur grade (Merck). For the potential control a FHI2740 potentiostat is used. Illumination of the electrode is provided by a 15 mW HeNe laser with a wavelength of 632.8 nm (Thorlabs HNL150L). The illumination intensity is controlled by an SLM (Hamamatsu x1046806). The ellipsometric signal is recorded with a JAICVA50 CCD camera. A background correction of the video data is performed according to
where \(\xi (\vec{x})\) is the corrected ellipsometric signal at \(\vec{x}\), \(\xi {(\vec{x})}_{{{raw}}}\) is the raw data of the ellipsometric signal at \(\vec{x}\), \({\overline{\xi (\vec{x})}}_{{{raw}}}\) is the temporal average of the raw data and \(\langle {\overline{\xi (\vec{x})}}_{{{raw}}}\rangle\) denotes the spatial average of the temporal average of the raw data. The homogenous mode is subtracted after the background correction is performed.
The experimental data presented are obtained from experiments under the following conditions: The electrolyte used for Fig. 9a, b had a pH of 2.3 and a fluoride concentration c_{F} = 50 mM. For (a) the applied potential was U = 5.65 V vs SHE, the external resistance times electrode area R_{ext}A_{el} = 0 kΩcm^{2} and the illumination intensity I_{ill} = 0.67 mW/cm^{2}. For (b) U = 6.65 V vs SHE, R_{ext}A_{el} = 3.84 kΩcm^{2}, I_{ill} = 0.57 mW/cm^{2}. The electrolyte used for (c) and (d) had a pH of 1 and a fluoride concentration of c_{F} = 75 mM. For (c) the applied potential was U = 8.65 V vs SHE, the external resistance times electrode area R_{ext}A_{el} = 0.81 kΩcm^{2} and the illumination intensity I_{ill} = mW/cm^{2}. For (d) U = 8.65 V vs SHE, R_{ext}A_{el} = 0.54 kΩcm^{2}, I_{ill} = 0.57 mW/cm^{2}. The illumination on the working electrode was homogeneous at any time.
Data availability
The numerical and experimental data generated in this study have been deposited in the database of the TUM University Library under the accession code https://doi.org/10.14459/2021mp1618587.
Code availability
The code is available as free and open source software under the GPL version 3 or later. It has been deposited in the database of the TUM University Library under the accession code https://doi.org/10.14459/2021mp1618587.
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Acknowledgements
We thank Felix P. Kemeth, Maximilian Patzauer and Seungjae Lee for fruitful discussions. Financial support from the Studienstiftung des deutschen Volkes and the Deutsche Forschungsgemeinschaft, project KR1189/18 “Chimera States and Beyond”, is gratefully acknowledged.
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S.W.H. carried out the simulations and analysed the numerical data. A.T. performed the experiments and analysed the experimental data. All three authors discussed the results and wrote the paper. K.K. supervised the project.
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Haugland, S.W., Tosolini, A. & Krischer, K. Between synchrony and turbulence: intricate hierarchies of coexistence patterns. Nat Commun 12, 5634 (2021). https://doi.org/10.1038/s41467021259077
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DOI: https://doi.org/10.1038/s41467021259077
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