Solvable model for chimera states of coupled oscillators
Networks of identical, symmetrically coupled oscillators can spontaneously split into synchronized and desynchronized sub-populations. Such chimera states were discovered in 2002, but are not well understood theoretically. Here we obtain the first exact results about the stability, dynamics, and bifurcations of chimera states by analyzing a minimal model consisting of two interacting populations of oscillators. Along with a completely synchronous state, the system displays stable chimeras, breathing chimeras, and saddle-node, Hopf and homoclinic bifurcations of chimeras.
Many creatures sleep with only half their brain at a time rattenborg (). Such unihemispheric sleep was first reported in dolphins and other sea mammals, and has now been seen in birds and inferred in lizards mathews (). When brain waves are recorded, the awake side of the brain shows desynchronized electrical activity, corresponding to millions of neurons oscillating out of phase, whereas the sleeping side is highly synchronized.
From a physicist’s perspective, unihemispheric sleep suggests the following (admittedly, extremely idealized) problem: What’s the simplest system of two oscillator populations, loosely analogous to the two hemispheres, such that one synchronizes while the other does not?
Our work in this direction was motivated by a series of recent findings in nonlinear dynamics kuramoto2002 (); abrams (); shima (); omelchenko (); sethia (); PREs (). In 2002, Kuramoto and Battogtokh reported that arrays of nonlocally coupled oscillators could spontaneously split into synchronized and desynchronized sub-populations kuramoto2002 (). The existence of such “chimera states” came as a surprise, given that the oscillators were identical and symmetrically coupled. On a one-dimensional ring kuramoto2002 (); abrams () the chimera took the form of synchronized domain next to a desynchronized one. In two dimensions, it appeared as a strange new kind of spiral wave shima (), with phase-locked oscillators in its arms coexisting with phase-randomized oscillators in its core—a circumstance made possible only by the nonlocality of the coupling. These phenomena were unprecedented in studies of pattern formation hoyle () and synchronization syncrefs () in physics, chemistry, and biology, and remain poorly understood.
Previous mathematical studies of chimera states have assumed that they are statistically stationary kuramoto2002 (); shima (); abrams (); omelchenko (); sethia (). What has been lacking is an analysis of their dynamics, stability, and bifurcations.
In this Letter we obtain the first such results by considering the simplest model that supports chimera states: a pair of oscillator populations in which each oscillator is coupled equally to all the others in its group, and less strongly to those in the other group. For this model we solve for the stationary chimeras and delineate where they exist in parameter space. An unexpected finding is that chimeras need not be stationary. They can breathe. Then the phase coherence in the desynchronized population waxes and wanes, while the phase difference between the two populations begins to wobble.
The governing equations for the model are
where and is the number of oscillators in population . The oscillators are assumed identical, so the frequency and phase lag are the same for all of them. The strength of the coupling from oscillators in onto those in is given by . To facilitate comparison with earlier work, we suppose that , and , with . Thus, the coupling within a group is stronger than the coupling between groups. This corresponds to the assumption kuramoto2002 (); abrams (); shima () of a nonlocal coupling that decreases with distance. By rescaling time, we may set . It also proves useful to define the parameters and , because, as we’ll show, chimeras exist only if these quantities are small enough.
Simulations of Eq. (1) display two types of behavior. For many initial conditions, the system approaches the synchronized state where all ’s are equal. Otherwise it evolves to a chimera state (Fig. 1). The oscillators in group 1 are in sync; those in group 2 are not.
Figure 2 illustrates the dynamics of chimera states. We plot the phase coherence of the desynchronized population, as quantified by the order parameter , where the angle brackets denote an average over all oscillators in population . In Fig. 2(a) the order parameter remains constant, except for slight fluctuations due to finite-size effects. Thus, this chimera is stable and statistically stationary. However, if we increase (the coupling within a population) relative to (the coupling between populations), the stationary state can lose stability. Now the order parameter pulsates, and the chimera starts to breathe (Fig. 2(b)). The breathing cycle lengthens as we increase the disparity between the couplings (Fig. 2(c)). At a critical disparity, the breathing period becomes infinite. Beyond that, the chimera disappears and the synchronized state becomes a global attractor.
where is the probability density of oscillators in population , and is their velocity, given by
(Note that we dropped the superscripts on to ease the notation. Thus, means and means .) If we define a complex order parameter
then simplifies to
where the denotes complex conjugate.
Following Ott and Antonsen ott (), we now consider a special class of density functions that have the form of a Poisson kernel. The remarkable fact that Ott and Antonsen discovered is that such kernels satisfy the governing equations exactly, if a certain low-dimensional system of ODEs is satisfied. In other words, for this family of densities, the dynamics reduce from infinite dimensional to finite (and low) dimensional. (Numerical evidence suggests that all attractors lie in this family, but proving this remains an open problem.)
What’s special here is that we use the same function in all the Fourier harmonics, except that is raised to the power in the harmonic. Inserting this into the governing equations, one finds that this is an exact solution, as long as
Instead of infinitely many amplitude equations, we have just one. (It’s the same equation for all .)
by orthogonality. Thus the amplitude equations become
and similarly for , with 1’s and 2’s interchanged.
Rewrite the amplitude equations using polar coordinates and , defined by . (The negative sign is included in the definition of so that the Poisson kernel converges to , not , as from below. Thus can be interpreted as the “center” of the density , and measures how sharply peaked it is.) Then Eq. (9) becomes
and similarly for and , with 1’s and 2’s interchanged.
Now consider the case of chimera states, for which one population is in sync while the other is not. Taking to be the synchronized population, we set , corresponding to a -function for that group. Note that satisfies the governing equations for all time, since when . Hence the condition defines an invariant manifold, on which the dynamics reduce to
where we’ve defined and . The dynamics decouple, yielding a 2- system given by
This system has a trivial fixed point (perfectly synchronized state) which always exists. The non-trivial fixed points correspond to stationary chimera states, in which the local order parameters and remain constant, as does the phase difference , despite the fact that the individual microscopic oscillators in population continue to move in a desynchronized fashion.
Figure 3 plots typical phase portraits for (Solvable model for chimera states of coupled oscillators). Figure 3(a) shows a stable chimera state coexisting with the stable synchronized state; the basin boundary between them is defined by the stable manifold of a saddle chimera. As we increase the disparity between the couplings and , the chimera becomes less stable and eventually undergoes a supercritical Hopf bifurcation, creating a stable limit cycle (Fig. 3(b)), the counterpart of the breathing chimera of Fig. 2(b).
Additional phase plane analysis (not shown) reveals two other bifurcations. With further increases in , the limit cycle expands and approaches the saddle. Meanwhile, its period lengthens, which accounts for the behavior seen earlier in Fig. 2(c). At sufficiently large the cycle touches the saddle point and destroys itself in a homoclinic bifurcation. On the other hand, if is decreased from its value in Fig. 3(a), the stable and saddle chimeras in Fig. 3(a) approach each other, and eventually coalesce and annihilate in a saddle-node bifurcation.
Figure 4 summarizes the bifurcations and stability regions for the system. In the rest of the paper we outline the analysis leading to these results.
To calculate the fixed points for Eq. (Solvable model for chimera states of coupled oscillators), we set and (since group is desynchronized) and obtain . Substituting and into and solving for yields
at a fixed point. Then imposing and using the expression for above, we find
Equations (13) and (14) together parametrize all the fixed points. They define a surface, or equivalently, a two-parameter family. Sweeping and yields the corresponding and values. The resulting surface has two sheets that join along a fold; its projection onto the plane defines the curve of saddle-node bifurcations.
To calculate the saddle-node curve, we linearize (Solvable model for chimera states of coupled oscillators) about a fixed point, and set the determinant of the Jacobian to zero. This implies
which matches the numerical curve shown in Fig. 4.
To find the Hopf curve, we set the trace of the Jacobian to zero, which gives Repeating the procedure above leads to an exact parametric equation for the Hopf curve. Its leading order behavior is
Future work should investigate whether breathing chimeras exist for the one- and two-dimensional arrays of oscillators studied previously kuramoto2002 (); shima (); abrams (); omelchenko (); sethia (). Are the stability diagrams for such systems similar to Fig. 4? Do chimeras also exist if the oscillators are non-identical syncrefs (); ott (); barreto () or arranged in complex networks netrefs ()? It would also be worth looking for experimental examples of chimera states. Candidate systems include arrays of lasers laser () and chemical chemical () or electrochemical electrochemical () oscillators.
We thank E. Ott and T. M. Antonsen for sending us their preprint. Research supported by NSF grant DMS-0412757 to S.H.S. and a Mathematical Sciences Postdoctoral Research fellowship to D.M.A.
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