Controlling chimeras

Coupled phase oscillators model a variety of dynamical phenomena in nature and technological applications. Non-local coupling gives rise to chimera states which are characterized by a distinct part of phase-synchronized oscillators while the remaining ones move incoherently. Here, we apply the idea of control to chimera states: using gradient dynamics to exploit drift of a chimera, it will attain any desired target position. Through control, chimera states become functionally relevant; for example, the controlled position of localized synchrony may encode information and perform computations. Since functional aspects are crucial in (neuro-)biology and technology, the localized synchronization of a chimera state becomes accessible to develop novel applications. Based on gradient dynamics, our control strategy applies to any suitable observable and can be generalized to arbitrary dimensions. Thus, the applicability of chimera control goes beyond chimera states in non-locally coupled systems.

Collective behavior emerges in a broad range of oscillatory systems in nature and technological applications.Examples include flashing fireflies, superconducting Josephson junctions, oscillations in neural circuits and chemical reactions, and many others [1].Phase coupled oscillators serve as paradigmatic models to study the dynamics of such systems [2].Remarkably, localized synchronization-in contrast to global synchrony-may arise in non-locally coupled systems where the coupling depends on the spatial distance between two oscillators.Dynamical states consisting of locally phase-coherent and incoherent parts are called chimera states [3,4], alluding to the fire-breathing Greek mythological creature composed of incongruous parts from different animals.Chimera states are relevant in a range of systems; chimeras have been observed experimentally in mechanical, (electro-)chemical, and laser systems [5,6], and related localized activity may play a role in neural dynamics [7][8][9][10].By definition, local synchrony is tied to a spatial position that may directly relate to function: in a neural network, for example, different neurons encode different information [11].In non-locally coupled phase oscillator rings, the spatial position of partial synchrony not only depends strongly on the initial conditions [3], but it also is subject to pseudo-random fluctuations [12].These fluctuations are particularly strong for persistent chimeras in networks of just few oscillators [13] as commonly found in experimental setups.This naturally leads to the question whether it is possible to pin the coherent part of a chimera state to a (desired) spatial position.
In this article, we derive a control scheme to modulate the spatial position of a chimera state dynamically.In contrast to other recent applications of control to chimera states [13], the goal here is to control the chimera state itself by imposing a target spatial location.Our control mechanism for a ring of non-locally coupled phase oscillators exploits drift induced by breaking the symmetry of the coupling [14] to move the chimera along the ring.This control approach may not only serve to control a chimera's spatial position in an application, but it may also elucidate how position is maintained in systems where spatial localization of synchrony plays a crucial functional role and that are subject to noise.Moreover, applications of control to dynamical states have led to intriguing applications in their own right [15].In terms of the analogy to the Greek mythological creature: what would you be able to do if you could control a fire-breathing chimera?
Chimeras in Non-Locally Coupled Rings-Rings of non-locally coupled phase oscillators provide a well studied model in which chimera states may occur [4].Let S := R/Z be the unit interval with endpoints identified and let T := R/2πZ denote the unit circle.Let d be a distance function on S, h : R → R be a positive function, and α ∈ T, ω ∈ R be parameters.The dynamics of the oscillator at position x ∈ S on the ring is given by The coupling kernel h determines the interaction strength between two oscillators depending on their mutual distance.The system evolves on the torus S × T where x ∈ S is the position of an oscillator on the ring and ϕ(x, t) ∈ T its phase at time t.
Chimera states are characterized by a region of local phase coherence while the rest of the oscillators rotate incoherently.Let φ ∈ Φ := {φ : S → T} denote a configuration of phases on the ring.The local order parameter is an observable which encodes the local level of synchrony of φ at x ∈ S. That is, its absolute value R(x, φ) = |Z(x, φ)| is close to zero if the oscillators are locally spread out and attains its maximum if the phases are phase synchronized close to x.A chimera state is a solution ϕ(x, t) of (1) which consists of locally synchronized and locally incoherent parts.The value of the local order parameter yields local properties of a chimera.The local order parameter obtains its maximum at the center of the phase synchronized region and its minimum at the center of the incoherent region; cf. Figure 1 for a finite dimensional approximation.
Chimera Control-Is it possible to dynamically move a state to a desired position by exploiting drift properties?Before considering chimera states, we concentrate on general solutions moving along the spatial direction.A solution of (1) may be seen as a one-parameter family of functions ϕ t ∈ Φ which assign a phase to each spatial position.Let Q : S × Φ → R n be differentiable.
Think of Q as an observable of the system which depends on the spatial position.A solution ϕ t of (1) with initial condition ϕ 0 ∈ Φ is called Q-traveling along S if there are suitably smooth functions y(t) and q : S → R n such that Q(x, ϕ t ) = q(x − y(t)) for all t; in particular, a solution is Hence, the temporal evolution of a Q-traveling solutions in terms of the observable Q is a shift along S.
Our control scheme is based on gradient dynamics and applies to general observables Q.Let ∂ z f (z)| z 0 denote the partial derivative of a function f with respect to z at z 0 , let f ′ denote its total derivative, and ż the temporal derivative of z(t).Let ϕ t be a Q-traveling solution with q(x) and y(t) such that Q(x, ϕ t ) = q(x − y(t)).Fix a target x 0 ∈ S and assume that q is differentiable with all critical points being extrema.The function q(x 0 − y) is maximized in y if y is subject to the gradient dynamics ẏ = γ∂ y q(x 0 −y) for γ > 0 (assuming that the initial condition is not a local minimum).Note that then the function Q(x, ϕ t ) will attain a maximum at x = x 0 in the limit of t → ∞.
Given a suitable family of coupling kernels, there exist solutions which maximize an observable Q at a given point.Assume that for a given observable Q there is a family h a of coupling kernels, indexed by a ∈ A, and an invertible map ν : A → R such that ϕ t is a Q-traveling solution at constant speed v = ν(a) of (1) with coupling kernel h a .In other words, we assume that a coupling kernel h a yields a Q-traveling solution moving at constant speed along S, i.e., Q(x, ϕ t ) = q(x − ν(a)t).The map ν now allows to use a as a control parameter.Equation ( 3) is a dynamical equation for the speed to have Q obtain a maximum at a given point x 0 ∈ S for large times.Note that ẏ = v(t) and thus (3) yields a direct relationship between the traveling solution and the parameter a.More precisely, choosing a time dependent control parameter a according to (4) yields a traveling solution whose dynamics maximize the observable Q at x 0 .
To control chimeras we apply this general control scheme to the absolute value R of the local order parameter.Since it encodes the local level of synchrony, dynamics that maximize the local order parameter through R-traveling chimera solutions yield a chimera moving to a specified target position.Note that R(x, ϕ t ) = r(x) of a chimera state ϕ t is stationary [4] so it is R-traveling at constant speed zero.Here we further assume that there is a family of coupling kernels h a that lead to R-traveling solutions at nonzero speed ν(a).The control parameter dynamics (4) for the observable R are Hence, choosing a time dependent control parameter a according to ( 5) is equivalent to gradient dynamics to maximize the local order parameter at x 0 .
Implementation in Finite Dimensional Rings-We implement chimera control in an approximation of the continuous equations (1) by a system of N phase oscillators.Suppose that ι : {1, . . ., N} → S, ι(k) = k/N assigns a position on the ring S to the kth oscillator.The evolution of each oscillator is for k = 1, . . ., N. Here, d(x, y) = x − y + 1 2 mod 1 − 1 2 is a signed distance function on S. The local order parameter of the discretized system is defined for ϕ = (ϕ 1 , . . ., ϕ N ) as and its absolute value R d (x, ϕ) encodes the local level of synchrony; cf. Figure 1.
To implement the chimera control scheme (4), the assumption of a monotonic relationship ν between a system parameter and the chimera's drift speed has to be satisfied.To this end, we employ the recent observation that breaking the symmetry of the coupling kernel results in the drift of the chimera [14].Here we consider a family of exponential coupling kernels for a ∈ (−1, 1), where a determines the symmetry of the coupling kernel.There is indeed a monotonic relationship ν(a), independent of the system's dimension (not shown).For non-zero values of a the resulting drifting chimeras are in good approximation R-traveling with constant speed.However, these states seem to be transient and break down quickly in particular for larger asymmetry (|a| 0.015).We discuss the implications of this breakdown on the control below but refer to a forthcoming article [14] for more details on drifting chimera states in systems with asymmetric coupling kernels.
The relationship between asymmetry parameter a and the drift speed now allows for a straightforward implementation of the control scheme.The control rule (5) acts as feedback control through the asymmetry parameter.If the chimera is off target, the nonzero asymmetry yields a drift of the chimera towards the target according to the derivative of the local order parameter at the target position.Once the target is approached, the control subsequently reduces the asymmetry and acts a corrective term keeping the randomly drifting chimera [12] on target.In the finite ring define a discrete derivative at x 0 ∈ S for a given δ ∈ (0, 0.5) by For small δ we have Since traveling chimeras are robust only for sufficiently small asymmetry of the coupling kernel, we employ a sigmoidal function to ensure an upper bound for the asymmetry.Let a max > 0 be a suitable bound for the asymmetry parameter, λ(x) = 2(1 + exp(−x)) −1 − 1 and K > 0 be a constant.Given a target position x 0 ∈ S, an approximation of (5) for control is These dynamics will maximize the local order parameter at x 0 .In other words, a chimera ϕ(t) will move along the ring until its synchronized part is centered at x 0 .
Solving the dynamical equations subject to control numerically shows that the chimera adjusts to the imposed target position.Figure 2 shows a simulation for N = 256 phase oscillators with K = 100, and a time dependent target position x 0 (t).The asymmetry parameter updates every ∆t = 1 time units according to (10).The chimera tracks the changes of the target position and adjusts to match a new control target.
Successful Control Despite Breakdown-A chimera moving along S will eventually break down if the asymmetry of the coupling kernel is too large.The system then converges to either the fully phase synchronized state ϕ 1 = • • • = ϕ N or the "splay state" with uniformly distributed phases.At the same time, a monotonic relationship ν(a) between the asymmetry parameter and the chimera's drift speed implies that the larger asymmetry, the faster the chimera will move along the ring.Note that the absolute value of the asymmetry parameter in a system subject to chimera control is only large until the target position is reached.Thus, as long as a chimera attains its target faster than it breaks down, control is successful.Specifically, let d max < ∞ be the distance between antipodal points on S. If chimeras drift reliably for a distance d max then control is always successful since any target position can be reached without a breakdown along the way.
To determine whether control is applicable despite the transient nature of traveling chimeras, we calculated the average distance it travels before breaking down.Note that the quantity I(t) = 1 0 R(y, ϕ t )dy is conserved for a R-traveling solution ϕ t .This means that the corresponding quantity I d (t) = 1 N N k=1 R(ι(k), ϕ(t)) for the discretized dynamics fluctuates around a constant value.To calculate the revolutions along the spatial dimension S before breakdown, we first sampled I d (t) during a chimera transient without asymmetry to determine its mean I d and standard deviation σ.After breaking the symmetry of the coupling the chimera is said to lose shape at the smallest t > 0 with I d (t) − I d > 2.5σ.This yields a rather conservative criterion in good agreement with individual trajectories; it is typically triggered before the transient drifting passes and the trajectory enters the immediate vicinity of the asymptotic state (not shown).For a traveling chimera at asymmetry a loosing shape at t 0 , the revolutions until breakdown (RUB) are given by RUB(a) = t 0 ν(a).
Remarkably, even for large values of the asymmetry parameter, our chimera control scheme remains viable.With the unit distance defined above, control is successful if a chimera reliably travels a distance d max = 1/2.Figure 3 shows that the RUBs typically exceed d max even for large values of control parameters.When large asymmetry is allowed, a chimera quickly reaches the target position and symmetric coupling is restored.Thus, the control scheme is robust even for large asymmetry despite a potential chimera breakdown.
Discussion-The chimera control presented here allows the dynamical modulation of the spatial position of a chimera state in real time.Such control is relevant for implementation in experimental setups.If the coupling is computer mediated [6] then an implementation is straightforward.
However, the applicability of our control goes beyond such computer dependent experiments: the coupling in ( 8) is motivated by coupling through a common external medium [16] subjected to drift as we detail elsewhere [14].In such a setup, control can be realized by modulating the drift speed in the coupling medium.Hence, we anticipate our control strategy to find direct application in experimental setups.
Effectively, the control can be seen as a coupling of the dynamical equations to a function of the local order parameter.In contrast to systems with symmetric order parameter-dependent interaction [17], in chimera control the order parameter induces a time-dependent asymmetry (5) to the nonlocal coupling to realize directed motion [14].As a result, the chimera drifts along a subspace defined by the symmetry of the uncontrolled system to achieve the target position.Our control is noninvasive in the sense that the control signal vanishes on average upon attaining the target position; cf.Equation (2).Note that our chimera control is not limited to the control of the spatial position of a chimera.Control may be applied if for a suitable observable there is a relationship between a control parameter and directed motion of a solution.Thus, we anticipate that a similar approach extends for example to control of chimeras in higher dimensions.
In summary, chimera control is a robust control scheme to control the spatial position of a chimera state.Remarkably, the control remains effective even if chimeras moving along the ring do not persist.It is worth noting that the breakdown is different than the spontaneous breakdown observed for chimeras in finite oscillator systems with symmetric coupling [18] because of its transient behavior.At the same time, gradient dynamics is just one approach to maximize an objective function.Here it serves as a proof of principle to show that chimera states themselves can be controlled.Applying the presented control scheme to experimental setups and studying its relevance in biological settings provides exciting directions for future research.

Figure 1 .
Figure1.Chimera state for a ring of N = 256 oscillators with exponential coupling kernel h 0 as defined in(8).The upper panel displays the oscillator phase φ(x) on the circle S, and the lower panel the magnitude of the local order parameter, |Z d (x, φ)|, defined in(7).The maximum indicates the center of the synchronized region, the minimum the position of the incoherent part.

Figure 2 .
Figure 2. The position of the chimera adjusts to the imposed target for the control scheme applied to N = 256 oscillators.The top panel displays the phase evolution of the chimera, with the gray shading indicating individual oscillator phases in the co-moving frame defined by the synchronized region with maximal order parameter.The black line shows the desired target position.The bottom panel displays the time evolution of the asymmetry parameter according to(10).The asymmetry parameter |a| is bounded by a max = 0.015 and a tends to stay near zero once the target position is reached.

Figure 3 .
Figure 3. Chimera control is successful even for large asymmetry values a. Revolutions until breakdown(RUB) quantify how many times the chimera travels around the circle S before it breaks down; errorbars depict mean and standard deviation across 10 runs.For every value of a, the chimera typically travels further than the minimal distance of 1 2 (dashed line) required for successful control.