Spiral wave chimera states in large populations of coupled chemical oscillators

Letters https://doi.org/10.1038/s41567-017-0005-8 Spiral wave chimera states in large populations of coupled chemical oscillators Jan Frederik Totz *, Julian Rode1, Mark R. Tinsley2, Kenneth Showalter 1 The coexistence of coherent and incoherent dynamics in a population of identically coupled oscillators is known as a chimera state1,2. Discovered in 20023, this counterintuitive dynamical behaviour has inspired extensive theoretical and experimental activity4–15. The spiral wave chimera is a particularly remarkable chimera state, in which an ordered spiral wave rotates around a core consisting of asynchronous oscillators. Spiral wave chimeras were theoretically predicted in 200416 and numerically studied in a variety of systems17–23. Here, we report their experimental verification using large populations of nonlocally coupled Belousov–Zhabotinsky chemical oscillators10,18 in a two-dimensional array. We characterize previously unreported spatiotemporal dynamics, including erratic motion of the asynchronous spiral core, growth and splitting of the cores, as well as the transition from the chimera state to disordered behaviour. Spiral wave chimeras are likely to occur in other systems with long-range interactions, such as cortical tissues24, cilia carpets25, SQUID metamaterials26 and arrays of optomechanical oscillators9. Experiments were carried out with a network of up to 1,600 photosensitive chemical oscillators, arranged in a 40 × 40 grid, photochemically coupled by specific illumination of each oscillator. The discrete micro-oscillators are catalyst-loaded ion-exchange beads, placed in catalyst-free Belousov–Zhabotinsky (BZ) solution10,18. The experimental set-up (Fig. 1a) allowed the current state of each micro-oscillator to be monitored by using a camera to record its fluorescence intensity, which is linearly dependent on the concentration of the reduced form of the catalyst, Ru(dmbpy)23 + (see Methods and Supplementary Section 'Experimental Setup'). The light intensity projected onto each oscillator was independently controlled using a spatial light modulator (SLM). The initial conditions for the experiment were set by individually forcing each oscillator with a periodic illumination intensity to align all of the oscillator phases to a desired phase distribution containing a phase singularity17. Once the desired phase alignment was attained, light-mediated nonlocal coupling was initiated between the oscillators in the network based on their state according to equation (1): j+l I j , k = I0 + K k+ l ∑ ∑ m= j − l n = k− l e −κr g m , n(t −τ )−g j , k(t )   (1) The light intensity projected on oscillator (j, k), at the centre of a square region of side length 2l +  1, is Ij,k. It is linearly dependent on the difference between the grey values gj,k and gm,n of oscillator (j, k) and the other oscillators (m, n) in the square region at times t and (t −  τ), respectively (see Methods). This difference is weighted with a nonlocal coupling kernel that decays exponentially with * and Harald Engel 2 * 1 distance r = (m−j )2 + (n−k)2 between oscillators (j, k) and (m, n). Parameters κ and K determine the coupling range and coupling strength3, respectively, and I0 is the background illumination intensity. The time delay τ plays a role similar to the phase frustration parameter in the Kuramoto model2,7. A spiral wave chimera exhibiting the characteristic coexistence of coherent and incoherent oscillators is shown in Fig. 1, with an ordered spiral wave rotating around a core made up of asynchronous oscillators. Figure 1b shows a snapshot of the grey values and Fig. 1c shows the phase of each oscillator determined from the grey values. The periods of the incoherent oscillators in the core and the coherent oscillators in the spiral wave are plotted in Fig. 1d. For a time delay of τ = 2.0 s, the rotation period Tspiral of the spiral arm is larger than the spatially averaged period Tcore of the core oscillators. In contrast, at lower values of τ = 1.0 s (Fig. 2b), the rotation period Tspiral is smaller than Tcore. The space–time plot in Fig. 1e shows the spiral wave propagating out from the asynchronous core along the cross-section j = 20 in Fig. 1b during approximately five rotational periods of the spiral. The disordered core region exhibits a low degree of synchronization, as measured by the two-dimensional local Kuramoto order parameter Rj,k, defined as R j ,k = 1 (2δ + 1)2 j+δ k+ δ ∑ ∑ e iϕm, n (2) m= j − δ n = k− δ where φm,n represents the phases of oscillators (m, n) in a square region of side length 2δ + 1 with oscillator (j, k) in the centre. We define a region of oscillators as asynchronous or incoherent in terms of the local order parameter, with Rj,k < 0.4; the value of Rj,k in these regions, however, is typically lower. Over the course of the experiment, the core expands at ~300 s after its formation, doubling its size from ~20 to ~40 oscillators, and drifts until it collides with the upper boundary of the oscillator array and disappears at 1,040 s, or approximately 30 rotational periods (Supplementary Video 2). Figure 2a–c shows the behaviour of a spiral wave chimera with a smaller value of time delay, τ = 1.0 s. The erratic motion of the core can be characterized from the grey value data by calculating the trajectory of the core centre, defined as the location where the local order parameter Rj,k reaches its weighted minimum value within the core (see Methods). A typical trajectory is shown in Fig. 2c. This erratic motion is in contrast to rigid or compound rotation known for spiral waves in reaction–diffusion systems27. With increasing delay time, τ = 5.0 s, the core of the spiral wave chimera increases in size and eventually becomes unstable, leading to splitting of the core of asynchronous oscillators. An experimental example of core splitting in a spiral wave chimera is displayed in Institut für Theoretische Physik, EW 7-1, TU Berlin, Hardenbergstr. 36, 10623 Berlin, Germany. 2C. Eugene Bennett Department of Chemistry, West Virginia University, Morgantown, WV 26506-6045, USA. *e-mail: ; ; 1 282 Nature Physics | VOL 14 | MARCH 2018 | 282–285 | www.nature.com/naturephysics © 2018 Macmillan Publishers Limited, part of Springer Nature. All rights reserved. Letters NaTUre PHySIcS a b Neutral filter 1.0 40 30 Fluorescence filter k Projector Reactor with chemical oscillators Camera 0.5 20 10 Computer 2л 40 d л 20 10 1 e 30 k k 30 30 40 25 20 10 1 10 20 30 40 0 1 1 10 20 j 30 40 0.0 1.0 40 30 k c 1 0.5 20 10 1 10 20 j 30 40 20 1 500 575 650 0.0 t (s) j Fig. 1 | Experimental set-up and spiral wave chimera. a, The camera records fluorescent light (λ >500 nm) emitted by the reduced form of the BZ catalyst (Ru(dmbpy)23 +). The grey values corresponding to the concentration of the oxidized catalyst (Ru(dmbpy)33+) are used to determine the illumination intensity Ij,k of oscillator (...truncated)