Abstract
Networks of nonlocally coupled phase oscillators^{1} can support chimera states in which identical oscillators evolve into distinct groups that exhibit coexisting synchronous and incoherent behaviours despite homogeneous coupling^{2,3,4,5,6}. Similar nonlocal coupling topologies implemented in networks of chaotic iterated maps also yield dynamical states exhibiting coexisting spatial domains of coherence and incoherence^{7,8}. In these discretetime systems, the phase is not a continuous variable, so these states are generalized chimeras with respect to a broader notion of incoherence. Chimeras continue to be the subject of intense theoretical investigation, but have yet to be realized experimentally^{6,9,10,11,12,13,14,15,16}. Here we show that these chimeras can be realized in experiments using a liquidcrystal spatial light modulator to achieve optical nonlinearity in a spatially extended iterated map system. We study the coherence–incoherence transition that gives rise to these chimera states through experiment, theory and simulation.
Main
Our system is an experimental realization of a coupledmap lattice (CML), a class of systems that has received sustained theoretical interest for the past three decades. Although the dynamics and statistical physics of CML systems have been theoretically explored, very few (if any) experimental realizations exist^{17,18,19,20,21,22}. In our experiments, we create CML dynamics by using a liquidcrystal spatial light modulator (SLM) to control the polarization properties of an optical wavefront. We may electronically introduce any desired coupling topology including nearest neighbour, nonlocal, small world and scale free. In this work, we impose periodic boundary conditions for both onedimensional (1D) and 2D nonlocally coupled maps. Thus, we have developed a powerful experimental technique to observe the parallel evolution of the dynamics of arrays of coupled maps numbering up to thousands or more depending on the goals of the experiment.
Figure 1 shows the experimental setup of the optical CML. Polarization optics create a nonlinear relationship between the spatially dependent phase shift applied by the SLM and the intensity of the light falling on the camera: . The operation of the experimental apparatus is described in the Methods. Both the SLM and the camera frames are partitioned into an M×M array of square regions. These regions correspond to nodes in the network of coupled maps. Time evolution of the network is achieved by iteratively updating the phase applied by each region of the SLM in a way that depends on the intensity measured by the camera.
We present results for two different coupling schemes shown schematically in Fig. 1b,c. In the 1D configuration, the elements in the array are arranged as a ring with periodic boundary conditions. The SLM is treated as a 1D lattice with the elements coupled in a rasterordered arrangement. If ϕ_{i}^{n} is the phase of the ith element in the ring at the nth iteration, and I(ϕ_{i}^{n}) is the intensity measured by the corresponding region in the camera, then the phase is updated according to
where i = 0,...,N−1 in the 1D ring configuration and the index i is periodic modulo N. Thus, each element is coupled diffusively to all of the elements within a distance R on either side, and describes the strength of the coupling. The parameter a controls the temporal dynamics of an isolated map. We choose a = 0.85 such that the local map ϕ^{n + 1} = f(ϕ^{n})≡2π a I(ϕ^{n}) is chaotic with a Lyapunov exponent λ≈0.58.
We also examine a 2D coupling scheme. In this configuration, each element is coupled to its neighbours within a square region, and the boundary conditions are periodic in both dimensions.
Figure 2a,b characterizes the dependence of the dynamics of the 1D and the 2D systems on coupling strength and range by plotting a measure of temporal entropy (see Methods). The colour maps were obtained through numerical simulation of equations (1) and (2). A period2 profile will have an entropy of 1 bit, and a chaotic profile will have an entropy close to 9 bits, because 512 bins were used. We see spatial profiles that are periodic in time, as well as profiles that are chaotic in time. There is a rich variety of both coherent and incoherent spatial structures. The thin vertical regions marked K = 1,2,3 in Fig. 2a indicate conditions for which the dynamical state is periodic in time and forms a spatially periodic standingwave pattern with a spatial wavenumber of K. Within each of these regions, there is a transition from spatial coherence to incoherence with decreasing coupling strength , as predicted in refs 7, 8. Equations (1) and (2) admit a globally synchronous solution. The hatched region in Fig. 2a,b indicates the region in parameter space for which global synchrony is stable, which is identical between the 1D and 2D systems.
A comparison of the 1D and 2D systems reveals that these systems show equivalent behaviour within a limited region in parameter space. In the 2D system there is a prominent region marked K = 1, which contains profiles that are homogeneous in one direction. Every solution of the 1D problem corresponds to a solution of the 2D problem that is uniform in one direction, that is, ϕ_{i,j} = ϕ_{i}. Although K = 2,3... profiles also exist in principle, only the K = 1 profile is observed in simulation and experiment in the 2D system within the evolution time we consider. Simulations also reveal two smaller regions that contain complex profiles with spatial structures that cannot be realized in one dimension.
Figure 2c shows experimental and numerical realizations of three profiles from the K = 1 region. The sequence A, B, C shows a progression from spatial coherence to spatial incoherence. All profiles have a temporal period of 2. At a coupling radius r≡R/N = 70/256≈0.27 and = 0.75 (A), the profile has a smooth spatial variation. At r = 105/256≈0.41 and = 0.44 (B), the profile shows two large domains of coherent, synchronized behaviour separated by narrow but finite regions of incoherent behaviour highlighted in yellow. Numerical evidence indicates that the width of these regions as a fraction of the system size remains constant as N increases, and hence this is not a boundary effect (see Supplementary Information). Although the entire profile is periodic in time, the dynamics of the coherent and incoherent regions are qualitatively different in their degree of spatial coherence: within the incoherent intervals there are numerous admissible combinations of upper and lower states whose multiplicity scales exponentially with the system size^{7}. This mixture of qualitatively different behaviours is analogous to the chimera states discussed in refs 2, 3, 4, 9, 10, 11, where an array of identical oscillators splits into two domains: one coherent and phase locked, the other incoherent and desynchronized. Unlike the chimeras in continuoustime phase oscillators^{2,3}, the generalized chimeras here and in refs 7, 8 have two coherent parts (with high and low intensity, respectively) and two incoherent parts, as a result of their mechanism of nascence from the completely coherent K = 1spatial profile. Finally, at r = 115/256≈0.45 and = 0.375 (C), the profile is completely incoherent. The same scenario occurs in the K = 2 and K = 3 regions and can be interpreted through a spatial rescaling described below.
Figure 2d shows snapshots of experimental and numerical realizations of the 2D system in a 32×32 lattice, obtained with similar parameter values to those used in Fig. 2b. All of these realizations are periodic in time with period 2, except (E), which has a period of 4 in the experimental realization shown. As for the 1D system, the 2D system undergoes a coherence–incoherence transition as is decreased. At r = 9/32≈0.28 and = 0.75, the system exhibits a smooth profile (D). For r = 13/32≈0.41 and = 0.44, there is a chimeralike coexistence of coherence and incoherence (E). Finally, at r = 14/32≈0.44 and = 0.375, the system is fully incoherent (F).
The transition from coherence to incoherence can be explained analytically. We derive the critical coupling strength _{c} at which the smooth profile breaks up giving rise to the incoherent regions (B) of Fig. 2c. Considering the spatially continuous version of equation (1) and solutions with wavenumber K = 1 and period2 dynamics in time, the system dynamics are given by alternating profiles ϕ^{0}(x) and ϕ^{1}(x) for even and odd time steps, respectively. By evaluating the spatial derivative of this equation with the profiles ϕ^{0}(x) and ϕ^{1}(x) at the positions where the smooth profiles break up, we obtain two equations that, when multiplied, yield the condition (see Supplementary Information)
Taking into account the local dynamics, we define the following function as a deviation from equation (3)
where we can use numerically obtained profiles ϕ^{0}(x) and ϕ^{1}(x)for given coupling parameters r and .
The critical coupling strength _{c} at which coherence is broken can be determined by applying the condition G(x) = 0 at the crossing points x_{1} and x_{2} where ϕ^{0}(x) = ϕ^{1}(x)≡ϕ^{*} as shown in Fig. 3b. Applying this condition to equation (1) yields
The 1D local map f has three fixed points for a = 0.85: ϕ_{0}^{*} = 0, ϕ_{1}^{*}≈0.79 and ϕ_{2}^{*}≈4.13. The fixed point ϕ_{2}^{*} plays the role of a saddle (unstable fixed point), which separates the attractor basins of the two stable fixed points of the twiceiterated local map, which correspond to the period2 solution of the local map (see Supplementary Fig. S5). Approximating ϕ^{*} by ϕ_{2}^{*} in equation (4) we obtain _{c}≈0.55, which is close to the numerical result _{c} = 0.54 shown in Fig. 3a,b.
One can further obtain an approximate full analytical solution (for a > 0.6) by Taylor expanding the local map f about π using cosϕ^{*}≈−1 + 1/2ψ^{2}, where ψ = ϕ^{*}−π. This yields an equation for the fixed point π + ψ≈π a(2−1/2ψ^{2}), and we find .
This gives ψ≈0.96, ϕ^{*}≈4.10, and through equation (4) _{c} = 0.54. Figure 3a shows the function G for _{c} = 0.54 (solid line), = 0.675 (dashed) and = 0.475 (dotted). Figure 3b depicts the corresponding snapshots ϕ_{i}^{0} and ϕ_{i}^{1} for _{c} = 0.54. The profiles ϕ_{i}^{0} and ϕ_{i}^{1} cross at ϕ^{*}≈4.10. The experimental results are added as light red and grey dots in Fig. 3b.
In addition we analytically demonstrate the invariance of the coherent profiles of different wavenumbers for equation (1) by a scaling relation (Methods). Figure 3c–e shows the scaling of profiles with K = 1,2,3 obtained from the experimental realization with N = 256 for coupling ranges R_{3} = 22 (Fig. 3c), R_{2} = 33 (Fig. 3d) and R_{1} = 66 (Fig. 3e) as black dots. The numerical profile from equation (1) for K = 1 and the rescaled profiles are depicted in red in Fig. 3c,d,e, respectively.
In summary we have constructed a versatile experimental system to explore the spatiotemporal dynamics of arbitrary networks of coupled maps. The nodes in the network are nonlocally and homogeneously coupled, and we observe the formation of coexisting spatially coherent and incoherent domains as the coupling parameters are varied. This behaviour is observed in both 1D and 2D systems.
Methods
Experimental apparatus.
Figure 1 shows the experimental setup of the optical CML. The SLM (Boulder Nonlinear Systems P5121550), with an active area of 7.68×7.68 mm^{2} and 512×512 pixels, is illuminated by collimated 1,550 nm light from a fibrecoupled superluminescent diode. The light passes through a polarizing beam splitter and a quarterwave plate oriented at a 45° angle to the direction of linear polarization of the incident light, is reflected by the SLM with a relative phase shift between the fast and slow axes, and passes again through the quarterwave plate and the beam splitter. We use the computercontrolled SLM to apply an arbitrary spatially dependent phase modulation to the optical wavefront using a birefringent liquid crystal sandwiched between an array of reflective electrodes and a transparent cover glass. Each of the electrodes acts as an independent pixel that can impose an arbitrary phase shift from 0 to 2π between the two polarization components of the incoming light by applying an electric field to reorient the liquid crystals^{23}. The polarization optics create a nonlinear relationship between the phase shift applied by the SLM and the intensity of light falling on the camera (Goodrich SU320KTSW1.7RT/RS170), with an active area of 8×6.4 mm^{2} and 320×256 pixels. A selected square area of 256×256 pixels was used. The phase shift ϕ_{i,j} and intensity I for a given pixel (i,j) are related by
The intensity has been normalized to be between 0 and 1. A lens is used to project an image of the SLM onto an infrared camera, which records the 2D intensity pattern I(ϕ_{i,j}). We construct a network of iterated maps by using the computer to communicate between the camera and the SLM. Both the SLM and the camera screens are partitioned into an M×M array of square regions. These regions correspond to nodes in the network of coupled maps. Feedback is achieved by iteratively updating the phase applied by each region on the SLM in a way that depends on the intensity measured by the camera.
Entropy calculation.
In Fig. 2a,b, the colour of each point corresponds to a single numerical simulation. Lattice sizes of 256 for the 1D case and 128×128 for the 2D case were used. Initially, the phase of each node is random and uniformly distributed between 0 and 2π. In each simulation we discard 50,000 transient iterations. Using the next 5,000 iterations, we construct one histogram for each of the N nodes in the network, binning the values the phase achieves in these iterations. For each lattice site, 512 bins were used. Thus, we estimate p_{i}^{n}, the fraction of the time that node i will spend in the nth bin. From these statistics we obtain the entropy, which is then averaged over the N nodes in the system
We note that for this system the temporal behaviour of the whole network is either chaotic or periodic, and the spatial average of the entropy will not differ substantially from the entropy of any given node. We also note that this calculation characterizes only the temporal behaviour of the system, and does not distinguish between coherence and incoherence.
Scaling of coherent profiles.
As a second analytic finding we demonstrate the scaling relation for coherent profiles of different wavenumbers. Considering the spatially continuous version of the 1D system in equation (1) (see Supplementary Information),
where the coupling strength is fixed, and f(ϕ) = 2π a I(ϕ), we assume that there exist two solutions ϕ_{K′}(x) and ϕ_{K}(x) with spatial periods K′ and K, respectively. It follows that the dynamics of ϕ_{K′}(x)is identical to the dynamics of at a rescaled spatial position for a correspondingly rescaled coupling radius This demonstrates the invariance of the solutions ϕ_{K} for appropriate rescaling of the coupling range (see Supplementary Information).
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Acknowledgements
We thank X. Li for her help in developing the early stages of the experiment. A.M.H., T.E.M. and R.R. acknowledge support by DOD MURI Grant No. ONR N000140710734. P.H. and I.O. acknowledge support by the BMBF under grant no. 01GQ1001B (Förderkennzeichen). E.S. acknowledges support by Deutsche Forschungsgemeinschaft in the framework of SFB910.
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A.M.H. developed the experimental system under the guidance of T.E.M. and R.R. and performed measurements and numerical simulations to characterize the parameter space of the 1D and 2D systems. P.H., I.O. and E.S. conducted theoretical and numerical studies of the scaling relation and critical coupling strength. All authors discussed the results and wrote the manuscript.
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Hagerstrom, A., Murphy, T., Roy, R. et al. Experimental observation of chimeras in coupledmap lattices. Nature Phys 8, 658–661 (2012). https://doi.org/10.1038/nphys2372
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DOI: https://doi.org/10.1038/nphys2372
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