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Patterns on a Triangular Lattice:  On a triangular lattice, the optimallocal pattern of nearest-neighbor oscillators locking into a p phase difference is incompatible with thegeometry.   Results from ournumerical simulations show that the frequencies synchronize beyond a criticalcoupling strength (for a finite size system), however the phase pattern remainsdisordered with domains of different helicities frozen in (Fig. 1). Experimentsperformed in the Fraden group show clear patterns of domains with differenthelicities, and the structures are qualitatively similar to our numerical results.  The experimental system, however, has acoupling that is quite different from the sine function in the Kuramotomodel.  Specifically, the effectivecoupling, measured experimentally, is always positive and can be modeled as aquadratic function cutoff at 2p.  We are now extending ournumerical studies to analyze domain formation in oscillators that are locallycoupled through this purely positive interaction.

Lattice Models of Reaction-Diffusion Systems: Theoretical studies of chemical oscillations broadlyfall into two categories: (1) a rate equation approach in which the reactantsare assumed to be well mixed, and (2) a partial differential equation approachin which spatial dependence is taken into account by adding diffusion terms tothe rate equations. These two approaches qualitatively explain experimentalmost of the phenomena observed in the microfluidic droplets.   It is, however, known fromtheoretical studies of individual level models of reaction diffusion systems(Butler and Goldenfeld, Phys. Rev. E 80, 030902 (R) (2009)) in the context ofpopulation dynamics, that the intrinsic noise in these systems can change thephase diagram, and enhance pattern formation.  The parameter that controls the fluctuations is the volumeof a well-mixed patch.  This volumecan be estimated as, where D is a typical diffusion constant, R is a typicalreaction rate and d is the spatial dimension of the system.  This parameter can be controlled in theBZ microfluidic array tuning the concentration of the reactants.  We have initiated a study of aone-dimensional lattice in which each site has chemical reactions correspondingto a single Brusselator, and the activator and inhibitor can hop from one latticesite to another with specified diffusion rates.

Results from a 1D lattice of stochastic Brusselators: Fig. 3 shows the mean-field phase diagram of aone-dimensional Brusselator model with the inhibitory species represented by y and the activator species represented by x.  Therate R₀ in Fig. 3 represents the creation rate of theactivator.  We studied thestochastic model over a range of parameters.  Fig. 4 shows space-time plots at  R₀ = 3000, and D_y/D_xvarying from 1 to 6.   There is a clear signature of emergenceof Turing patterns in the mean-field turing-oscillatory regime.  Even at D_y/D_x =1, the space-time plot shows clear deviations from a pure oscillatorybehavior.  We are in the process ofcarefully analyzing the pattern formation in the one-dimensional stochasticmodel.  In addition, we haveinitiated studies of an inhomogeneous lattice that is meant to model the BZmicrofluidic array.  In this lattersystem, the diffusion constants are different in different parts of the latticethat represent the oil and water regions. Preliminary results show an even richer phase diagram of the stochasticmodel in this spatially structured medium.

		Fig 4: Space-time plots of the stochastic Brusselator model on one-dimensional lattice.  The horizontal axis represents the location, i, on the lattice, and the vertical axis represents time.  The left panels shows signatures of intermittency while the right panel shows a clear Turing pattern.