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Pattern formation and spatial chaos in spatially discrete evolution equations. (English) Zbl 0883.58020

Authors’ abstract: “We consider an array of scalar nonlinear dynamical systems \(\dot u= -f(u)\), arranged on the sites of a spatial lattice, for example on the integer lattice \(\mathbb{Z}^2\) in the plane \(\mathbb{R}^2\). We impose a coupling between nearest neighbors, and also between next-nearest neighbors, in the form of discrete Laplacians with \(+\)- and \(\times\)-shaped stencils. These couplings can be of any strength, and of either sign (positive or negative), and the resulting infinite systems of ODE’s need not be near a PDE continuum limit.
We study stable equilibria for such systems, from the point of view of pattern formation and spatial chaos, where these terms mean that the spatial entropy of the set of stable equilibria is zero, respectively, positive. In particular, for an idealized class of nonlinearities \(f\) corresponding to a “double obstacle” at \(u=\pm 1\) with \(f(u)= \gamma u\) in between, it is natural to consider “mosaic solutions”, namely equilibria which assume only the values \(u_{i,j} \in \{-1,0,1\}\) at each \((i,j)\in \mathbb{Z}^2\). This in turn leads to explicit combinatorial criteria for the existence and stability of these equilibria. Rigorous upper and lower bounds on the spatial entropy of such stable mosaic solutions are obtained for a wide range of the coupling coefficients.
The systems we study here are perhaps the simplest nontrivial systems involving both local dynamics and near-neighbor coupling in a higher-dimensional lattice. Nevertheless, they exhibit a strikingly rich array of phenomena. They serve as a paradigm for more complex lattice systems arising in various areas of science, in particular material science, image processing, and biology”.

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
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