Mosaic solutions and entropy for spatially discrete Cahn-Hilliard equations

In this article, we present a lattice differential equation model for a class of neural networks. A subset of the equilibrium solutions called mosaic equilibrium solutions is defined. Existence and stability theorems are proved for mosaic equilibrium solutions. Regions of stability are defined and spatial entropy calculations, as a measure of the complexity of the system, are presented that give insights into the effects of spatial coupling.

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“…jL (1) ( + 1)jj (1) +1 j) B = ?jL 2 ( )j(jL (2) ( ? 1)jj (2) ?1 j + jL (2) ( + 1)jj (2) +1 j) C = (jL (1) ( )j ? 1)( (1) + (j(L (1) ( ?…”

Section: Middle Zerosmentioning

confidence: 99%

“…(1; 1; 1) is a valid transfer. Using this in the formula above with L de ned in 2, The stability region is 1 + 2 < ?j (1) ?1 j ? j (1) +1 j ?…”

Section: Middle Zerosmentioning

confidence: 99%

Mosaic Solutions and Spatial Entropy for a Class of Neural Network Models

Jennings

Vleck

2000

Int. J. Bifurcation Chaos

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“…These systems have been used as effective models to describe a multitude of phenomena, from phase separation in binary alloys and glasses, the excitation of travelling pulses in myelinated nerve axons, to pattern formation in cellular networks. Characteristic examples are the discrete Allen-Cahn type equations with monostable or bistable nonlinearities, the discrete Cahn-Hiliard, and the discrete Swift-Hohenberg equations, [6][7][8][9][10][11][12][13][14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

The Lefever–Lejeune nonlinear lattice: Convergence dynamics and the structure of equilibrium states

Karachalios

Kyriazopoulos

2020

Physica D: Nonlinear Phenomena

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We consider the Lefever-Lejeune nonlinear lattice, a spatially discrete propagation-inhibition model describing the growth of vegetation densities in dry-lands. We analytically identify parametric regimes distinguishing between decay (associated with spatial extinction of vegetation patches) and potentially non-trivial time-asymptotics. To gain insight on the convergence dynamics, a stability analysis of spatially uniform states is performed, revealing the existence of a threshold for the discretization parameter which depends on the lattice parameters, below which their destabilization occurs and spatially non-uniform equilibrium states may emerge. Direct numerical simulations justified that the analytical stability criteria and parametric thresholds effectively describe the above transition dynamics and revealed the rich structure of the equilibrium set. Connections with the continuous sibling Lefever-Lejeune partial differential equation are also discussed.In the above discrete set-up, the lattice (1) can be viewed as a discretization of the Lefever-Lejeune (LL) partial differential equation (PDE), which in a non-dimensional form reads asTo value the lattice (1), let us recall some information on the continuous LL model. Equation (6) is a spatially continuous propagation-inhibition model describing the growth of vegetation density in dry-lands, and is the formal continuum limit of the lattice (1), as h → 0. In such resource poor environments, spatial patterns of vegetation are observed, and LL attempts to explain their formation attributing it to a short-range cooperative and long-range competitive spatial mechanism. It should be remarked, that the original LL model [19], is a spatially non-local integral-differential equation which involves a continuous redistribution-kernel convoluted with a density dependent nonlinearity, encapsulating the dispersal and spatial interactions of individuals. Although the kernel-based models [20] are considered as more realistic, since they capture accurately enough global spatial-interactions involving kernel shapes that are common in nature, the Lefever-Lejeune PDE is a biharmonic approximation which is commonly preferred. The reason is two-fold: it successfully derives the qualitative behavior of plant community systems, and offers a simpler template for numerical investigations and mathematical analysis. In the differential form, the shortrange corporative interplay among plants is expressed by a linear diffusion term and the non-linear local growth term, while non-linear biharmonic and Laplacian diffusion term with negative coefficient imprint the long-range competition for resources. Numerical and analytical studies in one and two spatial dimensions, have revealed the pattern forming potential of the spatially continuous LL equation [19]. Besides the existence of Turing periodic patterns, the LL produces localized solutions such as isolated spots of vegetation, or groups of spots confined by the homogeneous zero vegetation [21]. Furthermore, the self-replication capabili...

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Optimal distributed control of nonlinear Cahn–Hilliard systems with computational realization

Wang

2011

J Math Sci

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Mosaic solutions and entropy for spatially discrete Cahn-Hilliard equations

Cited by 5 publications

References 12 publications

Mosaic Solutions and Spatial Entropy for a Class of Neural Network Models

Mosaic Solutions and Spatial Entropy for a Class of Neural Network Models

The Lefever–Lejeune nonlinear lattice: Convergence dynamics and the structure of equilibrium states

Optimal distributed control of nonlinear Cahn–Hilliard systems with computational realization

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