Pattern formation arising from interactions between Turing and wave instabilities

Yang, Lingfa; Dolnik, Miloš; Zhabotinsky, Anatol M.; Epstein, Irving R.

doi:10.1063/1.1507110

Cited by 117 publications

(105 citation statements)

References 28 publications

(30 reference statements)

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“…Experiments with 2D BZ systems have shown that under the right conditions they can serve as writable memory devices [48], but less work has been done with 1D models. Steady state solutions can be achieved using specially selected parameters [51], but it appears that the chance of discovering such configurations is significantly less than in the Gray-Scott case.…”

Section: Other Reaction-diffusion Systemsmentioning

confidence: 99%

The Evolution of Reaction-Diffusion Controllers for Minimally Cognitive Agents

Dale

Husbands

2010

Artificial Life

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This article describes work carried out to investigate whether a classic reaction-diffusion (RD) system could be used to control a minimally cognitive animat. The RD system chosen was that first described by Gray and Scott, and the minimally cognitive behaviors were those used by Beer et al. involving the fixation and discrimination of diamond and circle shapes by a whiskered animat. A further task was added, which required the RD controllers to maintain and use a chemical memory. The parameters of these controllers were evolved using an evolutionary, or genetic, algorithm.

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Section: Other Reaction-diffusion Systemsmentioning

confidence: 99%

The Evolution of Reaction-Diffusion Controllers for Minimally Cognitive Agents

Dale

Husbands

2010

Artificial Life

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“…Very recently, a new type of optical refractionnegative refraction, where both incident and refractory lines locate on the same side of the normal line of the interface, has been theoretically predicted [1] and experimentally observed [2] when refraction occurs between a right-handed material and a left-handed material where optical waves propagate inward to the wave source. In addition, some new observations have been reported in nonlinear wave propagation, and one particularly interesting phenomenon is antispiral waves [3,4,5,6,7,8]. In contrast to spirals (which are familiar to people) where waves propagate outwards from the tips, antispiral waves propagate inward to the spiral tip, the wave source.…”

mentioning

confidence: 99%

Negative refraction in nonlinear wave systems

Cao¹,

Zhang²,

Hu³

2007

Europhys. Lett.

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People have been familiar with the phenomenon of wave refraction for several centuries. Recently, a novel type of refraction, i.e., negative refraction, where both incident and refractory lines locate on the same side of the normal line, has been predicted and realized in the context of linear optics in the presence of both right-and left-handed materials. In this work, we reveal, by theoretical prediction and numerical verification, negative refraction in nonlinear oscillatory systems. We

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“…Codimension-two bifurcations correspond to particular combinations of the parameter values where two types of instability appear simultaneously. In the proximity of such points, the associated dynamics have been extensively studied in the case of TuringHopf codimension-two bifurcation [7][8][9][10], and analyzed for the Turing-wave codimension-two bifurcation in a few instances [11,12]. In particular, in the latter case, two instabilities appear simultaneously with two character- istic spatial scales, which can be very different depending on the parameter values.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, in the latter case, two instabilities appear simultaneously with two character- istic spatial scales, which can be very different depending on the parameter values. Such bifurcations appear in the BZ reaction [11], in catalytic surfaces with promotors [13], as well as in models of lipid domain formation in biomembranes [14,15]. The interaction between different types of phospholipids and proteins on the membrane of living cells induces a spatial instability of the homogeneous state with a short characteristic spatial scale [14].…”

Section: Introductionmentioning

confidence: 99%

Spatio-temporal dynamics induced by competing instabilities in two asymmetrically coupled nonlinear evolution equations

Schuler

Alonso

Torcini

et al. 2014

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Pattern formation often occurs in spatially extended physical, biological and chemical systems due to an instability of the homogeneous steady state. The type of the instability usually prescribes the resulting spatio-temporal patterns and their characteristic length scales. However, patterns resulting from the simultaneous occurrence of instabilities cannot be expected to be simple superposition of the patterns associated with the considered instabilities. To address this issue we design two simple models composed by two asymmetrically coupled equations of non-conserved (Swift-Hohenberg equations) or conserved (Cahn-Hilliard equations) order parameters with different characteristic wave lengths. The patterns arising in these systems range from coexisting static patterns of different wavelengths to traveling waves. A linear stability analysis allows to derive a two parameter phase diagram for the studied models, in particular revealing for the Swift-Hohenberg equations a co-dimension two bifurcation point of Turing and wave instability and a region of coexistence of stationary and traveling patterns. The nonlinear dynamics of the coupled evolution equations is investigated by performing accurate numerical simulations. These reveal more complex patterns, ranging from traveling waves with embedded Turing patterns domains to spatio-temporal chaos, and a wide hysteretic region, where waves or Turing patterns coexist. For the coupled Cahn-Hilliard equations the presence of an weak coupling is sufficient to arrest the coarsening process and to lead to the emergence of purely periodic patterns. The final states are characterized by domains with a characteristic length, which diverges logarithmically with the coupling amplitude.Some chemical and biological systems exhibit competing pattern forming instabilities with different characteristic wave numbers. Often such a phenomenon is caused by the presence of different physical processes that appear on different length scales and cause patterns with different wavelengths. Here, we investigate two coupled Swift-Hohenberg (CH) equations as well as two coupled Cahn-Hilliard (CH) equations as minimal models for such multiscale pattern formation. The CH and the SH equations are partial differential equations describing the evolution of a conserved and a a non-conserved order parameter, respectively. While the spatial domains in the SH equation self-organize into stationary periodic structures, for the CH equation the domains exhibit a coarsening dynamics that finally yield a single large domain. The competition between two instabilities with different wavelengths λ 1 and λ 2 is analyzed for coupled SH equations as well as for coupled CH equations. In both cases, the coupling of equations with stationary instabilities (Turing or phase separation) can lead to wave dynamics. Moreover, coupled SH equations exhibit a region of coexistence of Turing and traveling patterns as well as more complex patterns. The coupling of two CH equations leads to the arrest of coarsening and to the e...

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Pattern formation arising from interactions between Turing and wave instabilities

Cited by 117 publications

References 28 publications

The Evolution of Reaction-Diffusion Controllers for Minimally Cognitive Agents

The Evolution of Reaction-Diffusion Controllers for Minimally Cognitive Agents

Negative refraction in nonlinear wave systems

Spatio-temporal dynamics induced by competing instabilities in two asymmetrically coupled nonlinear evolution equations

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