Necessary condition of the Turing instability

Szili, L.; Tóth, János

doi:10.1103/physreve.48.183

Cited by 25 publications

(13 citation statements)

References 19 publications

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“…More recently, it was demonstrated that another necessary condition for the appearance of Turing structures is the presence of cross-inhibition [16], that is an intermediate species inhibits the increase of another intermediate species' concentration.…”

Section: What Are Turing Structures?mentioning

confidence: 99%

Turing structures in dc gas discharges

Popescu

2006

Europhys. Lett.

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Self-organized spatial plasma structures exhibit many similarities with Turing structures obtained in biology and chemistry. Using an analytical mesoscopic approach it is shown that plasma balls of fire belong to the same class of Turing structures like the Brusselator. It is also mathematically proved that the existence of these self-organized plasma structures is related with a negative differential resistance. There are also established the mathematical conditions which the negative differential resistance must satisfy for obtaining a stationary ball of fire at the anode of a plasma diode.

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Section: What Are Turing Structures?mentioning

confidence: 99%

Turing structures in dc gas discharges

Popescu

2006

Europhys. Lett.

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“…The forms and variations of patterns generated by Turing systems have been studied analytically and numerically by investigating the conditions for instability [18], assuming inhomogeneous diffusion coefficients [19], by introducing domain curvature [20] or growth [21], and both homogeneous [22] and periodic suppression [23]. In addition, symmetries in Turing systems are of great interest, since they may have biological relevance; see, e.g., Refs.…”

Section: Introductionmentioning

confidence: 99%

Morphological transitions and bistability in Turing systems

Leppänen¹,

Karttunen²,

Barrio

et al. 2004

Phys. Rev. E

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It is well known that in two dimensions Turing systems produce spots, stripes and labyrinthine patterns, and in three dimensions lamellar and spherical structures, or their combinations, are observed. In this paper we study transitions between these states in both two and three dimensions. First, we derive the regions of stability for different patterns using nonlinear bifurcation analysis. Then, we apply large scale computer simulations to analyze the pattern selection in a bistable system by studying the effect of parameter selection on morphological clustering and the appearance of topological defects. The method elaborated in this paper presents a probabilistic approach for studying pattern selection in a bistable reaction-diffusion system.

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“…However, we cannot ensure that the real part of the eigenvalue of the zero eigenmode is smaller than the eigenvalue of the eigenmode Condition vi''') gives the necessary condition f Tóth, [11], and Satnoianu et al, [12].…”

Section: T G ∩mentioning

confidence: 99%

“…In applied contexts, it is sometimes suggested that the existence of Turing patterns depends on the action-inhibition features of the local dynamics, [5], [6], [11] and [13]. To be more precise, if …”

Section: T G ∩mentioning

confidence: 99%

Turing instabilities and patterns near a Hopf bifurcation

Dilão

2005

Applied Mathematics and Computation

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We derive a necessary and sufficient condition for Turing instabilities to occur in twocomponent systems of reaction-diffusion equations with Neumann boundary conditions. We apply this condition to reaction-diffusion systems built from vector fields with one fixed point and a supercritical Hopf bifurcation. For the Brusselator and the GinzburgLandau reaction-diffusion equations, we obtain the bifurcation diagrams associated with the transition between time periodic solutions and asymptotically stable solutions (Turing patterns). In two-component systems of reaction-diffusion equations, we show that the existence of Turing instabilities is neither necessary nor sufficient for the existence of Turing pattern type solutions. Turing patterns can exist on both sides of the Hopf bifurcation associated to the local vector field, and, depending on the initial conditions, time periodic and stable solutions can coexist.

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Necessary condition of the Turing instability

Cited by 25 publications

References 19 publications

Turing structures in dc gas discharges

Turing structures in dc gas discharges

Morphological transitions and bistability in Turing systems

Turing instabilities and patterns near a Hopf bifurcation

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