Interaction of Turing and Hopf bifurcations in chemical systems

Rovinsky, Arkady B.; Menzinger, Michael

doi:10.1103/physreva.46.6315

Cited by 94 publications

(57 citation statements)

References 17 publications

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“…Such spatio-temporal structures have been observed in chemical experiments (Perraud et al (1993); Rüdiger et al (2003)). Theoretical studies prove their existence for the Brusselator model (Rovinsky and Menzinger (1992)), in optical systems (Tlidi et al (1997)) and in semiconductor heterostructures (Just et al (2001)). It has been shown that spatio-temporal patterns are very likely to be found in the neighborhood of Turing-Hopf bifurcations.…”

Section: Introductionmentioning

confidence: 89%

Instabilities in spatially extended predator–prey systems: Spatio-temporal patterns in the neighborhood of Turing–Hopf bifurcations

Baurmann

Groß

Feudel

2007

Journal of Theoretical Biology

327

218

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We investigate the emergence of spatio-temporal patterns in ecological systems. In particular we study a generalized predator-prey system on a spatial domain. On this domain diffusion is considered as the principal process of motion. We derive the conditions for Hopf and Turing instabilities without specifying the predatorprey functional responses and discuss their biological implications. Furthermore, we identify the codimension-2 Turing-Hopf bifurcation and the codimension-3 TuringTakens-Bogdanov bifurcation. These bifurcations give rise to complex pattern formation processes in their neighborhood. Our theoretical findings are illustrated with a specific model. In simulations a large variety of different types of long-term behavior, including homogenous distributions, stationary spatial patterns and complex spatio-temporal patterns is observed.

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Section: Introductionmentioning

confidence: 89%

Instabilities in spatially extended predator–prey systems: Spatio-temporal patterns in the neighborhood of Turing–Hopf bifurcations

Baurmann

Groß

Feudel

2007

Journal of Theoretical Biology

327

218

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“…The FHN model, like a number of other reaction-diffusion models [RoM92], may undergo both Turing and Hopf bifurcations. Figs.…”

Section: The Development Of Fronts Far Beyond Onsetmentioning

confidence: 99%

Pattern formation in non-gradient reaction-diffusion systems: the effects of front bifurcations

1994

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Patterns in reaction-diffusion systems often contain two spatial scales; a long scale determined by a typical wavelength or domain size, and a short scale pertaining to front structures separating different domains. Such patterns naturally develop in bistable and excitable systems, but may also appear far beyond Hopf and Turing bifurcations. The global behavior of domain patterns strongly depends on the fronts' inner structures. In this paper we study a symmetry breaking front bifurcation expected to occur in a wide class of reaction-diffusion systems, and the effects it has on pattern formation and pattern dynamics. We extend previous works on this type of front bifurcation and clarify the relations among them. We show that the appearance of front multiplicity beyond the bifurcation point allows the formation of persistent patterns rather than transient ones. In a different parameter regime, we find that the front bifurcation outlines a transition from oscillating (or breathing) patterns to traveling ones. Near a boundary we find that fronts beyond the bifurcation can reflect, while those below it either bind to the boundary or disappear.-2-

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“…The coupling between these two kinds of instabilities was first theoretically investigated in the 90's in the framework of the Lengyel-Epstein model of the CIMA reaction [56]. Since then it has become a key topic for pattern formation in the reaction-diffusion framework and has been extensively investigated in chemistry, physics and biology [2,3,24,33,48,49,55].…”

Section: Introductionmentioning

confidence: 99%

Spatio-temporal organization in a morphochemical electrodeposition model: Hopf and Turing instabilities and their interplay

2014

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In this paper, we investigate from a theoretical point of view the 2D reaction-diffusion system for electrodeposition coupling morphology and surface chemistry, presented and experimentally validated in Bozzini et al. (2013 J. Solid State Electr. 17, 467-479). We analyse the mechanisms responsible for spatio-temporal organization. As a first step, spatially uniform dynamics is discussed and the occurrence of a supercritical Hopf bifurcation for the local kinetics is proved. In the spatial case, initiation of morphological patterns induced by diffusion is shown to occur in a suitable region of the parameter space. The intriguing interplay between Hopf and Turing instability is also considered, by investigating the spatio-temporal behaviour of the system in the neighbourhood of the codimensiontwo Turing-Hopf bifurcation point. An ADI (Alternating Direction Implicit) scheme based on high-order finite differences in space is applied to obtain numerical approximations of Turing patterns at the steady state and for the simulation of the oscillating Turing-Hopf dynamics.

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Interaction of Turing and Hopf bifurcations in chemical systems

Cited by 94 publications

References 17 publications

Instabilities in spatially extended predator–prey systems: Spatio-temporal patterns in the neighborhood of Turing–Hopf bifurcations

Instabilities in spatially extended predator–prey systems: Spatio-temporal patterns in the neighborhood of Turing–Hopf bifurcations

Pattern formation in non-gradient reaction-diffusion systems: the effects of front bifurcations

Spatio-temporal organization in a morphochemical electrodeposition model: Hopf and Turing instabilities and their interplay

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