Turing Instabilities at Hopf Bifurcation

Ricard, M Rodríguez; Mischler, Stéphane

doi:10.1007/s00332-009-9041-6

Cited by 54 publications

(70 citation statements)

References 24 publications

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“…Inspired by [55], we investigate the formation of spatio-temporal patterns at the onset of TH instabilities.…”

Section: Turing-hopf Instabilities Of the Hopf Cyclementioning

confidence: 99%

“…where z(t) is an unknown function, then it is possible to show that the integration of system (6.1) can be reduced to the integration of a second order differential equation in the variable z(t) [55]. Moreover, if f and g in (6.1) are polynomial functions of degree M, in the neighbourhood of the origin the transformation H is close enough to the matrix transformation Γ [55].…”

Section: Turing-hopf Instabilities Of the Hopf Cyclementioning

confidence: 99%

“…Moreover, if f and g in (6.1) are polynomial functions of degree M, in the neighbourhood of the origin the transformation H is close enough to the matrix transformation Γ [55]. The main problem is how to choose the matrix transformation Γ .…”

Section: Turing-hopf Instabilities Of the Hopf Cyclementioning

confidence: 99%

“…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%

“…In [55] this mechanism is referred to as TH instability or diffusive instability of the Hopf limit cycle and represents a different way in which Turing and Hopf instabilities can interact. We will exploit the analytical approach developed in [55], which employs perturbation expansions, to find appropriate normal modes useful for the study of the TH instability of the Hopf limit cycle. We show that, in the neighbourhood of the TH point, our system can exhibit both a classical TH bistability regime and a TH instability of the Hopf cycle which possibly results in oscillatory inhomogeneous patterns.…”

Section: Introductionmentioning

confidence: 99%

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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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“…Inspired by [55], we investigate the formation of spatio-temporal patterns at the onset of TH instabilities.…”

Section: Turing-hopf Instabilities Of the Hopf Cyclementioning

confidence: 99%

Section: Turing-hopf Instabilities Of the Hopf Cyclementioning

confidence: 99%

Section: Turing-hopf Instabilities Of the Hopf Cyclementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

2014

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High‐lateral resolution X‐ray fluorescence microspectroscopy and dynamic mathematical modelling as tools for the study of electrodeposited electrocatalysts

et al. 2015

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In this paper, we report on the use of high-space resolution soft X-ray fluorescence microspectroscopy for the study of electrodeposited composites containing catalytic ternary metal nanostructures. X-ray fluorescence maps are interpreted in terms of a dynamic mathematical model of the electrode morphology and metal space distribution, allowing to reproduce the observed space patterns and electrochemical transients by assigning an appropriate set of electrokinetic parameters. The discussed materials-science case is the electrochemical growth of a Mn-Mg-Cu-polypyrrole nanocomposite electrocatalyst material -free of expensive Pt and environmentally unfriendly Co -with promising performance for fuel-cell oxygen electrodes. The synergy of high-resolution compositional mapping with electrokinetic modelling not only provides the general rationale for quantitative use of potentially large compositional distribution datasets but also yields unprecedented insight into the specific catalyst synthesis process. The expounded application is just a prototypical case study of a more general approach, which can be employed for the understanding of electrochemical material science processes, both in situ and ex situ, as well as for the characterisation of the corresponding products, with no other limitations in principle than X-ray transmission and beam damage.

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