Front propagation in one-dimensional spatially periodic bistable media

Löber, Jakob; Bär, Markus; Engel, Harald

doi:10.1103/physreve.86.066210

Cited by 16 publications

(21 citation statements)

References 33 publications

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“…In this limit, the front is well approximated by an iso-concentration line and one can assume that its velocity instantaneously adapts when traveling through the corrugated channel. Then, the average front velocity is correctly predicted by the harmonic mean velocity [39] which tends to c 0 for L/l → ∞. With decreasing ratio L/l, i.e., either increasing the diffusion constant D u or decreasing the period length L, the average propagation velocity lessens until it attains its minimum value.…”

Section: Schlögl Modelmentioning

confidence: 80%

“…Aiming at deriving an equation of motion for the front position in corrugated channels, we apply asymptotic perturbation analysis in a geometric parameter [27,36] and projection techniques [37][38][39][40] to the problem. Our goal is to analyze how spatial variations of the channel's cross-section affect front propagation and, in particular, to determine the dependence of the propagation velocity on the characteristic length scales in the system.…”

Section: Introductionmentioning

confidence: 99%

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Front propagation in channels with spatially modulated cross section

2015

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The problem of front propagation in a three-dimensional channel with spatially varying crosssection is reduced to an equivalent reaction-diffusion-advection equation with boundary-induced advection term. Treating the advection term as a weak perturbation, an equation of motion for the front position is derived. We analyze channels whose cross-sections vary periodically with L along the propagation direction of the front. Taking the Schlögl model as representative example, we calculate analytically the nonlinear dependence of the front velocity on the ratio L/l where l denotes the intrinsic front width. Our analytical results agree well with the results obtained by numerical simulations. In particular, the peculiarity of boundary-induced propagation failure for a finite range of L/l values is predicted by analytical calculations. Lastly, we demonstrate that the front velocity is determined by the suppressed diffusivity of the reactants for L l.

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Section: Schlögl Modelmentioning

confidence: 80%

Section: Introductionmentioning

confidence: 99%

Front propagation in channels with spatially modulated cross section

2015

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“…As a result, heterogeneities may induce disruption and subsequent reentry, as has been observed in very schematic models [170], in simplified ionic models [171], and in more complex models [172]. Homogenization has also been used to predict effective wave speeds in periodic media [42,173]. As long as the excitation front width is larger than the characteristic size of the heterogeneities the wave speed is approximately constant.…”

Section: Small-scale Heterogeneitiesmentioning

confidence: 99%

“…As long as the excitation front width is larger than the characteristic size of the heterogeneities the wave speed is approximately constant. Interestingly, intermediate periods of the heterogeneity length scale may yield conduction block of waves, whereas propagation is possible at small and large length scales [173].…”

Section: Small-scale Heterogeneitiesmentioning

confidence: 99%

Nonlinear physics of electrical wave propagation in the heart: a review

2016

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Abstract. The beating of the heart is a synchronized contraction of muscle cells (myocytes) that are triggered by a periodic sequence of electrical waves (action potentials) originating in the sino-atrial node and propagating over the atria and the ventricles. Cardiac arrhythmias like atrial and ventricular fibrillation (AF,VF) or ventricular tachycardia (VT) are caused by disruptions and instabilities of these electrical excitations, that lead to the emergence of rotating waves (VT) and turbulent wave patterns (AF,VF). Numerous simulation and experimental studies during the last 20 years have addressed these topics. In this review we focus on the nonlinear dynamics of wave propagation in the heart with an emphasis on the theory of pulses, spirals and scroll waves and their instabilities in excitable media and their application to cardiac modeling. After an introduction into electrophysiological models for action potential propagation, the modeling and analysis of spatiotemporal alternans, spiral and scroll meandering, spiral breakup and scroll wave instabilities like negative line tension and sproing are reviewed in depth and discussed with emphasis on their impact in cardiac arrhythmias.

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“…By multiple scale perturbation theory for small perturbation amplitude , the following equation of motion (EOM) for the position φ(t) of the perturbed front can be obtained, 34,35,[58][59][60][61][62] …”

Section: Position Control Of Traveling Front Solutions To the Schlöglmentioning

confidence: 99%

Control of Chemical Wave Propagation

Löber

Coles

Siebert

et al. 2014

Engineering of Chemical Complexity II

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Besides the well-known Turing patterns, reaction-diffusion (RD) systems possess a rich variety of spatio-temporal structures [Kuramoto (1984); Mikhailov (1990); Kapral and Showalter (1995)]. Spatially one-dimensional examples include traveling fronts, solitary pulses, and periodic pulse trains that are the building blocks of more complicated patterns in two-and three-dimensional active media as, e.g., spiral and scroll waves, respectively. Another important class of RD patterns forms stationary, breathing, moving or self-replicating localized spots. Labyrinthine patterns as well as phase turbulence, defect-mediated spiral and scroll wave turbulence are examples for more complex patterns. In the Belousov-Zhabotinsky (BZ) reaction in microemulsions the BZ inhibitor (bromide) is produced in nanodroplets and diffuses through the oil phase at a rate up to two orders of magnitude greater than that of the BZ activator (bromous acid). In this heterogeneous RD system, a variety of patterns including three-dimensional Turing patterns have been observed by computer tomography (see [Bánsági et al. (2011)] and references therein). Several control strategies have been developed for the purposeful manipulation of RD patterns. Below, we will differentiate between closed-loop or feedback control with and without nonlocal spatial coupling [Dahlem et al. (2008); Schneider et al. (2009); Siebert et al. (2014)] or time delay [Kim et al. (2001); Kyrychko et al. (2009)],and open-loop control that includes external spatio-temporal forcing, optimal control [Tröltzsch (2010)], control by imposed geometric constraints or heterogeneities, and others [Mikhailov and Showalter (2006); Schimansky-Geier et al. (2007);Vanag and Epstein (2008);Schöll and Schuster (2008)]. While feedback control relies on continuously running monitoring of the system's state, open-loop control is based on a detailed knowledge of the system's dynamics and its parameters. Feedback-mediated control has been applied quite successfully to the control of propagating one-dimensional (1D) waves as well as to spiral waves in 2D that are

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Front propagation in one-dimensional spatially periodic bistable media

Cited by 16 publications

References 33 publications

Front propagation in channels with spatially modulated cross section

Front propagation in channels with spatially modulated cross section

Nonlinear physics of electrical wave propagation in the heart: a review

Control of Chemical Wave Propagation

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