Experimental Studies on Long-Wavelength Instability and Spiral Breakup in a Reaction-Diffusion System

Zhou, Lü; Ouyang, Qi

doi:10.1103/physrevlett.85.1650

Cited by 98 publications

(54 citation statements)

References 19 publications

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“…It is now well known that supercritical modulations of traveling waves may result from a supercritical Eckhaus transition [54,7,14,66,37]. Since Eckhaus instability is a low-wavenumber instability, we discussed this behavior over the amplitude of the first mode K M = 2π/L [14].…”

Section: Supercritical Eckhaus Modulated Wave Patterns Near K Cmentioning

confidence: 99%

Nonlinear dynamics of waves and modulated waves in 1D thermocapillary flows. II. Convective/absolute transitions

Garnier¹,

Chiffaudel²,

Daviaud³

2003

Physica D: Nonlinear Phenomena

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We present experimental results on hydrothermal traveling-waves dynamics in long and narrow 1D channels. The onset of primary traveling-wave patterns is briefly presented for different fluid heights and for annular or bounded channels, i.e., within periodic or non-periodic boundary conditions. For periodic boundary conditions, by increasing the control parameter or changing the discrete mean-wavenumber of the waves, we produce modulated waves patterns. These patterns range from stable periodic phase-solutions, due to supercritical Eckhaus instability, to spatio-temporal defect-chaos involving traveling holes and/or counter-propagating-waves competition, i.e., traveling sources and sinks. The transition from non-linearly saturated Eckhaus modulations to transient pattern-breaks by traveling holes and spatiotemporal defects is documented. Our observations are presented in the framework of coupled complex Ginzburg-Landau equations with additional fourth and fifth order terms which account for the reflection symmetry breaking at high wave-amplitude far from onset. The second part of this paper [1] extends this study to spatially non-periodic patterns observed in both annular and bounded channel.

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Section: Supercritical Eckhaus Modulated Wave Patterns Near K Cmentioning

confidence: 99%

Nonlinear dynamics of waves and modulated waves in 1D thermocapillary flows. II. Convective/absolute transitions

Garnier¹,

Chiffaudel²,

Daviaud³

2003

Physica D: Nonlinear Phenomena

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“…If the above suggestion is correct, the phenomenon of spiral tip attraction should also occur when we locally increase the diffusion constant in Eqs. (1) and (2). In this part of work, we increase the diffusion coefficient in a small circular region Ω by D times and keep the system with regular connections.…”

Section: Compare With Increasing the Diffusion Coefficientmentioning

confidence: 99%

“…Spiral waves are characteristic structures of excitable media that have been observed in many extended systems such as reaction-diffusion media [1,2,3], aggregating colonies of slime mold [4], and heart tissues [5], where they are suspected to play an essential role in cardiac arrhythmia and fibrillation. Sudden cardiac death resulting from ventricular fibrillation is generated from the fragmenting or breakup of spiral waves [6,7,8].…”

Section: Introductionmentioning

confidence: 99%

Attraction of spiral waves by localized inhomogeneities with small-world connections in excitable media

Wang

Jiang

et al. 2004

Phys. Rev. E

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Trapping and un-trapping of spiral tips in a two-dimensional homogeneous excitable medium with local small-world connections is studied by numerical simulation. In a homogeneous medium which can be simulated with a lattice of regular neighborhood connections, the spiral wave is in the meandering regime. When changing the topology of a small region from regular connections to small-world connections, the tip of a spiral waves is attracted by the small-world region, where the average path length declines with the introduction of long distant connections. The "trapped" phenomenon also occurs in regular lattices where the diffusion coefficient of the small region is increased. The above results can be explained by the eikonal equation and the relation between core radius and diffusion coefficient.

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“…Spiral breakup is a phenomenologically different instability where, as in the meander instability prominent super-patterns occur in the farfield |x| → ∞. This time, however, perturbations grow in amplitude with distance from the center; see [BäEi93,BäOr99,Be&al97,ZhOu00]. Before spiral breakup, the perturbations of the spiral wave again take the form of superimposed spirals of compression and expansion of the local wavelength of the wavetrains.…”

Section: Phenomenology Of Spiral Instabilitiesmentioning

confidence: 99%

“…Both spiral breakup scenarios, farfield and core breakup, may occur as instabilities of rigidly rotating or meandering spirals [ZhOu00,BäOr99].…”

Section: Phenomenology Of Spiral Instabilitiesmentioning

confidence: 99%

Spatio-Temporal Dynamics of Reaction-Diffusion Patterns

Fiedler

Scheel

2003

Trends in Nonlinear Analysis

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In this survey we look at parabolic partial differential equations from a dynamical systems point of view. With origins deeply rooted in celestial mechanics, and many modern aspects traceable to the monumental influence of Poincaré, dynamical systems theory is mainly concerned with the global time evolution T (t)u 0 of points u 0 -and of sets of such points -in a more or less abstract phase space X. The success of dynamical concepts such as gradient flows, invariant manifolds, ergodicity, shift dynamics, etc. during the past century has been enormous -both as measured by achievement, and by vitality in terms of newly emerging questions and long-standing open problems.In parallel to this development, the applied horizon now reaches far beyond the classical sources of celestial and Hamiltonian mechanics. Applications areas today include physics, many branches of engineering, economy models, and mathematical biology, to name just a few. This influence can certainly be felt in several articles of this volume and cannot possibly be adequately summarized in our survey.Some resources on recent activities in the area of dynamics are the book series Dynamical Systems I -X of the Encyclopedia of Mathematical Sciences [AnAr88, Si89, Ar&al88, ArNo90, Ar94a, Ar93, ArNo94, Ar94b, An91, Ko02], the Handbook of Dynamical System [BrTa02, Fi02, KaHa02], the Proceedings [Fi&al00], and the fundamental books [ChHa82,GuHo83,Mo73,KaHa95].In the context of partial differential equations, the phase space X of solutions u = u(t, x) becomes infinite-dimensional: typically a Sobolev space of spatial profiles u(t) = u(t, ·). More specifically, the evolution of u(t) = T (t)u 0 ∈ X with time t is complemented by the behavior of the x-profiles x → u(t, x) of solutions u. Such spatial profiles could be monotone or oscillatory; in dim x = 1 they could define sharp fronts or peaks moving at constant or variable speeds, with possible collision or mutual repulsion. Target pat- 22Bernold Fiedler and Arnd Scheel terns or spirals can emerge in dim x = 2. Stacks of spirals, which Winfree called scroll waves, are possible in dim x = 3. For a first cursory orientation in such phenomena and their mathematical treatment we again refer to [Fi02] and the article of Fife in the present volume, as well as the many references there.For our present survey, we focus on the spatio-temporal dynamics of the following rather "simple", prototypical parabolic partial differential equation:(2.1)Here t ≥ 0 denotes time, x ∈ Ω ⊆ R m is space and u = u(t, x) ∈ R N is the solution vector. We consider C 1 -nonlinearities f and constant, positive, diagonal diffusion matrices D. This eliminates the beautiful pattern formation processes due to chemotaxis; see [JäLu92, St00], for example. Note that we will not always restrict f to be a pure reaction term, like f = f (u) or f = f (x, u). More general than that, we sometimes allow for a dependence of f on ∇u to include advection effects. The domain Ω will be assumed smooth and bounded, typically with Dirichlet or Neumann boundary c...

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Experimental Studies on Long-Wavelength Instability and Spiral Breakup in a Reaction-Diffusion System

Cited by 98 publications

References 19 publications

Nonlinear dynamics of waves and modulated waves in 1D thermocapillary flows. II. Convective/absolute transitions

Nonlinear dynamics of waves and modulated waves in 1D thermocapillary flows. II. Convective/absolute transitions

Attraction of spiral waves by localized inhomogeneities with small-world connections in excitable media

Spatio-Temporal Dynamics of Reaction-Diffusion Patterns

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