Core instability and spatiotemporal intermittency of spiral waves in oscillatory media

Aranson, Igor S.; Kramer, Lorenz; Weber, Andreas

doi:10.1103/physrevlett.72.2316

Cited by 57 publications

(56 citation statements)

References 20 publications

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“…(6). Substituting Φ = Φ(ξ ), we get the second-order ODE with boundary conditions (13). This problem can be easily solved numerically by matching stable and unstable one-dimensional manifolds of the fixed point.…”

Section: Phase Kinks As Natural Excitations In the Forced Systemmentioning

confidence: 99%

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Forcing oscillatory media: phase kinks vs. synchronization

Chaté

Pikovsky

Rudzick

1999

Physica D: Nonlinear Phenomena

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We consider the effect of periodic external forcing on the spatiotemporal dynamics of one-dimensional oscillatory media modelled by the complex Ginzburg-Landau equation (CGLE). We determine the domain of existence and linear stability of the spatially homogeneous synchronous solution found at strong forcing. Some of the synchronization scenarios observed are described, and the "turbulent synchronized states" encountered are detailed. We show that 2π -phase kinks are the ubiquitous objects mediating synchronization and study their nontrivial dynamics. In particular, we consider the processes of kinkbreeding and the spontaneous creation of kinks which are specific to our phase and amplitude description. The general consequences, at the statistical level, of breaking the gauge invariance of the CGLE by a periodic forcing are discussed.

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Section: Phase Kinks As Natural Excitations In the Forced Systemmentioning

confidence: 99%

“…[6,7]). The CGLE has attracted a large interest and is now quite well understood, at least in one and two space dimensions [8][9][10][11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

Forcing oscillatory media: phase kinks vs. synchronization

Chaté

Pikovsky

Rudzick

1999

Physica D: Nonlinear Phenomena

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“…We shall ignore isolated eigenvalues that belong to the point spectrum, instabilities caused by point eigenvalues lead to drifting waves, or to an unstable tip in excitable media and oscillation media [33,[38][39][40]. This phenomenon is not shown in the present paper.…”

Section: Stability Analysis Of Spiral Wavesmentioning

confidence: 74%

Self-organized wave pattern in a predator-prey model

Sun

Jin

et al. 2009

Nonlinear Dyn

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In this paper, pattern formation of a predatorprey model with spatial effect is investigated. We obtain the conditions for Hopf bifurcation and Turing bifurcation by mathematical analysis. When the values of the parameters can ensure a stable limit cycle of the no-spatial model, our study shows that the spatially extended models have spiral waves dynamics. Moreover, the stability of the spiral wave is given by the theory of essential spectrum. Furthermore, although the environment is heterogeneous, the system still exhibit spiral waves. The obtained results confirm that diffusion can form the population in the stable motion, which well enrich the finding of spatiotemporal dynamics in the predator-prey interactions and may well explain the field observed in some areas.

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“…After collision, new spirals form close to the core of the primary spiral, which break up immediately. This mechanism of instability leads to very incoherent spatio-temporal states, with many small spiral domains; see [Ar&al94,BäOr99]. We refer to this scenario of instability as core breakup.…”

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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Core instability and spatiotemporal intermittency of spiral waves in oscillatory media

Cited by 57 publications

References 20 publications

Forcing oscillatory media: phase kinks vs. synchronization

Forcing oscillatory media: phase kinks vs. synchronization

Self-organized wave pattern in a predator-prey model

Spatio-Temporal Dynamics of Reaction-Diffusion Patterns

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