Period-Doubling of Spiral Waves and Defects

Sandstede, Björn; Scheel, Arnd

doi:10.1137/060668158

Cited by 25 publications

(31 citation statements)

References 42 publications

(99 reference statements)

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“…The dynamical behavior of the spiral waves has been previously investigated for the diffusive Rössler model [18]. Here we reconsidered the effect of the time-delay and the cross-diffusion.…”

Section: Discussionmentioning

confidence: 97%

“…The break-up of spiral wave has been observed in experiments [11,12] and numerical simulations [13][14][15][16][17][18][19]. The break-up of Turing patterns has been observed in chemical experiments [20] and numerical simulations [21][22][23].…”

mentioning

confidence: 91%

“…For the system (1.1), the drift stability of the spiral wave was observed to vary with the parameter c in [18,19]. Apart from the interaction inducing the spiral wave, the effect of the time-delay [26][27][28] and cross-diffusion [29][30][31] can also generate the pattern formation.…”

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confidence: 98%

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Pattern dynamics in a diffusive Rössler model

Zhang

Tian

2014

Nonlinear Dyn

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This paper is concerned with the pattern formation and pattern dynamics of a diffusive Rössler model. We first show that the time-delay and the crossdiffusion can lead to Hopf bifurcation and Turing bifurcation, respectively, by computing Lyapunov characteristic exponent. Then by the calculation of the first Lyapunov number and weak nonlinear analysis, the dynamics of Hopf pattern and Turing pattern is investigated. Our results reveal that Hopf bifurcation generates the transient spiral wave, but the spiral wave will break up and becomes the terminate irregular pattern. Turing bifurcation generates a stable spotted pattern.

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Section: Discussionmentioning

confidence: 97%

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confidence: 91%

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Pattern dynamics in a diffusive Rössler model

Zhang

Tian

2014

Nonlinear Dyn

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“…[26]-identical to the one for the standard meandering transition-characterizes the spiral transition to alternans. Yet, in contrast to models of complex-oscillatory media, the transition itself does not seem to be a resonant 2 : 1 Hopf bifurcation but rather a nonresonant one since the former would lead to a straight drift of the core [26].…”

Section: Spiral Core Motionmentioning

confidence: 93%

“…Spirals with attached SDLs were first observed in numerical model studies of complexoscillatory media [17] and subsequently in experiments on the Belousov-Zhabotinsky reaction in a comparable regime [20][21][22][23][24][25]. The bifurcation from a regular or period-one spiral to a period-two spiral was later put on a solid mathematical foundation by Sandstede et al [26] and identified as a 2 : 1 resonant Hopf bifurcation for models of complex-oscillatory media. One of the important implications of their analytical findings is that period-two spirals drift with a finite velocity.…”

Section: Introductionmentioning

confidence: 98%

Induced spiral motion in cardiac tissue due to alternans

Cameron

Davidsen

2012

Phys. Rev. E

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Spiral wave meander is a typical feature observed in cardiac tissue and in excitable media in general. Here, we show for a simple model of excitable cardiac tissue that a transition to alternans--a beat-to-beat temporal alternation in the duration of cardiac excitation--can also induce a transition in the spiral core motion that is related to the presence of synchronization defect lines (SDLs) or nodal lines. While this is similar to what has been predicted and indeed observed for complex-oscillatory media close to onset, we find important qualitative differences. For example, single straight SDLs rotate and induce an additional nonresonant frequency characterizing the core motion of the attached spiral. We analyze this behavior quantitatively as a function of the steepness of the restitution curve and show that the velocity and the directionality of the core motion vary monotonically with the control parameter. Our findings agree with recent observations in rat heart tissue cultures indicating that the described behavior is of rather general nature. In particular, it could play an important role in the context of potentially life-threatening cardiac arrhythmias such as fibrillation for which alternans and spiral waves are known precursors.

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Spiraling Asymptotic Profiles of Competition‐Diffusion Systems

Terracini

Verzini

Zilio

2019

Comm Pure Appl Math

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This paper describes the structure of the nodal set of segregation profiles arising in the singular limit of planar, stationary, reaction‐diffusion systems with strongly competitive interactions of Lotka‐Volterra type, when the matrix of the interspecific competition coefficients is asymmetric and the competition parameter tends to infinity. Unlike the symmetric case, when it is known that the nodal set consists of a locally finite collection of curves meeting with equal angles at a locally finite number of singular points, the asymmetric case shows the emergence of spiraling nodal curves, still meeting at locally isolated points with finite vanishing order. © 2019 Wiley Periodicals, Inc.

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Period-Doubling of Spiral Waves and Defects

Cited by 25 publications

References 42 publications

Pattern dynamics in a diffusive Rössler model

Pattern dynamics in a diffusive Rössler model

Induced spiral motion in cardiac tissue due to alternans

Spiraling Asymptotic Profiles of Competition‐Diffusion Systems

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