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Sandstede, Björn; Scheel, Arnd

doi:10.1002/1522-2616(200112)232:1<39::aid-mana39>3.3.co;2-x

Cited by 4 publications

(3 citation statements)

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“…Our extensive simulations revealed that the simplest possible of the allowed configurations, where a single P1 SDL is attached to the core, is the unique asymptotic solution, even in the P4 and chaotic regimes where P2 SDL exist. The presence of the SDL breaks the rotational symmetry and thus, on general grounds, one expects the spiral core to move [18]. This is indeed what is observed, albeit this motion is very slow, taking typically several thousand oscillations to move by one wavelength.…”

supporting

confidence: 54%

From Ballistic to Brownian Vortex Motion in Complex Oscillatory Media

et al. 2004

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We show that the breaking of the rotation symmetry of spiral waves in two-dimensional complex (period-doubled or chaotic) oscillatory media by synchronization defect lines (SDLs) is accompanied by an intrinsic drift of the pattern. Single vortex motion changes from ballistic flights at a well-defined angle from the SDLs to Brownian-like diffusion when the turbulent character of the medium increases. It gives rise, in nonturbulent multispiral regimes, to a novel "vortex liquid."

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supporting

confidence: 54%

From Ballistic to Brownian Vortex Motion in Complex Oscillatory Media

et al. 2004

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“…where (x, y) ∈ R×Ω ′ and ∆ Ω ′ denotes the Laplacian on the cross-section Ω ′ ⊂ R n−1 . This idea first appeared in [10]; see also [1,3,8,9,11,13,14,15,16,17] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Exponential dichotomies for elliptic PDE on radial domains

Beck¹,

Cox²,

Jones³

et al. 2019

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It was recently shown by the authors that a semilinear elliptic equation can be represented as an infinite-dimensional dynamical system in terms of boundary data on a shrinking one-parameter family of domains. The resulting system is illposed, in the sense that solutions do not typically exist forward or backward in time.In this paper we consider a radial family of domains and prove that the linearized system admits an exponential dichotomy, with the unstable subspace corresponding to the boundary data of weak solutions to the linear PDE. This generalizes the spatial dynamics approach, which applies to infinite cylindrical (channel) domains, and also generalizes previous work on radial domains as we impose no symmetry assumptions on the equation or its solutions.

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“…For the sake of clarity, we pretend that these dimensions are both finite and refer to Ref. [13] for a more correct argument where dimensions are counted by means of a comparison to a reference equation. We demonstrated [11] that, for l to the right of the continuous spectrum, the dimensions of E s 1 and E u 2 add up to the dimension of the phase space, which is the space of 2p-periodic functions ͑y,ỹ͒ ͑w͒.…”

mentioning

confidence: 99%