Lamellar structures and self-replicating spots in a reaction-diffusion system

Lee, Kyoung J.; Swinney, Harry L.

doi:10.1103/physreve.51.1899

Cited by 157 publications

(159 citation statements)

References 54 publications

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“…Both perturbations lead to a coupling of the two degrees of freedom in the order parameter Eqs. (23) and allow for the nonsteady asymptotic motion of fronts.…”

Section: Front Reversal: Oscillations and Reboundmentioning

confidence: 99%

“…These motions can be driven by curvature [1,2], front interactions [3,4], convective instabilities [5,6], and external fields [7][8][9]. In some cases fronts reverse their direction of propagation, as for example in breathing pulses [10][11][12][13][14][15][16][17][18], where the reversal is periodic in time, and nucleation of spiral-vortex pairs, where the reversal is local along the extended front line [19][20][21][22][23].…”

Section: Introductionmentioning

confidence: 99%

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Order parameter equations for front transitions: Planar and circular fronts

et al. 1997

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Near a parity breaking front bifurcation, small perturbations may reverse the propagation direction of fronts. Often this results in nonsteady asymptotic motion such as breathing and domain breakup. Exploiting the time scale differences of an activator-inhibitor model and the proximity to the front bifurcation, we derive equations of motion for planar and circular fronts. The equations involve a translational degree of freedom and an order parameter describing transitions between left and right propagating fronts. Perturbations, such as a space dependent advective field or uniform curvature (axisymmetric spots), couple these two degrees of freedom. In both cases this leads to a transition from stationary to oscillating fronts as the parity breaking bifurcation is approached. For axisymmetric spots, two additional dynamic behaviors are found: rebound and collapse.

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“…Both perturbations lead to a coupling of the two degrees of freedom in the order parameter Eqs. (23) and allow for the nonsteady asymptotic motion of fronts.…”

Section: Front Reversal: Oscillations and Reboundmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Order parameter equations for front transitions: Planar and circular fronts

et al. 1997

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“…The similarities between these behaviors and chemical patterns arising in certain physical experiments are striking (cf. [24], [25]). These numerical and experimental studies have stimulated much theoretical work to classify steady-state and time-dependent spike behavior in the simpler case of one spatial dimension, including: spike-replication and dynamics in the weak-interaction regime (cf.…”

Section: Introductionmentioning

confidence: 99%

Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model

Kolokolnikov

Ward

Wei

2006

Interfaces Free Bound.

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Slow translational instabilities of symmetric k-spike equilibria for the one-dimensional singularly perturbed two-component Gray-Scott (GS) model are analyzed. These symmetric spike patterns are characterized by a common value of the spike amplitude. The GS model is studied on a finite interval in the semi-strong spike-interaction regime, where the diffusion coefficient of only one of the two chemical species is asymptotically small. Two distinguished limits for the GS model are considered: the low feed-rate regime and the intermediate regime. In the low feed-rate regime it is shown analytically that k−1 small eigenvalues, governing the translational stability of a symmetric kspike pattern, simultaneously cross through zero at precisely the same parameter value at which k −1 different asymmetric k-spike equilibria bifurcate off of the symmetric k-spike equilibrium branch. These asymmetric equilibria have the general form SBB . . . BS (neglecting the positioning of the B and S spikes in the overall spike sequence). For a one-spike equilibrium solution in the intermediate regime it is shown that a translational, or drift, instability can occur from a Hopf bifurcation in the spike-layer location when a reaction-time parameter τ is asymptotically large as ε → 0. Locally, this instability leads to small-scale oscillations in the spike-layer location. For a certain parameter range within the intermediate regime such a drift instability for the GS model is shown to be the dominant instability mechanism. Numerical experiments are performed to validate the asymptotic theory.

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“…To be specific, we focus on the Swift-Hohenberg equation, which has been proposed as a prototypical example for pattern forming systems, in areas as diverse as nonlinear optics [15], Rayleigh-Bénard convection [2], granular media [1], chemical reactions [14], liquid crystals and solidification; see [4] and references there. Most of those systems exhibit stationary stripe (or roll) patterns, that is, planar patterns which are independent of x and periodic in y, where x and y are coordinates in the plane of observation.…”

Section: Introductionmentioning

confidence: 99%

Interfaces between rolls in the Swift-Hohenberg equation

Hărăguş

Scheel²

2007

IJDSDE

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We study the existence of interfaces between stripe or roll solutions in the Swift-Hohenberg equation. We prove the existence of two different types of interfaces: corner-like interfaces, also referred to as knee solutions, and step-like interfaces. The analysis relies upon a spatial dynamics formulation of the existence problem and an equivariant center manifold reduction. In this setting, the interfaces are found as heteroclinic and homoclinic orbits of a reduced system of ODEs.

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Lamellar structures and self-replicating spots in a reaction-diffusion system

Cited by 157 publications

References 54 publications

Order parameter equations for front transitions: Planar and circular fronts

Order parameter equations for front transitions: Planar and circular fronts

Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model

Interfaces between rolls in the Swift-Hohenberg equation

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