264 research outputs found
Localized radial roll patterns in higher space dimensions
Localized roll patterns are structures that exhibit a spatially periodic profile in their center. When following such patterns in a system parameter in one space dimension, the length of the spatial interval over which these patterns resemble a periodic profile stays either bounded, in which case branches form closed bounded curves (“isolas”), or the length increases to infinity so that branches are unbounded in function space (“snaking”). In two space dimensions, numerical computations show that branches of localized rolls exhibit a more complicated structure in which both isolas and snaking occur. In this paper, we analyse the structure of branches of localized radial roll solutions in dimension 1+ε, with 0 < ε 1, through a perturbation analysis. Our analysis sheds light on some of the features visible in the planar case.http://math.bu.edu/people/mabeck/Bramburgeretal18.pdfFirst author draf
State selection in the noisy stabilized Kuramoto-Sivashinsky equation
In this work, we study the 1D stabilized Kuramoto Sivashinsky equation with
additive uncorrelated stochastic noise. The Eckhaus stable band of the
deterministic equation collapses to a narrow region near the center of the
band. This is consistent with the behavior of the phase diffusion constants of
these states. Some connections to the phenomenon of state selection in driven
out of equilibrium systems are made.Comment: 8 pages, In version 3 we corrected minor/typo error
Isolas of 2-Pulse Solutions in Homoclinic Snaking Scenarios
Homoclinic snaking refers to the bifurcation structure of symmetric localised roll patterns that are often found to lie on two sinusoidal “snaking” bifurcation curves, which are connected by an infinite number of “rung” segments along which asymmetric localised rolls of various widths exist. The envelopes of all these structures have a unique maximum and we refer to them as symmetric or asymmetric 1-pulses. In this paper, the existence of stationary 1D patterns of symmetric 2-pulses that consist of two well-separated 1-pulses is established. Corroborating earlier numerical evidence, it is shown that symmetric 2-pulses exist along isolas in parameter space that are formed by parts of the snaking curves and the rungs mentioned above
Spiral anchoring in anisotropic media with multiple inhomogeneities: a dynamical system approach
Various PDE models have been suggested in order to explain and predict the
dynamics of spiral waves in excitable media. In two landmark papers, Barkley
noticed that some of the behaviour could be explained by the inherent Euclidean
symmetry of these models. LeBlanc and Wulff then introduced forced Euclidean
symmetry-breaking (FESB) to the analysis, in the form of individual
translational symmetry-breaking (TSB) perturbations and rotational
symmetry-breaking (RSB) perturbations; in either case, it is shown that spiral
anchoring is a direct consequence of the FESB.
In this article, we provide a characterization of spiral anchoring when two
perturbations, a TSB term and a RSB term, are combined, where the TSB term is
centered at the origin and the RSB term preserves rotations by multiples of
, where is an integer. When
(such as in a modified bidomain model), it is shown that spirals
anchor at the origin, but when (such as in a planar
reaction-diffusion-advection system), spirals generically anchor away from the
origin.Comment: Revised versio
Inertialess multilayer film flow with surfactant: Stability and traveling waves
Multilayer film flow down an inclined plane in the presence of an insoluble surfactant is investigated with particular emphasis on determining flow stability and investigating the possibility of traveling-wave solutions. The investigation is conducted for two or three layers under conditions of Stokes flow and, separately, on the basis of a long-wave assumption. A normal mode linear stability analysis for Stokes flow shows that adding surfactant to one of the film surfaces can destabilize an otherwise stable flow configuration. For the long-wave system, periodic traveling-wave branches are detected and traced, revealing solutions with pulselike solitary waves on each film surface traveling in phase with each other, traveling waves with capillary ridge structures, and solutions with two of the film surfaces almost in contact. Time-periodic traveling-wave solutions are also found. The stability of the traveling waves is determined by solving initial-value problems and by computing eigenvalue spectra. Boundary element simulations for Stokes flow confirm the existence of traveling waves outside the long-wave regime
Lin's method for heteroclinic chains involving periodic orbits
We present an extension of the theory known as Lin's method to heteroclinic
chains that connect hyperbolic equilibria and hyperbolic periodic orbits. Based
on the construction of a so-called Lin orbit, that is, a sequence of continuous
partial orbits that only have jumps in a certain prescribed linear subspace,
estimates for these jumps are derived. We use the jump estimates to discuss
bifurcation equations for homoclinic orbits near heteroclinic cycles between an
equilibrium and a periodic orbit (EtoP cycles)
Depinning of three-dimensional drops from wettability defects
Substrate defects crucially influence the onset of sliding drop motion under
lateral driving. A finite force is necessary to overcome the pinning influence
even of microscale heterogeneities. The depinning dynamics of three-dimensional
drops is studied for hydrophilic and hydrophobic wettability defects using a
long-wave evolution equation for the film thickness profile. It is found that
the nature of the depinning transition explains the experimentally observed
stick-slip motion.Comment: 6 pages, 9 figures, submitted to ep
Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation
In a companion paper, we established nonlinear stability with detailed
diffusive rates of decay of spectrally stable periodic traveling-wave solutions
of reaction diffusion systems under small perturbations consisting of a
nonlocalized modulation plus a localized perturbation. Here, we determine
time-asymptotic behavior under such perturbations, showing that solutions
consist to leading order of a modulation whose parameter evolution is governed
by an associated Whitham averaged equation
Effects of steady state free precession parameters on cardiac mass, function, and volumes
G0400444/Medical Research Council/United Kingdom
Wellcome Trust/United Kingdo
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