617 research outputs found
Forced patterns near a Turing-Hopf bifurcation
We study time-periodic forcing of spatially-extended patterns near a
Turing-Hopf bifurcation point. A symmetry-based normal form analysis yields
several predictions, including that (i) weak forcing near the intrinsic Hopf
frequency enhances or suppresses the Turing amplitude by an amount that scales
quadratically with the forcing strength, and (ii) the strongest effect is seen
for forcing that is detuned from the Hopf frequency. To apply our results to
specific models, we perform a perturbation analysis on general two-component
reaction-diffusion systems, which reveals whether the forcing suppresses or
enhances the spatial pattern. For the suppressing case, our results explain
features of previous experiments on the CDIMA chemical reaction. However, we
also find examples of the enhancing case, which has not yet been observed in
experiment. Numerical simulations verify the predicted dependence on the
forcing parameters.Comment: 4 pages, 4 figure
Observation and inverse problems in coupled cell networks
A coupled cell network is a model for many situations such as food webs in
ecosystems, cellular metabolism, economical networks... It consists in a
directed graph , each node (or cell) representing an agent of the network
and each directed arrow representing which agent acts on which one. It yields a
system of differential equations , where the component
of depends only on the cells for which the arrow
exists in . In this paper, we investigate the observation problems in
coupled cell networks: can one deduce the behaviour of the whole network
(oscillations, stabilisation etc.) by observing only one of the cells? We show
that the natural observation properties holds for almost all the interactions
Bifurcations in nonlinear models of fluid-conveying pipes supported at both ends
Stationary bifurcations in several nonlinear models of fluid conveying pipes
fixed at both ends are analyzed with the use of Lyapunov-Schmidt reduction and
singularity theory. Influence of gravitational force, curvature and vertical
elastic support on various properties of bifurcating solutions are
investigated. In particular the conditions for occurrence of supercritical and
subcritical bifurcations are presented for the models of Holmes, Thurman and
Mote, and Paidoussis.Comment: to appear in Journal of Fluids and Structures; 6 figure
A torus bifurcation theorem with symmetry
Hopf bifurcation in the presence of symmetry, in situations where the normal form equations decouple into phase/amplitude equations is described. A theorem showing that in general such degeneracies are expected to lead to secondary torus bifurcations is proved. By applying this theorem to the case of degenerate Hopf bifurcation with triangular symmetry it is proved that in codimension two there exist regions of parameter space where two branches of asymptotically stable two-tori coexist but where no stable periodic solutions are present. Although a theory was not derived for degenerate Hopf bifurcations in the presence of symmetry, examples are presented that would have to be accounted for by any such general theory
Predictions of ultra-harmonic oscillations in coupled arrays of limit cycle oscillators
Coupled distinct arrays of nonlinear oscillators have been shown to have a
regime of high frequency, or ultra-harmonic, oscillations that are at multiples
of the natural frequency of individual oscillators. The coupled array
architectures generate an in-phase high-frequency state by coupling with an
array in an anti-phase state. The underlying mechanism for the creation and
stability of the ultra-harmonic oscillations is analyzed. A class of
inter-array coupling is shown to create a stable, in-phase oscillation having
frequency that increases linearly with the number of oscillators, but with an
amplitude that stays fairly constant. The analysis of the theory is illustrated
by numerical simulation of coupled arrays of Stuart-Landau limit cycle
oscillators.Comment: 24 pages, 9 figures, accepted to Phys. Rev. E, in pres
Stabilizing the Discrete Vortex of Topological Charge S=2
We study the instability of the discrete vortex with topological charge S=2
in a prototypical lattice model and observe its mediation through the central
lattice site. Motivated by this finding, we analyze the model with the central
site being inert. We identify analytically and observe numerically the
existence of a range of linearly stable discrete vortices with S=2 in the
latter model. The range of stability is comparable to that of the recently
observed experimentally S=1 discrete vortex, suggesting the potential for
observation of such higher charge discrete vortices.Comment: 4 pages, 4 figure
On the zero set of G-equivariant maps
Let be a finite group acting on vector spaces and and consider a
smooth -equivariant mapping . This paper addresses the question of
the zero set near a zero of with isotropy subgroup . It is known
from results of Bierstone and Field on -transversality theory that the zero
set in a neighborhood of is a stratified set. The purpose of this paper is
to partially determine the structure of the stratified set near using only
information from the representations and . We define an index
for isotropy subgroups of which is the difference of
the dimension of the fixed point subspace of in and . Our main
result states that if contains a subspace -isomorphic to , then for
every maximal isotropy subgroup satisfying , the zero
set of near contains a smooth manifold of zeros with isotropy subgroup
of dimension . We also present a systematic method to study
the zero sets for group representations and which do not satisfy the
conditions of our main theorem. The paper contains many examples and raises
several questions concerning the computation of zero sets of equivariant maps.
These results have application to the bifurcation theory of -reversible
equivariant vector fields
Homoclinic snaking in bounded domains
Homoclinic snaking is a term used to describe the back and forth oscillation of a branch of time-independent spatially localized states in a bistable, spatially reversible system as the localized structure grows in length by repeatedly adding rolls on either side. On the real line this process continues forever. In finite domains snaking terminates once the domain is filled but the details of how this occurs depend critically on the choice of boundary conditions. With periodic boundary conditions the snaking branches terminate on a branch of spatially periodic states. However, with non-Neumann boundary conditions they turn continuously into a large amplitude filling state that replaces the periodic state. This behavior, shown here in detail for the Swift-Hohenberg equation, explains the phenomenon of “snaking without bistability”, recently observed in simulations of binary fluid convection by Mercader, Batiste, Alonso and Knobloch (preprint)
Spontaneous chirality via long-range electrostatic forces
We consider a model for periodic patterns of charges constrained over a
cylindrical surface. In particular we focus on patterns of chiral helices,
achiral rings or vertical lamellae, with the constraint of global
electroneutrality. We study the dependence of the patterns' size and pitch
angle on the radius of the cylinder and salt concentration. We obtain a phase
diagram by using numerical and analytic techniques. For pure Coulomb
interactions, we find a ring phase for small radii and a chiral helical phase
for large radii. At a critical salt concentration, the characteristic domain
size diverges, resulting in macroscopic phase segregation of the components and
restoring chiral symmetry. We discuss possible consequences and generalizations
of our model.Comment: Revtex, 4 pages, 4 figure
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