1,988 research outputs found
Lyapunov-Schmidt Reduction for Unfolding Heteroclinic Networks of Equilibria and Periodic Orbits with Tangencies
This article concerns arbitrary finite heteroclinic networks in any phase
space dimension whose vertices can be a random mixture of equilibria and
periodic orbits. In addition, tangencies in the intersection of un/stable
manifolds are allowed. The main result is a reduction to algebraic equations of
the problem to find all solutions that are close to the heteroclinic network
for all time, and their parameter values. A leading order expansion is given in
terms of the time spent near vertices and, if applicable, the location on the
non-trivial tangent directions. The only difference between a periodic orbit
and an equilibrium is that the time parameter is discrete for a periodic orbit.
The essential assumptions are hyperbolicity of the vertices and transversality
of parameters. Using the result, conjugacy to shift dynamics for a generic
homoclinic orbit to a periodic orbit is proven. Finally,
equilibrium-to-periodic orbit heteroclinic cycles of various types are
considered
Patterns formation in axially symmetric Landau-Lifshitz-Gilbert-Slonczewski equations
The Landau-Lifshitz-Gilbert-Slonczewski equation describes magnetization
dynamics in the presence of an applied field and a spin polarized current. In
the case of axial symmetry and with focus on one space dimension, we
investigate the emergence of space-time patterns in the form of wavetrains and
coherent structures, whose local wavenumber varies in space. A major part of
this study concerns existence and stability of wavetrains and of front- and
domain wall-type coherent structures whose profiles asymptote to wavetrains or
the constant up-/down-magnetizations. For certain polarization the Slonczewski
term can be removed which allows for a more complete charaterization, including
soliton-type solutions. Decisive for the solution structure is the polarization
parameter as well as size of anisotropy compared with the difference of field
intensity and current intensity normalized by the damping
pde2path - A Matlab package for continuation and bifurcation in 2D elliptic systems
pde2path is a free and easy to use Matlab continuation/bifurcation package
for elliptic systems of PDEs with arbitrary many components, on general two
dimensional domains, and with rather general boundary conditions. The package
is based on the FEM of the Matlab pdetoolbox, and is explained by a number of
examples, including Bratu's problem, the Schnakenberg model, Rayleigh-Benard
convection, and von Karman plate equations. These serve as templates to study
new problems, for which the user has to provide, via Matlab function files, a
description of the geometry, the boundary conditions, the coefficients of the
PDE, and a rough initial guess of a solution. The basic algorithm is a one
parameter arclength continuation with optional bifurcation detection and
branch-switching. Stability calculations, error control and mesh-handling, and
some elementary time-integration for the associated parabolic problem are also
supported. The continuation, branch-switching, plotting etc are performed via
Matlab command-line function calls guided by the AUTO style. The software can
be downloaded from www.staff.uni-oldenburg.de/hannes.uecker/pde2path, where
also an online documentation of the software is provided such that in this
paper we focus more on the mathematics and the example systems
Reversing a granular flow on a vibratory conveyor
Experimental results are presented on the transport properties of granular
materials on a vibratory conveyor. For circular oscillations of the shaking
trough a non-monotonous dependence of the transport velocity on the normalized
acceleration is observed. Two maxima are separated by a regime, where the
granular flow is much slower and, in a certain driving range, even reverses its
direction. A similar behavior is found for a single solid body with a low
coefficient of restitution, whereas an individual glass bead of 1 mm diameter
is propagated in the same direction for all accelerations.Comment: 4 pages, 5 figures, submitted to Applied Physics Letter
pde2path - version 2.0: faster FEM, multi-parameter continuation, nonlinear boundary conditions, and periodic domains - a short manual
pdepath 2.0 is an upgrade of the continuation/bifurcation package pde2path
for elliptic systems of PDEs over bounded 2D domains, based on the Matlab
pdetoolbox. The new features include a more efficient use of FEM, easier
switching between different single parameter continuations, genuine
multi-parameter continuation (e.g., fold continuation), more efficient
implementation of nonlinear boundary conditions, cylinder and torus geometries
(i.e., periodic boundary conditions), and a general interface for adding
auxiliary equations like mass conservation or phase equations for continuation
of traveling waves. The package (library, demos, manuals) can be downloaded at
www.staff.uni-oldenburg.de/hannes.uecker/pde2pat
Unfolding symmetric Bogdanov-Takens bifurcations for front dynamics in a reaction-diffusion system
This manuscript extends the analysis of a much studied singularly perturbed
three-component reaction-diffusion system for front dynamics in the regime
where the essential spectrum is close to the origin. We confirm a conjecture
from a preceding paper by proving that the triple multiplicity of the zero
eigenvalue gives a Jordan chain of length three. Moreover, we simplify the
center manifold reduction and computation of the normal form coefficients by
using the Evans function for the eigenvalues. Finally, we prove the unfolding
of a Bogdanov-Takens bifurcation with symmetry in the model. This leads to
stable periodic front motion, including stable traveling breathers, and these
results are illustrated by numerical computations.Comment: 39 pages, 7 figure
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