3,925 research outputs found
Localised states in an extended Swift-Hohenberg equation
Recent work on the behaviour of localised states in pattern forming partial
differential equations has focused on the traditional model Swift-Hohenberg
equation which, as a result of its simplicity, has additional structure --- it
is variational in time and conservative in space. In this paper we investigate
an extended Swift-Hohenberg equation in which non-variational and
non-conservative effects play a key role. Our work concentrates on aspects of
this much more complicated problem. Firstly we carry out the normal form
analysis of the initial pattern forming instability that leads to
small-amplitude localised states. Next we examine the bifurcation structure of
the large-amplitude localised states. Finally we investigate the temporal
stability of one-peak localised states. Throughout, we compare the localised
states in the extended Swift-Hohenberg equation with the analogous solutions to
the usual Swift-Hohenberg equation
Instabilities and stickiness in a 3D rotating galactic potential
We study the dynamics in the neighborhood of simple and double unstable
periodic orbits in a rotating 3D autonomous Hamiltonian system of galactic
type. In order to visualize the four dimensional spaces of section we use the
method of color and rotation. We investigate the structure of the invariant
manifolds that we found in the neighborhood of simple and double unstable
periodic orbits in the 4D spaces of section. We consider orbits in the
neighborhood of the families x1v2, belonging to the x1 tree, and the z-axis
(the rotational axis of our system). Close to the transition points from
stability to simple instability, in the neighborhood of the bifurcated simple
unstable x1v2 periodic orbits we encounter the phenomenon of stickiness as the
asymptotic curves of the unstable manifold surround regions of the phase space
occupied by rotational tori existing in the region. For larger energies, away
from the bifurcating point, the consequents of the chaotic orbits form clouds
of points with mixing of color in their 4D representations. In the case of
double instability, close to x1v2 orbits, we find clouds of points in the four
dimensional spaces of section. However, in some cases of double unstable
periodic orbits belonging to the z-axis family we can visualize the associated
unstable eigensurface. Chaotic orbits close to the periodic orbit remain sticky
to this surface for long times (of the order of a Hubble time or more). Among
the orbits we studied we found those close to the double unstable orbits of the
x1v2 family having the largest diffusion speed.Comment: 29pages, 25 figures, accepted for publication in the International
Journal of Bifurcation and Chao
Quantifying loopy network architectures
Biology presents many examples of planar distribution and structural networks
having dense sets of closed loops. An archetype of this form of network
organization is the vasculature of dicotyledonous leaves, which showcases a
hierarchically-nested architecture containing closed loops at many different
levels. Although a number of methods have been proposed to measure aspects of
the structure of such networks, a robust metric to quantify their hierarchical
organization is still lacking. We present an algorithmic framework, the
hierarchical loop decomposition, that allows mapping loopy networks to binary
trees, preserving in the connectivity of the trees the architecture of the
original graph. We apply this framework to investigate computer generated
graphs, such as artificial models and optimal distribution networks, as well as
natural graphs extracted from digitized images of dicotyledonous leaves and
vasculature of rat cerebral neocortex. We calculate various metrics based on
the Asymmetry, the cumulative size distribution and the Strahler bifurcation
ratios of the corresponding trees and discuss the relationship of these
quantities to the architectural organization of the original graphs. This
algorithmic framework decouples the geometric information (exact location of
edges and nodes) from the metric topology (connectivity and edge weight) and it
ultimately allows us to perform a quantitative statistical comparison between
predictions of theoretical models and naturally occurring loopy graphs.Comment: 17 pages, 8 figures. During preparation of this manuscript the
authors became aware of the work of Mileyko at al., concurrently submitted
for publicatio
- ā¦