353 research outputs found
Spiralling dynamics near heteroclinic networks
There are few explicit examples in the literature of vector fields exhibiting
complex dynamics that may be proved analytically. We construct explicitly a
{two parameter family of vector fields} on the three-dimensional sphere
\EU^3, whose flow has a spiralling attractor containing the following: two
hyperbolic equilibria, heteroclinic trajectories connecting them {transversely}
and a non-trivial hyperbolic, invariant and transitive set. The spiralling set
unfolds a heteroclinic network between two symmetric saddle-foci and contains a
sequence of topological horseshoes semiconjugate to full shifts over an
alphabet with more and more symbols, {coexisting with Newhouse phenonema}. The
vector field is the restriction to \EU^3 of a polynomial vector field in
\RR^4. In this article, we also identify global bifurcations that induce
chaotic dynamics of different types.Comment: change in one figur
On Takens' Last Problem: tangencies and time averages near heteroclinic networks
We obtain a structurally stable family of smooth ordinary differential
equations exhibiting heteroclinic tangencies for a dense subset of parameters.
We use this to find vector fields -close to an element of the family
exhibiting a tangency, for which the set of solutions with historic behaviour
contains an open set. This provides an affirmative answer to Taken's Last
Problem (F. Takens (2008) Nonlinearity, 21(3) T33--T36). A limited solution
with historic behaviour is one for which the time averages do not converge as
time goes to infinity. Takens' problem asks for dynamical systems where
historic behaviour occurs persistently for initial conditions in a set with
positive Lebesgue measure.
The family appears in the unfolding of a degenerate differential equation
whose flow has an asymptotically stable heteroclinic cycle involving
two-dimensional connections of non-trivial periodic solutions. We show that the
degenerate problem also has historic behaviour, since for an open set of
initial conditions starting near the cycle, the time averages approach the
boundary of a polygon whose vertices depend on the centres of gravity of the
periodic solutions and their Floquet multipliers.
We illustrate our results with an explicit example where historic behaviour
arises -close of a -equivariant vector field
Dense heteroclinic tangencies near a Bykov cycle
This article presents a mechanism for the coexistence of hyperbolic and
non-hyperbolic dynamics arising in a neighbourhood of a Bykov cycle where
trajectories turn in opposite directions near the two nodes --- we say that the
nodes have different chirality. We show that in the set of vector fields
defined on a three-dimensional manifold, there is a class where tangencies of
the invariant manifolds of two hyperbolic saddle-foci occur densely. The class
is defined by the presence of the Bykov cycle, and by a condition on the
parameters that determine the linear part of the vector field at the
equilibria. This has important consequences: the global dynamics is
persistently dominated by heteroclinic tangencies and by Newhouse phenomena,
coexisting with hyperbolic dynamics arising from transversality. The
coexistence gives rise to linked suspensions of Cantor sets, with hyperbolic
and non-hyperbolic dynamics, in contrast with the case where the nodes have the
same chirality.
We illustrate our theory with an explicit example where tangencies arise in
the unfolding of a symmetric vector field on the three-dimensional sphere
Global bifurcations close to symmetry
Heteroclinic cycles involving two saddle-foci, where the saddle-foci share
both invariant manifolds, occur persistently in some symmetric differential
equations on the 3-dimensional sphere. We analyse the dynamics around this type
of cycle in the case when trajectories near the two equilibria turn in the same
direction around a 1-dimensional connection - the saddle-foci have the same
chirality. When part of the symmetry is broken, the 2-dimensional invariant
manifolds intersect transversely creating a heteroclinic network of Bykov
cycles.
We show that the proximity of symmetry creates heteroclinic tangencies that
coexist with hyperbolic dynamics. There are n-pulse heteroclinic tangencies -
trajectories that follow the original cycle n times around before they arrive
at the other node. Each n-pulse heteroclinic tangency is accumulated by a
sequence of (n+1)-pulse ones. This coexists with the suspension of horseshoes
defined on an infinite set of disjoint strips, where the first return map is
hyperbolic. We also show how, as the system approaches full symmetry, the
suspended horseshoes are destroyed, creating regions with infinitely many
attracting periodic solutions
Transitions of bifurcation diagrams of a forced heteroclinic cycle
A family of periodic perturbations of an attracting robust heteroclinic cycle
defined on the two-sphere is studied by reducing the analysis to that of a
one-parameter family of maps on a circle. The set of zeros of the family forms
a bifurcation diagram on the cylinder. The different bifurcation diagrams and
the transitions between them are obtained as the strength of attraction of the
cycle and the amplitude of the periodic perturbation vary. When the cycle is
weakly attracting further transitions are found giving rise to a frequency
locked invariant torus and to a frequency locked suspended horseshoe, arising
from heteroclinic tangencies in the family of maps. We determine a threshold in
the cycle's attraction strength above which there are no other transitions in
the bifurcation diagrams. Above this threshold and as the period of the
perturbation decreases, frequency locked periodic solutions with arbitrarily
long periods bifurcate from the cycle
Three-dimensional coherent X-ray diffraction imaging of a ceramic nanofoam: determination of structural deformation mechanisms
Ultra-low density polymers, metals, and ceramic nanofoams are valued for
their high strength-to-weight ratio, high surface area and insulating
properties ascribed to their structural geometry. We obtain the labrynthine
internal structure of a tantalum oxide nanofoam by X-ray diffractive imaging.
Finite element analysis from the structure reveals mechanical properties
consistent with bulk samples and with a diffusion limited cluster aggregation
model, while excess mass on the nodes discounts the dangling fragments
hypothesis of percolation theory.Comment: 8 pages, 5 figures, 30 reference
Efeito da temperatura e a participação do fitocromo no controle da germinação de sementes de embaúba
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