29,536 research outputs found
Comparing pion production models to MiniBooNE data
Predictions for neutrino-induced charged- and neutral-current single pion
production on CH_2 from theoretical models and Monte Carlo event generators are
compared with the cross section measurements from the MiniBooNE experiment.Comment: Presented at the Eigth International Workshop on Neutrino-Nucleus
Interactions in the Few-GeV Region (NuInt12
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
Complete set of invariants for a Bykov attractor
In this paper we consider an attracting heteroclinic cycle made by a
1-dimensional and a 2-dimensional separatrices between two hyperbolic saddles
having complex eigenvalues. The basin of the global attractor exhibits historic
behaviour and, from the asymptotic properties of these non-converging time
averages, we obtain a complete set of invariants under topological conjugacy in
a neighborhood of the cycle. These invariants are determined by the quotient of
the real parts of the eigenvalues of the equilibria, a linear combination of
their imaginary components and also the transition maps between two cross
sections on the separatrices.Comment: 23 pages, 4 figure
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
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
Marinari-Parisi and Supersymmetric Collective Field Theory
A field theoretic formulation of the Marinari-Parisi supersymmetric matrix
model is established and shown to be equivalent to a recently proposed
supersymmetrization of the bosonic collective string field theory. It also
corresponds to a continuum description of super-Calogero models. The
perturbation theory of the model is developed and, in this approach, an
infinite sequence of vertices is generated. A class of potentials is identified
for which the spectrum is that of a massless boson and Majorana fermion. For
the harmonic oscillator case, the cubic vertices are obtained in an oscillator
basis. For a rather general class of potentials it is argued that one cannot
generate from Marinari-Parisi models a continuum limit similar to that of the
d=1 bosonic string.Comment: 45 page
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
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