1,114 research outputs found
Canard Cycles and Poincar\'e Index of Non-Smooth Vector Fields on the Plane
This paper is concerned with closed orbits of non-smooth vector fields on the
plane. For a subclass of non-smooth vector fields we provide necessary and
sufficient conditions for the existence of canard kind solutions. By means of a
regularization we prove that the canard cycles are singular orbits of singular
perturbation problems which are limit periodic sets of a sequence of limit
cycles. Moreover, we generalize the Poincar\'e Index for non-smooth vector
fields.Comment: 20 pages, 25 figure
Melnikov analysis in nonsmooth differential systems with nonlinear switching manifold
We study the family of piecewise linear differential systems in the plane
with two pieces separated by a cubic curve. Our main result is that 7 is a
lower bound for the Hilbert number of this family. In order to get our main
result, we develop the Melnikov functions for a class of nonsmooth differential
systems, which generalizes, up to order 2, some previous results in the
literature. Whereas the first order Melnikov function for the nonsmooth case
remains the same as for the smooth one (i.e. the first order averaged function)
the second order Melnikov function for the nonsmooth case is different from the
smooth one (i.e. the second order averaged function). We show that, in this
case, a new term depending on the jump of discontinuity and on the geometry of
the switching manifold is added to the second order averaged function
Orbitally symmetric systems with applications to planar centers
We present a generalization of the most usual symmetries in differential equations known as the time-reversibility and the equivariance ones. We check that the typical properties are also valid for the new definition that unifies both. With it, we are able to present new families of planar polynomial vector fields having equilibrium points of center type. Moreover, we provide the highest lower bound for the local cyclicity of an equilibrium point of polynomial vector fields of degree 6, M(6) ≥ 48
Cyclicity Near Infinity in Piecewise Linear Vector Fields Having a Nonregular Switching Line
Altres ajuts: acords transformatius de la UABIn this paper we recover the best lower bound for the number of limit cycles in the planar piecewise linear class when one vector field is defined in the first quadrant and a second one in the others. In this class and considering a degenerated Hopf bifurcation near families of centers we obtain again at least five limit cycles but now from infinity, which is of monodromic type, and with simpler computations. The proof uses a partial classification of the center problem when both systems are of center type
Invariant curves and explosion of periodic Islands in systems of piecewise rotations
Copyright © 2005 Society for Industrial and Applied MathematicsInvertible piecewise isometric maps (PWIs) of the plane, in spite of their apparent simplicity, can show a remarkable number of dynamical features analogous to those found in nonlinear smooth area preserving maps. There is a natural partition of the phase space into an exceptional set, ⋶, consisting of the closure of the set of points whose orbits accumulate on discontinuities of the map, and its complement. In this paper we examine a family of noninvertible PWIs on the plane that consist of rotations on each of four atoms, each of which is a quadrant. We show that this family gives examples of global attractors with a variety of geometric structures. On some of these attractors, there appear to be nonsmooth invariant curves within ⋶ that form barriers to ergodicity of any invariant measure supported on ⋶. These invariant curves are observed to appear on perturbations of an “integrable” case where the exceptional set is a union of annuli and it decomposes into a one-dimensional family of interval exchange maps that may be minimal but nonergodic. We have no adequate theoretical explanation for the curves in the nonsmooth case, but they appear to come into existence at the same times as an explosion of periodic islands near where the interval exchanges used to be located. We exhibit another example—a piecewise rotation on the plane with two atoms that also appears to have nonsmooth invariant curves
Cutting and Shuffling a Line Segment: Mixing by Interval Exchange Transformations
We present a computational study of finite-time mixing of a line segment by
cutting and shuffling. A family of one-dimensional interval exchange
transformations is constructed as a model system in which to study these types
of mixing processes. Illustrative examples of the mixing behaviors, including
pathological cases that violate the assumptions of the known governing theorems
and lead to poor mixing, are shown. Since the mathematical theory applies as
the number of iterations of the map goes to infinity, we introduce practical
measures of mixing (the percent unmixed and the number of intermaterial
interfaces) that can be computed over given (finite) numbers of iterations. We
find that good mixing can be achieved after a finite number of iterations of a
one-dimensional cutting and shuffling map, even though such a map cannot be
considered chaotic in the usual sense and/or it may not fulfill the conditions
of the ergodic theorems for interval exchange transformations. Specifically,
good shuffling can occur with only six or seven intervals of roughly the same
length, as long as the rearrangement order is an irreducible permutation. This
study has implications for a number of mixing processes in which
discontinuities arise either by construction or due to the underlying physics.Comment: 21 pages, 10 figures, ws-ijbc class; accepted for publication in
International Journal of Bifurcation and Chao
Is Moodle Accessible for Visually Impaired People?
Proceedings of: 7th International Conference on Web Information Systems and Technologies. Noordwijkerhout, The Netherlands, May 6-9, 2011.Most educational centers are currently using e-learning tools to provide the pedagogical resources for the students, especially in higher education. Nevertheless, some students are not able to access to this information because these authoring tools are not as accessible as they should be. The main aim of this paper is to evaluate if one of the most widely e-learning tool used around the world, Moodle, is accessible for visually impaired people. The evaluation shows that the accessibility guidelines provided by the World Wide Web Consortium (W3C) are not accomplished by the tool. Moreover, it shows that people using screen readers are not able to access to the majority of the functionality of Moodle.The work presented in this paper has been partially founded by MA2VICMR (S2009/TIC-1542), GEMMA (TSI-020302-2010-141) and SAGAS (TSI-020100-2010-184) research projects.Publicad
Chains of infinite order, chains with memory of variable length, and maps of the interval
We show how to construct a topological Markov map of the interval whose
invariant probability measure is the stationary law of a given stochastic chain
of infinite order. In particular we caracterize the maps corresponding to
stochastic chains with memory of variable length. The problem treated here is
the converse of the classical construction of the Gibbs formalism for Markov
expanding maps of the interval
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