Imperfect Homoclinic Bifurcations

Experimental observations of an almost symmetric electronic circuit show complicated sequences of bifurcations. These results are discussed in the light of a theory of imperfect global bifurcations. It is shown that much of the dynamics observed in the circuit can be understood by reference to imperfect homoclinic bifurcations without constructing an explicit mathematical model of the system.Comment: 8 pages, 11 figures, submitted to PR

Periodic orbits of period 3 in the disc

Let f be an orientation preserving homeomorphism of the disc D2 which possesses a periodic point of period 3. Then either f is isotopic, relative the periodic orbit, to a homeomorphism g which is conjugate to a rotation by 2 pi /3 or 4 pi /3, or f has a periodic point of least period n for each n in N*.Comment: 7 page

Quasi-morphisms and L^p-metrics on groups of volume-preserving diffeomorphisms

Let M be a smooth compact connected oriented manifold of dimension at least two endowed with a volume form. We show that every homogeneous quasi-morphism on the identity component $Diff_0(M,vol)$ of the group of volume preserving diffeomorphisms of M, which is induced by a quasi-morphism on the fundamental group, is Lipschitz with respect to the L^p-metric on the group $Diff_0(M,vol)$. As a consequence, assuming certain conditions on the fundamental group, we construct bi-Lipschitz embeddings of finite dimensional vector spaces into $Diff_0(M,vol)$.Comment: This is a published versio

Efficient topological chaos embedded in the blinking vortex system

Periodic orbits forhomeomorphisms on the plane give mathematical braids, which are topologically classified into three types by Thurston-Nielsen (T-N) theory; (1) periodic, (2) reducible, and (3) pseudo-Anosov (pA). If the braid is pA, then the homeomorphism must have an infinitely many number of pe-riodic orbits of distinct periods. This kind of complexity induced by the pA braid is called “topological chaos”, which was introduced by Boyland et. al [4] recently. We investigate numerically the topological chaos embedded in the particle mixing by the blinking vortex system introduced by Aref [1]. It has already been known that the system generates the chaotic advection due to the homoclinic chaos, but the chaotic mixing region is restricted locally in the vicinity of the vortex points. In the present study, we propose an in-genious operation of the blinking vortex system that defines a mathematical braid of pA type. The operation not onlygenerates the chaotic mixing region due to the topological chaos, but also ensures global particle mixing in the whole plane. We give a mathematical explanation for the phenomenon by the T-N theory and some numerical evidences to support the explanation. More-over, we makemention of the relation between the topological chaos and the homoclinic chaos in the blinking vortex system

A phenomenological approach to normal form modeling: a case study in laser induced nematodynamics

An experimental setting for the polarimetric study of optically induced dynamical behavior in nematic liquid crystal films has allowed to identify most notably some behavior which was recognized as gluing bifurcations leading to chaos. This analysis of the data used a comparison with a model for the transition to chaos via gluing bifurcations in optically excited nematic liquid crystals previously proposed by G. Demeter and L. Kramer. The model of these last authors, proposed about twenty years before, does not have the central symmetry which one would expect for minimal dimensional models for chaos in nematics in view of the time series. What we show here is that the simplest truncated normal forms for gluing, with the appropriate symmetry and minimal dimension, do exhibit time signals that are embarrassingly similar to the ones found using the above mentioned experimental settings. The gluing bifurcation scenario itself is only visible in limited parameter ranges and substantial aspect of the chaos that can be observed is due to other factors. First, out of the immediate neighborhood of the homoclinic curve, nonlinearity can produce expansion leading to chaos when combined with the recurrence induced by the homoclinic behavior. Also, pairs of symmetric homoclinic orbits create extreme sensitivity to noise, so that when the noiseless approach contains a rich behavior, minute noise can transform the complex damping into sustained chaos. Leonid Shil'nikov taught us that combining global considerations and local spectral analysis near critical points is crucial to understand the phenomenology associated to homoclinic bifurcations. Here this helps us construct a phenomenological approach to modeling experiments in nonlinear dissipative contexts.Comment: 25 pages, 9 figure