1,989 research outputs found
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
On the use of blow up to study regularizations of singularities of piecewise smooth dynamical systems in
In this paper we use the blow up method of Dumortier and Roussarie
\cite{dumortier_1991,dumortier_1993,dumortier_1996}, in the formulation due to
Krupa and Szmolyan \cite{krupa_extending_2001}, to study the regularization of
singularities of piecewise smooth dynamical systems
\cite{filippov1988differential} in . Using the regularization
method of Sotomayor and Teixeira \cite{Sotomayor96}, first we demonstrate the
power of our approach by considering the case of a fold line. We quickly
recover a main result of Bonet and Seara \cite{reves_regularization_2014} in a
simple manner. Then, for the two-fold singularity, we show that the regularized
system only fully retains the features of the singular canards in the piecewise
smooth system in the cases when the sliding region does not include a full
sector of singular canards. In particular, we show that every locally unique
primary singular canard persists the regularizing perturbation. For the case of
a sector of primary singular canards, we show that the regularized system
contains a canard, provided a certain non-resonance condition holds. Finally,
we provide numerical evidence for the existence of secondary canards near
resonance.Comment: To appear in SIAM Journal of Applied Dynamical System
Hilbert's sixteenth problem for polynomial Liénard equations
This article reports on the survey talk 'Hilbert's Sixteenth Problem for Liénard equations,' given by the author at the Oberwolfach Mini-Workshop 'Algebraic and Analytic Techniques for Polynomial Vector Fields.' It is written in a way that it is accessible to a public with heterogeneous mathematical background. The article reviews recent developments and techniques used in the study of Hilbert's 16th problem where the main focus is put on the subclass of polynomial vector fields derived from the Liérd equations
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