14,520 research outputs found
Bifurcation analysis in an associative memory model
We previously reported the chaos induced by the frustration of interaction in
a non-monotonic sequential associative memory model, and showed the chaotic
behaviors at absolute zero. We have now analyzed bifurcation in a stochastic
system, namely a finite-temperature model of the non-monotonic sequential
associative memory model. We derived order-parameter equations from the
stochastic microscopic equations. Two-parameter bifurcation diagrams obtained
from those equations show the coexistence of attractors, which do not appear at
absolute zero, and the disappearance of chaos due to the temperature effect.Comment: 19 page
Bifurcation Analysis of the Watt Governor System
This paper pursues the study carried out by the authors in {\it Stability and
Hopf bifurcation in the Watt governor system} \cite{smb}, focusing on the
codimension one Hopf bifurcations in the centrifugal Watt governor differential
system, as presented in Pontryagin's book {\it Ordinary Differential
Equations}, \cite{pon}. Here are studied the codimension two and three Hopf
bifurcations and the pertinent Lyapunov stability coefficients and bifurcation
diagrams, illustrating the number, types and positions of bifurcating small
amplitude periodic orbits, are determined. As a consequence it is found a
region in the space of parameters where an attracting periodic orbit coexists
with an attracting equilibrium.Comment: 34 pages, 9 figure
Bifurcation analysis in a frustrated nematic cell
Using Landau-de Gennes theory to describe nematic order, we study a
frustrated cell consisting of nematic liquid crystal confined between two
parallel plates. We prove the uniqueness of equilibrium states for a small cell
width. Letting the cell width grow, we study the behaviour of this unique
solution. Restricting ourselves to a certain interval of temperature, we prove
that this solution becomes unstable at a critical value of the cell width.
Moreover, we show that this loss of stability comes with the appearance of two
new solutions: there is a symmetric pitchfork bifurcation. This picture agrees
with numerical simulations performed by P. Palffy-Muhorray, E.C. Gartland and
J.R. Kelly. Some of the methods that we use in the present paper apply to other
situations, and we present the proofs in a general setting. More precisely, the
paper contains the proof of a general uniqueness result for a class of
perturbed quasilinear elliptic systems, and general considerations about
symmetric solutions and their stability, in the spirit of Palais' Principle of
Symmetric Criticality
A bifurcation analysis for the Lugiato-Lefever equation
The Lugiato-Lefever equation is a cubic nonlinear Schr\"odinger equation,
including damping, detuning and driving, which arises as a model in nonlinear
optics. We study the existence of stationary waves which are found as solutions
of a four-dimensional reversible dynamical system in which the evolutionary
variable is the space variable. Relying upon tools from bifurcation theory and
normal forms theory, we discuss the codimension 1 bifurcations. We prove the
existence of various types of steady solutions, including spatially localized,
periodic, or quasi-periodic solutions
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