1,358 research outputs found
Hilbert's 16th problem for classical Liénard equations of even degree
AbstractClassical Liénard equations are two-dimensional vector fields, on the phase plane or on the Liénard plane, related to scalar differential equations x¨+f(x)x˙+x=0. In this paper, we consider f to be a polynomial of degree 2l−1, with l a fixed but arbitrary natural number. The related Liénard equation is of degree 2l. We prove that the number of limit cycles of such an equation is uniformly bounded, if we restrict f to some compact set of polynomials of degree exactly 2l−1. The main problem consists in studying the large amplitude limit cycles, of which we show that there are at most l
Canard-like phenomena in piecewise-smooth Van der Pol systems
We show that a nonlinear, piecewise-smooth, planar dynamical system can
exhibit canard phenomena. Canard solutions and explosion in nonlinear,
piecewise-smooth systems can be qualitatively more similar to the phenomena in
smooth systems than piecewise-linear systems, since the nonlinearity allows for
canards to transition from small cycles to canards ``with heads." The canards
are born of a bifurcation that occurs as the slow-nullcline coincides with the
splitting manifold. However, there are conditions under which this bifurcation
leads to a phenomenon called super-explosion, the instantaneous transition from
a globally attracting periodic orbit to relaxations oscillations. Also, we
demonstrate that the bifurcation---whether leading to canards or
super-explosion---can be subcritical.Comment: 17 pages, 11 figure
Fold-Saddle Bifurcation in Non-Smooth Vector Fields on the Plane
This paper presents results concerning bifurcations of 2D piecewise-smooth
dynamical systems governed by vector fields. Generic three parameter families
of a class of Non-Smooth Vector Fields are studied and its bifurcation diagrams
are exhibited. Our main result describes the unfolding of the so called
Fold-Saddle singularity
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
Mixed-mode oscillations in a multiple time scale phantom bursting system
In this work we study mixed mode oscillations in a model of secretion of GnRH
(Gonadotropin Releasing Hormone). The model is a phantom burster consisting of
two feedforward coupled FitzHugh-Nagumo systems, with three time scales. The
forcing system (Regulator) evolves on the slowest scale and acts by moving the
slow nullcline of the forced system (Secretor). There are three modes of
dynamics: pulsatility (transient relaxation oscillation), surge (quasi steady
state) and small oscillations related to the passage of the slow nullcline
through a fold point of the fast nullcline. We derive a variety of reductions,
taking advantage of the mentioned features of the system. We obtain two
results; one on the local dynamics near the fold in the parameter regime
corresponding to the presence of small oscillations and the other on the global
dynamics, more specifically on the existence of an attracting limit cycle. Our
local result is a rigorous characterization of small canards and sectors of
rotation in the case of folded node with an additional time scale, a feature
allowing for a clear geometric argument. The global result gives the existence
of an attracting unique limit cycle, which, in some parameter regimes, remains
attracting and unique even during passages through a canard explosion.Comment: 38 pages, 16 figure
The critical wave speed for the Fisher–Kolmogorov–Petrowskii–Piscounov equation with cut-off
Singular perturbation analysis of a regularized MEMS model
Micro-Electro Mechanical Systems (MEMS) are defined as very small structures
that combine electrical and mechanical components on a common substrate. Here,
the electrostatic-elastic case is considered, where an elastic membrane is
allowed to deflect above a ground plate under the action of an electric
potential, whose strength is proportional to a parameter . Such
devices are commonly described by a parabolic partial differential equation
that contains a singular nonlinear source term. The singularity in that term
corresponds to the so-called "touchdown" phenomenon, where the membrane
establishes contact with the ground plate. Touchdown is known to imply the
non-existence of steady state solutions and blow-up of solutions in finite
time. We study a recently proposed extension of that canonical model, where
such singularities are avoided due to the introduction of a regularizing term
involving a small "regularization" parameter . Methods from
dynamical systems and geometric singular perturbation theory, in particular the
desingularization technique known as "blow-up", allow for a precise description
of steady-state solutions of the regularized model, as well as for a detailed
resolution of the resulting bifurcation diagram. The interplay between the two
main model parameters and is emphasized; in particular,
the focus is on the singular limit as both parameters tend to zero
Tratamiento quirúrgico en las cifosis congénitas: Revisión de 14 pacientes
Los autores efectúan una revisión de 14 pacientes con cifosis congénita,
intervenidos entre los años 1979-1989, con un seguimiento medio de 7 años. La edad
media preoperatoria fue de 11 años (todos ellos mayores de 5 años), con una cifosis media
de 7 9 . En 6 casos se realizó una artrodesis posterior y en 8 una anterior combinada
con una fusión posterior. Inicialmente obtuvieron una corrección media de la curva de
18° con la artrodesis posterior y 20° con la artrodesis combinada. La pérdida postoperatoria
final fue de 10° y 8° respectivamente. En un caso, se produjo una pseudoartrosis
por fusión corta. Como complicaciones postoperatorias en 5 pacientes, una radiculopatÃa,
una infección superficial y cuatro protusiones de material que requirieron su extracción.
Los autores analizan los factores que han podido influir en los resultados obtenidos,
comparándolos posteriormente con los conseguidos por otros centros hospitalarios importantes.Fourteen patients with congenital kyphosis treated surgically between 1979-
1989 were reviewed. All had a follow-up of 2 years or more, with an average follow-up of 7 years.
The average age at surgery was 11 and the average kyphosis was 79°. Six cases had
posterior fusion only and eigth had combined anterior and posterior fusion. The results showed
an average correction of the curve at surgery of 18° with posterior arthrodesis and 20°
with combined arthrodesis. There was thus an average loss of 10° and 8° respectively from
the time of surgery in both types of treatment. Pseudoarthrosis by short fusion ocurred in
one case. Other complications after surgery were 1 radiculopathy, one wound infection and
four rod protusion (six patients). The factors that have influence in this results were analysed.
A comparison from the results of treatment at other medical centers was also carried
out
On the number of limit cycles of the Lienard equation
In this paper, we study a Lienard system of the form dot{x}=y-F(x),
dot{y}=-x, where F(x) is an odd polynomial. We introduce a method that gives a
sequence of algebraic approximations to the equation of each limit cycle of the
system. This sequence seems to converge to the exact equation of each limit
cycle. We obtain also a sequence of polynomials R_n(x) whose roots of odd
multiplicity are related to the number and location of the limit cycles of the
system.Comment: 10 pages, 5 figures. Submitted to Physical Review
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