118 research outputs found
Animated phase portraits of nonlinear and chaotic dynamical systems
The aim of this section is to present programs allowing to high- light the
slow-fast evolution of the solutions of nonlinear and chaotic dynamical systems
such as: Van der Pol, Chua and Lorenz models. These programs provide animated
phase portraits in dimension two and three, i.e. integration step by step"
which are useful tools enabling to understand the dynamic of such systems
Self-Excited Oscilations : from Poincar\'e to Andronov
In 1908 Henri Poincar\'e gave a series of 'forgotten lectures' on wireless
telegraphy in which he demonstrated the existence of a stable limit cycle in
the phase plane. In 1929 Aleksandr Andronov published a short note in the
Comptes Rendus in which he stated that there is a correspondence between the
periodic solution of self-oscillating systems and the concept of stable limit
cycles introduced by Poincar\'e. In this article Jean-Marc Ginoux describes
these two major contributions to the development of non-linear oscillation
theory and their reception in France
The First "lost" International Conference on Nonlinear Oscillations (I.C.N.O.)
From 28 to 30 January 1933 was held at the Institut Henri Poincar\'{e}
(Paris) the first International Conference of Nonlinear Oscillations organized
at the initiative of the Dutch physicist Balthazar Van der Pol and of the
Russian mathematician Nikola\"{i} Dmitrievich Papaleksi. The discovery of this
forgotten event, whose virtually no trace remains, was made possible thanks to
the report written by Papaleksi at his return in USSR. This document has
revealed, one the one hand, the list of participants who included French
mathematicians: Alfred Li\'{e}nard, \'{E}lie and Henri Cartan, Henri Abraham,
Eug\`{e}ne Bloch, L\'{e}on Brillouin, Yves Rocard, ... and, on the other hand
the content of presentations and discussions. The analysis of the minutes of
this conference presented here for the first time highlights the role and
involvement of the French scientific community in the development of the theory
of nonlinear oscillations.Comment: 8 pages, 1 figure, International Journal of Bifurcation and Chaos,
Vol. 22, No. 04, April 201
The Slow Invariant Manifold of the Lorenz--Krishnamurthy Model
During this last decades, several attempts to construct slow invariant
manifold of the Lorenz-Krishnamurthy five-mode model of slow-fast interactions
in the atmosphere have been made by various authors. Unfortunately, as in the
case of many two-time scales singularly perturbed dynamical systems the various
asymptotic procedures involved for such a construction diverge. So, it seems
that till now only the first-order and third-order approximations of this slow
manifold have been analytically obtained. While using the Flow Curvature Method
we show in this work that one can provide the eighteenth-order approximation of
the slow manifold of the generalized Lorenz-Krishnamurthy model and the
thirteenth-order approximation of the "conservative" Lorenz-Krishnamurthy
model. The invariance of each slow manifold is then established according to
Darboux invariance theorem
Poincare's forgotten conferences on wireless telegraphy
At the beginning of the twentieth century while Henri Poincar\'e (1854-1912)
was already deeply involved in the developments of wireless telegraphy, he was
invited, in 1908, to give a series of lectures at the \'Ecole Sup\'erieure des
Postes et T\'el\'egraphes (today Sup'T\'elecom). In the last part of his
presentation he established that the necessary condition for the existence of a
stable regime of maintained oscillations in a device of radio engineering
completely analogous to the triode: the singing arc, is the presence in the
phase plane of stable limit cycle. The aim of this work is to prove that the
correspondence highlighted by Andronov between the periodic solution of a
non-linear second order differential equation and Poincar\'e's concept of limit
cycle has been carried out by Poincar\'e himself, twenty years before in these
forgotten conferences of 1908
Differential Geometry and Mechanics Applications to Chaotic Dynamical Systems
The aim of this article is to highlight the interest to apply Differential
Geometry and Mechanics concepts to chaotic dynamical systems study. Thus, the
local metric properties of curvature and torsion will directly provide the
analytical expression of the slow manifold equation of slow-fast autonomous
dynamical systems starting from kinematics variables velocity, acceleration and
over-acceleration or jerk. The attractivity of the slow manifold will be
characterized thanks to a criterion proposed by Henri Poincar\'e. Moreover, the
specific use of acceleration will make it possible on the one hand to define
slow and fast domains of the phase space and on the other hand, to provide an
analytical equation of the slow manifold towards which all the trajectories
converge. The attractive slow manifold constitutes a part of these dynamical
systems attractor. So, in order to propose a description of the geometrical
structure of attractor, a new manifold called singular manifold will be
introduced. Various applications of this new approach to the models of Van der
Pol, cubic-Chua, Lorenz, and Volterra-Gause are proposed
The singing arc: the oldest memristor?
On April 30th 2008, the journal Nature announced that the missing circuit
element, postulated thirty-seven years before by Professor Leon O. Chua has
been found. Thus, after the capacitor, the resistor and the inductor, the
existence of a fourth fundamental element of electronic circuits called
"memristor" was established. In order to point out the importance of such a
discovery, the aim of this article is first to propose an overview of the
manner with which the three others have been invented during the past
centuries. Then, a comparison between the main properties of the singing arc,
i.e. a forerunner device of the triode used in Wireless Telegraphy, and that of
the memristor will enable to state that the singing arc could be considered as
the oldest memristor
Flow curvature manifolds for shaping chaotic attractors: Rossler-like systems
Poincar\'e recognized that phase portraits are mainly structured around fixed
points. Nevertheless, the knowledge of fixed points and their properties is not
sufficient to determine the whole structure of chaotic attractors. In order to
understand how chaotic attractors are shaped by singular sets of the
differential equations governing the dynamics, flow curvature manifolds are
computed. We show that the time dependent components of such manifolds
structure Rossler-like chaotic attractors and may explain some limitation in
the development of chaotic regimes
Canards Existence in FitzHugh-Nagumo and Hodgkin-Huxley Neuronal Models
In a previous paper we have proposed a new method for proving the existence
of "canard solutions" for three and four-dimensional singularly perturbed
systems with only one fast variable which improves the methods used until now.
The aim of this work is to extend this method to the case of four-dimensional
singularly perturbed systems with two slow and two fast variables. This method
enables to state a unique generic condition for the existence of "canard
solutions" for such four-dimensional singularly perturbed systems which is
based on the stability of folded singularities (pseudo singular points in this
case) of the normalized slow dynamics deduced from a well-known property of
linear algebra. This unique generic condition is identical to that provided in
previous works. Applications of this method to the famous coupled
FitzHugh-Nagumo equations and to the Hodgkin-Huxley model enables to show the
existence of "canard solutions" in such systems
Slow Manifold of a Neuronal Bursting Model
Comparing neuronal bursting models (NBM) with slow-fast autonomous dynamical
systems (S-FADS), it appears that the specific features of a (NBM) do not allow
a determination of the analytical slow manifold equation with the singular
approximation method. So, a new approach based on Differential Geometry,
generally used for (S-FADS), is proposed. Adapted to (NBM), this new method
provides three equivalent manners of determination of the analytical slow
manifold equation. Application is made for the three-variables model of
neuronal bursting elaborated by Hindmarsh and Rose which is one of the most
used mathematical representation of the widespread phenomenon of oscillatory
burst discharges that occur in real neuronal cells.Comment: arXiv admin note: text overlap with arXiv:1408.171
- …