Jean-Marc Ginoux Senior Lecturer
(A.S.R.) Doctor in Mathematics Doctor in History of Sciences EA
PROTEE n° 3819 I.U.T.
de Toulon, Université du Sud Génie
Mécanique Productique BP
20132, 83957 France
Research Domains:
Dynamical systems & History of Sciences
Scientific
collaborations:
·
Pr. Leon O. Chua,
·
Pr.
Aziz Alaoui, Cyrille Bertelle, Université du Havre
·
Pr.
Christophe Letellier, Université de Rouen
·
Pr. Jaume Llibre, Universita Autonoma
de Barcelona
·
Pr.
René Lozi, Université de Nice Sophia Antipolis
A. Dynamical
Systems: Flow Curvature
Method (Scholarpedia)
In the framework of Differential Geometry the trajectory
curve, integral of any n-dimensional
dynamical system may be considered as curve
in Euclidean n-space having local
metrics properties of curvatures.
The Flow Curvature Method is based on the
idea that if it is generally impossible to have a closed form of trajectory
curve it still possible to analytically compute its curvatures since it only involves its time derivatives.
The location of the points where the curvature of the trajectory curve vanishes defines a manifold called: flow curvature manifold.
In Differential
Geometry Applied to Dynamical Systems (World Scientific Series on Nonlinear Science, series A) it has been
stated that, since such manifold is defined starting from the time derivatives
of the velocity vector field and so, contains information about the dynamics of
the system, its only knowledge enables to find again the main features of the
dynamical system studied.
These features may be considered as the foundations of
Dynamical Systems Theory. There are
four of them: invariant sets, local bifurcations, slow-fast dynamical systems and integrability.
Thus,
· fixed points and
their stability,
· invariant manifolds
(straight lines, planes, hyperplanes),
· center manifold approximation,
· normal forms,
· slow invariant manifold analytical
equation,
· first integrals of
any n-dimensional dynamical systems
may be directly deduced from the flow curvature manifold.
Since
all the main features of Dynamical
Systems Theory may be found again according to the Flow Curvature Method, i.e., starting from the flow curvature manifold both Dynamical Systems Theory and Flow Curvature Method are consistent and
so Flow Curvature Method represents
an alternative geometric approach for the study of dynamical systems which may
be applied to autonomous as well as non-autonomous
n-dimensional dynamical systems.
This gallery proposes slow manifolds of singularly or non-singularly perturbed dynamical systems of dimension 3, 4, 5, 6…
and non-autonomous dynamical systems
as well as Mathematica Files which have enabled to
plot them.
In Differential
Geometry and Mechanics Applications to Chaotic Dynamical Systems, it
had been established that the flow curvature manifold directly provided the slow
manifold analytical equation of slow-fast autonomous dynamical systems of
dimension two and three, singularly
perturbed such as those of Van der Pol and Chua but also non-singularly
perturbed such as that of Lorenz.
In Slow Invariant Manifolds as
Curvature of the Flow of Dynamical Systems, the approach established in
Differential
Geometry and Mechanics Applications to Chaotic Dynamical Systems, has
been generalized to n-dimensional
dynamical systems. It has been demonstrated that the curvature of the flow, i.e., the curvature of the trajectory curves
directly provides the slow manifold
analytical equation of n-dimensional
slow-fast autonomous dynamical systems. Thus, the flow curvature method has enabled to obtain slow manifolds of many models singularly
perturbed or not of dimension 4, 5 & 6… Moreover, the flow curvature manifold invariance has
been demonstrated while using the concept of invariant manifolds introduced by G. Darboux
in 1878.
In Chaos
in a three dimensional Volterra-Gause model of
predator-prey type, a new model of predator-prey type elaborated from
the seminal works of Vito Volterra* and Giorgii
F. Gause has lead to chaotic attractor in the shape
of a snail (chaotic snail shell).
In Slow
Manifold of a Neuronal Bursting Model, application of the flow curvature
method has directly provided the slow manifold analytical equation of a
Neuronal Bursting Model (NBM).
In Invariant
Manifolds of Complex Systems, local
invariance of the flow curvature manifold analytical equation has been established in the case of
complex systems. Moreover, it has been demonstrated, under certain assumptions,
that such manifold is a local first integral.
|
2011 |
·
Flow curvature method applied to canard
explosion, J.M.
Ginoux & J. Llibre
Journal of Physics A:
Mathematical and Theoretical, 44(46), 465203 (2011) (IF
1,540).
·
Calculus
with MatLab, J.M. Ginoux
in Calculus with Applications, pp.
523-529, A.H Siddiqi, P. Manchanda,
M.Brokate, written under a project of ICTP,
|
2010 |
·
Connecting curves for
dynamical systems, R.
Gilmore, J.M. Ginoux,
T. Jones, C. Letellier & U. S. Freitas
Journal of Physics A:
Mathematical and Theoretical, 43(25), 255101 (2010) (IF
1,540).
·
Topological
analysis of chaotic solution of three-element memristive
circuit, J.M. Ginoux, C. Letellier & L. O. Chua
International Journal of
Bifurcation & Chaos, in press, 11, Vol. 20, pp. 3819-3827 (2010) (IF
0,870).
·
Flow Curvature
Method,
J.M. Ginoux
Scholarpedia,
2010: 10149~10149
|
2009 |
·
Flow
curvature manifolds for shaping chaotic attractors: I Rösslerlike
systems, J.M. Ginoux & C. Letellier
Journal of Physics A:
Mathematical and Theoretical, 42(28), 285101 (2009) (IF
1,540).
|
2008 |
·
Slow Invariant Manifolds as
Curvature of the Flow of Dynamical Systems, J.M. Ginoux,
B. Rossetto & L. Chua
International Journal of Bifurcation & Chaos, 11, Vol. 18, pp.
3409-3430, (2008) (IF 0,870).
·
Application de
Accreditation
to Supervise Research, Université du Sud, 19 juin 2008.
|
2007 |
·
Invariant
Manifolds of Complex Systems, J.M. Ginoux & B. Rossetto
Complex Systems and
Self-organization Modelling, Understanding Complex Systems. Springer-Verlag, Heidelberg, (2007)
|
2006 |
·
Differential Geometry and Mechanics
Applications to Chaotic Dynamical Systems, J.M. Ginoux
& B. Rossetto
International Journal of Bifurcation & Chaos, 4, Vol. 16, pp.
887-910, (2006) (IF 0,870).
·
Dynamical Systems Analysis Using
Differential Geometry, J.M.
Ginoux & B. Rossetto
Complex Computing-Networks,
Series: Springer
Proceedings in Physics, Vol. 104, (2006)
·
Slow Manifold of a Neuronal Bursting
Model, J.M.
Ginoux & B. Rossetto
Emergent Properties in Natural and Artificial Dynamical Systems,
Understanding Complex Systems. Springer-Verlag,
Heidelberg, (2006)
|
2005 |
·
Chaos in a three dimensional Volterra-Gause model of predator-prey type, J.M. Ginoux,
B. Rossetto & J.L. Jamet
International Journal of Bifurcation
& Chaos, 5, Vol. 15, pp. 1689-1708, (2005) (IF 0,870).
·
Stabilité des systèmes dynamiques chaotiques et variétés
singulières, J.M. Ginoux
Ph-D thesis in Applied Mathematics, Université
du Sud, 28 novembre
2005 (mention Très Honorable – Félicitations du jury).
Invited
lecturer
·
Etude de systèmes dynamiques de dimension n par la méthode de courbure
du flot
Invité au LIMSI par le Professeur Luc
PASTUR, 3 november 2011, Orsay, Paris, France.
·
Flow curvature method applied to
canard explosion
Invité au Centre de Recerca Matematica par le Professeur Jaume
LLIBRE, 28 october 2011, Barcelona, Spain.
·
The first “lost” international
conference on nonlinear oscillations (ICNO)
Invité a l’ENOC par le
Professeur Giuseppe REGA, 24-29 july
2011, Roma, Italy.
·
Henri Poincaré’s
legacy in Dynamical Systems
Invité au Dycoec: From Laser Dynamics to Toplogy of Chaos par le Professeur Christophe LETELLIER, 30 june
2011, Rouen.
·
L’histoire
des oscillations de relaxation
Invité aux Archives
Henri Poincaré par le Professeur Scott WALTER, 30 may 2011, Nancy, France.
·
Flow Curvature Method for Canard
Computation
Invité au Dynamics Days Europe par Mike JEFFREY & Mathieu DESROCHES, 6-10 september 2010, Bristol, U.K.
·
Slow Invariant Manifolds of a
Heartbeat Models
Invité à
·
Differential Geometry Applied to
Dynamical Systems
Invité au Mathematical Institute d’Oxford par le Professeur Irene MOROZ, 4 february 2009,
Oxford, U.K.
·
Curvature of the Flow of Dynamical
Systems
Invité au Centre de Recerca Matematica par le
Professeur Jaume LLIBRE, 10 march
2008, Barcelona, Spain.
·
Slow Invariant Manifolds of
Dynamical Systems
Invité au Bristol
Centre for Applied Nonlinear Mathematics par le Professeur
Bernd KRAUSKOPF, 15 february
2008, Bristol,
·
Vers une réduction de la complexité
Invité au séminaire
Dynamique et Interfaces du Laboratoire J.A. Dieudonné par le Professeur René
LOZI, 14 december 2007, Nice.
Reviewer
·
American Mathematical Society
·
European Journal of Applied Mathematics
·
Journal of Geometry and Physics
·
Journal of Mathematical Physics
·
Journal of Physics A: Mathematical
and Theoretical
·
Nonlinear Dynamics
·
IEEE Transactions on Circuits and
Systems
·
International Journal of Bifurcation
& Chaos
·
Proceedings of the Royal Society A
·
Nonlinearity
·
Chaos
·
Chaos, Solitons & Fractals
·
Dynamics of Continuous Discrete and
Impulsive Systems
B. History
of Sciences Les
conférences « oubliées
» d’Henri Poincaré sur la T.S.F. (Bibnum)
« Pour
prévoir l’avenir des mathématiques, la vraie méthode est d’étudier leur
histoire et leur état présent. » Henri Poincaré – Science et Méthode 1908, p. 19 –
· Pour en finir avec le mythe de
·
· Galilée et les expériences de la tour de Pise
· Les grands physiciens : Archimède de Syracuse, Héron d’Alexandrie, …
· Les paradoxes en Physique : le paradoxe d’Olbers,
le paradoxe E.P.R., …
· Récréations scientifiques
· Le modèle prédateur-proie de Vito Volterra
·
Henri Poincaré : une biographie au(x)
quotidien(s), J.M. Ginoux
& Ch. Gerini
Ce livre présente un portrait inédit du
mathématicien français Henri Poincaré à partir de ce qu’en disaient les
journaux de son temps. Un choix abondant de coupures de presse
permet en effet une approche originale du personnage : on y découvre les
faits les plus marquants de sa carrière mais aussi son rôle dans l’espace
public, tant du fait de ses multiples compétences scientifiques et
techniques que pour ses éclairages philosophiques. Doublement académicien, auteur d’ouvrages
largement diffusés, son aura dépassa le seul cercle des érudits pour
toucher le grand public dans les domaines les plus variés, société savante
et presse généraliste ayant fait de lui une sorte de référent dans la
plupart des champs de la connaissance et au-delà. Des anecdotes les plus insolites aux
publications méconnues, en passant par les diverses
polémiques dans lesquelles on l’entraîna souvent malgré lui, les journaux nous
dévoilent un Poincaré inattendu, qui se prêta au jeu de cette dialectique
entre espace savant et espace public et assumant ainsi de façon originale
une forme de « vulgarisation scientifique » comme un rôle d’éclaireur.
|
2012 |
·
The first international
conference on nonlinear oscillations (ICNO), J.M. Ginoux
International Journal of Bifurcation
& Chaos, in press (2012) (IF 0,870).
·
The Singing
Arc: The Oldest Memristor? J.M.
Ginoux & Bruno Rossetto
“Chaos, CNN, Memristors
and Beyond”, in press, (2012)
|
2011 |
·
Analyse mathématique des phénomènes oscillatoires non
linéaires, J.M. Ginoux
Ph-D thesis in History of Sciences, Université Pierre & Marie Curie, 15 mars 2011 à l’Institut Henri
Poincaré (mention Très Honorable).
|
2010 |
·
Poincaré’s forgotten conferences on wireless telegraphy, J.M. Ginoux
& L. Petitgirard
International Journal of
Bifurcation & Chaos, 11, Vol. 20, pp. 3617-3626, (2010) (IF
0,870).
|
2009 |
·
Development
of the nonlinear dynamical systems theory from radio engineering to
electronics, C. Letellier & J.M.
Ginoux
International Journal of Bifurcation
& Chaos, 7, Vol. 19, pp. 2131-2163, (2009) (IF 0,870).
Invited
lecturer
·
De l’expérience de Gérard-Lescuyer à la
triode
Invité au colloque Dycoec par le Professeur Christophe LETELLIER, 14-16
décembre 2009, Rouen
·
L’Histoire des oscillations de
relaxation : de Gérard-Lescuyer à Van der Pol
Invité aux Rencontres du Non-linéaire, 11
mars 2010, Institut Henri Poincaré, Paris
Mini-Symposium 2012 : A tribute to Henri Poincaré
To
celebrate the centenary of the death of Henri Poincaré
a conference is held at the Institut Henri Poincaré on march
14th 2012 in order to present some aspects of his scientific and philosophical works.
· Mathematics in 1st year
of Génie Mécanique et Productique
· Ph-D in Applied Mathematics
(2005), Université
du Sud, Laboratoire PROTEE
· Accreditation to supervise research in Applied Mathematics (2008), Université du Sud, Laboratoire PROTEE
· Ph-D in History of Sciences (2011), Université Pierre & Marie
Curie, Paris VI, Institut de Mathématiques de Jussieu.
Senior lecturer in Mathematics at
Mechanics Engineering department