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 Aaloui, 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
|
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