Jean-Marc
Ginoux EA PROTEE n° 3819 I.U.T. de Toulon, Université du Sud Génie Mécanique Productique BP 20132, 83957 France Phone: +334 94 14 24 88
Research Domains- Scientific collaborations
·
Leon O. Chua, · Aziz Aaloui,
Cyrille Bertelle, Université du Havre ·
Christophe Letellier,
Université de Rouen · René Lozi, Université de Nice
Sophia Antipolis ·
Dynamical Systems &
Differential Geometry ·
Complex Dynamical Systems ·
Population Dynamics &
Modelling ·
History of Sciences
Flow Curvature Method
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.
· Fixed points and their stability,
· Invariant manifolds (straight lines, planes, hyperplanes),
· Center manifold approximation,
· Normal forms,
· Slow invariant manifold analytical equation,
· First integral
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.
Dynamical Systems & Differential Geometry
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.
Population Dynamics & Modelling
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).
Complex Dynamical Systems
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.
History of Sciences
"La vraie méthode de prévision
du futur des mathématiques est d'étudier leur histoire et leur état
actuel"
Henri Poincaré – Science et Méthode
1908 –
Problematic:
Nonlinear Mathematical and Physical analysis in the 1920’s and the rediscover
of Henri Poincaré limit
cycles by A. A. Andronov.
(C. Gilain et D. Aubin, Institut de Mathématiques de Jussieu,
Paris)
· 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
· 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
· 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
· Invariant
Manifolds of Complex Systems, J.M. Ginoux & B. Rossetto
Complex Systems and Self-organization Modelling, Understanding Complex
Systems. Springer-Verlag, Heidelberg, in press, 2007
· 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
Invited lecturer
·
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
· Slow Invariant Manifolds of Dynamical Systems
Invité au Bristol Centre for Applied Nonlinear Mathematics
par le Professeur Bernd KRAUSKOPF, 15 february 2008, Bristol,
· Curvature of
the Flow of Dynamical Systems
Invité au Centre de Recerca Matematica par le
Professeur Jaume LLIBRE, 10 march
2008, Barcelona, Spain.
· Differential Geometry Applied to Dynamical Systems
Invité au World Congress of Nonlinear Analysis par le Professeur Valery GAIKO, 2-9 july
2008, Orlando, U.S.A.
·
Mathematics in 1st year of Génie Mécanique et Productique
History
of Sciences / Epistemology
· 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
Accreditation to supervise research in Applied Mathematics (2008)
Ph-D in Applied Mathematics (2005)
Senior lecturer in Mathematics
at Mechanics Engineering department