Research Domains: Dynamical Systems & History of Sciences

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


                                                                                                                                                                                                                                                               

 

 

 

 

 

 

 

 

 

 

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. 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 and 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, center manifold approximation, normal forms, slow invariant manifold analytical equation, first integrals of any n-dimensional dynamical systems).

 

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.

 

Publications NEW: All these publications are now in OPEN ACCESS. Just click on the title

 

 

2019

 

·       Comparaison de trois programmes de musculation chez des joueurs de squash de haut niveau,

    Sciberras P. & J.M. Ginoux, Science & Sport, Vol. 34(2), April 2019, 112-115, DOI: 10.1016/j.scispo.2019.01.001.

 

·       Chaos in a predator-prey-based mathematical model for illicit drug consumption,

J.M. Ginoux, R. Naeck, Y.B. Ruhomally, M.Z. Dauhoo & M. Perc,

    Applied Mathematics & Computation, Vol. 347, 15 April 2019, 502-513, DOI: 10.1016/j.amc.2018.10.089.

 

·       Implementing Poincaré Sections for a Chaotic Relaxation Oscillator,

S.  Euzzor, A. Di Garbo, J.M. Ginoux, F.T. Arecchi & R. Meucci,

    IEEE Transactions on Circuits and Systems II: Express Briefs, 24 June 2019, 1-7, DOI: 10.1109/TCSII.2019.2924475.

 

·       Canards Existence in The Hindmarsh-Rose Model, J.M. Ginoux, J. Llibre & K. Tchizawa,

    Mathematical Modelling of Natural Phenomena, Vol. 14(4), 12 July 2019, 1-21, DOI: 10.1109/TCSII.2019.2924475.

 

·       Delayed dynamics in an electronic relaxation oscillator,

A.  Di Garbo, S. Euzzor, J.M. Ginoux, F.T. Arecchi & R. Meucci, Physical Review E, in press.

 

·       Perimeter length determination of the eight-centered oval, J.M. Ginoux & J.C. Golvin,

       The Mathematical Intelligencer, in press

 

2018

 

·       Accurate prediction of continuous blood glucose based on support vector regression and differential evolution algorithm, T. Hamdi, B.A. Jaouher, V. Di Costanzo, F. Fnaiech, E. Moreau & J.M. Ginoux,

    Biocybernetics and Biomedical Engineering, Vol. 38(2), 2018, 362-372, DOI: 10.1016/j.bbe.2018.02.005.

 

·       Is type 1 diabetes a chaotic phenomenon?

    J.M., Ginoux, H. Ruskeepää, M. Perc, R. Naeck, V. Di Costanzo, M. Bouchouicha, F. Fnaiech, M. Sayadi & T. Hamdi,  

    Chaos Solitons & Fractals, Vol. 111, 2018, 198-205, DOI: 10.1016/j.chaos.2018.03.033.

 

·       Continuous Blood Glucose Level Prediction of Type 1 Diabetes Based on Artificial Neural Network,

    B.A. Jaouher, T. Hamdi, N. Fnaiech, V. Di Costanzo, F. Fnaiech & J.M. Ginoux,

    Biocybernetics and Biomedical Engineering, Vol. 38(4), 2018, 828-840, DOI: 10.1016/j.bbe.2018.06.005.

 

·       Torus Breakdown in a Uni Junction Memristor, J.M. Ginoux, R. Meucci, S. Euzzor & A. Di Garbo,

    International Journal of Bifurcation & Chaos, Vol. 28(10), September 2018, 1850128,

    DOI: 10.1142/S0218127418501286 (11 pages).

 

·       Sur la détermination du périmètre de l’ovale à huit centres, J.M. Ginoux & J.-C., Golvin

    Comptes Rendus de l’Académie des Sciences, Vol. 356 (11-12), November-December 2018, 1195-1202,

    DOI: 10.1016/j.crma.2018.10.009.

 

2017

 

       Mathematical Methods in the Applied Sciences, Vol. 40(18), December 2017, 7858-7866, DOI:10.1002/mma.4569.

 

       International Journal of Bifurcation & Chaos, Vol. 27(14), December 2017, 1750220,

       DOI: 10.1142/S0218127417502200 (12 pages).

 

2016

 

·       Canards Existence in FitzHugh-Nagumo and Hodgkin-Huxley Neuronal Models, J.M. Ginoux & J. Llibre,

Mathematical Problems in Engineering, Vol. 2015 (2015), Article ID 342010, 17 pages, DOI:10.1155/2015/342010

 

·       Mathematical Modelling of Sleep Fragmentation Diagnosis, E. Bouazizi, R. Naeck, D. D’Amore, M.-F. Matéo,

    A. Elias, J.-Ph. Suppini, R. Ali Ahmad, A. Raspopa, I. Cartacuzencu, J. Grapperon, O. Tible O. & J.M. Ginoux,

        Biomedical Signal Processing and Control Vol. 24, February 2016, 83-92, DOI:10.1016/j.bspc.2015.10.001.

 

2015

 

        Qualitative Theory of Dynamical Systems, September 2015 (51 pages), DOI:10.1007/s12346-015-0160-1.

 

2014

 

    Sensors (Basel). 2014 Aug 20,14(8):15371-86. DOI: 10.3390/s140815371.

 

2013

 

·       The Slow Invariant Manifold of the Lorenz-Krishnamurthy Model, J.M. Ginoux

         Qualitative Theory of Dynamical Systems, September 2013 (17 pages), DOI: 10.1007/s12346-013-0104-6.

·       Canards from Chua’s circuit, J.M. Ginoux, J. Llibre & L.O. Chua

    International Journal of Bifurcation & Chaos. Vol. 23 (4), 1330010 (2013) (13 pages)

·       Slow invariant manifold of hearbeat model, J.M. Ginoux & B. Rossetto

    Journal of Nonlinear Systems & Applications, vol. 4 (1), pp. 30-35.

·       Mathematical convergences of biodiversity indices, B. Bandeira, J.L. Jamet, D. Jamet & J.M. Ginoux

Ecological Indicators, vol. 29, pp. 522-528.

 

2012

 

·       Is AIS under 20-30° a chaotic dynamical system? J.C. de Mauroy & J.M. Ginoux

    in Studies in Health Technology and Informatics, Volume 176, 2012,
    Research into Spinal Deformities 8, Edited by Tomasz Kotwicki, Theodoros B. Grivas, ISBN 978-1-61499-066-6.).

 

2011

 

·       Flow curvature method applied to canard explosion, J.M. Ginoux & J. Llibre

    Journal of Physics A: Mathematical and Theoretical, 44(46), 465203 (2011).

·       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,  

         Trieste, Italy, August 2011, IK International publisher.

 

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).

·       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).

·       Flow Curvature Method, J.M. Ginoux

    Scholarpedia, 2010: 10149~10149 

 

2009

 

·       Differential Geometry Applied to Dynamical Systems, J.M. Ginoux

     World Scientific Series on Nonlinear Science, series A, Vol. 66, April 2009, with CD Rom, pp. 340.

·       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).

 

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).

·       Application de la Géométrie Différentielle à l’étude des Systèmes Dynamiques, J.M. Ginoux

      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).

·       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).

·       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.

 

Slow Manifold Gallery

 

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.

 

 

B. History of Sciences Poincaré’s forgotten conferences on wireless telegraphy (Scholarpedia)

 

« 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 –

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Conferences

 

 

 

 

 

 

 

 

 

 

Publications

 

·       Enseignement scientifique : histoire, enjeux et débats, J.M. Ginoux, Ellipses (2019)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


·        Pour en finir avec le mythe d’Albert Einstein, J.M. Ginoux, Hermann (2019)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


·       Petites chroniques scientifiques extraordinaires , J.M. Ginoux, Ellipses (2019)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


·       Les Grandes Découvertes de l’Histoire de la Physique, J.M. Ginoux, Ellipses (2018)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


·       History of Nonlinear Oscillations Theory (1880-1940), J.M. Ginoux, Springer-Verlag (2017)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


·       Albert Einstein: a biography through the Time(s), J.M. Ginoux, Hermann (2016)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


·       Histoire de la Théorie des Oscillations Non Linéaires, J.M. Ginoux, forthcoming book, Hermann (2015)

 

Préface Christian Mira

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


·       Henri Poincaré : a biography through the daily papers, J.M. Ginoux & Ch. Gerini, World Scientific (2013)

 

Preface Léon Chua

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


·       Henri Poincaré : une biographie au(x) quotidien(s), J.M. Ginoux & Ch. Gerini, Éditions Ellipses (3 juillet 2012)

 

Préface de Cédric Villani (Médaille Fields, 2010) et Dédicace de Nicolas Poincaré (arrière petit-fils d’Henri Poincaré)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


·       Henri Poincaré par Paul Appell, Ch. Gerini & J.M. Ginoux, Éditions Ellipses (décembre 2013)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


2018

·       From Branly Coherer to Chua Memristor, J.M. Ginoux & T. Cuff

       Fractional Dynamics, Anomalous Transport and Plasma Science,

       Springer International Publishing, Christos Skiadas (Ed.), (2018)

 

2017

·       The Paradox of Vito Volterra’s predator-prey model, J.M. Ginoux

   Lettera Matematica International, December 2017, Vol. 5(4), pp. 305-311, Springer-Milan.

 

2016

·        From Nonlinear Oscillations to Chaos Theory, J.M. Ginoux

    The Foundations of Chaos Revisited : From Poincaré to Recent Advancements,

    Springer International Publishing, Christos Skiadas (Ed.), (2016)

 

2013

·        The Singing Arc: The Oldest Memristor?  J.M. Ginoux & Bruno Rossetto

         “Chaos, CNN, Memristors and Beyond”, World Scientific Publishing, Adamatsky A. & Chen G. (Eds.), (2013)

 

2012

·        Henri Poincaré et l’émergence du concept de cycle limite, J.M. Ginoux

        Quadrature, Novembre 2012, Hors-Série n° 1, pp. 1-7.

·       Self-excited oscillations : from Poincaré to Andronov, J.M. Ginoux

        Nieuw Archief voor Wiskunde (New Archive for Mathematics)

        Royal Dutch Mathematical Society (Koninklijk Wiskundig Genootschap), 5(13) n°3, pp. 170-177 September (2012).

·       Poincaré et la rotation de la Terre, J.M. Ginoux & Ch. Gerini

         Pour la Science, July 2012, n° 417, pp. 78-81.

·       Blondel et les oscillations auto-entretenues, J.M. Ginoux & R. Lozi

            Archive for History of Exact Sciences (2012): 1-46, May 17, 2012

·       The First “Lost” International Conference on Nonlinear Oscillations (I.C.N.O.), J.M. Ginoux

         International Journal of Bifurcation & Chaos, 4, Vol. 22, pp. 3617-3626, (2012) 1250097 (8 pages).

·       Van der Pol and the history of relaxation oscillations: Toward the emergence of a concepts,

    J.M. Ginoux & C. Letellier, Chaos 22, 023120 (2012); http://dx.doi.org/10.1063/1.3670008 (15 pages)

 

 

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).

 

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).                                                                                                            

 

 

Lecture notes

 

·       Mathematics in 1st year of Génie Mécanique et Productique

 

 

Curriculum Vitae

 

·       Ph-D in Applied Mathematics (2005), Université de Toulon, Laboratoire PROTEE

·       Accreditation to supervise research in Applied Mathematics (2008), Université de Toulon, 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