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

 

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

 

·       Sleep Diversity Index for Sleep Fragmentation Analysis, R. Naeck, D. D’Amore, M.F. Mateo, A. Elias, J.P. Suppini, A. Rabat, P. Arlotto, M. Grimaldi, E. Moreau & J.M. Ginoux, Journal of Nonlinear Systems & Applications (in press).

·       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) (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,  

         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) (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

 

·       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) (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 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) (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).

 

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

 

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

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


·       Albert Einstein : une biographie à travers le temps, J.M. Ginoux, Hermann (2016)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


·       History of Nonlinear Oscillations Theory (1880-1940), J.M. Ginoux forthcoming book (2016)

 

This book aims studying the French scientific contribution to the process of developing the mathematical theory of nonlinear oscillations, particularly in the period between the wars.

It is shown that, contrary to what is often written, this contribution appears to be of great importance, both through the work of French scholars caught in all their diversity (mathematicians, physicists, engineers) and the role of a real crossroads Science then played by France.

However, to understand this situation, the period of this study has been chronologically extended to the period before the First World War. The mainstreaming of diverse literature sources such as periodicals on electricity has then allowed to highlight a very important text of Henri Poincaré from 1908, remained unknown. In this work, he applied the concept of limit cycle, introduced in 1882 in his own works, to study the stability of the oscillations of a device for radio engineering. The “discovery” of this text led in particular to modify the classical point of view of the historiography, which hitherto attributed to the Russian mathematician Andronov credit for having established this correspondence, in 1929. In this text of Poincaré, as in most of those who were the subject of this work, there appears a strong interaction between science and technology or, more precisely, between mathematical analysis and radio engineering. This feature is one of the main components of the process of developing the theory of nonlinear oscillations.

 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


        Download free Chapter already available: The early self-oscillating systems

 

·       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)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


2013

 

·       Les inattendus de la Science, J.M. Ginoux

 

    Chronique mensuelle de Métropole Var depuis Juin 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) journal published by the 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) (IF 0,870).

·       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)

·       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).                                                                                            

 

 

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