arXiv subjects

Jean-Marc Ginoux

Publications and source records attributed to Jean-Marc Ginoux.

At least 19 recordsLinked to original sources

arXiv2025

Comments on Weinstein's comments arXiv: 2509.09361 & 2510.03793

In 2024, after thirty years of research on this subject, I published a book entitled: \textit{Poincaré, Einstein and the discovery of special relativity. An end to the controversy} \cite{Ginoux2024}. In September 2025, Galina Weinstein published a review of this book entitled: \textit{Convergences and Divergences: Einstein Poincaré and Special Relativity} (arXiv:2509.09361) in which she harshly criticized my work in an unfair and error filled manner. The, she published a second comment (arXiv: 2510.03793) in which she added she added insults about me to all her mistakes, falsehoods and misleading criticisms. So I've decided to reply to her comments, in an academic way (as she would normally have done), in order to demonstrate that her allegedly ``novel way'' of reconstructing the history of the theory of special relativity is purely based on her own interpretation of the facts and not on the facts themselves. To this aim, I will follow the structure of each Weinstein's comment (arXiv: 2509.09361 \& 2510.03793) and I will highlight section by section all the erroneous things she has reported and repeated.

physics.hist-ph↗

arXiv2021

A physical memristor based Muthuswamy-Chua-Ginoux system

In 1976, Leon Chua showed that a thermistor can be modeled as a memristive device. Starting from this statement we designed a circuit that has four circuit elements: a linear passive inductor, a linear passive capacitor, a nonlinear resistor and a thermistor, that is, a nonlinear "locally active" memristor. Thus, the purpose of this work was to use a physical memristor, the thermistor, in a Muthuswamy-Chua chaotic system (circuit) instead of memristor emulators. Such circuit has been modeled by a new three-dimensional autonomous dynamical system exhibiting very particular properties such as the transition from torus breakdown to chaos. Then, mathematical analysis and detailed numerical investigations have enabled to establish that such a transition corresponds to the so-called route to Shilnikov spiral chaos but gives rise to a "double spiral attractor".

math.DS↗

arXiv2021

Generalized Liénard systems, singularly perturbed systems, Flow Curvature Method

In his famous book entitled \textit{Theory of Oscillations}, Nicolas Minorsky wrote: "\textit{each time the system absorbs energy the curvature of its trajectory decreases} and \textit{vice versa}". According to the \textit{Flow Curvature Method}, the location of the points where the \textit{curvature of trajectory curve}, integral of such planar \textit{singularly dynamical systems}, vanishes directly provides a first order approximation in $\varepsilon$ of its \textit{slow invariant manifold} equation. By using this method, we prove that, in the $\varepsilon$-vicinity of the \textit{slow invariant manifold} of generalized Liénard systems, the \textit{curvature of trajectory curve} increases while the \textit{energy} of such systems decreases. Hence, we prove Minorsky's statement for the generalized Liénard systems. Then, we establish a relationship between \textit{curvature} and \textit{energy} for such systems. These results are then exemplified with the classical Van der Pol and generalized Liénard \textit{singularly perturbed systems}.

math.DS↗

arXiv2020

Slow Invariant Manifolds of Slow-Fast Dynamical Systems

Slow-fast dynamical systems, i.e., singularly or non-singularly perturbed dynamical systems possess slow invariant manifolds on which trajectories evolve slowly. Since the last century various methods have been developed for approximating their equations. This paper aims, on the one hand, to propose a classification of the most important of them into two great categories: singular perturbation-based methods and curvature-based methods, and on the other hand, to prove the equivalence between any methods belonging to the same category and between the two categories. Then, a deep analysis and comparison between each of these methods enable to state the efficiency of the Flow Curvature Method which is exemplified with paradigmatic Van der Pol singularly perturbed dynamical system and Lorenz slow-fast dynamical system.

nlin.CD↗

arXiv2020

Minimal universal model for chaos in laser with feedback

We revisit the model of the laser with feedback and the minimal nonlinearity leading to chaos. Although the model has its origin in laser physics, with peculiarities related to the CO2 laser, it belongs to the class of the three dimensional paradigmatic nonlinear oscillator models giving chaos. The proposed model contains three key nonlinearities, two of which are of the type xy, where x and y are the fast and slow variables. The third one is of the type xz^2, where z is an intermediate feedback variable. We analytically demonstrate that it is essential for producing chaos via local or global homoclinic bifurcations. Its electronic implementation in the range of kilo Hertz region confirms its potential in describing phenomena evolving on different time scales.

physics.optics↗

arXiv2019

On the perimeter length determination of the eight-centered oval

On the perimeter length determination of the eight-centered oval. Several studies have shown that an eight-centered oval coincides almost perfectly with the ellipse constructed on the same axes and can be considered as a representation of the latter provided that the radii of the arcs of circles that compose it had been suitably chosen. Its perimeter's computation is then reduced to the simple sum of arc lengths of circles. However, it doesnot seem to us that this calculation, which could prove to be useful, has never been performed nor published. This note aims thus to present a geometric demonstration of the perimeter length determination of the eight-centered oval.

math.MG↗

arXiv2019

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

Recently, a mathematical model describing the illicit drug consumption in a population consisting of drug users and non-users has been proposed. The model describes the dynamics of non-users, experimental users, recreational users, and addict users within a population. The aim of this work is to propose a modified version of this model by analogy with the classical predator-prey models, in particular considering non-users as prey and users as predator. Hence, our model includes a stabilizing effect of the growth rate of the prey, and a destabilizing effect of the predator saturation. Functional responses of Verhulst and of Holling type II have been used for modeling these effects. To forecast the marijuana consumption in the states of Colorado and Washington, we used data from Hanley (2013) and a genetic algorithm to calibrate the parameters in our model. Assuming that the population of non-users increases in proportion with the demography, and following the seminal works of Sir Robert May (1976), we use the growth rate of non-users as the main bifurcation parameter. For the state of Colorado, the model first exhibits a limit cycle, which agrees quite accurately with the reported periodic data in Hanley (2013). By further increasing the growth rate of non-users, the population then enters into two chaotic regions, within which the evolution of the variables becomes unpredictable. For the state of Washington, the model also exhibits a periodic solution, which is again in good agreement with observed data. A chaotic region for Washington is likewise observed in the bifurcation diagram. Our research confirms that mathematical models can be a useful tool for better understanding illicit drug consumption, and for guiding policy-makers towards more effective policies to contain this epidemic.

math.DS↗

arXiv2019

Is type 1 diabetes a chaotic phenomenon?

A database of ten type 1 diabetes patients wearing a continuous glucose monitoring device has enabled to record their blood glucose continuous variations every minute all day long during fourteen consecutive days. These recordings represent, for each patient, a time series consisting of 1 value of glycaemia per minute during 24 hours and 14 days, i.e., 20,160 data point. Thus, while using numerical methods, these time series have been anonymously analyzed. Nevertheless, because of the stochastic inputs induced by daily activities of any human being, it has not been possible to discriminate chaos from noise. So, we have decided to keep only the 14 nights of these ten patients. Then, the determination of the time delay and embedding dimension according to the delay coordinate embedding method has allowed us to estimate for each patient the correlation dimension and the maximal Lyapunov exponent. This has led us to show that type 1 diabetes could indeed be a chaotic phenomenon. Once this result has been confirmed by the determinism test, we have computed the Lyapunov time and found that the limit of predictability of this phenomenon is nearly equal to half the 90-minutes sleep-dream cycle. We hope that our results will prove to be useful to characterize and predict blood glucose variations.

q-bio.OT↗

arXiv2019

Torus Breakdown in a Uni Junction Memristor

Experimental study of a uni junction transistor (UJT) has enabled to show that this electronic component has the same features as the so-called "memristor". So, we have used the memristor's direct current (DC) vM--iM characteristic for modeling the UJT's DC current--voltage characteristic. This has led us to confirm on the one hand, that the UJT is a memristor and, on the other hand, to propose a new four-dimensional autonomous dynamical system allowing to describe experimentally observed phenomena such as the transition from a limit cycle to torus breakdown.

eess.SY↗

arXiv2018

The Slow Invariant Manifold of the Lorenz--Krishnamurthy Model

During this last decades, several attempts to construct slow invariant manifold of the Lorenz-Krishnamurthy five-mode model of slow-fast interactions in the atmosphere have been made by various authors. Unfortunately, as in the case of many two-time scales singularly perturbed dynamical systems the various asymptotic procedures involved for such a construction diverge. So, it seems that till now only the first-order and third-order approximations of this slow manifold have been analytically obtained. While using the Flow Curvature Method we show in this work that one can provide the eighteenth-order approximation of the slow manifold of the generalized Lorenz-Krishnamurthy model and the thirteenth-order approximation of the "conservative" Lorenz-Krishnamurthy model. The invariance of each slow manifold is then established according to Darboux invariance theorem.

math.DS↗

arXiv2018

Canards Existence in Memristor's Circuits

The aim of this work is to propose an alternative method for determining the condition of existence of "canard solutions" for three and four-dimensional singularly perturbed systems with only one fast variable in the folded saddle case. This method enables to state a unique generic condition for the existence of "canard solutions" for such three and four-dimensional singularly perturbed systems which is based on the stability of folded singularities of the normalized slow dynamics deduced from a well-known property of linear algebra. This unique generic condition is perfectly identical to that provided in previous works. Application of this method to the famous three and four-dimensional memristor canonical Chua's circuits for which the classical piecewise-linear characteristic curve has been replaced by a smooth cubic nonlinear function according to the least squares method enables to show the existence of "canard solutions" in such Memristor Based Chaotic Circuits.

math.DS↗

arXiv2018

The paradox of Vito Volterra's predator-prey model

This article is dedicated to the late Giorgio Israel. R{é}sum{é}. The aim of this article is to propose on the one hand a brief history of modeling starting from the works of Fibonacci, Robert Malthus, Pierre Francis Verhulst and then Vito Volterra and, on the other hand, to present the main hypotheses of the very famous but very little known predator-prey model elaborated in the 1920s by Volterra in order to solve a problem posed by his son-in-law, Umberto D'Ancona. It is thus shown that, contrary to a widely-held notion, Volterra's model is realistic and his seminal work laid the groundwork for modern population dynamics and mathematical ecology, including seasonality, migration, pollution and more. 1. A short history of modeling 1.1. The Malthusian model. If the rst scientic view of population growth seems to be that of Leonardo Fibonacci [2], also called Leonardo of Pisa, whose famous sequence of numbers was presented in his Liber abaci (1202) as a solution to a population growth problem, the modern foundations of population dynamics clearly date from Thomas Robert Malthus [20]. Considering an ideal population consisting of a single homogeneous animal species, that is, neglecting the variations in age, size and any periodicity for birth or mortality, and which lives alone in an invariable environment or coexists with other species without any direct or indirect inuence, he founded in 1798, with his celebrated claim Population, when unchecked, increases in a geometrical ratio, the paradigm of exponential growth. This consists in assuming that the increase of the number N (t) of individuals of this population, during a short interval of time, is proportional to N (t). This translates to the following dierential equation : (1) dN (t) dt = $ε$N (t) where $ε$ is a constant factor of proportionality that represents the growth coe-cient or growth rate. By integrating (1) we obtain the law of exponential growth or law of Malthusian growth (see Fig. 1). This law, which does not take into account the limits imposed by the environment on growth and which is in disagreement with the actual facts, had a profound inuence on Charles Darwin's work on natural selection. Indeed, Darwin [1] founded the idea of survival of the ttest on the 1. According to Frontier and Pichod-Viale [3] the correct terminology should be population kinetics, since the interaction between species cannot be represented by forces. 2. A population is dened as the set of individuals of the same species living on the same territory and able to reproduce among themselves.

math.HO↗

arXiv2018

Canards Existence in FitzHugh-Nagumo and Hodgkin-Huxley Neuronal Models

In a previous paper we have proposed a new method for proving the existence of "canard solutions" for three and four-dimensional singularly perturbed systems with only one fast variable which improves the methods used until now. The aim of this work is to extend this method to the case of four-dimensional singularly perturbed systems with two slow and two fast variables. This method enables to state a unique generic condition for the existence of "canard solutions" for such four-dimensional singularly perturbed systems which is based on the stability of folded singularities (pseudo singular points in this case) of the normalized slow dynamics deduced from a well-known property of linear algebra. This unique generic condition is identical to that provided in previous works. Applications of this method to the famous coupled FitzHugh-Nagumo equations and to the Hodgkin-Huxley model enables to show the existence of "canard solutions" in such systems.

nlin.CD↗

arXiv2015

Henri Poincaré et l'émergence du concept de cycle limite

The concept of "limit cycle" was introduced by Henri Poincaré in his second memoir "On curves defined by a differential equation" in 1882. From the point of view of physics, a stable limit cycle (or attractive) is the representation of the periodic solution of a (mechanical or electrical) dissipative system whose oscillations are maintained by the system itself. Conversely, the existence of a stable limit cycle ensures the maintenance of the oscillations. So far, the historiography considered that the Russian mathematician Aleksandr Andronov was the first to establish such a correspondence between the periodic solution of a self-oscillating system and Poincaré's concept of limit cycle. The recent discovery of a series of lectures given by Henri Poincaré in 1908 at the School of Post and Telegraph (now Telecom Paris Tech) proves that he had already used his limit cycle concept to establish the existence of a regime of maintained waves in a wireless device of radiotechnics. This article aims on the one hand to trace the emergence of this concept since its creation by Poincaré and the other hand to highlight its importance and role in the history of non-linear oscillations.

nlin.CD↗

arXiv2015

Self-Excited Oscilations : from Poincaré to Andronov

In 1908 Henri Poincaré gave a series of 'forgotten lectures' on wireless telegraphy in which he demonstrated the existence of a stable limit cycle in the phase plane. In 1929 Aleksandr Andronov published a short note in the Comptes Rendus in which he stated that there is a correspondence between the periodic solution of self-oscillating systems and the concept of stable limit cycles introduced by Poincaré. In this article Jean-Marc Ginoux describes these two major contributions to the development of non-linear oscillation theory and their reception in France.

nlin.CD↗

arXiv2014

Blondel et les oscillations auto-entretenues

In 1893, the "physicist-engineer" André Blondel invents the oscilloscope for displaying voltage and current variables. With this powerful means of investigation, he first studies the phenomena of the arc then used for the coastal and urban lighting and then, the singing arc used as a transmitter of radio waves in wireless telegraphy. In 1905, he highlights a new type of non-sinusoidal oscillations in the singing arc. Twenty years later, Balthasar van der Pol will recognize that such oscillations were in fact "relaxation oscillations". To explain this phenomenon, he uses a representation in the phase plane and shows that its evolution takes the form of small cycles. During World War I the triode gradually replaces the singing arc in transmission systems. At the end of the war, using analogy, Blondel transposes to the triode most of the results he had obtained for the singing arc. In April 1919, he publishes a long memoir in which he introduces the terminology "self-sustained oscillations" and proposes to illustrate this concept starting from the example of the Tantalus cup which is then picked up by Van der Pol and Philippe Le Corbeiller. He then provides the definition of a self sustained system which is quite similar to that given later by Aleksandr Andronov and Van der Pol. To study the stability of oscillations sustained by the triode and by the singing arc he uses, this time, a representation in the complex plane and he expresses the amplitude in polar coordinates. He then justifies the maintaining of oscillations by the existence cycles which nearly present all the features of Poincaré's limit cycles. Finally, in November 1919, Blondel performs, a year before Van der Pol, the setting in equation of the triode oscillations. In March 1926, Blondel establishes the differential equation characterizing the oscillations of the singing arc, completely similar to that obtained concomitantly by Van der Pol for the triode. Thus, throughout his career, Blondel, has made fundamental and relatively unknown contributions to the development of the theory of nonlinear oscillations. The purpose of this article is to analyze his main work in this area and to measure their importance or influence by placing them in the perspective of the development of this theory.

physics.hist-ph↗

arXiv2014

Slow invariant manifold of heartbeat model

A new approach called Flow Curvature Method has been recently developed in a book entitled Differential Geometry Applied to Dynamical Systems. It consists in considering the trajectory curve, integral of any n-dimensional dynamical system as a curve in Euclidean n-space that enables to analytically compute the curvature of the trajectory - or the flow. Hence, it has been stated on the one hand that the location of the points where the curvature of the flow vanishes defines a manifold called flow curvature manifold and on the other hand that such a manifold associated with any n-dimensional dynamical system directly provides its slow manifold analytical equation the invariance of which has been proved according to Darboux theory. The Flow Curvature Method has been already applied to many types of autonomous dynamical systems either singularly perturbed such as Van der Pol Model, FitzHugh-Nagumo Model, Chua's Model, ...) or non-singularly perturbed such as Pikovskii-Rabinovich-Trakhtengerts Model, Rikitake Model, Lorenz Model,... More- over, it has been also applied to non-autonomous dynamical systems such as the Forced Van der Pol Model. In this article it will be used for the first time to analytically compute the slow invariant manifold analytical equation of the four-dimensional Unforced and Forced Heartbeat Model. Its slow invariant manifold equation which can be considered as a "state equation" linking all variables could then be used in heart prediction and control according to the strong correspondence between the model and the physiological cardiovascular system behavior.

math.DS↗

arXiv2014

Van der Pol and the history of relaxation oscillations: toward the emergence of a concept

Relaxation oscillations are commonly associated with the name of Balthazar van der Pol via his eponymous paper (Philosophical Magazine, 1926) in which he apparently introduced this terminology to describe the nonlinear oscillations produced by self-sustained oscillating systems such as a triode circuit. Our aim is to investigate how relaxation oscillations were actually discovered. Browsing the literature from the late 19th century, we identified four self-oscillating systems in which relaxation oscillations have been observed: i) the series dynamo machine conducted by Gérard-Lescuyer (1880), ii) the musical arc discovered by Duddell (1901) and investigated by Blondel (1905), iii) the triode invented by de Forest (1907) and, iv) the multivibrator elaborated by Abraham and Bloch (1917). The differential equation describing such a self-oscillating system was proposed by Poincaré for the musical arc (1908), by Janet for the series dynamo machine (1919), and by Blondel for the triode (1919). Once Janet (1919) established that these three self-oscillating systems can be described by the same equation, van der Pol proposed (1926) a generic dimensionless equation which captures the relevant dynamical properties shared by these systems. Van der Pol's contributions during the period of 1926-1930 were investigated to show how, with Le Corbeiller's help, he popularized the "relaxation oscillations" using the previous experiments as examples and, turned them into a concept.

physics.hist-ph↗