Jean-Marc Ginoux

An ultrasonic contactless sensor for breathing monitoring.

The monitoring of human breathing activity during a long period has multiple fundamental applications in medicine. In breathing sleep disorders such as apnea, the diagnosis is based on events during which the person stops breathing for several periods during sleep. In polysomnography, the standard for sleep disordered breathing analysis, chest movement and airflow are used to monitor the respiratory activity. However, this method has serious drawbacks. Indeed, as the subject should sleep overnight in a laboratory and because of sensors being in direct contact with him, artifacts modifying sleep quality are often observed. This work investigates an analysis of the viability of an ultrasonic device to quantify the breathing activity, without contact and without any perception by the subject. Based on a low power ultrasonic active source and transducer, the device measures the frequency shift produced by the velocity difference between the exhaled air flow and the ambient environment, i.e., the Doppler effect. After acquisition and digitization, a specific signal processing is applied to separate the effects of breath from those due to subject movements from the Doppler signal. The distance between the source and the sensor, about 50 cm, and the use of ultrasound frequency well above audible frequencies, 40 kHz, allow monitoring the breathing activity without any perception by the subject, and therefore without any modification of the sleep quality which is very important for sleep disorders diagnostic applications. This work is patented (patent pending 2013-7-31 number FR.13/57569).

Differential geometry applied to dynamical systems

Differential Equations Hartman-Grobman Theorem Liapounoff Stability Theorem Phase Portraits Poincare-Bendixson Theorem Attractors Strange Attractors Hamiltonian and Integrable Systems K A M Theorem Invariant Sets Global/Local Invariance Center Manifold Theorem Normal Form Theorem Local Bifurcations of Codimension 1 Hopf Bifurcation, Slow-Fast Dynamical Systems Geometric Singular Perturbation Theory Darboux Theory of Integrability Differential Geometry Generalized Frenet Moving Frame Curvatures of Trajectory Curves Flow Curvature Manifold Flow Curvature Method Van der Pol Model FitzHugh-Nagumo Model Pikovskii-Rabinovich-Trakhtengerts Model Rikitake Model Chuas Model Lorenz Model.

TOPOLOGICAL ANALYSIS OF CHAOTIC SOLUTION OF A THREE-ELEMENT MEMRISTIVE CIRCUIT

The simplest electronic circuit with a memristor was recently proposed. Chaotic attractors solution to this memristive circuit are topologically characterized and compared to Rossler-like attractors.

SLOW INVARIANT MANIFOLDS AS CURVATURE OF THE FLOW OF DYNAMICAL SYSTEMS

Considering trajectory curves, integral of n-dimensional dynamical systems, within the framework of Differential Geometry as curves in Euclidean n-space, it will be established in this article that the curvature of the flow, i.e. the curvature of the trajectory curves of any n-dimensional dynamical system directly provides its slow manifold analytical equation the invariance of which will be then proved according to Darboux theory. Thus, it will be stated that the flow curvature method, which uses neither eigenvectors nor asymptotic expansions but only involves time derivatives of the velocity vector field, constitutes a general method simplifying and improving the slow invariant manifold analytical equation determination of high-dimensional dynamical systems. Moreover, it will be shown that this method generalizes the Tangent Linear System Approximation and encompasses the so-called Geometric Singular Perturbation Theory. Then, slow invariant manifolds analytical equation of paradigmatic Chuas piecewise linear and cubic models of dimensions three, four and five will be provided as tutorial examples exemplifying this method as well as those of high-dimensional dynamical systems.

CHAOS IN A THREE-DIMENSIONAL VOLTERRA–GAUSE MODEL OF PREDATOR–PREY TYPE

The aim of this paper is to present results concerning a three-dimensional model including a prey, a predator and top-predator, which we have named the Volterra–Gause model because it combines the original model of V. Volterra incorporating a logisitic limitation of the P. F. Verhulst type on growth of the prey and a limitation of the G. F. Gause type on the intensity of predation of the predator on the prey and of the top-predator on the predator. This study highlights that this model has several Hopf bifurcations and a period-doubling cascade generating a snail shell-shaped chaotic attractor. With the aim of facilitating the choice of the simplest and most consistent model a comparison is established between this model and the so-called Rosenzweig–MacArthur and Hastings–Powell models. Many resemblances and differences are highlighted and could be used by the modellers. The exact values of the parameters of the Hopf bifurcation are provided for each model as well as the values of the parameters making it possible to carry out the transition from a typical phase portrait characterizing one model to another (Rosenzweig–MacArthur to Hastings–Powell and vice versa). The equations of the Volterra–Gause model cannot be derived from those of the other models, but this study shows similarities between the three models. In cases in which the top-predator has no effect on the predator and consequently on the prey, the models can be reduced to two dimensions. Under certain conditions, these models present slow–fast dynamics and their attractors are lying on a slow manifold surface, the equation of which is given.

The flow curvature method applied to canard explosion

The aim of this work is to establish that the bifurcation parameter value leading to a canard explosion in dimension two obtained by the so-called Geometric Singular Perturbation Method can be found according to the Flow Curvature Method. This result will be then exemplified with the classical Van der Pol oscillator.

Connecting curves for dynamical systems

We introduce one-dimensional sets to help describe and constrain the integral curves of an n-dimensional dynamical system. These curves provide more information about the system than zero-dimensional sets (fixed points). In fact, these curves pass through the fixed points. Connecting curves are introduced using two different but equivalent definitions, one from dynamical systems theory, the other from differential geometry. We describe how to compute these curves and illustrate their properties by showing the connecting curves for a number of dynamical systems.

DEVELOPMENT OF THE NONLINEAR DYNAMICAL SYSTEMS THEORY FROM RADIO ENGINEERING TO ELECTRONICS

Although initial results that contributed to the emergence of the nonlinear dynamical system theory arose from astronomy (the three-body problem), many subsequent developments were related to radio engineering and electronics. The path between the van der Pol equation and the Chua circuit is thus reviewed through main historical contributions.

Slow Manifold of a Neuronal Bursting Model

Comparing neuronal bursting models (NBM) with slow-fast autonomous dynamical systems (S-FADS), it appears that the specific features of a (NBM) do not allow a determination of the analytical slow manifold equation with the singular approximation method. So, a new approach based on Differential Geometry, generally used for (S-FADS), is proposed. Adapted to (NBM), this new method provides three equivalent manners of determination of the analytical slow manifold equation. Application is made for the three-variables model of neuronal bursting elaborated by Hindmarsh and Rose which is one of the most used mathematical representation of the widespread phenomenon of oscillatory burst discharges that occur in real neuronal cells.

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.