arXiv subjects

Bruno Rossetto

Publications and source records attributed to Bruno Rossetto.

6 recordsLinked to original sources

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

The singing arc: the oldest memristor?

On April 30th 2008, the journal Nature announced that the missing circuit element, postulated thirty-seven years before by Professor Leon O. Chua has been found. Thus, after the capacitor, the resistor and the inductor, the existence of a fourth fundamental element of electronic circuits called "memristor" was established. In order to point out the importance of such a discovery, the aim of this article is first to propose an overview of the manner with which the three others have been invented during the past centuries. Then, a comparison between the main properties of the singing arc, i.e. a forerunner device of the triode used in Wireless Telegraphy, and that of the memristor will enable to state that the singing arc could be considered as the oldest memristor.

physics.hist-ph↗

arXiv2014

Invariant Manifolds of Complex Systems

The aim of this work is to establish the existence of invariant manifolds in complex systems. Considering trajectory curves integral of multiple time scales dynamical systems of dimension two and three (predator-prey models, neuronal bursting models) it is shown that there exists in the phase space a curve (resp. a surface) which is invariant with respect to the flow of such systems. These invariant manifolds are playing a very important role in the stability of complex systems in the sense that they are "restoring" the determinism of trajectory curves.

math.DS↗

arXiv2014

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.

math.DS↗

arXiv2014

Differential Geometry and Mechanics Applications to Chaotic Dynamical Systems

The aim of this article is to highlight the interest to apply Differential Geometry and Mechanics concepts to chaotic dynamical systems study. Thus, the local metric properties of curvature and torsion will directly provide the analytical expression of the slow manifold equation of slow-fast autonomous dynamical systems starting from kinematics variables velocity, acceleration and over-acceleration or jerk. The attractivity of the slow manifold will be characterized thanks to a criterion proposed by Henri Poincaré. Moreover, the specific use of acceleration will make it possible on the one hand to define slow and fast domains of the phase space and on the other hand, to provide an analytical equation of the slow manifold towards which all the trajectories converge. The attractive slow manifold constitutes a part of these dynamical systems attractor. So, in order to propose a description of the geometrical structure of attractor, a new manifold called singular manifold will be introduced. Various applications of this new approach to the models of Van der Pol, cubic-Chua, Lorenz, and Volterra-Gause are proposed.

math.DS↗

arXiv2014

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 Chua's 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.

math.DS↗