Suite logistique

Résumé

En mathématiques, une suite logistique est une suite réelle simple, mais dont la récurrence n'est pas linéaire. Sa relation de récurrence est :x_{n+1} = \mu x_n(1 - x_n), ~x_0\in [0; 1]. Suivant la valeur du paramètre μ (dans [0; 4] pour assurer que les valeurs de x restent dans [0; 1]), elle engendre soit une suite convergente, soit une suite soumise à oscillations, soit une suite chaotique. Souvent citée comme exemple de la complexité de comportement pouvant surgir d'une relation non linéaire simple, cette suite fut popularisée par le biologiste Robert May en 1976. Une application de la suite logistique est la modélisation de la taille d'une population biologique au fil des générations. Elle est la solution en temps discret du modèle de Verhulst. Le terme « logistique » provient de l'ouvrage de Pierre François Verhulst qui appelle courbe logistique la solution en temps continu de son modèle. Il écrit en 1845 dans son ouvrage consacré à ce phénomène : . L'auteur n'exp

Source officielle

https://fr.wikipedia.org/wiki/Suite_logistique

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Introduction to local and global behavior of nonlinear dynamical systems arising from maps and ordinary differential equations. Theoretical and computational aspects studied.

The course provides students with the tools to approach the study of nonlinear systems and chaotic dynamics. Emphasis is given to concrete examples and numerical applications are carried out during the exercise sessions.

This introduction to Enviromental Engineering is meant to show the students how upcoming courses in mathematics, physics, chemistry, biology and other areas will be used to gain a scientific understanding of environmental problems and then help to solve them.

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Séances de cours associées (26)

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How to repair CSK using small perturbation control - Case study and performance analysis

Slobodan Kozic, Kumiko Oshima, Thomas Schimming

In this paper we study coded modulation schemes based on iterated chaotic maps. In particular, we consider

CSK

-like systems in which a chaotic map is switched depending on the payload. We show how small perturbation control can substantially improve the performance and we demonstrate using a minimum distance approach how absence/wrongly designed control can lead to bad performance.

2003

Chargement

Minimum distance properties of coded modulations based on iterated chaotic maps

Slobodan Kozic, Kumiko Oshima, Thomas Schimming

In this paper we introduce a method for analyzing the performance of coded modulation schemes based on iterated chaotic maps. In particular, it is known from coding theory that the minimum distance plays an important role, defining the behavior of the error probability performance at sufficiently high signal to noise ratios. We introduce the method and illustrate it for two examples of chaotic maps, the Bernoulli shift map and the tent map. We emphasize that the performance of the map can not be understood by a reasoning in terms of ergodic behavior of the map. The given simulation results confirm the relevance of the minimum distance for describing the performance of such a system.

2003

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Flow control of weakly non-parallel flows: application to trailing vortices

François Gallaire, Giacomo Valerio Iungo, Erica Pezzica, Francesco Viola

A general formulation is proposed to control the integral amplification factor of harmonic disturbances in weakly non-parallel amplifier flows. The sensitivity of the local spatial stability spectrum to a base-flow modification is first determined, generalizing the results of Bottaro et al. (J. Fluid Mech., vol. 476, 2003, pp. 293-302). This result is then used to evaluate the sensitivity of the overall spatial growth to a modification of the inlet flow condition. This formalism is applied to a non-parallel Batchelor vortex, which is a well-known model for trailing vortices generated by a lifting wing. The resulting sensitivity map indicates the optimal modification of the inlet flow condition enabling the stabilization of the helical modes. It is shown that the control, formulated using a single linearization of the flow dynamics carried out on the uncontrolled configuration, successfully reduces the total spatial amplification of all convectively unstable disturbances.

Cambridge Univ Press2017

Chargement

How to repair CSK using small perturbation control - Case study and performance analysis

Slobodan Kozic, Kumiko Oshima, Thomas Schimming

In this paper we study coded modulation schemes based on iterated chaotic maps. In particular, we consider

CSK

2003