Catégorie

Théorie du chaos

La théorie du chaos est une théorie scientifique rattachée aux mathématiques et à la physique qui étudie le comportement des systèmes dynamiques sensibles aux conditions initiales, un phénomène généralement illustré par l'effet papillon. Dans de nombreux systèmes dynamiques, des modifications infimes des conditions initiales entraînent des évolutions rapidement divergentes, rendant toute prédiction impossible à long terme. Bien que ce soient des systèmes déterministes, dont le comportement futur est déterminé par les conditions initiales, sans aucune intervention du hasard, ils sont imprévisibles (au moins dans le détail) car on ne peut pas connaître les conditions initiales avec une précision infinie. Ce comportement paradoxal est connu sous le nom de chaos déterministe, ou tout simplement de chaos. Le comportement chaotique est à la base de nombreux systèmes naturels, tels que la météo ou le climat. Ce comportement peut être étudié grâce à l'analyse par des modèles mathématiques chaotiques, ou par des techniques analytiques de récurrence et des applications de Poincaré. La théorie du chaos a des applications en météorologie, climatologie, sociologie, physique, informatique, ingénierie, économie, biologie et philosophie. La notion de chaos renvoie à un concept qui remonte à l'Antiquité, dans la perspective d'une explication du monde reposant sur le principe de l'harmonie et du cosmos. C'est un concept de philosophie avant d'être un concept des mathématiques. Un système dynamique est dit chaotique si une portion « significative » de son espace des phases présente simultanément les deux caractéristiques suivantes : le phénomène de sensibilité aux conditions initiales ; une forte récurrence. La présence de ces deux propriétés entraîne un comportement extrêmement désordonné, qualifié à juste titre de « chaotique ». Les systèmes chaotiques s'opposent notamment aux systèmes intégrables de la mécanique classique, qui furent longtemps les symboles d'une régularité toute puissante en physique théorique.

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Exercices associés (32)

Un flotteur en mouvement

xa0 The float of a fishing rod floats at the surface of water. This floater has a cylindrical shape with a radius r, a height h and with an homogeneous mass. The float is standing vertically in water and it moves from top to bottom while always staying partially submerged. In addition to its weight, the float is subject to the Archimedian buoyancy

\vec{P_A}

and at a viscous friction force F⃗u2004=u2004u2005−u2005kηv⃗. The volumic mass of the float has a value equal to two third of the one of water : $\rh

Questions conceptuelles sur la gravitation

Why doesn’t the Moon fall on the Earth like an apple falling from a tree?
This question was very important in Newton’s work, and the answer he gave allowed science to distance itself from the antiquated approach that celestial bodies and everyday objects did not obey the same physical laws. The Moon, like the apple, is subjected to an attracting force from the Earth. Is an apple, or the Moon, is released without any initial speed, then it will crash on Earth. But what hap

Point sur un cylindre

A material point moves on an infinitely long cylinder of radius R. The material point is attracted to a point O on the cylinder’s axes by a force that’s proportional to the distance between the material point and O. There is neither gravity nor friction.

List the forces acting on the material point. Draw these forces on a diagram, as well as the frame of reference associated with the chosen coordinate system.
Write the equations of motion and project them on the frame of reference.

Système pendule et ressorts

A simple pendulum of mass m, of length lu2004

Pendule perturbé par un ressort

A material point of mass m is restricted to move without friction along a vertical circle of radius R. It is subjected to two forces: its weight, and the force from a spring linking it to a fixed point F on the circle, at the same height as the circle’s center. The spring has spring constant k and null rest length.

Show that the material’s point potential energy as a function of θ is E_(pot)(θ)u2004=u2004u2005−u2005kR²cosu2006θu2005−u2005mgRsinu2006θu2005+u2005constant.
Show that the two e

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Notebooks associés (28)

Modeling Epidemics with Signal Processing Techniques

Epidemics and Signal Processing[1]¶

                                      [2]Paolo Prandoni, [3]LCAV, EPFL

In [1]: import matplotlib.pyplot as plt import numpy as np import scipy.signal as sp import seaborn as sns sns.set() In [2]: plt.rcParams["figure.figsize"] = (14,4) The basic reproduction number[4]¶ The propagation mechanism of an epidemic such as the one caused by SARS-CoV-2 virus can be roughly modeled, at least in its inital phase, as a process in which each infected individ

Exploring the Karplus-Strong Algorithm for Sound Synthesis

The Karplus-Strong Algorithm[1]¶

                                      [2]Paolo Prandoni, [3]LCAV, EPFL

The Karplus-Strong algorithm is a simple method to generate synthetic sounds with arbitrary pitch. In this notebook we will explore a few variations of the method to illustrate some basic notions of discrete-time signal processing. In [1]: %matplotlib inline import matplotlib import matplotlib.pyplot as plt import numpy as np import IPython In [2]: plt.rcParams["figure.figsize"]

Voice Transformation Techniques Using Digital Signal Processing

A Gallery of Voice Transformers[1]¶

                                      [2]Paolo Prandoni, [3]LCAV, EPFL

In this notebook we will explore various ways to implement a system that changes the voice of a speaker. Typical applications for these techniques, strictly sorted by plausibility, include international spy missions and prank telephone calls, and devices such as the one displayed here can be easily bought on the web... or in a novelty shop. No description has been provided for t

Illustrating Band Structures in Solids: Free Electrons and Atomic Orbitals

In [1]: import numpy as np from numpy import linalg as LA import matplotlib.pyplot as plt

from future import print_function

from ipywidgets import interact, interactive, fixed, interact_manual import ipywidgets as widgets %matplotlib widget The emergence of band structures[1]¶ If you have questions about this notebook, e-mail Oleg Malanyuk (oleg.malanyuk@epfl.ch) This jupyter notebook is intended to provide a illustration of the band structures in solides, starting from the points of v

Digital Simulation of Guitar Feedback in I Feel Fine by The Beatles

I Feel Fine, Digitally[1]¶ This notebook is a little musing inspired by the "internal mechanics" of a very famous audio snippet, namely the guitar sound at the beginning of the song "I Feel Fine", written and recorded by the Beatles in 1964. The historical significance of the record lies in the following claim by John Lennon:

 "I defy anybody to find a record... unless it is some old blues
 record from 1922... that uses feedback that way. So I claim it for
 the Beatles. Before H

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Exercices associés (32)

Un flotteur en mouvement

\vec{P_A}

and at a viscous friction force F⃗u2004=u2004u2005−u2005kηv⃗. The volumic mass of the float has a value equal to two third of the one of water : $\rh

Questions conceptuelles sur la gravitation

Why doesn’t the Moon fall on the Earth like an apple falling from a tree?
This question was very important in Newton’s work, and the answer he gave allowed science to distance itself from the antiquated approach that celestial bodies and everyday objects did not obey the same physical laws. The Moon, like the apple, is subjected to an attracting force from the Earth. Is an apple, or the Moon, is released without any initial speed, then it will crash on Earth. But what hap

Point sur un cylindre

List the forces acting on the material point. Draw these forces on a diagram, as well as the frame of reference associated with the chosen coordinate system.
Write the equations of motion and project them on the frame of reference.

Système pendule et ressorts

A simple pendulum of mass m, of length lu2004

Pendule perturbé par un ressort

Show that the material’s point potential energy as a function of θ is E_(pot)(θ)u2004=u2004u2005−u2005kR²cosu2006θu2005−u2005mgRsinu2006θu2005+u2005constant.
Show that the two e

Afficher plus

Théorie du chaos

Chattez avec Graph Search

from future import print_function

from future import print_function

OPF-based optimal power system network restoration considering frequency dynamics

Nonlinear periodic orbit solutions and their bifurcation structure at the origin of soliton hopping in coupled microresonators

Local and global bifurcations to large-scale oblique patterns in inclined layer convection