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Chaos

Part of the Progress in Mathematical Physics Book Series (volume 66)

Editors: Duplantier, Bertrand; Nonnenmacher, Stéphane; Rivasseau, Vincent

DOI: 10.1007/978-3-0348-0697-8

ISBN: 978-3-0348-0696-1

Publisher: Springer Basel - Springer Journals

Oct 18, 2013

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Chapter 1The Lorenz Attractor, a Paradigm for Chaos

Ghys, Étienne

Pages: 1 –54

[It is very unusual for a mathematical or physical idea to disseminate into the society at large. An interesting example is chaos theory, popularized by Lorenz’s butterfly effect: “does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?” A tiny cause can generate big...
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Chapter 2Chaotic Dynamos Generated by Fully Turbulent Flows

Fauve, Stéphan

Pages: 55 –95

[We report experimental results on the generation of a magnetic field by the motion of an electrically conducting liquid metal. We recall that this dynamo effect is a canonical example of an instability process that occurs on a strongly turbulent flow. Although the magnetic field is driven by a...
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Chapter 3Discrete Graphs – A Paradigm Model for Quantum Chaos

Smilansky, Uzy

Pages: 97 –124

[The research in Quantum Chaos attempts to uncover the fingerprints of classical chaotic dynamics in the corresponding quantum description. To get to the roots of this problem, various simplified models were proposed and used. Here a very simple model of a random walker on large d-regular graphs,...
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Chapter 4Quantum Chaos, Random Matrix Theory, and the Riemann ζ-function

Bourgade, Paul; Keating, Jonathan P.

Pages: 125 –168

[We review some connections between quantum chaos and the theory of the Riemann zeta function and the primes. Specifically, we give an overview of the similarities between the semiclassical trace formula that connects quantum energy levels and classical periodic orbits in chaotic systems and an...
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Chapter 5Chaos in Microwave Resonators

Stöckmann, Hans-Jürgen

Pages: 169 –191

[Chaotic billiards are a paradigm of quantum chaos studied theoretically in numerous papers. In flat microwave resonators with cross-sections mimicking the billiard shape there is a one-to-one correspondence between the stationary Schrödinger equation and the Helmholtz equation. This allows an...
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Chapter 6Anatomy of Quantum Chaotic Eigenstates

Nonnenmacher, Stéphane

Pages: 193 –238

[We present one aspect of “Quantum Chaos”, namely the description of high frequency eigenmodes of a quantum system, the classical limit of which is chaotic (we call such eigenmodes “chaotic eigenmodes”). A paradigmatic example is provided by the eigenmodes of the Laplace–Beltrami operator on a...
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Chapter 7Is the Solar System Stable?

Laskar, Jacques

Pages: 239 –270

[Since the formulation of the problem by Newton, and during three centuries, astronomers and mathematicians have sought to demonstrate the stability of the Solar System. As mentioned by Poincaré, several demonstrations of the stability of the Solar System have been published. By Laplace and...

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