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Quantifying Chaos

Robert C. Hilborn

in Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers

Published in print:
2000
Published Online:
January 2010
ISBN:
9780198507239
eISBN:
9780191709340
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198507239.003.0009
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter explores various means of quantifying the degree of chaotic behaviour of both numerically generated time series and time series from real experiments. These methods include Lyapunov ... More


Three-Dimensional State Space and Chaos

Robert C. Hilborn

in Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers

Published in print:
2000
Published Online:
January 2010
ISBN:
9780198507239
eISBN:
9780191709340
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198507239.003.0004
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter extends state space descriptions to systems with three (or more) dimensions. For systems described by differential equations, higher state space dimensions allow for more complex ... More


Iterated Maps

Robert C. Hilborn

in Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers

Published in print:
2000
Published Online:
January 2010
ISBN:
9780198507239
eISBN:
9780191709340
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198507239.003.0005
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Dynamical systems described by iterated map functions are used to explore the period-doubling route to chaos, Lyapunov exponents, and Feigenbaum numbers. A simple derivation of the numerical value of ... More


Ergodic Theory

Robert Alicki and Mark Fannes

in Quantum Dynamical Systems

Published in print:
2001
Published Online:
February 2010
ISBN:
9780198504009
eISBN:
9780191708503
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198504009.003.0007
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter defines and characterizes various degrees of ergodicity — both for classical and quantum systems — in terms of the Hilbert space formalism, Koopman formalism for classical systems, and ... More


The Butterfly Effect

David P. Feldman

in Chaos and Fractals: An Elementary Introduction

Published in print:
2012
Published Online:
December 2013
ISBN:
9780199566433
eISBN:
9780191774966
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199566433.003.0011
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter explores the phenomenon of sensitive dependence on initial conditions (SDIC), also known as the butterfly effect. SDIC is one of the criteria in the definition of chaos which highlights ... More


Nonlinear dynamics and chaos

Joseph F. Boudreau and Eric S. Swanson

in Applied Computational Physics

Published in print:
2017
Published Online:
February 2018
ISBN:
9780198708636
eISBN:
9780191858598
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198708636.003.0013
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Simple maps and dynamical systems are used to explore chaos in nature. The discussion starts with a review of the properties of nonlinear ordinary differential equations, including the useful ... More


Anderson Localization

Keith Slevin and Tomi Ohtsuki

in Conductor-Insulator Quantum Phase Transitions

Published in print:
2012
Published Online:
September 2012
ISBN:
9780199592593
eISBN:
9780191741050
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199592593.003.0003
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter reviews briefly the theory of the Anderson localisation of electrons. In disordered materials at low temperatures, quantum interference may lead to the suppression of diffusion. If this ... More


The Features of Chaos

Ray Huffaker, Marco Bittelli, and Rodolfo Rosa

in Nonlinear Time Series Analysis with R

Published in print:
2017
Published Online:
February 2018
ISBN:
9780198782933
eISBN:
9780191826153
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198782933.003.0004
Subject:
Physics, Theoretical, Computational, and Statistical Physics

In this chapter we introduce the features of Chaotic systems. We describe “sensitivity to initial conditions” and its quantitative measure, the Lyapunov exponent, which reflect the average rate of ... More


Nonlinear Dynamics and Chaos

David D. Nolte

in Introduction to Modern Dynamics: Chaos, Networks, Space, and Time

Published in print:
2019
Published Online:
November 2019
ISBN:
9780198844624
eISBN:
9780191880216
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198844624.003.0004
Subject:
Physics, Theoretical, Computational, and Statistical Physics

The geometric structure of state space is understood through phase portraits and stability analysis that classify the character of fixed points based on the properties of Lyapunov exponents. Limit ... More


Nonlinear systems∗

Chun Wa Wong

in Introduction to Mathematical Physics: Methods & Concepts

Published in print:
2013
Published Online:
May 2013
ISBN:
9780199641390
eISBN:
9780191747786
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199641390.003.0006
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Nonlinear systems show two very different faces: unpredictability and unusual coherence. Unpredictability is shown to arise from instabilities and bifurcations that are highly sensitive to small ... More


Causal images of time

George Jaroszkiewicz

in Images of Time: Mind, Science, Reality

Published in print:
2016
Published Online:
January 2016
ISBN:
9780198718062
eISBN:
9780191787553
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198718062.003.0009
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter discusses notions of causality, determinism, blame, teleology, and chaos, looking at views from Antiquity to recent times. It reviews the comments of Leibniz, Locke, and Mill on ... More


Variational Techniques of Many-Body Theory

Klaus Morawetz

in Interacting Systems far from Equilibrium: Quantum Kinetic Theory

Published in print:
2017
Published Online:
February 2018
ISBN:
9780198797241
eISBN:
9780191838743
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198797241.003.0011
Subject:
Physics, Condensed Matter Physics / Materials

The variational technique for nonequilibrium Green’s functions is derived resulting in the Hedin equations. This allows exploring of the high-density limit of diagrammatic expansions. Nonequilibrium ... More


Complex Oscillations and Chaos

Irving R. Epstein and John A. Pojman

in An Introduction to Nonlinear Chemical Dynamics: Oscillations, Waves, Patterns, and Chaos

Published in print:
1998
Published Online:
November 2020
ISBN:
9780195096705
eISBN:
9780197560815
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780195096705.003.0014
Subject:
Chemistry, Physical Chemistry

After studying the first seven chapters of this book, the reader may have come to the conclusion that a chemical reaction that exhibits periodic ... More


Dynamical systems on networks

Mark Newman

in Networks

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198805090
eISBN:
9780191843235
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198805090.003.0017
Subject:
Physics, Theoretical, Computational, and Statistical Physics

An introduction to the theory of dynamical systems on networks. This chapter starts with a short introduction to classical (non-network) dynamical systems theory, including linear stability analysis, ... More


Numerical Analysis of Conservative Maps: A Possible Foundation of Nonextensive Phenomena

Fulvio Baldovin

in Nonextensive Entropy: Interdisciplinary Applications

Published in print:
2004
Published Online:
November 2020
ISBN:
9780195159769
eISBN:
9780197562024
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780195159769.003.0010
Subject:
Earth Sciences and Geography, Atmospheric Sciences

We discuss the sensitivity to initial conditions and the entropy production of low-dimensional conservative maps, focusing on situations where the phase space presents ... More


Generalized Information Measures and the Analysis of Brain Electrical Signals

A. Plastino and M. T. Martin

in Nonextensive Entropy: Interdisciplinary Applications

Published in print:
2004
Published Online:
November 2020
ISBN:
9780195159769
eISBN:
9780197562024
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780195159769.003.0020
Subject:
Earth Sciences and Geography, Atmospheric Sciences

The traditional way of analyzing brain electrical activity, on the basis of electroencephalogram (EEG) records, relies mainly on visual inspection and years of ... More


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