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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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