*Richard M. Goodwin*

- Published in print:
- 1990
- Published Online:
- November 2003
- ISBN:
- 9780198283355
- eISBN:
- 9780191596315
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198283350.001.0001
- Subject:
- Economics and Finance, Macro- and Monetary Economics

This collection of short essays provides an application of chaotic dynamics to economic systems. Each chapter presents several economic models incorporating differential (or difference) equations ...
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This collection of short essays provides an application of chaotic dynamics to economic systems. Each chapter presents several economic models incorporating differential (or difference) equations such as the Rössler equations, which exhibit a chaotic attractor. Combining the insights of Schumpeter, Marx, and Keynes, the models endogenously generate irregular, wavelike growth. Goodwin therefore argues that the apparent unpredictability of economic systems is due to deterministic chaos as much as to exogeneous shocks. The book is aimed primarily at economists interested in theories of economic growth. However, readers with a general interest in the application of chaos theory to social sciences will also find it useful. Some mathematical knowledge of systems of differential equations is assumed.Less

This collection of short essays provides an application of chaotic dynamics to economic systems. Each chapter presents several economic models incorporating differential (or difference) equations such as the Rössler equations, which exhibit a chaotic attractor. Combining the insights of Schumpeter, Marx, and Keynes, the models endogenously generate irregular, wavelike growth. Goodwin therefore argues that the apparent unpredictability of economic systems is due to deterministic chaos as much as to exogeneous shocks. The book is aimed primarily at economists interested in theories of economic growth. However, readers with a general interest in the application of chaos theory to social sciences will also find it useful. Some mathematical knowledge of systems of differential equations is assumed.

*Iwo Białynicki-Birula and Iwona Białynicka-Birula*

- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198531005
- eISBN:
- 9780191713033
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198531005.003.0007
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Deterministic chaos sounds like an oxymoron but it is a well defined mathematical concept. Deterministic means that there are exact rules that govern the evolution and chaos here means only that it ...
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Deterministic chaos sounds like an oxymoron but it is a well defined mathematical concept. Deterministic means that there are exact rules that govern the evolution and chaos here means only that it is impossible to predict the outcome in the long run. The key to the resolution of this paradox is the practical limitation that is always present in our ability to process information. For chaotic systems the resources needed to make a prediction grow extremely fast with the extent of time for which the prediction is to be valid.Less

Deterministic chaos sounds like an oxymoron but it is a well defined mathematical concept. Deterministic means that there are exact rules that govern the evolution and chaos here means only that it is impossible to predict the outcome in the long run. The key to the resolution of this paradox is the practical limitation that is always present in our ability to process information. For chaotic systems the resources needed to make a prediction grow extremely fast with the extent of time for which the prediction is to be valid.

*Abraham Bers*

- Published in print:
- 2016
- Published Online:
- November 2016
- ISBN:
- 9780199295784
- eISBN:
- 9780191749063
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199295784.003.0029
- Subject:
- Physics, Nuclear and Plasma Physics, Particle Physics / Astrophysics / Cosmology

This chapter continues the discussion of nonlinear dynamics in plasmas by exploring nonlinear particle interactions with coherent waves, including soliton wave-wave interactions (WWI) and chaotic ...
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This chapter continues the discussion of nonlinear dynamics in plasmas by exploring nonlinear particle interactions with coherent waves, including soliton wave-wave interactions (WWI) and chaotic dynamics. It turns to results from computational physics which draws from examples developed for, and carried over from, solved problems in nonlinear physics and engineering of the past and present. These results are helpful in understanding solitons and deterministic chaos. The study here can be separated in two major parts, with the first part giving the treatment of a variety of simple models, each accounting for specific nonlinear plasma manifestations in both wave-particle interactions (WPI) and WWI. Also included is an analysis of nonlaminarity in boundary dynamics and a general and detailed derivation of the Manley–Rowe relations. The second part addresses two specific nonlinear evolutions in plasmas—solitons in nonlinear waves and WWI, and deterministic chaos in WPI.Less

This chapter continues the discussion of nonlinear dynamics in plasmas by exploring nonlinear particle interactions with coherent waves, including soliton wave-wave interactions (WWI) and chaotic dynamics. It turns to results from computational physics which draws from examples developed for, and carried over from, solved problems in nonlinear physics and engineering of the past and present. These results are helpful in understanding solitons and deterministic chaos. The study here can be separated in two major parts, with the first part giving the treatment of a variety of simple models, each accounting for specific nonlinear plasma manifestations in both wave-particle interactions (WPI) and WWI. Also included is an analysis of nonlaminarity in boundary dynamics and a general and detailed derivation of the Manley–Rowe relations. The second part addresses two specific nonlinear evolutions in plasmas—solitons in nonlinear waves and WWI, and deterministic chaos in WPI.

*Peter Mann*

- Published in print:
- 2018
- Published Online:
- August 2018
- ISBN:
- 9780198822370
- eISBN:
- 9780191861253
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198822370.003.0023
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter introduces the reader to canonical perturbation theory as a tool for studying near-integrable systems. Many problems in physics and chemistry do not have exact analytical solutions; ...
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This chapter introduces the reader to canonical perturbation theory as a tool for studying near-integrable systems. Many problems in physics and chemistry do not have exact analytical solutions; these systems are in direct opposition to integrable systems and action-angle variables. The chapter starts by considering tiny perturbations to integrable Hamiltonians. Poincaré in 1893 claimed this was the fundamental question of classical mechanics and, fittingly, Hamilton–Jacobi theory is the starting point. The chapter develops Poincaré’s fundamental equation as well as Delaunay’s small divisor problem. Resonant, near–resonant and non-resonant tori are investigated in the context of Poincaré’s theorem and KAM theory is described in detail. Chaos and Poincaré maps are presented before discussing determinism, deterministic chaos and Laplace’s demon.Less

This chapter introduces the reader to canonical perturbation theory as a tool for studying near-integrable systems. Many problems in physics and chemistry do not have exact analytical solutions; these systems are in direct opposition to integrable systems and action-angle variables. The chapter starts by considering tiny perturbations to integrable Hamiltonians. Poincaré in 1893 claimed this was the fundamental question of classical mechanics and, fittingly, Hamilton–Jacobi theory is the starting point. The chapter develops Poincaré’s fundamental equation as well as Delaunay’s small divisor problem. Resonant, near–resonant and non-resonant tori are investigated in the context of Poincaré’s theorem and KAM theory is described in detail. Chaos and Poincaré maps are presented before discussing determinism, deterministic chaos and Laplace’s demon.