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.
UteChristina Herzfeld
- Published in print:
- 1994
- Published Online:
- November 2020
- ISBN:
- 9780195085938
- eISBN:
- 9780197560525
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195085938.003.0023
- Subject:
- Computer Science, Software Engineering
"Fractals" and "chaos" have become increasingly popular in geology; however, the use of "fractal" methods is mostly limited to simple cases of selfsimilarity, often ...
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"Fractals" and "chaos" have become increasingly popular in geology; however, the use of "fractal" methods is mostly limited to simple cases of selfsimilarity, often taken as the prototype of a scaling property if not mistaken as equivalent to a fractal as such. Here; a few principles of fractal and chaos theory are clarified, an overview of geoscience applications is given, and possible pitfalls are discussed. An example from seafloor topography relates fractal dimension, self-similarity, and multifractal cascade scaling to traditional geostatistical and statistical concepts. While the seafloor has neither self-similar nor cascade scaling behavior, methods developed in the course of "fractal analysis" provide ways to quantitatively describe variability in spatial structures across scales arid yield geologically meaningful results. Upon hearing the slogan "the appleman reigns between order and chaos" in the early 1980's and seeing colorful computer-generated pictures, one was simply fascinated by the strangely beautiful figure of the "appleman" that, when viewed through a magnifying glass, has lots of parts that, are smaller, and smaller, and smaller applemen. The "appleman" is the recurrent feature of the Mandelbrot set, a self-similar fractal, and in a certain sense, the universal fractal (e.g., see Peitgen and Saupe, 1988, p. 195 ff.). Soon the realm of the appleman expanded, made possible by increasing availability of fast, cheap computer power and increasingly sophisticated computer graphics. In its first phase of popularity, when the Bremen working group traveled with their computer graphics display seeking public recognition through exhibits in the foyers of savings banks, the fractal was generally considered to be a contribution to modern art (Peitgen and Richter, The Beauty of Fractals, 1986). While the very title of Mandelbrot's famous book, The Fractal Geometry of Nature (1983), proclaims the discovery of the proper geometry to describe nature, long hidden by principals of Euclidean geometry, the "fractal" did not appeal to Earth scientists for well over two decades after its rediscovery by Mandelbrot (1964, 1965, 1967, 1974, 1975).
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"Fractals" and "chaos" have become increasingly popular in geology; however, the use of "fractal" methods is mostly limited to simple cases of selfsimilarity, often taken as the prototype of a scaling property if not mistaken as equivalent to a fractal as such. Here; a few principles of fractal and chaos theory are clarified, an overview of geoscience applications is given, and possible pitfalls are discussed. An example from seafloor topography relates fractal dimension, self-similarity, and multifractal cascade scaling to traditional geostatistical and statistical concepts. While the seafloor has neither self-similar nor cascade scaling behavior, methods developed in the course of "fractal analysis" provide ways to quantitatively describe variability in spatial structures across scales arid yield geologically meaningful results. Upon hearing the slogan "the appleman reigns between order and chaos" in the early 1980's and seeing colorful computer-generated pictures, one was simply fascinated by the strangely beautiful figure of the "appleman" that, when viewed through a magnifying glass, has lots of parts that, are smaller, and smaller, and smaller applemen. The "appleman" is the recurrent feature of the Mandelbrot set, a self-similar fractal, and in a certain sense, the universal fractal (e.g., see Peitgen and Saupe, 1988, p. 195 ff.). Soon the realm of the appleman expanded, made possible by increasing availability of fast, cheap computer power and increasingly sophisticated computer graphics. In its first phase of popularity, when the Bremen working group traveled with their computer graphics display seeking public recognition through exhibits in the foyers of savings banks, the fractal was generally considered to be a contribution to modern art (Peitgen and Richter, The Beauty of Fractals, 1986). While the very title of Mandelbrot's famous book, The Fractal Geometry of Nature (1983), proclaims the discovery of the proper geometry to describe nature, long hidden by principals of Euclidean geometry, the "fractal" did not appeal to Earth scientists for well over two decades after its rediscovery by Mandelbrot (1964, 1965, 1967, 1974, 1975).
Christof Koch
- Published in print:
- 1998
- Published Online:
- November 2020
- ISBN:
- 9780195104912
- eISBN:
- 9780197562338
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195104912.003.0021
- Subject:
- Computer Science, Mathematical Theory of Computation
The majority of experiments in neurophysiology are based upon spike trains recorded from individual or multiple nerve cells. If all the action potentials are taken to ...
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The majority of experiments in neurophysiology are based upon spike trains recorded from individual or multiple nerve cells. If all the action potentials are taken to be identical and only the times at which they are generated are considered, the experimentalist obtains a discrete series of time events {t1,···, tn}, where t¡ corresponds to the occurrence of the i th spike, characterizing the spike train. This spike train is transmitted down the axon to all of the target cells of the neuron, and it is this spike train that contains all of the relevant information that the cell is representing (assuming no dendro-dendritic connections). As alluded to in the preceding chapter, there are two opposing views of neuronal coding, with many interim shades. One view holds that it is the firing rate, averaged over a suitable temporal window (Eqs. 14.1 or 14.2), that is relevant for information processing. The dissenting view, correlation coding, argues that the interactions among spikes, at the single cell as well as between multiple cells, encodes information. A key property of spike trains is their seemingly stochastic or random nature, quite in contrast to switching in digital computers. This randomness is apparent in the highly irregular discharge pattern of a central neuron to a sensory stimulus whose details are rarely reproducible from one trial to the next. The apparent lack of reproducible spike patterns has been one of the principal arguments in favor of the hypothesis that neurons only care about the firing frequency averaged over very long time windows. Such a mean rate code is very robust to “sloppy” hardware but is also relatively inefficient in terms of transmitting the maximal amount of information per spike. Encoding information in the intervals between spikes is obviously much more efficient, in particular if correlated across multiple neurons. Such a scheme does place a premium on postsynaptic neurons that can somehow decode this information. Because little or no information can be encoded into a stream of regularly spaced action potentials, this raises the question of how variable neuronal firing really is.
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The majority of experiments in neurophysiology are based upon spike trains recorded from individual or multiple nerve cells. If all the action potentials are taken to be identical and only the times at which they are generated are considered, the experimentalist obtains a discrete series of time events {t1,···, tn}, where t¡ corresponds to the occurrence of the i th spike, characterizing the spike train. This spike train is transmitted down the axon to all of the target cells of the neuron, and it is this spike train that contains all of the relevant information that the cell is representing (assuming no dendro-dendritic connections). As alluded to in the preceding chapter, there are two opposing views of neuronal coding, with many interim shades. One view holds that it is the firing rate, averaged over a suitable temporal window (Eqs. 14.1 or 14.2), that is relevant for information processing. The dissenting view, correlation coding, argues that the interactions among spikes, at the single cell as well as between multiple cells, encodes information. A key property of spike trains is their seemingly stochastic or random nature, quite in contrast to switching in digital computers. This randomness is apparent in the highly irregular discharge pattern of a central neuron to a sensory stimulus whose details are rarely reproducible from one trial to the next. The apparent lack of reproducible spike patterns has been one of the principal arguments in favor of the hypothesis that neurons only care about the firing frequency averaged over very long time windows. Such a mean rate code is very robust to “sloppy” hardware but is also relatively inefficient in terms of transmitting the maximal amount of information per spike. Encoding information in the intervals between spikes is obviously much more efficient, in particular if correlated across multiple neurons. Such a scheme does place a premium on postsynaptic neurons that can somehow decode this information. Because little or no information can be encoded into a stream of regularly spaced action potentials, this raises the question of how variable neuronal firing really is.
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.