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1993 | Buch | 2. Auflage

Introduction Kapitel lesen Erstes Kapitel lesen

Nonlinear Dynamical Economics and Chaotic Motion

verfasst von: Dr. Hans-Walter Lorenz

Verlag: Springer Berlin Heidelberg

Enthalten in: Professional Book Archive

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Über dieses Buch

Usually, the first edition of a book still contains a multiplicity of typographic, con­ ceptional, and computational errors even if one believes the opposite at the time of publication. As this book did not represent a counterexample to this rule, the current second edition offers a chance to remove at least the known shortcomings. The book has been partly re-organized. The previously rather long Chapter 4 has been split into two separate chapters dealing with discrete-time and continuous­ time approaches to nonlinear economic dynamics. The short summary of basic properties of linear dynamical systems has been banned to an appendix because the line of thought in the chapter seems to have been unnecessarily interrupted by these technical details and because the book concentrates on nonlinear systems. This appendix, which mainly deals with special formal properties of dynamical sys­ tems, also contains some new material on invariant subspaces and center-manifold reductions. A brief introduction into the theory of lags and operators is followed by a few remarks on the relation between the 'true' properties of dynamical systems and their behavior observable in numerical experiments. Additional changes in the main part of the book include a re-consideration of Popper's determinism vs. inde­ terminism discussion in the light of chaotic properties of deterministic, nonlinear systems in Chapter 1. An investigation of a simultaneous price-quantity adjustment process, a more detailed inquiry into the uniqueness property of limit cycles, and a short presentation of relaxation oscillations are included in Chapter 2.

Inhaltsverzeichnis

Frontmatter
Introduction
Abstract
The history of economic science abounds in examples of the emergence and decline of fashionable trends in economic thought. Basic and paradigmatic attitudes toward the conceptual understanding of an economy, concentrations on specific classes of economic models which are believed to be an optimal reflection of economic reality, or the usage of formal or verbal techniques whose applications are believed to provide new insights into existing paradigms have rarely gained lasting serious attention over the decades. It is this transitoriness which allows to assign many textbooks and monographs to a certain era.
Hans-Walter Lorenz
Chapter 1. Economic Dynamics, Linearities, and the Classical Mechanistic Worldview
Abstract
Nonlinear economic dynamics may be considered just a collection of models with essentially nonlinear ingredients that require the use of a particular set of (relatively new) mathematical tools. As such, nonlinear economic dynamics has a rank comparable to that of game theory, optimal control, or many other innovations in economic theory made during the last 50 years. However, nonlinear dynamical systems emerging in several fields have never been evaluated only from an exclusive formal point of view. The potential complexity and impredictability of nonlinear dynamical systems have almost immediately initiated a discussion of basic science-theoretic themes. Popular treaties of the subject occasionally talk of a scientific revolution or employ similar spectacular expressions. However, it seems as if in several examples of these inquiries the scientific environment which is supposed to encounter such a revolution is not always described with a sufficient accuracy. The following remarks do not (and cannot, actually) attempt to provide a completely satisfactory account of the origin of economic theorizing and the extend to which nonlinear dynamics might contribute to a change in the attitude toward economic dynamic processes. The sole purpose of the following notes consists in encouraging further reflections on the role of dynamical systems in the modeling of dynamic economic processes.
Hans-Walter Lorenz
Chapter 2. Nonlinearities and Economic Dynamics
Abstract
If the world is not linear (and there is no qualitative reason to assume the contrary), it should be natural to model dynamic economic phenomena in the form of nonlinear dynamical systems. However, there will not always exist an advantage in such a modelling. It depends crucially on the kind of nonlinearity in a model and sometimes on the subject of the investigation whether techniques appropriate to nonlinear systems provide new insights into the dynamic behavior of an economic system. Nonlinearities may be so weak that linear approximations do not constitute an essential error in answering qualitative questions about the system, e.g., whether or not the system converges to an equilibrium state. While this is certainly true for many low-dimensional systems, the effects of nonlinearities in higher-dimensional systems cannot always be anticipated with preciseness, implying that linear approximations should be treated with skepticism especially when the nonlinearities obviously diverge from linear structures.
Hans-Walter Lorenz
Chapter 3. Bifurcation Theory and Economic Dynamics
Abstract
This chapter deals with a subject that has become a major focus of research in economic dynamics during the last decade, namely bifurcation theory. Central to this topic is the question whether the qualitative properties of a dynamical system change when one or more of the exogenous parameters are changing. In contrast to the physical sciences, it is usually impossible to assign a definitive, once-and-for-all valid number to most parameters occurring in dynamical systems in economics. Parameters are introduced into an economic model in order to reflect the influence of exogenous forces which are either beyond the scope of pure economic explanation or which are intentionally considered as being exogenously given from the point of view of partial theorizing. It is desirable to determine whether the qualitative behavior of a dynamical system persists under variations in the parameter space. Thus, the results of bifurcation theory are especially important to dynamic modelling in economics.
Hans-Walter Lorenz
Chapter 4. Chaotic Dynamics in Discrete-Time Economic Models
Abstract
The presentation of nonlinear dynamical systems in the preceding two chapters uncovered a variety of mathematical concepts which allow one to establish endogenous oscillations in economic applications. In these models, cyclical behavior can prevail for large ranges of the parameters while persistent oscillations in linear dynamical systems usually occur only for a particular parameter constellation. It seems natural, therefore, to refer to nonlinear approaches when cyclical motion is to be modeled in economics. In other words, cyclical behavior is synonymous with the presence of nonlinearities in most cases.
Hans-Walter Lorenz
Chapter 5. Chaotic Dynamics in Continuous-Time Economic Models
Abstract
Most existing economic models dealing with the chaos property are discrete-time models and can be reduced to a one-dimensional dynamical system. The main reason for this concentration on one-dimensional systems can probably be found in the relative ease with which chaotic motion can be established in these systems and because the two-dimensional case is already much more difficult to handle. However, chaos does not occur only in discrete-time models, but may be a property of continuous-time models as well.
Hans-Walter Lorenz
Chapter 6. Numerical Tools
Abstract
The theoretical results presented above allow to establish the existence of chaotic trajectories in several dynamical systems, which fulfill the assumptions of the appropriate theorems. For example, when the difference equation is unimodal, it is possible to apply the Li/Yorke theorem or Sarkovskii’s theorem and to establish the existence of chaos (defined in the sense of one of the definitions provided in the previous chapters). However, in many cases it may be difficult or analytically impossible to detect a period-three cycle, and for most differential equation systems there are no theoretical results at all. Experiments show that even for cycles of a relatively low period it may be impossible to distinguish regular time series from completely chaotic series by simple visual inspection.
Hans-Walter Lorenz
Chapter 7. Catastrophe Theory and Economic Dynamics
Abstract
This final chapter deals with catastrophe theory and its role in economic dynamics. Catastrophe theory was very popular in the 1970s and was considered a promising technique for the modeling of discontinuous jumps in the state variables of a dynamical system. In applications of the theory such interesting empirical topics like the abrupt emergence of aggression in the behavior of various species, stock-market crashes, the capsizing of ships, etc. were studied. All these examples describe phenomena which are characterized by an immediate, discontinuous change in a variable.
Hans-Walter Lorenz
Chapter 8. Concluding Remarks
Abstract
The foregoing presentation has, hopefully, made it evident that dynamical economics can be enriched by incorporating recent developments in the theory of nonlinear dynamical systems. However, a few final remarks seem to be in order. The general tendency in all mathematical theorems and economic applications presented in this short survey of nonlinear dynamics is that even the simplest dynamical systems may involve intuitively unexpected phenomena and highly complicated motions of the state variables. While traditional investigations of an evolving economy (especially in business-cycle theory) have concentrated on regularity aspects, and while recent revivals of (new-) classical macroeconomics scroll the recognized irregularities back to the noneconomic exogenous world, nonlinear dynamical systems allow for an entirely new theoretical attitude toward an understanding of cyclical motion which must not necessarily be irregular or chaotic. By an appropriate choice of nonlinearities it is almost always possible to model a particular dynamical phenomenon which is believed to prevail in reality.
Hans-Walter Lorenz
Backmatter
Metadaten
Titel
Nonlinear Dynamical Economics and Chaotic Motion
verfasst von
Dr. Hans-Walter Lorenz
Copyright-Jahr
1993
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-78324-1
Print ISBN
978-3-642-78326-5
DOI
https://doi.org/10.1007/978-3-642-78324-1