Nonlinear Difference Equations: Theory with Applications to Social Science Models

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Springer Science & Business Media, Mar 14, 2013 - Mathematics - 388 pages
It is generally acknowledged that deterministic formulations of dy namical phenomena in the social sciences need to be treated differently from similar formulations in the natural sciences. Social science phe nomena typically defy precise measurements or data collection that are comparable in accuracy and detail to those in the natural sciences. Con sequently, a deterministic model is rarely expected to yield a precise description of the actual phenomenon being modelled. Nevertheless, as may be inferred from a study of the models discussed in this book, the qualitative analysis of deterministic models has an important role to play in understanding the fundamental mechanisms behind social sci ence phenomena. The reach of such analysis extends far beyond tech nical clarifications of classical theories that were generally expressed in imprecise literary prose. The inherent lack of precise knowledge in the social sciences is a fun damental trait that must be distinguished from "uncertainty. " For in stance, in mathematically modelling the stock market, uncertainty is a prime and indispensable component of a model. Indeed, in the stock market, the rules are specifically designed to make prediction impossible or at least very difficult. On the other hand, understanding concepts such as the "business cycle" involves economic and social mechanisms that are very different from the rules of the stock market. Here, far from seeking unpredictability, the intention of the modeller is a scientific one, i. e.
 

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Contents

PRELIMINARIES
11
VECTOR DIFFERENCE EQUATIONS
79
B The basic dynamics
87
Fiber bifurcations
94
Invariants
105
F Notes
111
Rsemiconjugate maps and chaos
124
Notes
139
Additional equations
195
A Weak contractions and stability
201
B Weak expansions and instability
210
The equation an E1 XXoaian + g XXo biani
224
B Boundedness and stability
226
vii
235
F Notes
236
CHAOS AND STABILITY IN SOME MODELS
243

Continuity revisited
149
Mode structures
155
Notes
163
Permanence
185
ADDITIONAL MODELS
339
Bibliography
367
Index
385
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