Nonlinear Difference Equations: Theory with Applications to Social Science Models
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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VECTOR DIFFERENCE EQUATIONS
B The basic dynamics
Rsemiconjugate maps and chaos
A Weak contractions and stability
B Weak expansions and instability
The equation an E1 XXoaian + g XXo biani
B Boundedness and stability
CHAOS AND STABILITY IN SOME MODELS
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Nonlinear Difference Equations: Theory with Applications to Social Science ...
No preview available - 2010
2-cycle absorbing interval an+1 apply assume b)-semiconjugate map bounded chaotic behavior compact consider contains continuous maps converges Corollary curve cycle of length D C R decreasing define definition derivative difference equation dimensional dynamics eigenvalues ejector cycle Example existence fiber H Figure fived point follows globally asymptotically stable globally attracting graph Hence holds homeomorphism implies induction inequality initial values integer invariant fiber Lemma Let F Liapunov function limit cycle limit point linear maps link map logistic map map F mode monotonically non-decreasing nonempty nontrivial obtain orbit origin parameter particular periodic points persistent oscillations point of F polymodal system polynomial positive fixed point positive integer possible proof of Theorem R-semiconjugate real numbers Remark result satisfies Section Sedaghat segment semiconjugacy semipermanent sequence shows snap-back repeller solution of 4.1b stable fixed point Suppose tangent bifurcation topologically conjugate transcritical bifurcation unique fixed point vector zero