By G. Abraham-Frois (auth.), Prof. Gilbert Abraham-Frois (eds.)
Considerable paintings has been performed on chaotic dynamics within the box of financial progress and dynamic macroeconomic versions over the last twenty years. This booklet considers various new advancements: creation of infrastructure in progress types, heterogeneity of brokers, hysteresis platforms, overlapping types with "pay-as-you-go" platforms, keynesian techniques with finance issues, interactions among rest cycles and chaotic dynamics, methodological matters, lengthy reminiscence tactics and fractals... A quantity of contributions which exhibits the relevance and fruitfulness of non-linear research for the reason of complicated dynamics in financial systems.
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Extra info for Non-Linear Dynamics and Endogenous Cycles
Let us note hk(a) this second polynomial for any integer k. Then, the bifurcation points correspond to the roots of hk(a) = 0 (for a proof of this result the reader is referred to Myberg ). We already know, from the lemma above that the iterates of the map P(Xt, IL) can be written as the sum of two polynomials, respectively in x and IL. L is a solution of Gk(lL) = O. L», k = 1,2, ... L)1/2, IL <0 (33) This last equation can be rewritten: (34) from which: (35) One finally obtains: (36) or: 38 q is the limit of a series antecedents of rank k, defined at Xt = O.
Of course, this depends of the initial point. The closer the initial capital-labor ratio is from the stationary state, the higher is the likelihood to achieve the stationary state quickly and the lesser to observe transient erratic fluctuations. These point are now examined in the following section. 3 Sensitivity to parameters and transient chaos in logistic family equations To investigate the property of sensitivity to parameters in logistic family equations, we present a simple analytical method that enables to show that the subsequent bifurcation points can be ordered in sets which have a "boxes within boxes" structure.
981-1001. , (1996), "CapitalLabor substitution and Competitive Nonlinear Endogenous Business Cycles", Cepremap working paper, forthcoming. 406-414. , (1993), Nonlinear Dynamical Economics and Chaotic Motion, Springer-Verlag, Heidelberg. , (1987), "Microeconomic Foundations of Cyclical Irregularities or Chaos", Mathematical Social Sciences, 13, 183-214. 419-423. , "sur k'iteration des polynomes reels quadratiques", Journal de Mathematiques Pures Appliquees, 41, 339-351. 985-992. 271-273. 61-71 Part 2 Business Cycles Hysteresis Arising from Individual Inertia and Behavorial Heterogeneity Reiner FRANKE 1 Introduction Appeal to the concept of hysteresis has become widespread in explanations of a variety of economic phenomena.