Introduction
The aim of this work is to try to offer a stimulating environment for the study of complex or chaotic nonlinear Dynamics. The topicality of this type of dynamics results from widely different scientific disciplines. And although keeping an economic or financial prevalence, the assigned objective can only be approached by an opening to the other disciplines related to the subject. Economic models have long been elaborated from constructions whose algebraic nature was of a linear order. This...
Info Ike
It is possible to write that P implies -R and C -C and H implies -R -H implies -C. A.6.3 Affine and Projective Planes A.6.3.1 Affine Planes Definition A.75 Affine plane of incidence . An affine plane of incidence is any set of points and lines which verifies A1 For any pair of distinct points A and B, there exists one and only one line l incident to A and B. A2 For any line l there exists at least one point A which is not incident to l. A3 Given a line l and a point D which is non incident to...
A101 Lagrange Method
This method is also called Lagrange interpolation polynomial. The interpolation polynomial associated with the points Pp xp,yp , can be written in the form Calculations resulting from this method are often rather long. Furthermore, if one adds new points to the set to be interpolated, it is necessary to restart all calculations. Unlike Lagrange method, the Newton-Gregory method avoids such a problem.
A8 Distribution Theory
Distribution theory, created in the 1950s by Laurent Schwartz, made possible to make rigorous certain heuristic process i.e. symbolic calculation ofHeaviside, delta of Dirac , to clarify the notion of weak solution of a partial differential equation PDE , and lastly to provide a general framework to Fourier transform. Distribution theory is a vast generalization of the function notion of several variables The fundamental idea is that of duality Distributions are, by definition, linear forms on...
Info Wvp
Fig. 1.31 Archetype of strange attractor and an associated Poincare section Fig. 1.31 Archetype of strange attractor and an associated Poincare section 2. The dimension D of the attractor is fractal non-integer 2 lt D lt n, where n is the dimension of the phase space. 3. There is sensitive dependence on initial conditions SDIC two trajectories of the attractor initially very close will diverge. An illustration of strange attractors can be given by means of the well-known James Gleick picture...
Info Cnv
The numerical value of a J thus plays an important part in the nature of solutions of the system. The approach of these numerical values opened developments concerning the algebraic number concept, which will not be evoked here. In the framework of this paragraph, we will not describe either the case where the rotation number a J p q is rational, for which there is resonance between p and q, which leads to the description of groups of Poincare-Birkhoffpoints with hyperbolic or elliptic forms,...
Info Sec
1.7.2 Transitions Stemming from the Linear Stability Loss in Dissipative Systems The study of eigenvalues of the matrix M Df x allows a first approach of the bifurcation concept. For a fixed point x, there are three ways of losing its stability 1 If the unit circle is crossed by 1 then occurs a saddle-node bifurcation. The solution is not only unstable but disappears completely, then just beyond the bifurcation threshold, the system generates a specific regime called type-1 inter-mittency,...
Smale Horseshoe Structural Stability
Stephen Smale 1965 built a diffeomorphism f R2 R2, with very complex dynamics, which admits an infinity of periodic orbits of arbitrarily large periods. To illustrate it as simply as possible, we are only interested in a diffeomorphism of the rectangle A 0,1 2 on its image. The construction is carried out see figure which follows by a composition f p o E of a hyperbolic linear map E x, y 3x, y 3 with a nonlinear transformation p. These elements are defined such that f A0 Ao, f ao x,y 3x,y 3...
SmaleBirkhoff Homoclinic Theorem
Closely related to the Smale horseshoe map topology43 and to the hyperbolicity concept, it is interesting to present the following fundamental theorem. Theorem 1.11 Smale-Birkhoff homoclinic theorem . Let f be a diffeomor-phism C1 and suppose p is a hyperbolic fixed point. A homoclinic point is a point q p which is in the stable and unstable manifold. If the stable and unstable manifold intersect transversally44 at q, then q is called transverse. This implies that there is a homoclinic orbit y...
Nonlinear Theory
In the general introduction we observed that the irruption of the nonlinear led to a profound transformation of a great number of scientific fields. The behaviors resulting from the nonlinear make it possible to better understand the natural phenomena considered as complex. The nonlinear introduced a set of concepts and tools, i.e. analysis and investigation instruments of dynamics generated by the nonlinear. We try to gather these investigation tools, knowing that today it is possible to say...
Springer
Dr. Thierry Vialar Pr. Alain Goergen University of Paris X cole normale sup rieure de Cachan Dept. of Economics Dept. of Economics 200, ave de la R publique 61, ave du Pr sident Wilson 92001 Nanterre Cedex 94235 Cachan Cedex ISBN 978-3-540-85977-2 e-ISBN 978-3-540-85978-9 Library of Congress Control Number 2008937818 2009 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights...