By Thierry Vialar
Complex dynamics represent a becoming and more and more vital quarter as they give a powerful strength to provide an explanation for and formalize ordinary, actual, monetary and monetary phenomena.
This publication pursues the formidable objective to collect an in depth physique of information concerning advanced dynamics from a variety of educational disciplines. past its concentrate on economics and finance, together with for example the evolution of macroeconomic progress versions in the direction of nonlinear constructions in addition to sign processing purposes to inventory markets, primary elements of the e-book are dedicated to using nonlinear dynamics in arithmetic, information, sign thought and processing.
Numerous examples and purposes, nearly seven hundred illustrations and numerical simulations in keeping with using Matlab make the ebook a necessary reference for researchers and scholars from many alternative disciplines who're drawn to the nonlinear box. An appendix recapitulates the fundamental mathematical strategies required to exploit the book.
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Additional info for Complex and Chaotic Nonlinear Dynamics. Advances in Economics and Finance, Mathematics and Statistics
The behaviors resulting from the “nonlinear” make it possible to better understand the natural phenomena considered as complex. 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 that there exists a kind of “conceptual unification” through the notions of attractors, period-doublings, subharmonic cascades, bifurcations, Lyapunov exponent, sensitive dependence on initial conditions, etc.
For μ > 0, r˙ = 0 for r = (μ / |a1 |)1/2 or for r = 0. Then we have for μ > 0 a stable limit cycle, of radius proportional to μ 1/2 , which surrounds a hyperbolic, unstable focus at the origin. This is a supercritical Hopf bifurcation (see Fig. 6a). • Suppose a1 > 0, the limit cycle occurs for μ < 0, it is unstable and surrounds a stable fixed point. When μ increases, the radius of the cycle decreases to zero at μ = 0, where the fixed point at the origin becomes a (weakly) unstable focus. If μ > 0, (x1 , x2 )T = 0 is unstable and hyperbolic.
In such a case, there are three possibilities according the manner whose eigenvalue of the Floquet matrix crosses the unit circle of the complex plane. (1) If the crossing is carried out by +1, the theory is not radically different from that of fixed point bifurcations. The normal form, topologically equivalent to a node-saddle bifurcation is now written: x˙ = x + μ − x2 , the term x is introduced by the eigenvalue +1. Similarly, there are normal forms which lead to pitchfork bifurcations or stability exchanges according to the considered problem.
Complex and Chaotic Nonlinear Dynamics. Advances in Economics and Finance, Mathematics and Statistics by Thierry Vialar