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Open Access item Quadratic discrete dynamics associated to quadratic maps in the plane

Authors:Durán Díaz, Raúl
Hernández Encinas, Luis
Muñoz Masqué, Jaime
Keywords:Bilinear symmetric map, Fractal subset, Linear group, Quadratic discrete system., MSC 2000, Primary 68U05, Secondary 28A80, 13A50, 15A72, 37M30, 65D18
Issue Date:Jun-2009
Abstract:Dynamical systems allow one to model many phenomena, not only in Physics and Mathematics, but also in Biology, Engineering, Economics, etc. A bidimensional discrete dynamical system consists in iterating a mapping $f\colon \mathbb{R}^2\to \mathbb{R}^2$. The evolution of a point $p\in \mathbb{R}^2$ is described by the sequence of its iterated points, i.e.,$f^{n}(p)=f(f^{n-1}(p)),\dotsc,$ $n=0$, $n=1$, $n=2,\dotsc,$ showing the state of $p$ at the times $t=0,t=1,t=2,\dotsc,t=n$, etc. Such sequences replace the flux of a vector field in a continuous system and they are the object of Discrete Dynamics. The best known bidimensional discrete dynamical systems are the quadratic ones. In order to study them it is important to work with normal forms. This is achieved by means of invariants, which is a function depending on the system parameters remaining unchanged when a change of coordinates, is made. We study the fractal associated to any bidimensional regular homogeneous quadratic system (each of these terms defined below), which is the analogous to the Mandelbrot fractal associated to the map $z=x+yi\mapsto z^2$.
Description:In: International Conference on Computational and Mathematical Methods on Science and Engineering, (CMMSE-2009), vol. II, Proceedings 437-448. J. Vigo-Aguiar (Ed.), Junio-Julio, 2009.
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