On the Unresolved Cases of Convergence of Bidimensional Slow Discrete Dynamical Systems
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On the Unresolved Cases of Convergence of Bidimensional Slow Discrete Dynamical Systems Francisco J. Solis1
· Alina Sotolongo1
© Foundation for Scientific Research and Technological Innovation 2017
Abstract In this work we complete the analysis of convergence for nonhyperbolic fixed points of two dimensional discrete dynamical systems. We analyze all different scenarios of systems with isolated equilibria whose linearized part has eigenvalues of norm one, except for those with real eigenvalues equal to one, which have been studied previously. We also include the non-diagonalizable case and the case of mixing eigenvalues, where only one real eigenvalue has absolute value equal to one. Different techniques have been applied depending on the hyperbolic scenario. Finally, many two dimensional examples are presented in order to illustrate the diverse orbit behaviors and the number of iterations required to converge. Keywords Nonhyperbolic fixed points · Convergence · Slow discrete dynamical systems · Eigenvalues Mathematics Subject Classification 37M99 · 65P40
Introduction Convergence for nonhyperbolic fixed points of discrete dynamical systems is an interesting and challenging topic. Nonhyperbolic equilibria are disregarded and sometimes rejected in the literature by claiming that they are nongeneric. Although hyperbolicity is a generic property, there are many theoretical situations where nonhyperbolicity is important by its intrinsic value, for example there are dynamical systems where the conjunction of two cycles in a system may create a nonhyperbolic cycle, see [1–3]. Moreover, there are many applications where nonhyperbolicity appears in a natural way, see [4–7]. However, the analysis of systems
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Francisco J. Solis [email protected] Alina Sotolongo [email protected]
1
CIMAT, Guanajuato, GTO 36000, Mexico
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Differ Equ Dyn Syst
with nonhyperbolic attractors remains as a difficult task, since such systems can not be analyzed by linearization; there are directions on the phase space which neither contract nor expand. So far, for one dimensional discrete dynamical systems, nonhyperbolicity has been completely analyzed [8,9]. For two dimensional systems, convergence has been analyzed only partially. The aim of this work is to complete the analysis of convergence of two dimensional discrete dynamical systems with isolated equilibria whose linearized part has eigenvalues with norm one. These systems are called slow discrete dynamical systems. Since for these systems the diagonalizable case with eigenvalues equal to one has been analyzed in [10], we will focus here on the remaining cases assuming without loss in generality that the isolated equilibrium is located at the origin. This paper is organized as follows: In “Problem Formulation and Preliminaries”, we formulate the problem of interest and provide a review of the main results for the case of equilibria whose linearized part has eigenvalues equal to one. In “Eigenvalues One and Minus One”, we analyze the case with real eigenvalues of norm one with
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