September  2006, 5(3): 537-550. doi: 10.3934/cpaa.2006.5.537

On a criterium of global attraction for discrete dynamical systems

1. 

Dept. de Matemàtiques i Informàtica, Universitat de les Illes Balears, Escola Politècnica Superior, 07122-Palma de Mallorca, Spain, Spain

2. 

Dept. de Matemàtiques, Universitat Autónoma de Barcelona, Edifici C, 08193 Bel-laterra, Barcelona

Received  April 2005 Revised  January 2006 Published  June 2006

Consider that the origin is a fix point of a discrete dynamical system $x^{(n+1)}=F(x^{(n)})$, defined in the whole $\mathbb R^m.$ LaSalle, in his book of 1976, [13], proposes to study several conditions which might imply global attraction. One of his suggestions is to write $F(x)=A(x)x$, where $A(x)$ is a real $m\times m$ matrix, and to assume that all the eigenvalues of eigenvalues of $A(x)$, for all $x\in \mathbb R^m$, have modulus smaller than one. In the paper [4], Cima et al. show that, when $m\ge2$, such hypothesis does not guarantee that the origin is a global attractor, even for polynomial maps $F$. From the observation that the decomposition of $F(x)$ as $A(x)x$ is not unique, in this paper we wonder whether LaSalle condition, for a special and canonical choice of $A,$ forces the origin to be a global attractor. This canonical choice is given by $A_c(x)=\int_0^1 DF(sx) ds,$ where the integration of the matrix $DF(x)$ is made term by term. In fact, we prove that LaSalle condition for $A_c(x)$ is a sufficient condition to get the global attraction of the origin when $m=1,$ or when $m=2$ and $F$ is polynomial. We also show that this is no more true for $m=2$ when $F$ is a rational map or when $m\ge3.$ Finally we consider the equivalent question for ordinary differential equations.
Citation: B. Coll, A. Gasull, R. Prohens. On a criterium of global attraction for discrete dynamical systems. Communications on Pure & Applied Analysis, 2006, 5 (3) : 537-550. doi: 10.3934/cpaa.2006.5.537
[1]

Anatoli F. Ivanov. On global dynamics in a multi-dimensional discrete map. Conference Publications, 2015, 2015 (special) : 652-659. doi: 10.3934/proc.2015.0652

[2]

Richard A. Norton, G. R. W. Quispel. Discrete gradient methods for preserving a first integral of an ordinary differential equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 1147-1170. doi: 10.3934/dcds.2014.34.1147

[3]

Antonio Garijo, Xavier Jarque. The secant map applied to a real polynomial with multiple roots. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020133

[4]

Sebastián Buedo-Fernández. Global attraction in a system of delay differential equations via compact and convex sets. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3171-3181. doi: 10.3934/dcdsb.2020056

[5]

Victor Kozyakin. Polynomial reformulation of the Kuo criteria for v- sufficiency of map-germs. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 587-602. doi: 10.3934/dcdsb.2010.14.587

[6]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2020094

[7]

Yuriy Golovaty, Anna Marciniak-Czochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reaction-diffusion equation coupled to the system of ordinary differential equations. Communications on Pure & Applied Analysis, 2012, 11 (1) : 229-241. doi: 10.3934/cpaa.2012.11.229

[8]

Jaume Giné, Maite Grau, Jaume Llibre. Polynomial and rational first integrals for planar quasi--homogeneous polynomial differential systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4531-4547. doi: 10.3934/dcds.2013.33.4531

[9]

Bin Wang, Arieh Iserles. Dirichlet series for dynamical systems of first-order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (1) : 281-298. doi: 10.3934/dcdsb.2014.19.281

[10]

Atanas Stefanov. On the Lipschitzness of the solution map for the 2 D Navier-Stokes system. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1471-1490. doi: 10.3934/dcds.2010.26.1471

[11]

Mourad Bellassoued, David Dos Santos Ferreira. Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map. Inverse Problems & Imaging, 2011, 5 (4) : 745-773. doi: 10.3934/ipi.2011.5.745

[12]

Daniel Fusca. The Madelung transform as a momentum map. Journal of Geometric Mechanics, 2017, 9 (2) : 157-165. doi: 10.3934/jgm.2017006

[13]

Lluís Alsedà, Michał Misiurewicz. Semiconjugacy to a map of a constant slope. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3403-3413. doi: 10.3934/dcdsb.2015.20.3403

[14]

Richard Evan Schwartz. Outer billiards and the pinwheel map. Journal of Modern Dynamics, 2011, 5 (2) : 255-283. doi: 10.3934/jmd.2011.5.255

[15]

Valentin Ovsienko, Richard Schwartz, Serge Tabachnikov. Quasiperiodic motion for the pentagram map. Electronic Research Announcements, 2009, 16: 1-8. doi: 10.3934/era.2009.16.1

[16]

John Erik Fornæss, Brendan Weickert. A quantized henon map. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 723-740. doi: 10.3934/dcds.2000.6.723

[17]

Zenonas Navickas, Rasa Smidtaite, Alfonsas Vainoras, Minvydas Ragulskis. The logistic map of matrices. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 927-944. doi: 10.3934/dcdsb.2011.16.927

[18]

Hunseok Kang. Dynamics of local map of a discrete Brusselator model: eventually trapping regions and strange attractors. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 939-959. doi: 10.3934/dcds.2008.20.939

[19]

Lukáš Adam, Jiří Outrata. On optimal control of a sweeping process coupled with an ordinary differential equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2709-2738. doi: 10.3934/dcdsb.2014.19.2709

[20]

Tomás Caraballo, Renato Colucci, Luca Guerrini. Bifurcation scenarios in an ordinary differential equation with constant and distributed delay: A case study. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2639-2655. doi: 10.3934/dcdsb.2018268

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]