American Institute of Mathematical Sciences

Advanced
July  2009, 12(1): 109-131. doi: 10.3934/dcdsb.2009.12.109

Numerical computation of dichotomy rates and projectors in discrete time

Thorsten Hüls 1,

1. 

Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld

Received  September 2008 Revised  December 2008 Published  May 2009

We introduce a characterization of exponential dichotomies for linear difference equations that can be tested numerically and enables the approximation of dichotomy rates and projectors with high accuracy. The test is based on computing the bounded solutions of a specific inhomogeneous difference equation. For this task a boundary value and a least squares approach is applied. The results are illustrated using Hénon's map. We compute approximations of dichotomy rates and projectors of the variational equation, along a homoclinic orbit and an orbit on the attractor as well as for an almost periodic example. For the boundary value and the least squares approach, we analyze in detail errors that occur, when restricting the infinite dimensional problem to a finite interval.
Keywords: Boundary value problem, Dichotomy projectors, Exponential dichotomy, Least squares solution..
Mathematics Subject Classification: Primary: 34D09, 37C75; Secondary: 34B0.
Citation: Thorsten Hüls. Numerical computation of dichotomy rates and projectors in discrete time. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 109-131. doi: 10.3934/dcdsb.2009.12.109
[1]

Mihail Megan, Adina Luminiţa Sasu, Bogdan Sasu. Discrete admissibility and exponential dichotomy for evolution families. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 383-397. doi: 10.3934/dcds.2003.9.383
[2]

Christian Pötzsche. Dichotomy spectra of triangular equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 423-450. doi: 10.3934/dcds.2016.36.423
[3]

Éder Rítis Aragão Costa. An extension of the concept of exponential dichotomy in Fréchet spaces which is stable under perturbation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 845-868. doi: 10.3934/cpaa.2019041
[4]

Piotr Kowalski. The existence of a solution for Dirichlet boundary value problem for a Duffing type differential inclusion. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2569-2580. doi: 10.3934/dcdsb.2014.19.2569
[5]

Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319
[6]

Yu Tian, John R. Graef, Lingju Kong, Min Wang. Existence of solutions to a multi-point boundary value problem for a second order differential system via the dual least action principle. Conference Publications, 2013, 2013 (special) : 759-769. doi: 10.3934/proc.2013.2013.759
[7]

António J.G. Bento, Nicolae Lupa, Mihail Megan, César M. Silva. Integral conditions for nonuniform $μ$-dichotomy on the half-line. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3063-3077. doi: 10.3934/dcdsb.2017163
[8]

Kristin Dettmers, Robert Giza, Rafael Morales, John A. Rock, Christina Knox. A survey of complex dimensions, measurability, and the lattice/nonlattice dichotomy. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 213-240. doi: 10.3934/dcdss.2017011
[9]

Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167
[10]

Xiaoyun Cai, Liangwen Liao, Yongzhong Sun. Global strong solution to the initial-boundary value problem of a 2-D Kazhikhov-Smagulov type model. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 917-923. doi: 10.3934/dcdss.2014.7.917
[11]

Peng Jiang. Unique global solution of an initial-boundary value problem to a diffusion approximation model in radiation hydrodynamics. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3015-3037. doi: 10.3934/dcds.2015.35.3015
[12]

Yunhai Xiao, Soon-Yi Wu, Bing-Sheng He. A proximal alternating direction method for $\ell_{2,1}$-norm least squares problem in multi-task feature learning. Journal of Industrial & Management Optimization, 2012, 8 (4) : 1057-1069. doi: 10.3934/jimo.2012.8.1057
[13]

Sunghan Kim, Ki-Ahm Lee, Henrik Shahgholian. Homogenization of the boundary value for the Dirichlet problem. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-22. doi: 10.3934/dcds.2019234
[14]

Ciprian Preda. Discrete-time theorems for the dichotomy of one-parameter semigroups. Communications on Pure & Applied Analysis, 2008, 7 (2) : 457-463. doi: 10.3934/cpaa.2008.7.457
[15]

Thai Son Doan, Martin Rasmussen, Peter E. Kloeden. The mean-square dichotomy spectrum and a bifurcation to a mean-square attractor. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 875-887. doi: 10.3934/dcdsb.2015.20.875
[16]

Bassam Fayad, A. Windsor. A dichotomy between discrete and continuous spectrum for a class of special flows over rotations. Journal of Modern Dynamics, 2007, 1 (1) : 107-122. doi: 10.3934/jmd.2007.1.107
[17]

Álvaro Castañeda, Gonzalo Robledo. Dichotomy spectrum and almost topological conjugacy on nonautonomus unbounded difference systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (5) : 2287-2304. doi: 10.3934/dcds.2018094
[18]

Jingli Ren, Dandan Zhu, Haiyan Wang. Spreading-vanishing dichotomy in information diffusion in online social networks with intervention. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1843-1865. doi: 10.3934/dcdsb.2018240
[19]

Juan Dávila, Louis Dupaigne, Marcelo Montenegro. The extremal solution of a boundary reaction problem. Communications on Pure & Applied Analysis, 2008, 7 (4) : 795-817. doi: 10.3934/cpaa.2008.7.795
[20]

Ya-Xiang Yuan. Recent advances in numerical methods for nonlinear equations and nonlinear least squares. Numerical Algebra, Control & Optimization, 2011, 1 (1) : 15-34. doi: 10.3934/naco.2011.1.15

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences