Smooth roughness of exponential dichotomies, revisited

Previous Article
On Lyapunov exponents of difference equations with random delay
DCDS-B Home
This Issue
Next Article
Stochastic dynamics in a fluid--plate interaction model with the only longitudinal deformations of the plate

May 2015, 20(3): 853-859. doi: 10.3934/dcdsb.2015.20.853

Smooth roughness of exponential dichotomies, revisited

Christian Pötzsche ^1,

1.	Institut für Mathematik, Alpen-Adria Universität Klagenfurt, Universitätsstraße 65-67, 9020 Klagenfurt, Austria

Received October 2013 Revised April 2014 Published January 2015

Abstract
Related Papers

As a direct consequence of well-established proof techniques, we establish that the invariant projectors of exponential dichotomies for parameter-dependent nonautonomous difference equations are as smooth as their right-hand sides. For instance, this guarantees that the saddle-point structure in the vicinity of hyperbolic solutions inherits its differentiability properties from the particular given equation.

Keywords: roughness, Exponential dichotomy, smooth parameter-dependence, nonautonomous dynamical system..

Mathematics Subject Classification: Primary: 34D09; Secondary: 37B55, 39A3.

Citation: Christian Pötzsche. Smooth roughness of exponential dichotomies, revisited. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 853-859. doi: 10.3934/dcdsb.2015.20.853

References:

[1]	B. Aulbach and N. Van Minh, The concept of spectral dichotomy for linear difference equations II, J. Difference Equ. Appl., 2 (1996), 251-262. doi: 10.1080/10236199608808060.
[2]	B. Aulbach and T. Wanner, The Hartman-Grobman theorem for Carathéodory-type differential equations in Banach spaces, Nonlin. Analysis (TMA), 40 (2000), 91-104. doi: 10.1016/S0362-546X(00)85006-3.
[3]	_______, Topological simplification of nonautonomous difference equations, J. Difference Equ. Appl., 12 (2006), 283-296. doi: 10.1080/10236190500489384.
[4]	R. H. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, Second edition, Applied Mathematical Sciences, 75, Springer, Berlin etc., 1988. doi: 10.1007/978-1-4612-1029-0.
[5]	L. Barreira and C. Valls, Smooth robustness of exponential dichotomies, Proc. Am. Math. Soc., 139 (2011), 999-1012. doi: 10.1090/S0002-9939-2010-10531-2.
[6]	A. Carvalho, J. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems, Applied Mathematical Sciences, 182, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.
[7]	S.-N. Chow and J. K. Hale, Methods of Bifurcation Theory, Grundlehren der mathematischen Wissenschaften, 251, Springer, Berlin etc., 1996.
[8]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes Math., 840, Springer, Berlin etc., 1981.
[9]	T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.
[10]	P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176, AMS, Providence, RI, 2011. doi: 10.1090/surv/176.
[11]	K. J. Palmer, Transversal heteroclinic points and Cherry's example of a nonintegrable Hamiltonian system, J. Differ. Equations, 65 (1986), 321-360. doi: 10.1016/0022-0396(86)90023-9.
[12]	_______, A perturbation theorem for exponential dichotomies, Proc. R. Soc. Edinb. Section A, 106 (1987), 25-37. doi: 10.1017/S0308210500018175.
[13]	_______, Exponential dichotomies, the shadowing lemma and transversal homoclinic points, in Dynamics Reported. Vol. 1 (eds. U. Kirchgraber and H.-O. Walther), B.G. Teubner and John Wiley and Sons, Stuttgart/Chichester etc., 1988, 265-306.
[14]	C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lect. Notes Math., 2002, Springer, Berlin etc., 2010. doi: 10.1007/978-3-642-14258-1.
[15]	_______, Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach, Discrete Contin. Dyn. Syst. (Series B), 14 (2010), 739-776. doi: 10.3934/dcdsb.2010.14.739.
[16]	_______, Nonautonomous continuation of bounded solutions, Commun. Pure Appl. Anal., 10 (2011), 937-961. doi: 10.3934/cpaa.2011.10.937.
[17]	B. Sandstede, Verzweigungstheorie Homokliner Verdopplungen, Ph.D. thesis, Universität Stuttgart, Germany, 1993.
[18]	K. Sakamoto, Estimates on the strength of exponential dichotomies and application to integral manifolds, J. Differ. Equations, 107 (1994), 259-279. doi: 10.1006/jdeq.1994.1012.
[19]	Y. Yi, A generalized integral manifold theorem, J. Differ. Equations, 102 (1993), 153-187. doi: 10.1006/jdeq.1993.1026.
[20]	K. Yosida, Functional Analysis, Grundlehren der mathematischen Wissenschaften, 123, Springer, Berlin etc., 1980.

show all references

References:

[1]	B. Aulbach and N. Van Minh, The concept of spectral dichotomy for linear difference equations II, J. Difference Equ. Appl., 2 (1996), 251-262. doi: 10.1080/10236199608808060.
[2]	B. Aulbach and T. Wanner, The Hartman-Grobman theorem for Carathéodory-type differential equations in Banach spaces, Nonlin. Analysis (TMA), 40 (2000), 91-104. doi: 10.1016/S0362-546X(00)85006-3.
[3]	_______, Topological simplification of nonautonomous difference equations, J. Difference Equ. Appl., 12 (2006), 283-296. doi: 10.1080/10236190500489384.
[4]	R. H. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, Second edition, Applied Mathematical Sciences, 75, Springer, Berlin etc., 1988. doi: 10.1007/978-1-4612-1029-0.
[5]	L. Barreira and C. Valls, Smooth robustness of exponential dichotomies, Proc. Am. Math. Soc., 139 (2011), 999-1012. doi: 10.1090/S0002-9939-2010-10531-2.
[6]	A. Carvalho, J. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems, Applied Mathematical Sciences, 182, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.
[7]	S.-N. Chow and J. K. Hale, Methods of Bifurcation Theory, Grundlehren der mathematischen Wissenschaften, 251, Springer, Berlin etc., 1996.
[8]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes Math., 840, Springer, Berlin etc., 1981.
[9]	T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.
[10]	P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176, AMS, Providence, RI, 2011. doi: 10.1090/surv/176.
[11]	K. J. Palmer, Transversal heteroclinic points and Cherry's example of a nonintegrable Hamiltonian system, J. Differ. Equations, 65 (1986), 321-360. doi: 10.1016/0022-0396(86)90023-9.
[12]	_______, A perturbation theorem for exponential dichotomies, Proc. R. Soc. Edinb. Section A, 106 (1987), 25-37. doi: 10.1017/S0308210500018175.
[13]	_______, Exponential dichotomies, the shadowing lemma and transversal homoclinic points, in Dynamics Reported. Vol. 1 (eds. U. Kirchgraber and H.-O. Walther), B.G. Teubner and John Wiley and Sons, Stuttgart/Chichester etc., 1988, 265-306.
[14]	C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems, Lect. Notes Math., 2002, Springer, Berlin etc., 2010. doi: 10.1007/978-3-642-14258-1.
[15]	_______, Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach, Discrete Contin. Dyn. Syst. (Series B), 14 (2010), 739-776. doi: 10.3934/dcdsb.2010.14.739.
[16]	_______, Nonautonomous continuation of bounded solutions, Commun. Pure Appl. Anal., 10 (2011), 937-961. doi: 10.3934/cpaa.2011.10.937.
[17]	B. Sandstede, Verzweigungstheorie Homokliner Verdopplungen, Ph.D. thesis, Universität Stuttgart, Germany, 1993.
[18]	K. Sakamoto, Estimates on the strength of exponential dichotomies and application to integral manifolds, J. Differ. Equations, 107 (1994), 259-279. doi: 10.1006/jdeq.1994.1012.
[19]	Y. Yi, A generalized integral manifold theorem, J. Differ. Equations, 102 (1993), 153-187. doi: 10.1006/jdeq.1993.1026.
[20]	K. Yosida, Functional Analysis, Grundlehren der mathematischen Wissenschaften, 123, Springer, Berlin etc., 1980.

[1]	Junya Nishiguchi. On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5657-5679. doi: 10.3934/dcds.2016048
[2]	Mihail Megan, Adina Luminiţa Sasu, Bogdan Sasu. Discrete admissibility and exponential dichotomy for evolution families. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 383-397. doi: 10.3934/dcds.2003.9.383
[3]	Ciprian Preda. Discrete-time theorems for the dichotomy of one-parameter semigroups. Communications on Pure and Applied Analysis, 2008, 7 (2) : 457-463. doi: 10.3934/cpaa.2008.7.457
[4]	Radosław Kurek, Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1955-1981. doi: 10.3934/dcds.2018079
[5]	Yejuan Wang, Chengkui Zhong, Shengfan Zhou. Pullback attractors of nonautonomous dynamical systems. Discrete and Continuous Dynamical Systems, 2006, 16 (3) : 587-614. doi: 10.3934/dcds.2006.16.587
[6]	Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Approximation of attractors of nonautonomous dynamical systems. Discrete and Continuous Dynamical Systems - B, 2005, 5 (2) : 215-238. doi: 10.3934/dcdsb.2005.5.215
[7]	Chunqiu Li, Desheng Li, Xuewei Ju. On the forward dynamical behavior of nonautonomous systems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 473-487. doi: 10.3934/dcdsb.2019190
[8]	Lei Wang, Jinlong Yuan, Yingfang Li, Enmin Feng, Zhilong Xiu. Parameter identification of nonlinear delayed dynamical system in microbial fermentation based on biological robustness. Numerical Algebra, Control and Optimization, 2014, 4 (2) : 103-113. doi: 10.3934/naco.2014.4.103
[9]	Giselle A. Monteiro, Milan Tvrdý. Generalized linear differential equations in a Banach space: Continuous dependence on a parameter. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 283-303. doi: 10.3934/dcds.2013.33.283
[10]	Kaizhi Wang, Jun Yan. Lipschitz dependence of viscosity solutions of Hamilton-Jacobi equations with respect to the parameter. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1649-1659. doi: 10.3934/dcds.2016.36.1649
[11]	Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part II: The nonlinear system.. Evolution Equations and Control Theory, 2014, 3 (1) : 83-118. doi: 10.3934/eect.2014.3.83
[12]	Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part I: The linearized system.. Evolution Equations and Control Theory, 2014, 3 (1) : 59-82. doi: 10.3934/eect.2014.3.59
[13]	Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. I. Invariant torus and its normal hyperbolicity. Journal of Geometric Mechanics, 2012, 4 (4) : 443-467. doi: 10.3934/jgm.2012.4.443
[14]	Björn Schmalfuss. Attractors for nonautonomous and random dynamical systems perturbed by impulses. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 727-744. doi: 10.3934/dcds.2003.9.727
[15]	David Cheban. Global attractors of nonautonomous quasihomogeneous dynamical systems. Conference Publications, 2001, 2001 (Special) : 96-101. doi: 10.3934/proc.2001.2001.96
[16]	Hongyong Cui, Peter E. Kloeden, Meihua Yang. Forward omega limit sets of nonautonomous dynamical systems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (4) : 1103-1114. doi: 10.3934/dcdss.2020065
[17]	Ralf Spatzier, Lei Yang. Exponential mixing and smooth classification of commuting expanding maps. Journal of Modern Dynamics, 2017, 11: 263-312. doi: 10.3934/jmd.2017012
[18]	Éder Rítis Aragão Costa. An extension of the concept of exponential dichotomy in Fréchet spaces which is stable under perturbation. Communications on Pure and Applied Analysis, 2019, 18 (2) : 845-868. doi: 10.3934/cpaa.2019041
[19]	Jianhua Huang, Wenxian Shen. Pullback attractors for nonautonomous and random parabolic equations on non-smooth domains. Discrete and Continuous Dynamical Systems, 2009, 24 (3) : 855-882. doi: 10.3934/dcds.2009.24.855
[20]	John R. Graef, Lingju Kong. Uniqueness and parameter dependence of positive solutions of third order boundary value problems with $p$-laplacian. Conference Publications, 2011, 2011 (Special) : 515-522. doi: 10.3934/proc.2011.2011.515

2020 Impact Factor: 1.327

Tools

Metrics

Other articles
by authors

on AIMS
Christian Pötzsche
on Google Scholar
Christian Pötzsche

[Back to Top]