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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

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 & 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.  doi: 10.1080/10236199608808060.  Google Scholar

[2]

B. Aulbach and T. Wanner, The Hartman-Grobman theorem for Carathéodory-type differential equations in Banach spaces,, Nonlin. Analysis (TMA), 40 (2000), 91.  doi: 10.1016/S0362-546X(00)85006-3.  Google Scholar

[3]

_______, Topological simplification of nonautonomous difference equations,, J. Difference Equ. Appl., 12 (2006), 283.  doi: 10.1080/10236190500489384.  Google Scholar

[4]

R. H. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications,, Second edition, (1988).  doi: 10.1007/978-1-4612-1029-0.  Google Scholar

[5]

L. Barreira and C. Valls, Smooth robustness of exponential dichotomies,, Proc. Am. Math. Soc., 139 (2011), 999.  doi: 10.1090/S0002-9939-2010-10531-2.  Google Scholar

[6]

A. Carvalho, J. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems,, Applied Mathematical Sciences, (2013).  doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[7]

S.-N. Chow and J. K. Hale, Methods of Bifurcation Theory,, Grundlehren der mathematischen Wissenschaften, (1996).   Google Scholar

[8]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lect. Notes Math., (1981).   Google Scholar

[9]

T. Kato, Perturbation Theory for Linear Operators,, Reprint of the 1980 edition, (1980).   Google Scholar

[10]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems,, Mathematical Surveys and Monographs, (2011).  doi: 10.1090/surv/176.  Google Scholar

[11]

K. J. Palmer, Transversal heteroclinic points and Cherry's example of a nonintegrable Hamiltonian system,, J. Differ. Equations, 65 (1986), 321.  doi: 10.1016/0022-0396(86)90023-9.  Google Scholar

[12]

_______, A perturbation theorem for exponential dichotomies,, Proc. R. Soc. Edinb. Section A, 106 (1987), 25.  doi: 10.1017/S0308210500018175.  Google Scholar

[13]

_______, Exponential dichotomies, the shadowing lemma and transversal homoclinic points,, in Dynamics Reported. Vol. 1 (eds. U. Kirchgraber and H.-O. Walther), (1988), 265.   Google Scholar

[14]

C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems,, Lect. Notes Math., (2002).  doi: 10.1007/978-3-642-14258-1.  Google Scholar

[15]

_______, Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach,, Discrete Contin. Dyn. Syst. (Series B), 14 (2010), 739.  doi: 10.3934/dcdsb.2010.14.739.  Google Scholar

[16]

_______, Nonautonomous continuation of bounded solutions,, Commun. Pure Appl. Anal., 10 (2011), 937.  doi: 10.3934/cpaa.2011.10.937.  Google Scholar

[17]

B. Sandstede, Verzweigungstheorie Homokliner Verdopplungen,, Ph.D. thesis, (1993).   Google Scholar

[18]

K. Sakamoto, Estimates on the strength of exponential dichotomies and application to integral manifolds,, J. Differ. Equations, 107 (1994), 259.  doi: 10.1006/jdeq.1994.1012.  Google Scholar

[19]

Y. Yi, A generalized integral manifold theorem,, J. Differ. Equations, 102 (1993), 153.  doi: 10.1006/jdeq.1993.1026.  Google Scholar

[20]

K. Yosida, Functional Analysis,, Grundlehren der mathematischen Wissenschaften, (1980).   Google Scholar

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.  doi: 10.1080/10236199608808060.  Google Scholar

[2]

B. Aulbach and T. Wanner, The Hartman-Grobman theorem for Carathéodory-type differential equations in Banach spaces,, Nonlin. Analysis (TMA), 40 (2000), 91.  doi: 10.1016/S0362-546X(00)85006-3.  Google Scholar

[3]

_______, Topological simplification of nonautonomous difference equations,, J. Difference Equ. Appl., 12 (2006), 283.  doi: 10.1080/10236190500489384.  Google Scholar

[4]

R. H. Abraham, J. E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications,, Second edition, (1988).  doi: 10.1007/978-1-4612-1029-0.  Google Scholar

[5]

L. Barreira and C. Valls, Smooth robustness of exponential dichotomies,, Proc. Am. Math. Soc., 139 (2011), 999.  doi: 10.1090/S0002-9939-2010-10531-2.  Google Scholar

[6]

A. Carvalho, J. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems,, Applied Mathematical Sciences, (2013).  doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[7]

S.-N. Chow and J. K. Hale, Methods of Bifurcation Theory,, Grundlehren der mathematischen Wissenschaften, (1996).   Google Scholar

[8]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lect. Notes Math., (1981).   Google Scholar

[9]

T. Kato, Perturbation Theory for Linear Operators,, Reprint of the 1980 edition, (1980).   Google Scholar

[10]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems,, Mathematical Surveys and Monographs, (2011).  doi: 10.1090/surv/176.  Google Scholar

[11]

K. J. Palmer, Transversal heteroclinic points and Cherry's example of a nonintegrable Hamiltonian system,, J. Differ. Equations, 65 (1986), 321.  doi: 10.1016/0022-0396(86)90023-9.  Google Scholar

[12]

_______, A perturbation theorem for exponential dichotomies,, Proc. R. Soc. Edinb. Section A, 106 (1987), 25.  doi: 10.1017/S0308210500018175.  Google Scholar

[13]

_______, Exponential dichotomies, the shadowing lemma and transversal homoclinic points,, in Dynamics Reported. Vol. 1 (eds. U. Kirchgraber and H.-O. Walther), (1988), 265.   Google Scholar

[14]

C. Pötzsche, Geometric Theory of Discrete Nonautonomous Dynamical Systems,, Lect. Notes Math., (2002).  doi: 10.1007/978-3-642-14258-1.  Google Scholar

[15]

_______, Nonautonomous bifurcation of bounded solutions I: A Lyapunov-Schmidt approach,, Discrete Contin. Dyn. Syst. (Series B), 14 (2010), 739.  doi: 10.3934/dcdsb.2010.14.739.  Google Scholar

[16]

_______, Nonautonomous continuation of bounded solutions,, Commun. Pure Appl. Anal., 10 (2011), 937.  doi: 10.3934/cpaa.2011.10.937.  Google Scholar

[17]

B. Sandstede, Verzweigungstheorie Homokliner Verdopplungen,, Ph.D. thesis, (1993).   Google Scholar

[18]

K. Sakamoto, Estimates on the strength of exponential dichotomies and application to integral manifolds,, J. Differ. Equations, 107 (1994), 259.  doi: 10.1006/jdeq.1994.1012.  Google Scholar

[19]

Y. Yi, A generalized integral manifold theorem,, J. Differ. Equations, 102 (1993), 153.  doi: 10.1006/jdeq.1993.1026.  Google Scholar

[20]

K. Yosida, Functional Analysis,, Grundlehren der mathematischen Wissenschaften, (1980).   Google Scholar

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