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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), 251-262. 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), 91-104. doi: 10.1016/S0362-546X(00)85006-3.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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S.-N. Chow and J. K. Hale, Methods of Bifurcation Theory, Grundlehren der mathematischen Wissenschaften, 251, Springer, Berlin etc., 1996. Google Scholar

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D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes Math., 840, Springer, Berlin etc., 1981.  Google Scholar

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T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar

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

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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.  Google Scholar

[12]

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[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.  Google Scholar

[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.  Google Scholar

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_______, 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.  Google Scholar

[16]

_______, Nonautonomous continuation of bounded solutions, Commun. Pure Appl. Anal., 10 (2011), 937-961. doi: 10.3934/cpaa.2011.10.937.  Google Scholar

[17]

B. Sandstede, Verzweigungstheorie Homokliner Verdopplungen, Ph.D. thesis, Universität Stuttgart, Germany, 1993. Google Scholar

[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.  Google Scholar

[19]

Y. Yi, A generalized integral manifold theorem, J. Differ. Equations, 102 (1993), 153-187. doi: 10.1006/jdeq.1993.1026.  Google Scholar

[20]

K. Yosida, Functional Analysis, Grundlehren der mathematischen Wissenschaften, 123, Springer, Berlin etc., 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), 251-262. 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), 91-104. doi: 10.1016/S0362-546X(00)85006-3.  Google Scholar

[3]

_______, Topological simplification of nonautonomous difference equations, J. Difference Equ. Appl., 12 (2006), 283-296. doi: 10.1080/10236190500489384.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

S.-N. Chow and J. K. Hale, Methods of Bifurcation Theory, Grundlehren der mathematischen Wissenschaften, 251, Springer, Berlin etc., 1996. Google Scholar

[8]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes Math., 840, Springer, Berlin etc., 1981.  Google Scholar

[9]

T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar

[10]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, Mathematical Surveys and Monographs, 176, AMS, Providence, RI, 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), 321-360. 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), 25-37. 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), B.G. Teubner and John Wiley and Sons, Stuttgart/Chichester etc., 1988, 265-306.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

_______, Nonautonomous continuation of bounded solutions, Commun. Pure Appl. Anal., 10 (2011), 937-961. doi: 10.3934/cpaa.2011.10.937.  Google Scholar

[17]

B. Sandstede, Verzweigungstheorie Homokliner Verdopplungen, Ph.D. thesis, Universität Stuttgart, Germany, 1993. Google Scholar

[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.  Google Scholar

[19]

Y. Yi, A generalized integral manifold theorem, J. Differ. Equations, 102 (1993), 153-187. doi: 10.1006/jdeq.1993.1026.  Google Scholar

[20]

K. Yosida, Functional Analysis, Grundlehren der mathematischen Wissenschaften, 123, Springer, Berlin etc., 1980.  Google Scholar

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