October 2017, 22(8): 3023-3042. doi: 10.3934/dcdsb.2017161

Generalized exponential behavior and topological equivalence

1. 

Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal

2. 

Department of Mathematics and Informatics, Oradea University, Str. Universitatii Nr.1, Oradea, 410087, Romania

* Corresponding author: Luis Barreira

Received  May 2015 Revised  August 2015 Published  June 2017

Fund Project: L.B. and C.V. were supported by FCT/Portugal through UID/MAT/04459/2013

We discuss the topological equivalence between evolution families with a generalized exponential dichotomy. These can occur for example when all Lyapunov exponents are infinite or all Lyapunov exponents are zero. In particular, we show that any evolution family admitting a generalized exponential dichotomy is topologically equivalent to a certain normal form, in the which the exponential behavior in the stable and unstable directions are multiples of the identity. Moreover, we show that the topological equivalence between two evolution families admitting generalized exponential dichotomies, possibly with different growth rates, can be completely characterized in terms of a new notion of equivalence between these rates.

Citation: Luis Barreira, Liviu Horia Popescu, Claudia Valls. Generalized exponential behavior and topological equivalence. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3023-3042. doi: 10.3934/dcdsb.2017161
References:
[1]

L. Barreira and C. Valls, Growth rates and nonuniform hyperbolicity, Discrete Contin. Dyn. Syst., 22 (2008), 509-528. doi: 10.3934/dcds.2008.22.509.

[2]

L. Barreira and C. Valls, Robustness of general dichotomies, J. Funct. Anal., 257 (2009), 464-484. doi: 10.1016/j.jfa.2008.11.018.

[3]

Ju. Dalec'kiǐ and M. Kreǐn, Stability of Solutions of Differential Equations in Banach Space Translations of Mathematical Monographs, 43, Amer. Math. Soc. , 1974.

[4]

L. Jiang, Generalized exponential dichotomy and global linearization, J. Math. Anal. Appl., 315 (2006), 474-490. doi: 10.1016/j.jmaa.2005.05.042.

[5]

R. Naulin and M. Pinto, Dichotomies and asymptotic solutions of nonlinear differential sys\-tems, Nonlinear Anal., 23 (1994), 871-882. doi: 10.1016/0362-546X(94)90125-2.

[6]

K. Palmer, A characterization of exponential dichotomy in terms of topological equivalence, J. Math. Anal. Appl., 69 (1979), 8-16. doi: 10.1016/0022-247X(79)90175-6.

[7]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer, 1983. doi: 10.1007/978-1-4612-5561-1.

[8]

L. H. Popescu, A topological classification of linear differential equations on Banach spaces, J. Differential Equations, 203 (2004), 28-37. doi: 10.1016/j.jde.2004.03.038.

show all references

References:
[1]

L. Barreira and C. Valls, Growth rates and nonuniform hyperbolicity, Discrete Contin. Dyn. Syst., 22 (2008), 509-528. doi: 10.3934/dcds.2008.22.509.

[2]

L. Barreira and C. Valls, Robustness of general dichotomies, J. Funct. Anal., 257 (2009), 464-484. doi: 10.1016/j.jfa.2008.11.018.

[3]

Ju. Dalec'kiǐ and M. Kreǐn, Stability of Solutions of Differential Equations in Banach Space Translations of Mathematical Monographs, 43, Amer. Math. Soc. , 1974.

[4]

L. Jiang, Generalized exponential dichotomy and global linearization, J. Math. Anal. Appl., 315 (2006), 474-490. doi: 10.1016/j.jmaa.2005.05.042.

[5]

R. Naulin and M. Pinto, Dichotomies and asymptotic solutions of nonlinear differential sys\-tems, Nonlinear Anal., 23 (1994), 871-882. doi: 10.1016/0362-546X(94)90125-2.

[6]

K. Palmer, A characterization of exponential dichotomy in terms of topological equivalence, J. Math. Anal. Appl., 69 (1979), 8-16. doi: 10.1016/0022-247X(79)90175-6.

[7]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer, 1983. doi: 10.1007/978-1-4612-5561-1.

[8]

L. H. Popescu, A topological classification of linear differential equations on Banach spaces, J. Differential Equations, 203 (2004), 28-37. doi: 10.1016/j.jde.2004.03.038.

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