American Institute of Mathematical Sciences

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December  2012, 32(12): 4111-4131. doi: 10.3934/dcds.2012.32.4111

Noninvertible cocycles: Robustness of exponential dichotomies

Luis Barreira 1, and  Claudia Valls 2,

1. 

Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa
2. 

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

Received  June 2010 Revised  May 2012 Published  August 2012

For the dynamics defined by a sequence of bounded linear operators in a Banach space, we establish the robustness of the notion of exponential dichotomy. This means that an exponential dichotomy persists under sufficiently small linear perturbations. We consider the general cases of a nonuniform exponential dichotomy, which requires much less than a uniform exponential dichotomy, and of a noninvertible dynamics or, more precisely, of a dynamics that may not be invertible in the stable direction.
Keywords: Exponential dichotomies, linear perturbations, noninvertible dynamics, robustness., nonuniform exponential dichotomies.
Mathematics Subject Classification: Primary: 34D99, 37C7.
Citation: Luis Barreira, Claudia Valls. Noninvertible cocycles: Robustness of exponential dichotomies. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4111-4131. doi: 10.3934/dcds.2012.32.4111
References:
[1]

L. Barreira and Ya. Pesin, "Nonuniform Hyperbolicity,", Encyclopedia of Math. and Its Appl. 115, 115 (2007). Google Scholar

[2]

L. Barreira and C. Valls, Stability theory and Lyapunov regularity,, J. Differential Equations, 232 (2007), 675. doi: 10.1016/j.jde.2006.09.021. Google Scholar

[3]

L. Barreira and C. Valls, Robustness of nonuniform exponential dichotomies in Banach spaces,, J. Differential Equations, 244 (2008), 2407. doi: 10.1016/j.jde.2008.02.028. Google Scholar

[4]

L. Barreira and C. Valls, "Stability of Nonautonomous Differential Equations,", Lect. Notes in Math. 1926, 1926 (2008). Google Scholar

[5]

L. Barreira and C. Valls, Robustness of discrete dynamics via Lyapunov sequences,, Comm. Math. Phys., 290 (2009), 219. doi: 10.1007/s00220-009-0762-z. Google Scholar

[6]

L. Barreira and C. Valls, Robust nonuniform dichotomies and parameter dependence,, J. Math. Anal. Appl., 373 (2011), 690. doi: 10.1016/j.jmaa.2010.08.026. Google Scholar

[7]

C. Chicone and Yu. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations,", Mathematical Surveys and Monographs 70, 70 (1999). Google Scholar

[8]

S.-N. Chow and H. Leiva, Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces,, J. Differential Equations, 120 (1995), 429. doi: 10.1006/jdeq.1995.1117. Google Scholar

[9]

W. Coppel, Dichotomies and reducibility,, J. Differential Equations, 3 (1967), 500. Google Scholar

[10]

W. Coppel, "Dichotomies in Stability Theory,", Lect. Notes in Math. 629, 629 (1978). Google Scholar

[11]

Ju. Dalec$'$kiĭ and M. Kreĭn, "Stability of Solutions of Differential Equations in Banach Space,", Translations of Mathematical Monographs 43, 43 (1974). Google Scholar

[12]

J. Hale, "Asymptotic Behavior of Dissipative Systems,", Mathematical Surveys and Monographs 25, 25 (1988). Google Scholar

[13]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lect. Notes in Math. 840, 840 (1981). Google Scholar

[14]

N. Huy, Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line,, J. Funct. Anal., 235 (2006), 330. doi: 10.1016/j.jfa.2005.11.002. Google Scholar

[15]

J. Massera and J. Schäffer, Linear differential equations and functional analysis. I,, Ann. of Math., 67 (1958), 517. doi: 10.2307/1969871. Google Scholar

[16]

J. Massera and J. Schäffer, "Linear Differential Equations and Function Spaces,", Pure and Applied Mathematics, 21 (1966). Google Scholar

[17]

R. Naulin and M. Pinto, Stability of discrete dichotomies for linear difference systems,, J. Differ. Equations Appl., 3 (1997), 101. Google Scholar

[18]

R. Naulin and M. Pinto, Admissible perturbations of exponential dichotomy roughness,, Nonlinear Anal., 31 (1998), 559. doi: 10.1016/S0362-546X(97)00423-9. Google Scholar

[19]

O. Perron, Die Stabilit\"atsfrage bei Differentialgleichungen,, Math. Z., 32 (1930), 703. doi: 10.1007/BF01194662. Google Scholar

[20]

V. Pliss and G. Sell, Robustness of exponential dichotomies ininfinite-dimensional dynamical systems,, J. Dynam. Differential Equations, 11 (1999), 471. doi: 10.1023/A:1021913903923. Google Scholar

[21]

L. Popescu, Exponential dichotomy roughness on Banach spaces,, J. Math. Anal. Appl., 314 (2006), 436. doi: 10.1016/j.jmaa.2005.04.011. Google Scholar

[22]

A. Sasu, Exponential dichotomy and dichotomy radius for difference equations,, J. Math. Anal. Appl., 344 (2008), 906. doi: 10.1016/j.jmaa.2008.03.019. Google Scholar

[23]

B. Sasu and A. Sasu, Input-output conditions for the asymptotic behavior of linear skew-product flows and applications,, Commun. Pure Appl. Anal., 5 (2006), 551. Google Scholar

[24]

G. Sell and Y. You, "Dynamics of Evolutionary Equations,", Applied Mathematical Sciences 143, 143 (2002). Google Scholar

show all references

References:
[1]

L. Barreira and Ya. Pesin, "Nonuniform Hyperbolicity,", Encyclopedia of Math. and Its Appl. 115, 115 (2007). Google Scholar

[2]

L. Barreira and C. Valls, Stability theory and Lyapunov regularity,, J. Differential Equations, 232 (2007), 675. doi: 10.1016/j.jde.2006.09.021. Google Scholar

[3]

L. Barreira and C. Valls, Robustness of nonuniform exponential dichotomies in Banach spaces,, J. Differential Equations, 244 (2008), 2407. doi: 10.1016/j.jde.2008.02.028. Google Scholar

[4]

L. Barreira and C. Valls, "Stability of Nonautonomous Differential Equations,", Lect. Notes in Math. 1926, 1926 (2008). Google Scholar

[5]

L. Barreira and C. Valls, Robustness of discrete dynamics via Lyapunov sequences,, Comm. Math. Phys., 290 (2009), 219. doi: 10.1007/s00220-009-0762-z. Google Scholar

[6]

L. Barreira and C. Valls, Robust nonuniform dichotomies and parameter dependence,, J. Math. Anal. Appl., 373 (2011), 690. doi: 10.1016/j.jmaa.2010.08.026. Google Scholar

[7]

C. Chicone and Yu. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations,", Mathematical Surveys and Monographs 70, 70 (1999). Google Scholar

[8]

S.-N. Chow and H. Leiva, Existence and roughness of the exponential dichotomy for skew-product semiflow in Banach spaces,, J. Differential Equations, 120 (1995), 429. doi: 10.1006/jdeq.1995.1117. Google Scholar

[9]

W. Coppel, Dichotomies and reducibility,, J. Differential Equations, 3 (1967), 500. Google Scholar

[10]

W. Coppel, "Dichotomies in Stability Theory,", Lect. Notes in Math. 629, 629 (1978). Google Scholar

[11]

Ju. Dalec$'$kiĭ and M. Kreĭn, "Stability of Solutions of Differential Equations in Banach Space,", Translations of Mathematical Monographs 43, 43 (1974). Google Scholar

[12]

J. Hale, "Asymptotic Behavior of Dissipative Systems,", Mathematical Surveys and Monographs 25, 25 (1988). Google Scholar

[13]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations,", Lect. Notes in Math. 840, 840 (1981). Google Scholar

[14]

N. Huy, Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line,, J. Funct. Anal., 235 (2006), 330. doi: 10.1016/j.jfa.2005.11.002. Google Scholar

[15]

J. Massera and J. Schäffer, Linear differential equations and functional analysis. I,, Ann. of Math., 67 (1958), 517. doi: 10.2307/1969871. Google Scholar

[16]

J. Massera and J. Schäffer, "Linear Differential Equations and Function Spaces,", Pure and Applied Mathematics, 21 (1966). Google Scholar

[17]

R. Naulin and M. Pinto, Stability of discrete dichotomies for linear difference systems,, J. Differ. Equations Appl., 3 (1997), 101. Google Scholar

[18]

R. Naulin and M. Pinto, Admissible perturbations of exponential dichotomy roughness,, Nonlinear Anal., 31 (1998), 559. doi: 10.1016/S0362-546X(97)00423-9. Google Scholar

[19]

O. Perron, Die Stabilit\"atsfrage bei Differentialgleichungen,, Math. Z., 32 (1930), 703. doi: 10.1007/BF01194662. Google Scholar

[20]

V. Pliss and G. Sell, Robustness of exponential dichotomies ininfinite-dimensional dynamical systems,, J. Dynam. Differential Equations, 11 (1999), 471. doi: 10.1023/A:1021913903923. Google Scholar

[21]

L. Popescu, Exponential dichotomy roughness on Banach spaces,, J. Math. Anal. Appl., 314 (2006), 436. doi: 10.1016/j.jmaa.2005.04.011. Google Scholar

[22]

A. Sasu, Exponential dichotomy and dichotomy radius for difference equations,, J. Math. Anal. Appl., 344 (2008), 906. doi: 10.1016/j.jmaa.2008.03.019. Google Scholar

[23]

B. Sasu and A. Sasu, Input-output conditions for the asymptotic behavior of linear skew-product flows and applications,, Commun. Pure Appl. Anal., 5 (2006), 551. Google Scholar

[24]

G. Sell and Y. You, "Dynamics of Evolutionary Equations,", Applied Mathematical Sciences 143, 143 (2002). Google Scholar

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