Noninvertible cocycles: Robustness of exponential dichotomies

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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

Abstract
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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 and Continuous Dynamical Systems, 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, Cambridge Univ. Press, 2007.
[2]	L. Barreira and C. Valls, Stability theory and Lyapunov regularity, J. Differential Equations, 232 (2007), 675-701. doi: 10.1016/j.jde.2006.09.021.
[3]	L. Barreira and C. Valls, Robustness of nonuniform exponential dichotomies in Banach spaces, J. Differential Equations, 244 (2008), 2407-2447. doi: 10.1016/j.jde.2008.02.028.
[4]	L. Barreira and C. Valls, "Stability of Nonautonomous Differential Equations," Lect. Notes in Math. 1926, Springer, 2008.
[5]	L. Barreira and C. Valls, Robustness of discrete dynamics via Lyapunov sequences, Comm. Math. Phys., 290 (2009), 219-238. doi: 10.1007/s00220-009-0762-z.
[6]	L. Barreira and C. Valls, Robust nonuniform dichotomies and parameter dependence, J. Math. Anal. Appl., 373 (2011), 690-708. doi: 10.1016/j.jmaa.2010.08.026.
[7]	C. Chicone and Yu. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations," Mathematical Surveys and Monographs 70, Amer. Math. Soc., 1999.
[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), 429-477. doi: 10.1006/jdeq.1995.1117.
[9]	W. Coppel, Dichotomies and reducibility, J. Differential Equations, 3 (1967), 500-521.
[10]	W. Coppel, "Dichotomies in Stability Theory," Lect. Notes in Math. 629, Springer, 1978.
[11]	Ju. Dalec$'$kiĭ and M. Kreĭn, "Stability of Solutions of Differential Equations in Banach Space," Translations of Mathematical Monographs 43, Amer. Math. Soc., 1974.
[12]	J. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs 25, Amer. Math. Soc., 1988.
[13]	D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lect. Notes in Math. 840, Springer, 1981.
[14]	N. Huy, Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line, J. Funct. Anal., 235 (2006), 330-354. doi: 10.1016/j.jfa.2005.11.002.
[15]	J. Massera and J. Schäffer, Linear differential equations and functional analysis. I, Ann. of Math., 67 (1958), 517-573. doi: 10.2307/1969871.
[16]	J. Massera and J. Schäffer, "Linear Differential Equations and Function Spaces," Pure and Applied Mathematics, 21, Academic Press, 1966.
[17]	R. Naulin and M. Pinto, Stability of discrete dichotomies for linear difference systems, J. Differ. Equations Appl., 3 (1997), 101-123.
[18]	R. Naulin and M. Pinto, Admissible perturbations of exponential dichotomy roughness, Nonlinear Anal., 31 (1998), 559-571. doi: 10.1016/S0362-546X(97)00423-9.
[19]	O. Perron, Die Stabilit\"atsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728. doi: 10.1007/BF01194662.
[20]	V. Pliss and G. Sell, Robustness of exponential dichotomies ininfinite-dimensional dynamical systems, J. Dynam. Differential Equations, 11 (1999), 471-513. doi: 10.1023/A:1021913903923.
[21]	L. Popescu, Exponential dichotomy roughness on Banach spaces, J. Math. Anal. Appl., 314 (2006), 436-454. doi: 10.1016/j.jmaa.2005.04.011.
[22]	A. Sasu, Exponential dichotomy and dichotomy radius for difference equations, J. Math. Anal. Appl., 344 (2008), 906-920. doi: 10.1016/j.jmaa.2008.03.019.
[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), 551-569.
[24]	G. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences 143, Springer, 2002.

show all references

References:

[1]	L. Barreira and Ya. Pesin, "Nonuniform Hyperbolicity," Encyclopedia of Math. and Its Appl. 115, Cambridge Univ. Press, 2007.
[2]	L. Barreira and C. Valls, Stability theory and Lyapunov regularity, J. Differential Equations, 232 (2007), 675-701. doi: 10.1016/j.jde.2006.09.021.
[3]	L. Barreira and C. Valls, Robustness of nonuniform exponential dichotomies in Banach spaces, J. Differential Equations, 244 (2008), 2407-2447. doi: 10.1016/j.jde.2008.02.028.
[4]	L. Barreira and C. Valls, "Stability of Nonautonomous Differential Equations," Lect. Notes in Math. 1926, Springer, 2008.
[5]	L. Barreira and C. Valls, Robustness of discrete dynamics via Lyapunov sequences, Comm. Math. Phys., 290 (2009), 219-238. doi: 10.1007/s00220-009-0762-z.
[6]	L. Barreira and C. Valls, Robust nonuniform dichotomies and parameter dependence, J. Math. Anal. Appl., 373 (2011), 690-708. doi: 10.1016/j.jmaa.2010.08.026.
[7]	C. Chicone and Yu. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations," Mathematical Surveys and Monographs 70, Amer. Math. Soc., 1999.
[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), 429-477. doi: 10.1006/jdeq.1995.1117.
[9]	W. Coppel, Dichotomies and reducibility, J. Differential Equations, 3 (1967), 500-521.
[10]	W. Coppel, "Dichotomies in Stability Theory," Lect. Notes in Math. 629, Springer, 1978.
[11]	Ju. Dalec$'$kiĭ and M. Kreĭn, "Stability of Solutions of Differential Equations in Banach Space," Translations of Mathematical Monographs 43, Amer. Math. Soc., 1974.
[12]	J. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs 25, Amer. Math. Soc., 1988.
[13]	D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lect. Notes in Math. 840, Springer, 1981.
[14]	N. Huy, Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line, J. Funct. Anal., 235 (2006), 330-354. doi: 10.1016/j.jfa.2005.11.002.
[15]	J. Massera and J. Schäffer, Linear differential equations and functional analysis. I, Ann. of Math., 67 (1958), 517-573. doi: 10.2307/1969871.
[16]	J. Massera and J. Schäffer, "Linear Differential Equations and Function Spaces," Pure and Applied Mathematics, 21, Academic Press, 1966.
[17]	R. Naulin and M. Pinto, Stability of discrete dichotomies for linear difference systems, J. Differ. Equations Appl., 3 (1997), 101-123.
[18]	R. Naulin and M. Pinto, Admissible perturbations of exponential dichotomy roughness, Nonlinear Anal., 31 (1998), 559-571. doi: 10.1016/S0362-546X(97)00423-9.
[19]	O. Perron, Die Stabilit\"atsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728. doi: 10.1007/BF01194662.
[20]	V. Pliss and G. Sell, Robustness of exponential dichotomies ininfinite-dimensional dynamical systems, J. Dynam. Differential Equations, 11 (1999), 471-513. doi: 10.1023/A:1021913903923.
[21]	L. Popescu, Exponential dichotomy roughness on Banach spaces, J. Math. Anal. Appl., 314 (2006), 436-454. doi: 10.1016/j.jmaa.2005.04.011.
[22]	A. Sasu, Exponential dichotomy and dichotomy radius for difference equations, J. Math. Anal. Appl., 344 (2008), 906-920. doi: 10.1016/j.jmaa.2008.03.019.
[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), 551-569.
[24]	G. Sell and Y. You, "Dynamics of Evolutionary Equations," Applied Mathematical Sciences 143, Springer, 2002.

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