American Institute of Mathematical Sciences

Advanced
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

[1]

Luis Barreira, Claudia Valls. Nonuniform exponential dichotomies and admissibility. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 39-53. doi: 10.3934/dcds.2011.30.39
[2]

Luis Barreira, Claudia Valls. Characterization of stable manifolds for nonuniform exponential dichotomies. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1025-1046. doi: 10.3934/dcds.2008.21.1025
[3]

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
[4]

Luis Barreira, Claudia Valls. Admissibility versus nonuniform exponential behavior for noninvertible cocycles. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1297-1311. doi: 10.3934/dcds.2013.33.1297
[5]

Luis Barreira, Claudia Valls. Delay equations and nonuniform exponential stability. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 219-223. doi: 10.3934/dcdss.2008.1.219
[6]

Luis Barreira, Davor Dragičević, Claudia Valls. From one-sided dichotomies to two-sided dichotomies. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2817-2844. doi: 10.3934/dcds.2015.35.2817
[7]

Feng-Yu Wang. Exponential convergence of non-linear monotone SPDEs. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5239-5253. doi: 10.3934/dcds.2015.35.5239
[8]

Jinhuo Luo, Jin Wang, Hao Wang. Seasonal forcing and exponential threshold incidence in cholera dynamics. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2261-2290. doi: 10.3934/dcdsb.2017095
[9]

A. Rodríguez-Bernal. Perturbation of the exponential type of linear nonautonomous parabolic equations and applications to nonlinear equations. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 1003-1032. doi: 10.3934/dcds.2009.25.1003
[10]

Michael Scheutzow. Exponential growth rate for a singular linear stochastic delay differential equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1683-1696. doi: 10.3934/dcdsb.2013.18.1683
[11]

Eduardo Cerpa, Emmanuelle Crépeau. Rapid exponential stabilization for a linear Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 655-668. doi: 10.3934/dcdsb.2009.11.655
[12]

Salim A. Messaoudi, Abdelfeteh Fareh. Exponential decay for linear damped porous thermoelastic systems with second sound. Discrete & Continuous Dynamical Systems - B, 2015, 20 (2) : 599-612. doi: 10.3934/dcdsb.2015.20.599
[13]

J. Húska, Peter Poláčik. Exponential separation and principal Floquet bundles for linear parabolic equations on $R^N$. Discrete & Continuous Dynamical Systems - A, 2008, 20 (1) : 81-113. doi: 10.3934/dcds.2008.20.81
[14]

Monica Conti, Elsa M. Marchini, Vittorino Pata. Exponential stability for a class of linear hyperbolic equations with hereditary memory. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1555-1565. doi: 10.3934/dcdsb.2013.18.1555
[15]

Vittorino Pata. Exponential stability in linear viscoelasticity with almost flat memory kernels. Communications on Pure & Applied Analysis, 2010, 9 (3) : 721-730. doi: 10.3934/cpaa.2010.9.721
[16]

Jin Zhang, Yonghai Wang, Chengkui Zhong. Robustness of exponentially κ-dissipative dynamical systems with perturbations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3875-3890. doi: 10.3934/dcdsb.2017198
[17]

Cecilia Cavaterra, M. Grasselli. Robust exponential attractors for population dynamics models with infinite time delay. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1051-1076. doi: 10.3934/dcdsb.2006.6.1051
[18]

Jin Zhang, Peter E. Kloeden, Meihua Yang, Chengkui Zhong. Global exponential κ-dissipative semigroups and exponential attraction. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3487-3502. doi: 10.3934/dcds.2017148
[19]

Christian Pötzsche, Stefan Siegmund, Fabian Wirth. A spectral characterization of exponential stability for linear time-invariant systems on time scales. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1223-1241. doi: 10.3934/dcds.2003.9.1223
[20]

Xiaojie Wang. Weak error estimates of the exponential Euler scheme for semi-linear SPDEs without Malliavin calculus. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 481-497. doi: 10.3934/dcds.2016.36.481

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences