# American Institute of Mathematical Sciences

2011, 30(1): 55-76. doi: 10.3934/dcds.2011.30.55

## Regularity of center manifolds under nonuniform hyperbolicity

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

Received  December 2009 Revised  May 2010 Published  February 2011

We construct $C^k$ invariant center manifolds for differential equations $u'=A(t)u+f(t,u)$ obtained from sufficiently small perturbations of a nonuniform exponential trichotomy. We emphasize that our results are optimal, in the sense that the invariant manifolds are as regular as the vector field. In addition, we can also consider linear perturbations with the same method.
Citation: Luis Barreira, Claudia Valls. Regularity of center manifolds under nonuniform hyperbolicity. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 55-76. doi: 10.3934/dcds.2011.30.55
##### References:
 [1] L. Barreira and Ya. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory,", University Lecture Series \textbf{23}, 23 (2002). [2] L. Barreira and Ya. Pesin, "Nonuniform Hyperbolicity,", Encyclopedia of Math. and Its Appl. \textbf{115}, 115 (2007). [3] L. Barreira and J. Schmeling, Sets of "non-typical" points have full topological entropy and full Hausdorff dimension,, Israel J. Math., 116 (2000), 29. doi: 10.1007/BF02773211. [4] L. Barreira and C. Valls, Center manifolds for non-uniformly partially hyperbolic trajectories,, Ergodic Theory Dynam. Systems, 26 (2006), 1707. doi: 10.1017/S0143385706000654. [5] L. Barreira and C. Valls, A Grobman-Hartman theorem for nonuniformly hyperbolic dynamics,, J. Differential Equations, 228 (2006), 285. doi: 10.1016/j.jde.2006.04.001. [6] L. Barreira and C. Valls, Reversibility and equivariance in center manifolds of nonautonomous dynamics,, Discrete Contin. Dyn. Syst., 18 (2007), 677. doi: 10.3934/dcds.2007.18.677. [7] L. Barreira and C. Valls, Smooth center manifolds for nonuniformly partially hyperbolic trajectories,, J. Differential Equations, 237 (2007), 307. doi: 10.1016/j.jde.2007.03.020. [8] L. Barreira and C. Valls, "Stability of Nonautonomous Differential Equations,", Lect. Notes in Math. \textbf{1926}, 1926 (2008). [9] L. Barreira and C. Valls, Robustness of nonuniform exponential trichotomies in Banach spaces,, J. Math. Anal. Appl., 351 (2009), 373. doi: 10.1016/j.jmaa.2008.10.030. [10] L. Barreira and C. Valls, Optimal regularity of robustness for parameterized perturbations,, Bull. Sci. Math., 134 (2010), 767. doi: 10.1016/j.bulsci.2009.12.003. [11] M. Brin, On dynamical coherence,, Ergodic Theory Dynam. Systems, 23 (2003), 395. doi: 10.1017/S0143385702001499. [12] J. Carr, "Applications of Centre Manifold Theory,", Applied Mathematical Sciences \textbf{35}, 35 (1981). [13] C. Chicone, "Ordinary Differential Equations with Applications,", Texts in Applied Mathematics \textbf{34}, 34 (2006). [14] C. Chicone and Yu. Latushkin, Center manifolds for infinite dimensional nonautonomous differential equations,, J. Differential Equations, 141 (1997), 356. doi: 10.1006/jdeq.1997.3343. [15] S.-N. Chow, W. Liu and Y. Yi, Center manifolds for invariant sets,, J. Differential Equations, 168 (2000), 355. doi: 10.1006/jdeq.2000.3890. [16] S.-N. Chow, W. Liu and Y. Yi, Center manifolds for smooth invariant manifolds,, Trans. Amer. Math. Soc., 352 (2000), 5179. doi: 10.1090/S0002-9947-00-02443-0. [17] J. Lamb and J. Roberts, Time-reversal symmetry in dynamical systems: a survey,, in Time-Reversal Symmetry in Dynamical Systems (Coventry, 112 (1998), 1. doi: 10.1016/S0167-2789(97)00199-1. [18] A. Mielke, A reduction principle for nonautonomous systems in infinite-dimensional spaces,, J. Differential Equations, 65 (1986), 68. doi: 10.1016/0022-0396(86)90042-2. [19] V. Oseledets, A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems,, Trans. Moscow Math. Soc., 19 (1968), 197. [20] Ya. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents,, Math. USSR-Izv., 10 (1976), 1261. doi: 10.1070/IM1976v010n06ABEH001835. [21] J. Roberts and R. Quispel, Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems,, Phys. Rep., 216 (1992), 63. doi: 10.1016/0370-1573(92)90163-T. [22] M. Sevryuk, "Reversible Systems,", Lect. Notes in Math. \textbf{1211}, 1211 (1986). [23] A. Vanderbauwhede, Centre manifolds, normal forms and elementary bifurcations,, in, 2 (1989), 89. [24] A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions,, in, 1 (1992), 125. [25] A. Vanderbauwhede and S. van Gils, Center manifolds and contractions on a scale of Banach spaces,, J. Funct. Anal., 72 (1987), 209. doi: 10.1016/0022-1236(87)90086-3.

show all references

##### References:
 [1] L. Barreira and Ya. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory,", University Lecture Series \textbf{23}, 23 (2002). [2] L. Barreira and Ya. Pesin, "Nonuniform Hyperbolicity,", Encyclopedia of Math. and Its Appl. \textbf{115}, 115 (2007). [3] L. Barreira and J. Schmeling, Sets of "non-typical" points have full topological entropy and full Hausdorff dimension,, Israel J. Math., 116 (2000), 29. doi: 10.1007/BF02773211. [4] L. Barreira and C. Valls, Center manifolds for non-uniformly partially hyperbolic trajectories,, Ergodic Theory Dynam. Systems, 26 (2006), 1707. doi: 10.1017/S0143385706000654. [5] L. Barreira and C. Valls, A Grobman-Hartman theorem for nonuniformly hyperbolic dynamics,, J. Differential Equations, 228 (2006), 285. doi: 10.1016/j.jde.2006.04.001. [6] L. Barreira and C. Valls, Reversibility and equivariance in center manifolds of nonautonomous dynamics,, Discrete Contin. Dyn. Syst., 18 (2007), 677. doi: 10.3934/dcds.2007.18.677. [7] L. Barreira and C. Valls, Smooth center manifolds for nonuniformly partially hyperbolic trajectories,, J. Differential Equations, 237 (2007), 307. doi: 10.1016/j.jde.2007.03.020. [8] L. Barreira and C. Valls, "Stability of Nonautonomous Differential Equations,", Lect. Notes in Math. \textbf{1926}, 1926 (2008). [9] L. Barreira and C. Valls, Robustness of nonuniform exponential trichotomies in Banach spaces,, J. Math. Anal. Appl., 351 (2009), 373. doi: 10.1016/j.jmaa.2008.10.030. [10] L. Barreira and C. Valls, Optimal regularity of robustness for parameterized perturbations,, Bull. Sci. Math., 134 (2010), 767. doi: 10.1016/j.bulsci.2009.12.003. [11] M. Brin, On dynamical coherence,, Ergodic Theory Dynam. Systems, 23 (2003), 395. doi: 10.1017/S0143385702001499. [12] J. Carr, "Applications of Centre Manifold Theory,", Applied Mathematical Sciences \textbf{35}, 35 (1981). [13] C. Chicone, "Ordinary Differential Equations with Applications,", Texts in Applied Mathematics \textbf{34}, 34 (2006). [14] C. Chicone and Yu. Latushkin, Center manifolds for infinite dimensional nonautonomous differential equations,, J. Differential Equations, 141 (1997), 356. doi: 10.1006/jdeq.1997.3343. [15] S.-N. Chow, W. Liu and Y. Yi, Center manifolds for invariant sets,, J. Differential Equations, 168 (2000), 355. doi: 10.1006/jdeq.2000.3890. [16] S.-N. Chow, W. Liu and Y. Yi, Center manifolds for smooth invariant manifolds,, Trans. Amer. Math. Soc., 352 (2000), 5179. doi: 10.1090/S0002-9947-00-02443-0. [17] J. Lamb and J. Roberts, Time-reversal symmetry in dynamical systems: a survey,, in Time-Reversal Symmetry in Dynamical Systems (Coventry, 112 (1998), 1. doi: 10.1016/S0167-2789(97)00199-1. [18] A. Mielke, A reduction principle for nonautonomous systems in infinite-dimensional spaces,, J. Differential Equations, 65 (1986), 68. doi: 10.1016/0022-0396(86)90042-2. [19] V. Oseledets, A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems,, Trans. Moscow Math. Soc., 19 (1968), 197. [20] Ya. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents,, Math. USSR-Izv., 10 (1976), 1261. doi: 10.1070/IM1976v010n06ABEH001835. [21] J. Roberts and R. Quispel, Chaos and time-reversal symmetry. Order and chaos in reversible dynamical systems,, Phys. Rep., 216 (1992), 63. doi: 10.1016/0370-1573(92)90163-T. [22] M. Sevryuk, "Reversible Systems,", Lect. Notes in Math. \textbf{1211}, 1211 (1986). [23] A. Vanderbauwhede, Centre manifolds, normal forms and elementary bifurcations,, in, 2 (1989), 89. [24] A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions,, in, 1 (1992), 125. [25] A. Vanderbauwhede and S. van Gils, Center manifolds and contractions on a scale of Banach spaces,, J. Funct. Anal., 72 (1987), 209. doi: 10.1016/0022-1236(87)90086-3.
 [1] 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 [2] 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 [3] Luis Barreira, Claudia Valls. Center manifolds for nonuniform trichotomies and arbitrary growth rates. Communications on Pure & Applied Analysis, 2010, 9 (3) : 643-654. doi: 10.3934/cpaa.2010.9.643 [4] 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 [5] 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 [6] 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 [7] 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 [8] Luis Barreira, Claudia Valls. Reversibility and equivariance in center manifolds of nonautonomous dynamics. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 677-699. doi: 10.3934/dcds.2007.18.677 [9] Martin Golubitsky, Claire Postlethwaite. Feed-forward networks, center manifolds, and forcing. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2913-2935. doi: 10.3934/dcds.2012.32.2913 [10] Ricardo Rosa. Approximate inertial manifolds of exponential order. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 421-448. doi: 10.3934/dcds.1995.1.421 [11] Redouane Qesmi, Hans-Otto Walther. Center-stable manifolds for differential equations with state-dependent delays. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1009-1033. doi: 10.3934/dcds.2009.23.1009 [12] Roland Schnaubelt. Center manifolds and attractivity for quasilinear parabolic problems with fully nonlinear dynamical boundary conditions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1193-1230. doi: 10.3934/dcds.2015.35.1193 [13] Constantinos Siettos. Equation-free computation of coarse-grained center manifolds of microscopic simulators. Journal of Computational Dynamics, 2014, 1 (2) : 377-389. doi: 10.3934/jcd.2014.1.377 [14] Yuri Latushkin, Jan Prüss, Ronald Schnaubelt. Center manifolds and dynamics near equilibria of quasilinear parabolic systems with fully nonlinear boundary conditions. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 595-633. doi: 10.3934/dcdsb.2008.9.595 [15] Alin Pogan, Kevin Zumbrun. Stable manifolds for a class of singular evolution equations and exponential decay of kinetic shocks. Kinetic & Related Models, 2019, 12 (1) : 1-36. doi: 10.3934/krm.2019001 [16] 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 [17] Luis Barreira, Claudia Valls. Growth rates and nonuniform hyperbolicity. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 509-528. doi: 10.3934/dcds.2008.22.509 [18] Yakov Pesin. On the work of Dolgopyat on partial and nonuniform hyperbolicity. Journal of Modern Dynamics, 2010, 4 (2) : 227-241. doi: 10.3934/jmd.2010.4.227 [19] Jana Rodriguez Hertz. Genericity of nonuniform hyperbolicity in dimension 3. Journal of Modern Dynamics, 2012, 6 (1) : 121-138. doi: 10.3934/jmd.2012.6.121 [20] António J.G. Bento, Nicolae Lupa, Mihail Megan, César M. Silva. Integral conditions for nonuniform $μ$-dichotomy on the half-line. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3063-3077. doi: 10.3934/dcdsb.2017163

2017 Impact Factor: 1.179