Abstract
For nonautonomous linear equations x′ = A(t)x, we give a complete characterization of the existence of exponential behavior in terms of Lyapunov functions. In particular, we obtain an inverse theorem giving explicitly Lyapunov functions for each exponential dichotomy. The main novelty of our work is that we consider a very general type of nonuniform exponential dichotomy. This includes for example uniform exponential dichotomies, nonuniform exponential dichotomies and polynomial dichotomies. We also consider the case of different growth rates for the uniform and the nonuniform parts of the dichotomy. As an application of our work, we establish in a very direct manner the robustness of nonuniform exponential dichotomies under sufficiently small linear perturbations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Barreira L., Chu J., C. Valls: Robustness of nonuniform dichotomies with different growth rates, São Paulo. J. Math. Sci. 5, 1–2 (2011)
L. Barreira and Ya. Pesin, Nonuniform Hyperbolicity, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, 2007.
Barreira L., J. Schmeling: Sets of “non-typical" points have full topological entropy and full Hausdorff dimension. Israel J. Math. 116, 29–70 (2000)
L. Barreira and C. Valls, Robustness of nonuniform exponential dichotomies in Banach spaces, J. Differential Equations 244 (2008), 2407–2447.
L. Barreira and C. Valls, Stability of Nonautonomous Differential Equations, Lect. Notes. in Math. 1926, Springer, 2008.
L. Barreira and C. Valls, Robustness via Lyapunov functions, J. Differential Equations 246 (2009), 2891–2907.
A. Bento and C. Silva, Generalized nonuniform dichotomies and local stable manifolds, preprint.
N. Bhatia and G. Szegö, Stability Theory of Dynamical Systems, Grundlehren der mathematischen Wissenschaften 161, Springer, 1970.
C. Chicone and Yu. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, Mathematical Surveys and Monographs 70, Amer. Math. Soc., 1999.
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.
W. Coppel, Dichotomies in Stability Theory, Lect. Notes in Math. 629, Springer, 1978.
Ju. Dalec′kiĭ and M. Kreĭn, Stability of Solutions of Differential Equations in Banach Space, Translations of Mathematical Monographs 43, Amer. Math. Soc. 1974.
W. Hahn, Stability of Motion, Grundlehren der mathematischen Wissenschaften 138, Springer, 1967.
J. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25, Amer. Math. Soc., 1988.
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes in Math. 840, Springer, 1981.
J. LaSalle and S. Lefschetz, Stability by Liapunov’s Direct Method, with Applications, Mathematics in Science and Engineering 4, Academic Press, 1961.
X.-B. Lin, Exponential dichotomies and homoclinic orbits in functional-differential equations, J. Differential Equations 63 (1986), 227–254.
A. Lyapunov, The General Problem of the Stability of Motion, Taylor & Francis, 1992.
A. Maĭzel’, On stability of solutions of systems of differential equations, Ural. Politehn. Inst. Trudy 51 (1954), 20–50.
J. Massera and J. Schäffer, Linear differential equations and functional analysis. I, Ann. of Math. (2) 67 (1958), 517–573.
Yu. Mitropolsky, A. Samoilenko and V. Kulik, Dichotomies and Stability in Nonautonomous Linear Systems, Stability and Control: Theory, Methods and Applications 14, Taylor & Francis, 2003.
R. Naulin and M. Pinto, Admissible perturbations of exponential dichotomy roughness, Nonlinear Anal. 31 (1998), 559–571.
K. Palmer, Exponential dichotomies and transversal homoclinic points, J. Differential Equations 55 (1984), 225–256.
O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z. 32 (1930), 703–728.
V. Pliss and G. Sell, Robustness of exponential dichotomies in infinite-dimensional dynamical systems, J. Dynam. Differential Equations 11 (1999), 471–513.
L. Popescu, Exponential dichotomy roughness on Banach spaces, J. Math. Anal. Appl. 314 (2006), 436–454.
G. Sell and Y. You, Dynamics of Evolutionary Equations, Applied Mathematical Sciences 143, Springer, 2002.
Author information
Authors and Affiliations
Corresponding author
Additional information
Luis Barreira and Claudia Valls were partially supported by FCT through CAMGSD, Lisbon. Jifeng Chu was supported by the National Natural Science Foundation of China (Grants No. 10, No. 2 and No. 21333), the Program for New Century Excellent Talents in University (Grant No. NCET–032), China Postdoctoral Science Foundation funded project (Grant No. 20T0431).
Rights and permissions
About this article
Cite this article
Barreira, L., Chu, J. & Valls, C. Lyapunov Functions for General Nonuniform Dichotomies. Milan J. Math. 81, 153–169 (2013). https://doi.org/10.1007/s00032-013-0198-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00032-013-0198-y