Abstract
The main objective of the present paper is to formulate sufficient conditions under which a nonautonomous dynamics acting on a arbitrary Fréchet space exhibits shadowing property and (partial) linearization. These conditions require that the linear part is hyperbolic (in the sense of the concept recently introduced by Aragão Costa) and that the nonlinear part is Lipschitz. Our results extend those previously known in the setting of nonautonomous dynamics acting on Banach space. We consider both the case of discrete and continuous time dynamics.
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References
Aragão Costa, E.R.: An extension of the concept of exponential dichotomy in Fréchet spaces which is stable under perturbation. Commun. Pure Appl. Anal. 18, 845–868 (2019)
Backes, L., Dragičević, D.: A general approach to nonautonomous shadowing for nonlinear dynamics. Bull. Sci. Math. 170, 102996 (2021)
Backes, L., Dragičević, D.: Shadowing for infinite dimensional dynamics and exponential trichotomies. Proc. R. Soc. Edinb. Sect. A 151(3), 863–884 (2021)
Backes, L., Dragičević, D.: Smooth linearization of nonautonomous coupled systems. Discrete Contin. Dyn. Syst. Ser. B 28, 4497–4518 (2023)
Backes, L., Dragičević, D.: Multiscale linearization of nonautonomous systems. Proc. R. Soc. Edinb. Sect. A 153, 1609–1629 (2023)
Backes, L., Dragičević, D., Onitsuka, M., Pituk, M.: Conditional Lipschitz shadowing for ordinary differential equations. J. Dyn. Differ. Equat. (2023). https://doi.org/10.1007/s10884-023-10246-6
Backes, L., Dragičević, D., Zhang, W.: Smooth linearization of nonautonomous dynamics under polynomial behaviour. Preprint arXiv:2210.04804
Barbu, D., Buşe, C., Tabassum, A.: Hyers–Ulam stability and discrete dichotomy. J. Math. Anal. Appl. 423, 1738–1752 (2015)
Bernardes, N., Jr., Cirilo, P.R., Darji, U.B., Messaoudi, A., Pujals, E.R.: Expansivity and shadowing in linear dynamics. J. Math. Anal. Appl. 461, 796–816 (2018)
Bowen, R.: Markov partitions for axiom a diffeomorphisms. Am. J. Math. 92, 725–747 (1970)
Bowen, R.: Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lect. Notes in Math., vol. 470. Springer, London (1975)
Buşe, C., Lupulescu, V., O’Regan, D.: Hyers–Ulam stability for equations with differences and differential equations with time-dependent and periodic coefficients. Proc. R. Soc. Edinb. Sect. A 150, 2175–2188 (2020)
Buşe, C., O’Regan, D., Saierli, O., Tabassum, A.: Hyers–Ulam stability and discrete dichotomy for difference periodic systems. Bull. Sci. Math. 140, 908–934 (2016)
Castañeda, A., Robledo, G.: Differentiability of Palmer’s linearization theorem and converse result for density function. J. Differ. Equ. 259, 4634–4650 (2015)
Castañeda, A., Jara, N.: A note on the differentiability of discrete Palmer’s linearization. Proc. R. Soc. Edinb. Sect. A Math. (2023). https://doi.org/10.1017/prm.2023.25
Dragičević, D., Zhang, W.N., Zhang, W.M.: Smooth linearization of nonautonomous differential equations with a nonuniform dichotomy. Proc. Lond. Math. Soc. 121, 32–50 (2020)
Folland, G.B.: Real Analysis: Modern Techniques and Their Applications, 2nd edn. Wiley Intercience, London (1999)
Grobman, D.: Homeomorphism of systems of differential equations. Dokl. Akad. Nauk SSSR 128, 880–881 (1959)
Grobman, D.: Topological classification of neighborhoods of a singularity in \(n\)-space. Mat. Sb. (N.S.) 56, 77–94 (1962)
Hartman, P.: On local homeomorphisms of Euclidean spaces. Bol. Soc. Mat. Mexicana (2) 5, 220–241 (1960)
Hartman, P.: A lemma in the theory of structural stability of differential equations. Proc. Am. Math. Soc. 11, 610–620 (1960)
Hartman, P.: On the local linearization of differential equations. Proc. Am. Math. Soc. 14, 568–573 (1963)
Jara, N.: Smoothness of class \(C^2\) of nonautonomous linearization without spectral conditions. J. Dyn. Diff. Equ. (2022). https://doi.org/10.1007/s10884-022-10207-5
Palmer, K.J.: A generalization of Hartman’s linearization theorem. J. Math. Anal. Appl. 41, 753–758 (1973)
Palmer, K.: Shadowing in Dynamical Systems, Theory and Applications. Kluwer, Dordrecht (2000)
Pilyugin, S.Y.: Shadowing in Dynamical Systems. Lecture Notes in Mathematics, vol. 1706. Springer, Berlin (1999)
Pilyugin, S.Y.: Multiscale conditional shadowing. J. Dyn. Differ. Equ. (2021). https://doi.org/10.1007/s10884-021-10096-0
Pilyugin, S.Y., Sakai, K.: Shadowing and Hyperbolicity. Lecture Notes in Mathematics, vol. 2193. Springer, Cham (2017)
Acknowledgements
L. Backes was partially supported by a CNPq-Brazil PQ fellowship under Grant No. 307633/2021-7. D. Dragičević was supported in part by Croatian Science Foundation under the Project IP-2019-04-1239 and by the University of Rijeka under the Projects uniri-prirod-18-9 and uniri-prprirod-19-16.
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Backes, L., Dragičević, D. Stability of nonautonomous systems on Fréchet spaces. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 105 (2024). https://doi.org/10.1007/s13398-024-01606-y
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DOI: https://doi.org/10.1007/s13398-024-01606-y