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On the global linearization theorem of Charles Pugh
Published online by Cambridge University Press: 08 September 2025
Abstract
The global C0 linearization theorem on Banach spaces was first proposed by Pugh [26], but it requires that the nonlinear term is globally bounded. In the present paper, we discuss global linearization of semilinear autonomous ordinary differential equations on Banach spaces assuming that the linear part is hyperbolic (including contraction as a particular case) and that the nonlinear term is only Lipschitz with a sufficiently small Lipschitz constant. To overcome the difficulties arising in this problem, in this paper, we rely on a splitting lemma to decouple the hyperbolic system into a contractive system along the stable manifold and an expansive system along the unstable manifold. We then construct a transformation to linearize a contractive/expansive system, which is defined by the crossing time with respect to the unit sphere. To demonstrate the strength of our result, we apply our results to a nonlinear Duffing oscillator without external excitation.
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- © The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh
References
Backes, L., Dragičević, D. and Palmer, K.. Linearization and Hölder continuity for nonautonomous systems. J. Differential Equations. 297 (2021), 536–574.10.1016/j.jde.2021.06.035CrossRefGoogle Scholar
Belickii, G.. Equivalence and normal forms of germs of smooth mappings. Russian Math. Surveys. 33 (1978), 107–177.Google Scholar
Brjuno, A.. Analytical form of differential equations. Trans. Moscow Math. Soc. 25 (1971), 131–288.Google Scholar
Castañeda, A. and Robledo, G.. Differentiability of Palmer’s linearization theorem and converse result for density function. J. Differential Equations. 259 (2015), 4634–4650.10.1016/j.jde.2015.06.004CrossRefGoogle Scholar
Castañeda, A. and Robledo, G.. Dichotomy spectrum and almost topological conjugacy on nonautonomous unbounded difference systems. Discrete Contin. Dyn. Syst. 38 (2018), 2287–2304.10.3934/dcds.2018094CrossRefGoogle Scholar
Castañeda, N. and Rosa, R.. Optimal Estimates for the Uncoupling of Differential Equations. J. Dynam. Differential Equations. 8 (1996), 103–139.CrossRefGoogle Scholar
Deimling, K.. Ordinary Differential Equations in Banach spaces, Lecture Notes in Mathematics. (Springer-Verlag, 1977).Google Scholar
Dragičević, D., Zhang, W. and Zhang, W.. Smooth linearization of nonautonomous differential equations with a nonuniform dichotomy. Proc. London Math. Soc. 121 (2020), 32–50.10.1112/plms.12315CrossRefGoogle Scholar
ElBialy, M.. Local contractions of Banach spaces and spectral gap conditions. J. Funct. Anal. 182 (2001), 108–150.10.1006/jfan.2000.3723CrossRefGoogle Scholar
Gohberg, I, Goldberg, S. and Kaashoek, M. A.. Classes of Linear Operators, Vol. I, (Birkhäuser, Basel, 1990).10.1007/978-3-0348-7509-7CrossRefGoogle Scholar
Grobman, D.. Homeomorphisms of systems of differential equations. Dokl. Akad. Nauk SSSR. 128 (1959), 880–881.Google Scholar
Hartman, P.. A lemma in the theory of structural stability of differential equations. Proc. Amer. Math. Soc. 11 (1960), 610–620.10.1090/S0002-9939-1960-0121542-7CrossRefGoogle Scholar
Hein, M. and Prüss, J.. The Hartman-Grobman theorem for semilinear hyperbolic evolution equations. J. Differential Equations. 261 (2016), 4709–4727.10.1016/j.jde.2016.07.015CrossRefGoogle Scholar
Huerta, I. Linearization of a nonautonomous unbounded system with nonuniform contraction: A spectral approach. Discrete Contin. Dyn. Syst. 40 (2020), 5571–5590.10.3934/dcds.2020238CrossRefGoogle Scholar
Jiang, L.. Generalized exponential dichotomy and global linearization. J. Math. Anal. Appl. 315 (2006), 474–490.10.1016/j.jmaa.2005.05.042CrossRefGoogle Scholar
Lin, F.. Hartman’s linearization on nonautonomous unbounded system. Nonlinear Anal. 66 (2007), 38–50.10.1016/j.na.2005.11.007CrossRefGoogle Scholar
Lu, K.. A Hartman-Grobman theorem for scalar reaction diffusion equations. J. Differential Equations. 93 (1991), 364–394.10.1016/0022-0396(91)90017-4CrossRefGoogle Scholar
Lu, K., Zhang, W. and Zhang, W.. C 1 Hartman Theorem for random dynamical systems. Adv. Math. 375 (2020), .CrossRefGoogle Scholar
Mikusinki, J.. The Bochner integral, BirkhäUser Verlag. , (Basel-Stuttgart, 1978).10.1007/978-3-0348-5567-9CrossRefGoogle Scholar
Moser, J.. On a theorem of Anosov. J. Differential Equations. 5 (1969), 411–440.10.1016/0022-0396(69)90083-7CrossRefGoogle Scholar
Ott, E.. Chaos in Dynamical Systems. 2nd edn, (Cambridge University Press, Cambridge, UK, 2002).10.1017/CBO9780511803260CrossRefGoogle Scholar
Palis, J.. On the local structure of hyperbolic points in Banach spaces. An. Acad. Brasil. Ci. 40 (1968), 263–266.Google Scholar
Palmer, K.. A generalization of Hartman’s linearization theorem. J. Math. Anal. Appl. 41 (1973), 753–758.10.1016/0022-247X(73)90245-XCrossRefGoogle Scholar
Palmer, K.. A characterization of exponential dichotomy in terms of topological equivalence. J. Math. Anal. Appl. 69 (1979), 8–16.10.1016/0022-247X(79)90175-6CrossRefGoogle Scholar
Poincaré, H.. Sur le problème des trois corps et les équations de la dyanamique. Acta Math. 13 (1890), 1–270.Google Scholar
Pugh, C.. On a theorem of P. Hartman. Amer. J. Math. 91 (1969), 363–367.10.2307/2373513CrossRefGoogle Scholar
Reinfelds, A. and , D. Šteinberga, Dynamical equivalence of quasilinear equations. Int. J. Pure Appl. Math. 98 (2015), 355–364.10.12732/ijpam.v98i3.8CrossRefGoogle Scholar
Rodrigues, H. and , J. Solà-Morales, Smooth linearization for a saddle on Banach spaces. J. Dyn. Differ. Equ. 16 (2004), 767–793.10.1007/s10884-004-6116-9CrossRefGoogle Scholar
Siegel, C.. Iteration of analytic functions. Ann. of Math. 43 (1942), 607–612.10.2307/1968952CrossRefGoogle Scholar
Sternberg, S.. Local contractions and a theorem of Poincaré. Amer. J. Math. 79 (1957), 809–824.10.2307/2372437CrossRefGoogle Scholar
Sternberg, S.. On the structure of local homeomorphism of Euclidean n-space. Amer. J. Math. 80 (1958), 623–631.CrossRefGoogle Scholar
Tan, B.. σ-Hölder continuous linearization near hyperbolic fixed points in 
$\mathbb{R}^n$. J. Differential Equations. 162 (2000), 251–269.10.1006/jdeq.1999.3693CrossRefGoogle Scholar
$\mathbb{R}^n$. J. Differential Equations. 162 (2000), 251–269.10.1006/jdeq.1999.3693CrossRefGoogle Scholarvan Strien, S.. Smooth linearization of hyperbolic fixed points without resonance conditions. J. Differential Equations. 85 (1990), 66–90.10.1016/0022-0396(90)90089-8CrossRefGoogle Scholar
Wu, M. and Xia, Y.. Linearization of a nonautonomous unbounded system with hyperbolic linear part: A spectral approach. Discrete Contin. Dyn. Syst. 44 (2024), 491–522. 10.3934/dcds.2023112.10.3934/dcds.2023112CrossRefGoogle Scholar
Xia, Y., Wang, R., Kou, K. and O’Regan, D.. On the linearization theorem for nonautonomous differential equations. Bull. Sci. Math. 139 (2015), 829–846.10.1016/j.bulsci.2014.12.005CrossRefGoogle Scholar
Yoccoz, J.. Linéarisation des germes de difféomorphismes holomorphes
$d(\mathbb{C},0)$. C. R. Acad. Sci. Paris. 36 (1988), 55–58.Google Scholar
$d(\mathbb{C},0)$. C. R. Acad. Sci. Paris. 36 (1988), 55–58.Google ScholarZhang, W., Zhang, W. and Jarczyk, W.. Sharp regularity of linearization for
$C^{1,1}$ hyperbolic diffeomorphisms. Math. Ann. 358 (2014), 69–113.10.1007/s00208-013-0954-xCrossRefGoogle Scholar
$C^{1,1}$ hyperbolic diffeomorphisms. Math. Ann. 358 (2014), 69–113.10.1007/s00208-013-0954-xCrossRefGoogle ScholarZhang, W. and Zhang, W.. Sharpness for C 1 linearization of planar hyperbolic diffeomorphisms. J. Differential Equations. 257 (2014), 4470–4502.10.1016/j.jde.2014.08.014CrossRefGoogle Scholar
Zhang, W. and Zhang, W.. α-Hölder linearization of hyperbolic diffeomorphisms with resonance. Ergod. Theor. Dyn. Syst. 36 (2016), 310–334.10.1017/etds.2014.51CrossRefGoogle Scholar