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September  2020, 40(9): 5571-5590. doi: 10.3934/dcds.2020238

Linearization of a nonautonomous unbounded system with nonuniform contraction: A spectral approach

Departamento de Matemática, Universidad Técnica Federico Santa María, Casilla 110-V, Valparaíso, Chile

* Corresponding author: Ignacio Huerta

Received  January 2020 Revised  March 2020 Published  June 2020

Fund Project: This research has been partially supported FONDECYT Regular 1170968 and CONICYTPCHA/2015-21150270

For a nonautonomous linear system with nonuniform contraction, we construct a topological conjugacy between this system and an unbounded nonlinear perturbation. This topological conjugacy is constructed as a composition of homeomorphisms. The first one is set up by considering the fact that linear system is almost reducible to diagonal system with a small enough perturbation where the diagonal entries belong to spectrum of the nonuniform exponential dichotomy; and the second one is constructed in terms of the crossing times with respect to unit sphere of an adequate Lyapunov function associated to the linear system.

Citation: Ignacio Huerta. Linearization of a nonautonomous unbounded system with nonuniform contraction: A spectral approach. Discrete & Continuous Dynamical Systems, 2020, 40 (9) : 5571-5590. doi: 10.3934/dcds.2020238
References:
[1]

L. Barreira and C. Valls, Smoothness of invariant manifolds for nonautonomous equations, Comm. Math. Phys., 259 (2005), 639-677.  doi: 10.1007/s00220-005-1380-z.  Google Scholar

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B. F. Bylov, Almost reducible system of differential equations, Sibirsk. Mat. Ž., 3 (1962), 333–359.  Google Scholar

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Á. Castañeda and I. Huerta, Nonuniform almost reducibility of nonautonomous linear differential equations, J. Math. Anal. Appl., 485 (2020), 22pp. doi: 10.1016/j.jmaa.2019.123822.  Google Scholar

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Á. Castañeda and G. Robledo, Almost reducibility of linear difference systems from a spectral point of view, Commun. Pure Appl. Anal., 16 (2017), 1977-1988.  doi: 10.3934/cpaa.2017097.  Google Scholar

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Á. Castañeda and G. Robledo, Dichotomy spectrum and almost topological conjugacy on nonautonomous unbounded difference systems, Discrete Contin. Dyn. Syst., 38 (2018), 2287-2304.  doi: 10.3934/dcds.2018094.  Google Scholar

[6]

J. ChuF.-F. LiaoS. SiegmundY. Xia and W. Zhang, Nonuniform dichotomy spectrum and reducibility for nonautonomous equations, Bull. Sci. Math., 139 (2015), 538-557.  doi: 10.1016/j.bulsci.2014.11.002.  Google Scholar

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F.-F. Liao, Y. Jiang, and Z. Xie, A generalized nonuniform contraction and Lyapunov function, Abstr. Appl. Anal., 2012 (2012), 14pp. doi: 10.1155/2012/613038.  Google Scholar

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L. Jiang, Strongly topological linearization with generalized exponential dichotomy, Nonlinear Anal., 67 (2007), 1102-1110.  doi: 10.1016/j.na.2006.06.054.  Google Scholar

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F. Lin, Hartman's linearization on nonautonomous unbounded system, Nonlinear Anal., 66 (2007), 38-50.  doi: 10.1016/j.na.2005.11.007.  Google Scholar

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F. X. Lin, Spectrum sets and contractible sets of linear differential equations, Chinese Ann. Math. Ser. A, 11 (1990), 111-120.   Google Scholar

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K. J. Palmer, A generalization of Hartman's linearization theorem, J. Math. Anal. Appl., 41 (1973), 753-758.  doi: 10.1016/0022-247X(73)90245-X.  Google Scholar

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K. J. Palmer, A characterization of exponential dichotomy in terms of topological equivalence, J. Math. Anal. Appl., 69 (1979), 8-16.  doi: 10.1016/0022-247X(79)90175-6.  Google Scholar

[13]

K. J. Palmer, The structurally stable linear systems on the half-line are those with exponential dichotomies, J. Differential Equations, 33 (1979), 16-25.  doi: 10.1016/0022-0396(79)90076-7.  Google Scholar

[14]

C. Pugh, On a theorem of P. Hartman, Amer. J. Math., 91 (1969), 363-367.  doi: 10.2307/2373513.  Google Scholar

[15]

J. L. Shi and K. Q. Xiong, On Hartman's linearization theorem and Palmer's linearization theorem, J. Math. Anal. Appl., 192 (1995), 813-832.  doi: 10.1006/jmaa.1995.1205.  Google Scholar

[16]

Y. XiaY. Bai and D. O'Regan, A new method to prove the nonuniform dichotomy spectrum theorem in $\mathbb{R}^n$, Proc. Amer. Math. Soc., 147 (2019), 3905-3917.  doi: 10.1090/proc/14535.  Google Scholar

[17]

Y. XiaH. Huang and K. Kou, Hartman-Grobman theorem for the impulsive system with unbounded nonlinear term, Qual. Theory Dyn. Syst., 16 (2017), 705-730.  doi: 10.1007/s12346-016-0218-8.  Google Scholar

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X. Zhang, Nonuniform dichotomy spectrum and normal forms for nonautonomous differential systems, J. Funct. Anal., 267 (2014), 1889-1916.  doi: 10.1016/j.jfa.2014.07.029.  Google Scholar

show all references

References:
[1]

L. Barreira and C. Valls, Smoothness of invariant manifolds for nonautonomous equations, Comm. Math. Phys., 259 (2005), 639-677.  doi: 10.1007/s00220-005-1380-z.  Google Scholar

[2]

B. F. Bylov, Almost reducible system of differential equations, Sibirsk. Mat. Ž., 3 (1962), 333–359.  Google Scholar

[3]

Á. Castañeda and I. Huerta, Nonuniform almost reducibility of nonautonomous linear differential equations, J. Math. Anal. Appl., 485 (2020), 22pp. doi: 10.1016/j.jmaa.2019.123822.  Google Scholar

[4]

Á. Castañeda and G. Robledo, Almost reducibility of linear difference systems from a spectral point of view, Commun. Pure Appl. Anal., 16 (2017), 1977-1988.  doi: 10.3934/cpaa.2017097.  Google Scholar

[5]

Á. Castañeda and G. Robledo, Dichotomy spectrum and almost topological conjugacy on nonautonomous unbounded difference systems, Discrete Contin. Dyn. Syst., 38 (2018), 2287-2304.  doi: 10.3934/dcds.2018094.  Google Scholar

[6]

J. ChuF.-F. LiaoS. SiegmundY. Xia and W. Zhang, Nonuniform dichotomy spectrum and reducibility for nonautonomous equations, Bull. Sci. Math., 139 (2015), 538-557.  doi: 10.1016/j.bulsci.2014.11.002.  Google Scholar

[7]

F.-F. Liao, Y. Jiang, and Z. Xie, A generalized nonuniform contraction and Lyapunov function, Abstr. Appl. Anal., 2012 (2012), 14pp. doi: 10.1155/2012/613038.  Google Scholar

[8]

L. Jiang, Strongly topological linearization with generalized exponential dichotomy, Nonlinear Anal., 67 (2007), 1102-1110.  doi: 10.1016/j.na.2006.06.054.  Google Scholar

[9]

F. Lin, Hartman's linearization on nonautonomous unbounded system, Nonlinear Anal., 66 (2007), 38-50.  doi: 10.1016/j.na.2005.11.007.  Google Scholar

[10]

F. X. Lin, Spectrum sets and contractible sets of linear differential equations, Chinese Ann. Math. Ser. A, 11 (1990), 111-120.   Google Scholar

[11]

K. J. Palmer, A generalization of Hartman's linearization theorem, J. Math. Anal. Appl., 41 (1973), 753-758.  doi: 10.1016/0022-247X(73)90245-X.  Google Scholar

[12]

K. J. Palmer, A characterization of exponential dichotomy in terms of topological equivalence, J. Math. Anal. Appl., 69 (1979), 8-16.  doi: 10.1016/0022-247X(79)90175-6.  Google Scholar

[13]

K. J. Palmer, The structurally stable linear systems on the half-line are those with exponential dichotomies, J. Differential Equations, 33 (1979), 16-25.  doi: 10.1016/0022-0396(79)90076-7.  Google Scholar

[14]

C. Pugh, On a theorem of P. Hartman, Amer. J. Math., 91 (1969), 363-367.  doi: 10.2307/2373513.  Google Scholar

[15]

J. L. Shi and K. Q. Xiong, On Hartman's linearization theorem and Palmer's linearization theorem, J. Math. Anal. Appl., 192 (1995), 813-832.  doi: 10.1006/jmaa.1995.1205.  Google Scholar

[16]

Y. XiaY. Bai and D. O'Regan, A new method to prove the nonuniform dichotomy spectrum theorem in $\mathbb{R}^n$, Proc. Amer. Math. Soc., 147 (2019), 3905-3917.  doi: 10.1090/proc/14535.  Google Scholar

[17]

Y. XiaH. Huang and K. Kou, Hartman-Grobman theorem for the impulsive system with unbounded nonlinear term, Qual. Theory Dyn. Syst., 16 (2017), 705-730.  doi: 10.1007/s12346-016-0218-8.  Google Scholar

[18]

X. Zhang, Nonuniform dichotomy spectrum and normal forms for nonautonomous differential systems, J. Funct. Anal., 267 (2014), 1889-1916.  doi: 10.1016/j.jfa.2014.07.029.  Google Scholar

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