doi: 10.3934/dcdsb.2022055
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A note on differentiability of the conjugacy in a delayed version of Hartman-Grobman Theorem

Departamento de Matemática, Universidad del Bío-Bío, Concepción, 4051381, Chile

*Corresponding author: Adrian Gomez

Received  October 2021 Revised  February 2022 Early access March 2022

Fund Project: This research has been partially supported by FONDECYT 1181061 ANID, Chile (D. Carrasco and H. Elorreaga) and by Proyecto Regular Interno UBB 2021-2023 (A. Gomez)

In this work we study the differentiability properties of the conjugation in the Barreira-Valls version of the Hartman-Grobman Theorem for Non-autonomous and Delayed systems. Indeed, we show that the conjugacy in the Barreira-Valls Theorem is a $ C^1 $ diffeomorphism if we impose some extra hypothesis related with the decay of the perturbation.

Citation: Adrian Gomez, Dante Carrasco, Heli Elorreaga. A note on differentiability of the conjugacy in a delayed version of Hartman-Grobman Theorem. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022055
References:
[1]

L. Barreira and C. Valls, Hyperbolicity via admissibility in delay equations, Indiana Univ. Math. J., 69 (2020), 1155-1179.  doi: 10.1512/iumj.2020.69.7963.

[2]

L. Barreira and C. Valls, Perturbations of delay equations, J. Differential Equations, 269 (2020), 7015-7041.  doi: 10.1016/j.jde.2020.05.028.

[3]

G. R. Belicki$\mathop {\rm{i}}\limits^ \vee $, Functional equations, and conjugacy of local diffeomorphisms of finite smoothness class, (Russian) Funkcional. Anal. i Priložen., 7 (1973), 17–28.

[4]

G. R. Belicki$\mathop {\rm{i}}\limits^ \vee $, Equivalence and normal forms of germs of smooth mappings (Russian), Uspekhi Math. Nauk, 33 (1978), 95-155. 

[5]

Á. Castañeda and G. Robledo, Differentiability of Palmer's linearization theorem and converse result for density functions, J. Differential Equations, 259 (2015), 4634-4650.  doi: 10.1016/j.jde.2015.06.004.

[6]

K.-T. Chen, Equivalence and descomposition of vector fields about an elementary critical point, Amer. J. Math., 85 (1963), 693-722.  doi: 10.2307/2373115.

[7]

D. DragičevićW. Zhang and W. Zhang, Smooth linearization of nonautonomous differential equations with a nonuniform dichotomy, Proc. Lond. Math. Soc., 121 (2020), 32-50.  doi: 10.1112/plms.12315.

[8]

G. Farkas, A Hartman-Grobman result for retarded functional differential equations with an application to the numerics around hyperbolic equilibria, Z. Angew. Math. Phys., 52 (2001), 421-432.  doi: 10.1007/PL00001554.

[9]

D. M. Grobman, Topological classification of neighborhoods of a singularity in $n$-space, (Russian), Mat. Sb. (N.S.), 56 (1962), 77-94. 

[10]

M. GuysinskyB. Hasselblatt and V. Rayskin, Differentiability of the Hartman-Grobman linearization, Discrete Contin. Dyn. Syst., 9 (2003), 979-984.  doi: 10.3934/dcds.2003.9.979.

[11]

J. Hale and S. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, Springer, 1993. doi: 10.1007/978-1-4612-4342-7.

[12]

P. Hartman, On local homeomorphisms of Euclidean spaces, Bol. Soc. Mat. Mexicana, 5 (1960), 220-241. 

[13]

P. Hartman, On the local linearization of differential equations, Proc. Amer. Math. Soc., 14 (1963), 568-573.  doi: 10.1090/S0002-9939-1963-0152718-3.

[14]

P. Hartman, Ordinary Differential Equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964.

[15]

P. Hartman, A lemma in the theory of structural stability of differential equations, Proc. Amer. Math. Soc., 11 (1960), 610-620.  doi: 10.1090/S0002-9939-1960-0121542-7.

[16]

M. Naguno and K. Isé, On the normal forms of differential equations in the neighborhood of an equilibrium point, Osaka J. Math., 9 (1957), 221-234. 

[17]

E. Nelson, Topics in Dynamics. I: Flows, Mathematical Notes, Princenton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1969.

[18]

K. Palmer, A Generalization of Hartman's Linearization Theorem, J. Math. Anal. Appl., 41 (1973), 753-758.  doi: 10.1016/0022-247X(73)90245-X.

[19]

H. Poincaré, Sur le problème des trois corps et les équations de la dynamique, Acta Math., 13 (1890), 1-270. 

[20]

G. R. Sell, Smooth linearization near a fix point, Amer. J. Math., 107 (1985), 1035-1091.  doi: 10.2307/2374346.

[21]

C. L. Siegel, Über die Normalform analytischer Differentialgleichungen in der Nähe einer Gleichgewichtslösung, Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. Math.-Phys.-Chem. Abt., 1952 (1952), 21-30. 

[22]

S. Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math., 79 (1957), 809-824.  doi: 10.2307/2372437.

[23]

S. Sternberg, On the structure of local homeomorphisms of euclidean $n$-space. Ⅱ, Amer. J. Math., 80 (1958), 623-631.  doi: 10.2307/2372774.

[24]

N. Sternberg, A Hartman-Grobman theorem for a class of retarded functional differential equations, J. Math. Anal. Appl., 176 (1993), 156-165.  doi: 10.1006/jmaa.1993.1206.

[25]

S. V. Strien, Smooth linearizatin of hyperbolic fixed points without resonance conditions, J. Differential Equations, 85 (1990), 66-90.  doi: 10.1016/0022-0396(90)90089-8.

[26]

X.-B. Lin, Exponential Dichotomies and Homoclinic Orbits in Functional Differential Equations, J. Differential Equations, 63 (1986), 227-254.  doi: 10.1016/0022-0396(86)90048-3.

[27]

W. ZhangK. Lu and W. Zhang, Differentiability of the conjugacy in the Hartman-Grobman theorem, Trans. Amer. Math. Soc., 369 (2017), 4995-5030.  doi: 10.1090/tran/6810.

show all references

References:
[1]

L. Barreira and C. Valls, Hyperbolicity via admissibility in delay equations, Indiana Univ. Math. J., 69 (2020), 1155-1179.  doi: 10.1512/iumj.2020.69.7963.

[2]

L. Barreira and C. Valls, Perturbations of delay equations, J. Differential Equations, 269 (2020), 7015-7041.  doi: 10.1016/j.jde.2020.05.028.

[3]

G. R. Belicki$\mathop {\rm{i}}\limits^ \vee $, Functional equations, and conjugacy of local diffeomorphisms of finite smoothness class, (Russian) Funkcional. Anal. i Priložen., 7 (1973), 17–28.

[4]

G. R. Belicki$\mathop {\rm{i}}\limits^ \vee $, Equivalence and normal forms of germs of smooth mappings (Russian), Uspekhi Math. Nauk, 33 (1978), 95-155. 

[5]

Á. Castañeda and G. Robledo, Differentiability of Palmer's linearization theorem and converse result for density functions, J. Differential Equations, 259 (2015), 4634-4650.  doi: 10.1016/j.jde.2015.06.004.

[6]

K.-T. Chen, Equivalence and descomposition of vector fields about an elementary critical point, Amer. J. Math., 85 (1963), 693-722.  doi: 10.2307/2373115.

[7]

D. DragičevićW. Zhang and W. Zhang, Smooth linearization of nonautonomous differential equations with a nonuniform dichotomy, Proc. Lond. Math. Soc., 121 (2020), 32-50.  doi: 10.1112/plms.12315.

[8]

G. Farkas, A Hartman-Grobman result for retarded functional differential equations with an application to the numerics around hyperbolic equilibria, Z. Angew. Math. Phys., 52 (2001), 421-432.  doi: 10.1007/PL00001554.

[9]

D. M. Grobman, Topological classification of neighborhoods of a singularity in $n$-space, (Russian), Mat. Sb. (N.S.), 56 (1962), 77-94. 

[10]

M. GuysinskyB. Hasselblatt and V. Rayskin, Differentiability of the Hartman-Grobman linearization, Discrete Contin. Dyn. Syst., 9 (2003), 979-984.  doi: 10.3934/dcds.2003.9.979.

[11]

J. Hale and S. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, Springer, 1993. doi: 10.1007/978-1-4612-4342-7.

[12]

P. Hartman, On local homeomorphisms of Euclidean spaces, Bol. Soc. Mat. Mexicana, 5 (1960), 220-241. 

[13]

P. Hartman, On the local linearization of differential equations, Proc. Amer. Math. Soc., 14 (1963), 568-573.  doi: 10.1090/S0002-9939-1963-0152718-3.

[14]

P. Hartman, Ordinary Differential Equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964.

[15]

P. Hartman, A lemma in the theory of structural stability of differential equations, Proc. Amer. Math. Soc., 11 (1960), 610-620.  doi: 10.1090/S0002-9939-1960-0121542-7.

[16]

M. Naguno and K. Isé, On the normal forms of differential equations in the neighborhood of an equilibrium point, Osaka J. Math., 9 (1957), 221-234. 

[17]

E. Nelson, Topics in Dynamics. I: Flows, Mathematical Notes, Princenton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1969.

[18]

K. Palmer, A Generalization of Hartman's Linearization Theorem, J. Math. Anal. Appl., 41 (1973), 753-758.  doi: 10.1016/0022-247X(73)90245-X.

[19]

H. Poincaré, Sur le problème des trois corps et les équations de la dynamique, Acta Math., 13 (1890), 1-270. 

[20]

G. R. Sell, Smooth linearization near a fix point, Amer. J. Math., 107 (1985), 1035-1091.  doi: 10.2307/2374346.

[21]

C. L. Siegel, Über die Normalform analytischer Differentialgleichungen in der Nähe einer Gleichgewichtslösung, Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. Math.-Phys.-Chem. Abt., 1952 (1952), 21-30. 

[22]

S. Sternberg, Local contractions and a theorem of Poincaré, Amer. J. Math., 79 (1957), 809-824.  doi: 10.2307/2372437.

[23]

S. Sternberg, On the structure of local homeomorphisms of euclidean $n$-space. Ⅱ, Amer. J. Math., 80 (1958), 623-631.  doi: 10.2307/2372774.

[24]

N. Sternberg, A Hartman-Grobman theorem for a class of retarded functional differential equations, J. Math. Anal. Appl., 176 (1993), 156-165.  doi: 10.1006/jmaa.1993.1206.

[25]

S. V. Strien, Smooth linearizatin of hyperbolic fixed points without resonance conditions, J. Differential Equations, 85 (1990), 66-90.  doi: 10.1016/0022-0396(90)90089-8.

[26]

X.-B. Lin, Exponential Dichotomies and Homoclinic Orbits in Functional Differential Equations, J. Differential Equations, 63 (1986), 227-254.  doi: 10.1016/0022-0396(86)90048-3.

[27]

W. ZhangK. Lu and W. Zhang, Differentiability of the conjugacy in the Hartman-Grobman theorem, Trans. Amer. Math. Soc., 369 (2017), 4995-5030.  doi: 10.1090/tran/6810.

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