February  2021, 20(2): 511-532. doi: 10.3934/cpaa.2020278

Topological Equivalence of nonautonomous difference equations with a family of dichotomies on the half line

1. 

Departamento de Matemáticas, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile

2. 

Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibañez, Peñalolén, Santiago, Chile

* Corresponding author

Received  March 2020 Revised  September 2020 Published  February 2021 Early access  December 2020

Fund Project: This work was supported by FONDECYT Regular 1170968 and FONDECYT Regular 1200653

A linear system of difference equations and a nonlinear perturbation are considered, we obtain sufficient conditions to ensure the topological equivalence between them, namely, the linear part satisfies a property of dichotomy on the positive half–line while the nonlinearity has some boundedness and Lipschitz conditions. In addition, we provide new characterizations for the resulting homeomorphisms. When the linear system is asymptotically stable and the nonlinear system has a unique equilibrium, we deduce sharper results for the smoothness of the topological equivalence. Finally, we study the asymptotic stability and its preservation by topological equivalence.

Citation: Álvaro Castañeda, Pablo González, Gonzalo Robledo. Topological Equivalence of nonautonomous difference equations with a family of dichotomies on the half line. Communications on Pure and Applied Analysis, 2021, 20 (2) : 511-532. doi: 10.3934/cpaa.2020278
References:
[1]

B. Aulbach and S. Siegmund, A spectral theory for nonautonomous difference equations, in New Trends in Difference Equations, London, 2002.

[2]

M.G. Babutia and M. Megan, Nonuniform exponential dichotomy for discrete dynamical systems in Banach spaces, Mediterr. J. Math., 13 (2016) 1653–1667. doi: 10.1007/s00009-015-0605-4.

[3]

L. Barreira and C. Valls, A Grobman–Hartman theorem for nonuniformly hyperbolic dynamics, J. Differ. Equ., 228 (2006), 285-310.  doi: 10.1016/j.jde.2006.04.001.

[4]

L. BarreiraM. FanC. Valls and J. Zhang, Robustness of nonuniform polynomial dichotomies for difference equations, Topol. Methods Nonlinear Anal., 37 (2011), 357-376. 

[5]

L. BarreiraL. H. Popescu and C. Valls, Nonautonomous dynamics with discrete time and topological equivalence, Z. Anal. Anwend., 35 (2016), 21-39.  doi: 10.4171/ZAA/1553.

[6]

A. Bento and C. Silva, Nonuniform $(\mu, \nu)$–dichotomies and local dynamics of difference equations, Nonlinear Anal., 75 (2012), 78-90.  doi: 10.1016/j.na.2011.08.008.

[7]

Á. Castañeda and G. Robledo, A topological equivalence result for a family of nonlinear difference systems having generalized exponential dichotomy, J. Difference Equ. Appl., 22 (2016), 1271-1291.  doi: 10.1080/10236198.2016.1192161.

[8]

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.

[9]

Á. Castañeda, P. Monzón and G. Robledo, Smoothness of Topological Equivalence on the Half Line for Nonautonomous Systems, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 2484-2502. doi: 10.1017/prm.2019.32.

[10]

J. Chu, Robustness of nonuniform behavior for discrete dynamics, Bull. Sci. Math., 137 (2013), 1031-1047.  doi: 10.1016/j.bulsci.2013.03.003.

[11]

Ch. V Coffman and J. J. Schäfer, Dichotomies for linear difference equations, Math. Ann., 172 (1967), 139-166.  doi: 10.1007/BF01350095.

[12]

W. A. Coppel, Dichotomies in Stability Theory, Springer, Verlag, Berlin, 1978.

[13]

V. Crai and M. Aldescu, On $(h, k)$–dichotomy of linear discrete-time systems in Banach spaces, Difference equations, discrete dynamical systems and applications, Springer Proc. Math. Stat., 287, Springer, Cham, 2019,257–271. doi: 10.1007/978-3-030-20016-9_10.

[14]

D. Dragi$\check{c}$evićW. Zhang and W. Zhang, Smooth linearization of nonautonomous difference equations with a nonuniform dichotomy, Math. Z., 292 (2019), 1175-1193.  doi: 10.1007/s00209-018-2134-x.

[15]

S. Elaydi, An Introduction to Difference Equations, Springer, New York, 2005.

[16]

D. M. Grobman, Homeomorphism of systems of differential equations, Dokl. Akad. Nauk. SSSR, 128 (1959), 880-881. 

[17]

P. Hartman, On local homeomorphisms of Euclidean spaces, Bol. Soc. Mat. Mexicana (2), 5 (1960), 220-241. 

[18]

D. Hinrichsen and A. J. Pritchard, Mathematical Systems Theory I, Springer, Heidelberg, Berlin, 2010.

[19]

J. Kurzweil and G. Papaschinopoulos, Topological equivalence and structural stability for linear difference equations, J. Differ. Equ., 89 (1991), 89-94.  doi: 10.1016/0022-0396(91)90112-M.

[20]

Z. Lin and Y. X. Lin, Linear Systems Exponential Dichotomy and Structure of Sets of Hyperbolic Points, World Scientific, Singapore, 2000. doi: 10.1142/9789812793027.

[21]

J. Palis, On the local structure of hyperbolic points in Banach space, An. Acad. Brasil. Ci., 40 (1968), 263-266. 

[22]

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.

[23]

G. Papaschinopoulos, Criteria for an exponential dichotomy of difference equations, Czechoslovak Math. J., 35 (1985), 295-299. 

[24]

G. Papaschinopoulos and G. Schinas, Structural stability via the density of a class of linear discrete systems, J. Math. Anal. Appl., 127 (1987), 530-539.  doi: 10.1016/0022-247X(87)90127-2.

[25]

G. Papaschinopoulos, Some roughness results concerning reducibility for linear difference equations, Internat. J. Math. Sci., 11 (1988), 793-804.  doi: 10.1155/S0161171288000961.

[26]

G. Papaschinopoulos, A linearization result for a differential equation with piecewise constant argument, Analysis, 16 (1996), 161-170.  doi: 10.1524/anly.1996.16.2.161.

[27]

R. Plastock, Homeomorphisms between Banach spaces, T. Am. Math. Soc., 200 (1974), 169-183.  doi: 10.2307/1997252.

[28]

J. Popenda, Gronwall type inequalities, Z. Angew. Math. Mech., 75 (1995), 669-677.  doi: 10.1002/zamm.19950750903.

[29]

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

[30]

V. Rayskin, $\alpha-$H$\ddot{o}$lder linearization, J. Differ. Equ., 147 (1998), 271-284.  doi: 10.1006/jdeq.1997.3410.

[31]

A. Reinfelds, Global topological equivalence of nonlinear flows, Differencial'nye Uravnenija, 10 (1972), 1901-1903. 

[32]

A. Reinfelds, Grobman's–Hartman's theorem for time-dependent difference equations, Math. Differ. equ. (Russian), 9-13, Latv. Univ. Zinat. Raksti, 605, Latv. Univ., Riga, 1997.

[33]

A. Reinfelds and D. $\check{S}$teinberga., Dynamical equivalence of quasilinear equations, Internat. J. Pure Appl. Math. 98 (2015), 355-364. doi: 10.1515/tmmp-2015-0035.

[34]

J. Schinas and G. Papaschinopoulos, Topological equivalence via dichotomies and Lyapunov functions, Boll. Un. Mat. Ital. C (6), 4 (1985), 61-70. 

[35]

W. Zhou and W. Zhang, Admissibility and roughness of nonuniform exponential dichotomies for difference equations, J. Funct. Anal., 271 (2016), 1087-1129.  doi: 10.1016/j.jfa.2016.06.005.

show all references

References:
[1]

B. Aulbach and S. Siegmund, A spectral theory for nonautonomous difference equations, in New Trends in Difference Equations, London, 2002.

[2]

M.G. Babutia and M. Megan, Nonuniform exponential dichotomy for discrete dynamical systems in Banach spaces, Mediterr. J. Math., 13 (2016) 1653–1667. doi: 10.1007/s00009-015-0605-4.

[3]

L. Barreira and C. Valls, A Grobman–Hartman theorem for nonuniformly hyperbolic dynamics, J. Differ. Equ., 228 (2006), 285-310.  doi: 10.1016/j.jde.2006.04.001.

[4]

L. BarreiraM. FanC. Valls and J. Zhang, Robustness of nonuniform polynomial dichotomies for difference equations, Topol. Methods Nonlinear Anal., 37 (2011), 357-376. 

[5]

L. BarreiraL. H. Popescu and C. Valls, Nonautonomous dynamics with discrete time and topological equivalence, Z. Anal. Anwend., 35 (2016), 21-39.  doi: 10.4171/ZAA/1553.

[6]

A. Bento and C. Silva, Nonuniform $(\mu, \nu)$–dichotomies and local dynamics of difference equations, Nonlinear Anal., 75 (2012), 78-90.  doi: 10.1016/j.na.2011.08.008.

[7]

Á. Castañeda and G. Robledo, A topological equivalence result for a family of nonlinear difference systems having generalized exponential dichotomy, J. Difference Equ. Appl., 22 (2016), 1271-1291.  doi: 10.1080/10236198.2016.1192161.

[8]

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.

[9]

Á. Castañeda, P. Monzón and G. Robledo, Smoothness of Topological Equivalence on the Half Line for Nonautonomous Systems, Proc. Roy. Soc. Edinburgh Sect. A, 150 (2020), 2484-2502. doi: 10.1017/prm.2019.32.

[10]

J. Chu, Robustness of nonuniform behavior for discrete dynamics, Bull. Sci. Math., 137 (2013), 1031-1047.  doi: 10.1016/j.bulsci.2013.03.003.

[11]

Ch. V Coffman and J. J. Schäfer, Dichotomies for linear difference equations, Math. Ann., 172 (1967), 139-166.  doi: 10.1007/BF01350095.

[12]

W. A. Coppel, Dichotomies in Stability Theory, Springer, Verlag, Berlin, 1978.

[13]

V. Crai and M. Aldescu, On $(h, k)$–dichotomy of linear discrete-time systems in Banach spaces, Difference equations, discrete dynamical systems and applications, Springer Proc. Math. Stat., 287, Springer, Cham, 2019,257–271. doi: 10.1007/978-3-030-20016-9_10.

[14]

D. Dragi$\check{c}$evićW. Zhang and W. Zhang, Smooth linearization of nonautonomous difference equations with a nonuniform dichotomy, Math. Z., 292 (2019), 1175-1193.  doi: 10.1007/s00209-018-2134-x.

[15]

S. Elaydi, An Introduction to Difference Equations, Springer, New York, 2005.

[16]

D. M. Grobman, Homeomorphism of systems of differential equations, Dokl. Akad. Nauk. SSSR, 128 (1959), 880-881. 

[17]

P. Hartman, On local homeomorphisms of Euclidean spaces, Bol. Soc. Mat. Mexicana (2), 5 (1960), 220-241. 

[18]

D. Hinrichsen and A. J. Pritchard, Mathematical Systems Theory I, Springer, Heidelberg, Berlin, 2010.

[19]

J. Kurzweil and G. Papaschinopoulos, Topological equivalence and structural stability for linear difference equations, J. Differ. Equ., 89 (1991), 89-94.  doi: 10.1016/0022-0396(91)90112-M.

[20]

Z. Lin and Y. X. Lin, Linear Systems Exponential Dichotomy and Structure of Sets of Hyperbolic Points, World Scientific, Singapore, 2000. doi: 10.1142/9789812793027.

[21]

J. Palis, On the local structure of hyperbolic points in Banach space, An. Acad. Brasil. Ci., 40 (1968), 263-266. 

[22]

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.

[23]

G. Papaschinopoulos, Criteria for an exponential dichotomy of difference equations, Czechoslovak Math. J., 35 (1985), 295-299. 

[24]

G. Papaschinopoulos and G. Schinas, Structural stability via the density of a class of linear discrete systems, J. Math. Anal. Appl., 127 (1987), 530-539.  doi: 10.1016/0022-247X(87)90127-2.

[25]

G. Papaschinopoulos, Some roughness results concerning reducibility for linear difference equations, Internat. J. Math. Sci., 11 (1988), 793-804.  doi: 10.1155/S0161171288000961.

[26]

G. Papaschinopoulos, A linearization result for a differential equation with piecewise constant argument, Analysis, 16 (1996), 161-170.  doi: 10.1524/anly.1996.16.2.161.

[27]

R. Plastock, Homeomorphisms between Banach spaces, T. Am. Math. Soc., 200 (1974), 169-183.  doi: 10.2307/1997252.

[28]

J. Popenda, Gronwall type inequalities, Z. Angew. Math. Mech., 75 (1995), 669-677.  doi: 10.1002/zamm.19950750903.

[29]

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

[30]

V. Rayskin, $\alpha-$H$\ddot{o}$lder linearization, J. Differ. Equ., 147 (1998), 271-284.  doi: 10.1006/jdeq.1997.3410.

[31]

A. Reinfelds, Global topological equivalence of nonlinear flows, Differencial'nye Uravnenija, 10 (1972), 1901-1903. 

[32]

A. Reinfelds, Grobman's–Hartman's theorem for time-dependent difference equations, Math. Differ. equ. (Russian), 9-13, Latv. Univ. Zinat. Raksti, 605, Latv. Univ., Riga, 1997.

[33]

A. Reinfelds and D. $\check{S}$teinberga., Dynamical equivalence of quasilinear equations, Internat. J. Pure Appl. Math. 98 (2015), 355-364. doi: 10.1515/tmmp-2015-0035.

[34]

J. Schinas and G. Papaschinopoulos, Topological equivalence via dichotomies and Lyapunov functions, Boll. Un. Mat. Ital. C (6), 4 (1985), 61-70. 

[35]

W. Zhou and W. Zhang, Admissibility and roughness of nonuniform exponential dichotomies for difference equations, J. Funct. Anal., 271 (2016), 1087-1129.  doi: 10.1016/j.jfa.2016.06.005.

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