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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

Álvaro Castañeda 1, , Pablo González 2,, and  Gonzalo Robledo 1,

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  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.

Keywords: Difference equations, topological equivalence, dichotomies, diffeomorphisms.
Mathematics Subject Classification: Primary: 39A06; Secondary: 34D09.
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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

L. Barreira, M. Fan, C. Valls and J. Zhang, Robustness of nonuniform polynomial dichotomies for difference equations, Topol. Methods Nonlinear Anal., 37 (2011), 357-376.   Google Scholar

[5]

L. Barreira, L. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

[14]

D. Dragi$\check{c}$ević, W. 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.  Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

L. Barreira, M. Fan, C. Valls and J. Zhang, Robustness of nonuniform polynomial dichotomies for difference equations, Topol. Methods Nonlinear Anal., 37 (2011), 357-376.   Google Scholar

[5]

L. Barreira, L. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

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

[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.  Google Scholar

[14]

D. Dragi$\check{c}$ević, W. 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.  Google Scholar

[15]

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

[16]

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

[17]

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

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[27]

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

[28]

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

[29]

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

[30]

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

[31]

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

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[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.  Google Scholar

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