May  2018, 38(5): 2287-2304. doi: 10.3934/dcds.2018094

Dichotomy spectrum and almost topological conjugacy on nonautonomus unbounded difference systems

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

* Corresponding author: Gonzalo Robledo

Received  September 2017 Published  March 2018

Fund Project: This research has been partially supported by MATHAMSUD cooperation program (16-MATH-04 STADE) and FONDECYT Regular 1170968.

We construct a bijection between the solutions of a linear system of nonautonomous difference equations which is uniformly asymptotically stable and its unbounded perturbation. The key idea used to made this bijection is to consider the crossing times of the solutions with the unit sphere. This approach prompt us to introduce the concept of almost topological conjugacy in this nonautonomous framework. This task is carried out by simplifying both systems through a spectral approach of the notion of almost reducibility combined with suitable technical assumptions.

Citation: Álvaro Castañeda, Gonzalo Robledo. Dichotomy spectrum and almost topological conjugacy on nonautonomus unbounded difference systems. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2287-2304. doi: 10.3934/dcds.2018094
References:
[1]

B. Aulbach and S. Siegmund, A spectral theory for nonautonomous difference equations, in in New Trends in Difference Equations, Temuco, Chile, 2000 (eds. J. López-Fenner and M. Pinto), Taylor and Francis, (2002), 45-55.

[2]

B. Aulbach and S. Siegmund, The dichotomy spectrum for noninvertible systems of linear difference equations, J. Difference Equ. Appl., 7 (2001), 895-913.  doi: 10.1080/10236190108808310.

[3]

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.

[4]

L. BarreriraL. H. Popescu and C. Valls, Generalized exponential behavior and topological equivalence, Discrete Contin. Dyn. Syst. Ser. B., 22 (2017), 3023-3042.  doi: 10.3934/dcdsb.2017161.

[5]

B. Bylov, Almost reducible system of differential equations, Sibirsk. Mat. Z., 3 (1962), 333-359. 

[6]

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.

[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]

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.

[9]

W. A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics 629, Springer-Verlag, Berlin, 1978.

[10]

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

[11]

I. GohbergM. A. Kaashoek and J. Kos, Classification of linear time-varying difference equations under kinematic similarity, Integral Equations Operator Theory, 25 (1996), 445-480.  doi: 10.1007/BF01203027.

[12]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Mathematical Society, Providence, RI, 2011.

[13]

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

[14]

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

[15]

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.

[16]

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.

[17]

G. Papaschinopoulos, A linearization result for a differential equation with piecewise constant argument, Analysis, 16 (1996), 161-170. 

[18]

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

[19]

G. Papaschinopoulos, Dichotomies in terms of Lyapunov functions for linear difference equations, J. Math. Anal. Appl., 152 (1990), 524-535.  doi: 10.1016/0022-247X(90)90082-Q.

[20]

C. Pötzsche, Dichotomy spectra of triangular equations, Discrete Contin. Dyn. Syst., 36 (2016), 423-450. 

[21]

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

[22]

A. Reinfelds and D. Šteinberga, Dynamical equivalence of quasilinear equations, Int. J. Pure and Appl. Math., 98 (2015), 355-364. 

[23]

R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358.  doi: 10.1016/0022-0396(78)90057-8.

[24]

S. Siegmund, Block diagonalization of linear difference equations, J. Difference Equ. Appl., 8 (2002), 177-189.  doi: 10.1080/10236190211950.

[25]

R. L. Wheeden and A. Zygmund, Measure and Integral, Marcel Dekker Inc., New York-Basel, 1977.

show all references

References:
[1]

B. Aulbach and S. Siegmund, A spectral theory for nonautonomous difference equations, in in New Trends in Difference Equations, Temuco, Chile, 2000 (eds. J. López-Fenner and M. Pinto), Taylor and Francis, (2002), 45-55.

[2]

B. Aulbach and S. Siegmund, The dichotomy spectrum for noninvertible systems of linear difference equations, J. Difference Equ. Appl., 7 (2001), 895-913.  doi: 10.1080/10236190108808310.

[3]

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.

[4]

L. BarreriraL. H. Popescu and C. Valls, Generalized exponential behavior and topological equivalence, Discrete Contin. Dyn. Syst. Ser. B., 22 (2017), 3023-3042.  doi: 10.3934/dcdsb.2017161.

[5]

B. Bylov, Almost reducible system of differential equations, Sibirsk. Mat. Z., 3 (1962), 333-359. 

[6]

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.

[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]

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.

[9]

W. A. Coppel, Dichotomies in Stability Theory, Lecture Notes in Mathematics 629, Springer-Verlag, Berlin, 1978.

[10]

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

[11]

I. GohbergM. A. Kaashoek and J. Kos, Classification of linear time-varying difference equations under kinematic similarity, Integral Equations Operator Theory, 25 (1996), 445-480.  doi: 10.1007/BF01203027.

[12]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Mathematical Society, Providence, RI, 2011.

[13]

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

[14]

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

[15]

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.

[16]

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.

[17]

G. Papaschinopoulos, A linearization result for a differential equation with piecewise constant argument, Analysis, 16 (1996), 161-170. 

[18]

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

[19]

G. Papaschinopoulos, Dichotomies in terms of Lyapunov functions for linear difference equations, J. Math. Anal. Appl., 152 (1990), 524-535.  doi: 10.1016/0022-247X(90)90082-Q.

[20]

C. Pötzsche, Dichotomy spectra of triangular equations, Discrete Contin. Dyn. Syst., 36 (2016), 423-450. 

[21]

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

[22]

A. Reinfelds and D. Šteinberga, Dynamical equivalence of quasilinear equations, Int. J. Pure and Appl. Math., 98 (2015), 355-364. 

[23]

R. J. Sacker and G. R. Sell, A spectral theory for linear differential systems, J. Differential Equations, 27 (1978), 320-358.  doi: 10.1016/0022-0396(78)90057-8.

[24]

S. Siegmund, Block diagonalization of linear difference equations, J. Difference Equ. Appl., 8 (2002), 177-189.  doi: 10.1080/10236190211950.

[25]

R. L. Wheeden and A. Zygmund, Measure and Integral, Marcel Dekker Inc., New York-Basel, 1977.

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