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 & Continuous Dynamical Systems - A, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  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]

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

[9]

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

[10]

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

[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.  Google Scholar

[12]

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

[13]

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

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[20]

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

[21]

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

[22]

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

[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.  Google Scholar

[24]

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

[25]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  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]

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

[9]

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

[10]

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

[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.  Google Scholar

[12]

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

[13]

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

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[17]

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

[18]

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

[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.  Google Scholar

[20]

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

[21]

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

[22]

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

[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.  Google Scholar

[24]

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

[25]

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

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