November 2017, 16(6): 1977-1988. doi: 10.3934/cpaa.2017097

Almost reducibility of linear difference systems from a spectral point of view

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

* Corresponding author

Received  September 2016 Revised  May 2017 Published  July 2017

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

We prove that, under some conditions, a linear nonautonomous difference system is Bylov's almost reducible to a diagonal one whose terms are contained in the Sacker and Sell spectrum of the original system.

In the above context, we provide an example of the concept of diagonally significant system, recently introduced by Pötzsche. This example plays an essential role in the demonstration of our results.

Citation: Álvaro Castañeda, Gonzalo Robledo. Almost reducibility of linear difference systems from a spectral point of view. Communications on Pure & Applied Analysis, 2017, 16 (6) : 1977-1988. doi: 10.3934/cpaa.2017097
References:
[1]

B. AulbachN. Van Minh and P.P. Zabreiko, The concept of spectral dichotomy for linear difference equations, J. Math. Anal. Appl., 185 (1994), 275-287. doi: 10.1006/jmaa.1994.1248.

[2]

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

[3]

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.

[4]

F. Battelli and K. J. Palmer, Criteria for exponential dichotomy for triangular systems, J. Math. Anal. Appl., 428 (2015), 525-543. doi: 10.1016/j.jmaa.2015.03.029.

[5]

B. F. Bylov, Almost reducible system of differential equations, Sibirsk. Mat. Zh., 3 (1963), 333-359 (Russian).

[6]

S. Elaydi, An Introduction to Difference Equations 3rd edition, Springer-Verlag, New York, 2005.

[7]

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.

[8]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Mathematical Society, Providence RI, 2011. doi: 10.1090/surv/176.

[9]

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

[10]

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

[11]

K. J. Palmer, Exponential dichotomies, the shadowing lemma and transversal homoclinic points, in Dynamics Reported (eds. U. Kirchgraber and H. -O. Walther), John Wiley & Sons, Ltd. , Chichester; B. G. Teubner, Stuttgart, (1988), 265-306.

[12]

G. Papaschinopoulos and J. Schinas, Criteria for an exponential dichotomy of difference equations, Czechoslovak Math. J., 35 (1985), 295-299. doi: 10.1016/0022-247X(86)90216-7.

[13]

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.

[14]

M. Pinto, Discrete Dichotomies, Comput. Math. Appl., 28 (1994), 259-270. doi: 10.1016/0898-1221(94)00114-6.

[15]

C. Pötzsche, Dichotomy spectra of triangular equations, Discrete Contin. Dyn. Syst., 36 (2016), 423-450. doi: 10.3934/dcds.2016.36.423.

[16]

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.

[17]

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

show all references

References:
[1]

B. AulbachN. Van Minh and P.P. Zabreiko, The concept of spectral dichotomy for linear difference equations, J. Math. Anal. Appl., 185 (1994), 275-287. doi: 10.1006/jmaa.1994.1248.

[2]

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

[3]

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.

[4]

F. Battelli and K. J. Palmer, Criteria for exponential dichotomy for triangular systems, J. Math. Anal. Appl., 428 (2015), 525-543. doi: 10.1016/j.jmaa.2015.03.029.

[5]

B. F. Bylov, Almost reducible system of differential equations, Sibirsk. Mat. Zh., 3 (1963), 333-359 (Russian).

[6]

S. Elaydi, An Introduction to Difference Equations 3rd edition, Springer-Verlag, New York, 2005.

[7]

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.

[8]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Mathematical Society, Providence RI, 2011. doi: 10.1090/surv/176.

[9]

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

[10]

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

[11]

K. J. Palmer, Exponential dichotomies, the shadowing lemma and transversal homoclinic points, in Dynamics Reported (eds. U. Kirchgraber and H. -O. Walther), John Wiley & Sons, Ltd. , Chichester; B. G. Teubner, Stuttgart, (1988), 265-306.

[12]

G. Papaschinopoulos and J. Schinas, Criteria for an exponential dichotomy of difference equations, Czechoslovak Math. J., 35 (1985), 295-299. doi: 10.1016/0022-247X(86)90216-7.

[13]

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.

[14]

M. Pinto, Discrete Dichotomies, Comput. Math. Appl., 28 (1994), 259-270. doi: 10.1016/0898-1221(94)00114-6.

[15]

C. Pötzsche, Dichotomy spectra of triangular equations, Discrete Contin. Dyn. Syst., 36 (2016), 423-450. doi: 10.3934/dcds.2016.36.423.

[16]

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.

[17]

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

[1]

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

[2]

Roberta Fabbri, Sylvia Novo, Carmen Núñez, Rafael Obaya. Null controllable sets and reachable sets for nonautonomous linear control systems. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1069-1094. doi: 10.3934/dcdss.2016042

[3]

D. Motreanu, V. V. Motreanu, Nikolaos S. Papageorgiou. Nonautonomous resonant periodic systems with indefinite linear part and a nonsmooth potential. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1401-1414. doi: 10.3934/cpaa.2011.10.1401

[4]

Yejuan Wang, Chengkui Zhong, Shengfan Zhou. Pullback attractors of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 587-614. doi: 10.3934/dcds.2006.16.587

[5]

Bernd Aulbach, Martin Rasmussen, Stefan Siegmund. Approximation of attractors of nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 215-238. doi: 10.3934/dcdsb.2005.5.215

[6]

Massimo Tarallo. Fredholm's alternative for a class of almost periodic linear systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2301-2313. doi: 10.3934/dcds.2012.32.2301

[7]

Lars Grüne, Peter E. Kloeden, Stefan Siegmund, Fabian R. Wirth. Lyapunov's second method for nonautonomous differential equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (2&3) : 375-403. doi: 10.3934/dcds.2007.18.375

[8]

Janusz Mierczyński, Wenxian Shen. Time averaging for nonautonomous/random linear parabolic equations. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 661-699. doi: 10.3934/dcdsb.2008.9.661

[9]

Ioana Moise, Ricardo Rosa, Xiaoming Wang. Attractors for noncompact nonautonomous systems via energy equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 473-496. doi: 10.3934/dcds.2004.10.473

[10]

Tatsien Li, Zhiqiang Wang. A note on the exact controllability for nonautonomous hyperbolic systems. Communications on Pure & Applied Analysis, 2007, 6 (1) : 229-235. doi: 10.3934/cpaa.2007.6.229

[11]

Xiao-Qiang Zhao, Shengfan Zhou. Kernel sections for processes and nonautonomous lattice systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 763-785. doi: 10.3934/dcdsb.2008.9.763

[12]

Björn Schmalfuss. Attractors for nonautonomous and random dynamical systems perturbed by impulses. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 727-744. doi: 10.3934/dcds.2003.9.727

[13]

David Cheban. Global attractors of nonautonomous quasihomogeneous dynamical systems. Conference Publications, 2001, 2001 (Special) : 96-101. doi: 10.3934/proc.2001.2001.96

[14]

Zhengxin Zhou. On the Poincaré mapping and periodic solutions of nonautonomous differential systems. Communications on Pure & Applied Analysis, 2007, 6 (2) : 541-547. doi: 10.3934/cpaa.2007.6.541

[15]

A. Rodríguez-Bernal. Perturbation of the exponential type of linear nonautonomous parabolic equations and applications to nonlinear equations. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 1003-1032. doi: 10.3934/dcds.2009.25.1003

[16]

Andrew J. Steyer, Erik S. Van Vleck. Underlying one-step methods and nonautonomous stability of general linear methods. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2859-2877. doi: 10.3934/dcdsb.2018108

[17]

Marta Štefánková. Inheriting of chaos in uniformly convergent nonautonomous dynamical systems on the interval. Discrete & Continuous Dynamical Systems - A, 2016, 36 (6) : 3435-3443. doi: 10.3934/dcds.2016.36.3435

[18]

João Ferreira Alves, Michal Málek. Zeta functions and topological entropy of periodic nonautonomous dynamical systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 465-482. doi: 10.3934/dcds.2013.33.465

[19]

Luís Silva. Periodic attractors of nonautonomous flat-topped tent systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-8. doi: 10.3934/dcdsb.2018243

[20]

Simon Brendle, Rainer Nagel. PFDE with nonautonomous past. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 953-966. doi: 10.3934/dcds.2002.8.953

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (5)
  • HTML views (4)
  • Cited by (0)

Other articles
by authors

[Back to Top]