July  2014, 34(7): 2929-2962. doi: 10.3934/dcds.2014.34.2929

Discrete admissibility and exponential trichotomy of dynamical systems

1. 

Faculty of Mathematics and Computer Science, West University of Timişoara, V. Pârvan Blvd. No. 4, 300223 Timişoara, Romania

Received  March 2013 Revised  October 2013 Published  December 2013

The aim of this paper is to present a new and complex study on the asymptotic behavior of dynamical systems, providing necessary and sufficient conditions for the existence of the exponential trichotomy. We associate to a nonautonomous discrete dynamical system an input-output system and we define a new admissibility concept called $(l^\infty(\mathbb{Z}, X), l^1(\mathbb{Z}, X))$-admissibility. First, we prove that the admissibility is a sufficient condition for the existence of the trichotomy projections, for their uniform boundedness, for their compatibility with the coefficients of the initial dynamical system and for certain reversibility properties. Assuming that the associated input-output operators satisfy a natural boundedness condition, we deduce that the admissibility is a necessary and sufficient condition for the existence of the uniform exponential trichotomy. Next, based on admissibility arguments, we obtain, for the first time in the literature, that all the trichotomic properties of a nonautonomous system can be completely recovered from the trichotomic behavior of the associated discrete dynamical system. Finally, we apply the main results in order to obtain a new characterization for uniform exponential trichotomy of evolution families in terms of discrete admissibility.
Citation: Adina Luminiţa Sasu, Bogdan Sasu. Discrete admissibility and exponential trichotomy of dynamical systems. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2929-2962. doi: 10.3934/dcds.2014.34.2929
References:
[1]

A. I. Alonso, J. Hong and R. Obaya, Exponential dichotomy and trichotomy for difference equations, Comput. Math. Appl., 38 (1999), 41-49. doi: 10.1016/S0898-1221(99)00167-4.

[2]

L. Barreira and C. Valls, Center manifolds for nonuniformly partially hyperbolic diffeomorphisms, J. Math. Pures Appl. (9), 84 (2005), 1693-1715. doi: 10.1016/j.matpur.2005.07.005.

[3]

L. Barreira and C. Valls, Center manifolds for infinite delay, J. Differential Equations, 247 (2009), 1297-1310. doi: 10.1016/j.jde.2009.04.006.

[4]

L. Barreira and C. Valls, Robustness of nonuniform exponential trichotomies in Banach spaces, J. Math. Anal. Appl., 351 (2009), 373-381. doi: 10.1016/j.jmaa.2008.10.030.

[5]

L. Barreira and C. Valls, Lyapunov functions for trichotomies with growth rates, J. Differential Equations, 248 (2010), 151-183. doi: 10.1016/j.jde.2009.07.001.

[6]

L. Barreira and C. Valls, Nonuniform exponential dichotomies and admissibility, Discrete Contin. Dyn. Syst., 30 (2011), 39-53. doi: 10.3934/dcds.2011.30.39.

[7]

L. Barreira and C. Valls, Admissibility versus nonuniform exponential behavior for noninver-tible cocycles, Discrete Contin. Dyn. Syst., 33 (2013), 1297-1311.

[8]

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

[9]

S.-N. Chow and H. Leiva, Existence and roughness of the exponential dichotomy for linear skew-product semiflows in Banach spaces, J. Differential Equations, 120 (1995), 429-477. doi: 10.1006/jdeq.1995.1117.

[10]

Ju. L. Dalec'kiĭ and M. G. Kreĭn, Stability of Solutions of Differential Equations in Banach Spaces, Trans. Math. Monographs, Vol. 43, Amer. Math. Soc., Providence R.I., 1974.

[11]

S. Elaydi and O. Hájek, Some remarks on nonlinear dichotomy and trichotomy, in Non-linear Analysis and Applications (Arlington, Tex., 1986), Lect. Notes Pure Appl. Math., 109, Dekker, New York, (1987), 175-178.

[12]

S. Elaydi and O. Hájek, Exponential trichotomy of differential systems, J. Math. Anal. Appl., 129 (1988), 362-374. doi: 10.1016/0022-247X(88)90255-7.

[13]

S. Elaydi and O. Hájek, Exponential dichotomy and trichotomy of nonlinear differential equations, Differ. Integral Equ., 3 (1990), 1201-1224.

[14]

S. Elaydi and K. Janglajew, Dichotomy and trichotomy of difference equations, J. Differ. Equations Appl., 3 (1998), 417-448. doi: 10.1080/10236199708808113.

[15]

N. T. Huy and N. Van Minh, Exponential dichotomy of difference equations and applications to evolution equations on the half-line, Comput. Math. Appl., 42 (2001), 301-311. doi: 10.1016/S0898-1221(01)00155-9.

[16]

J. López-Fenner and M. Pinto, $(h,k)$-trichotomies and asymptotics of nonautonomous difference equations, Comp. Math. Appl., 33 (1997), 105-124. doi: 10.1016/S0898-1221(97)00080-1.

[17]

J. L. Massera and J. J. Schäffer, Linear Differential Equations and Function Spaces, Pure and Applied Mathematics, Vol. 21, Academic Press, New York-London, 1966.

[18]

M. Megan, A. L. Sasu and B. Sasu, Discrete admissibility and exponential dichotomy for evolution families, Discrete Contin. Dyn. Syst., 9 (2003), 383-397.

[19]

N. Van Minh and N. T. Huy, Characterizations of dichotomies of evolution equations on the half-line, J. Math. Anal. Appl., 261 (2001), 28-44. doi: 10.1006/jmaa.2001.7450.

[20]

O. Perron, Die Stabilitätsfrage bei Differentialgleischungen, Math. Z., 32 (1930), 703-728. doi: 10.1007/BF01194662.

[21]

K. J. Palmer, Exponential dichotomy and expansivity, Ann. Mat. Pura Appl. (4), 185 (2006), S171-S185. doi: 10.1007/s10231-004-0141-5.

[22]

V. A. Pliss and G. R. Sell, Robustness of the exponential dichotomy in infinite-dimensional dynamical systems, J. Dynam. Differential Equations, 11 (1999), 471-513. doi: 10.1023/A:1021913903923.

[23]

R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splittings for linear differential systems, III, J. Differential Equations, 22 (1976), 497-522. doi: 10.1016/0022-0396(76)90043-7.

[24]

B. Sasu and A. L. Sasu, Exponential dichotomy and $(l^p, l^q)$-admissibility on the half-line, J. Math. Anal. Appl., 316 (2006), 397-408. doi: 10.1016/j.jmaa.2005.04.047.

[25]

B. Sasu, Uniform dichotomy and exponential dichotomy of evolution families on the half-line, J. Math. Anal. Appl., 323 (2006), 1465-1478. doi: 10.1016/j.jmaa.2005.12.002.

[26]

B. Sasu and A. L. Sasu, Exponential trichotomy and p-admissibility for evolution families on the real line, Math. Z., 253 (2006), 515-536. doi: 10.1007/s00209-005-0920-8.

[27]

A. L. Sasu, Exponential dichotomy and dichotomy radius for difference equations, J. Math. Anal. Appl., 344 (2008), 906-920. doi: 10.1016/j.jmaa.2008.03.019.

[28]

A. L. Sasu and B. Sasu, Exponential trichotomy for variational difference equations, J. Difference Equ. Appl., 15 (2009), 693-718. doi: 10.1080/10236190802285118.

[29]

A. L. Sasu and B. Sasu, Input-output admissibility and exponential trichotomy of difference equations, J. Math. Anal. Appl., 380 (2011), 17-32. doi: 10.1016/j.jmaa.2011.02.045.

[30]

B. Sasu and A. L. Sasu, On the dichotomic behavior of discrete dynamical systems on the half-line, Discrete Contin. Dyn. Syst., 33 (2013), 3057-3084. doi: 10.3934/dcds.2013.33.3057.

[31]

X. Liu, Exponential trichotomy and homoclinic bifurcation with saddle-center equilibrium, Appl. Math. Lett., 23 (2010), 409-416. doi: 10.1016/j.aml.2009.11.008.

[32]

D. Zhu, Exponential trichotomy and heteroclinic bifurcations, Nonlinear Anal., 28 (1997), 547-557. doi: 10.1016/0362-546X(95)00164-Q.

show all references

References:
[1]

A. I. Alonso, J. Hong and R. Obaya, Exponential dichotomy and trichotomy for difference equations, Comput. Math. Appl., 38 (1999), 41-49. doi: 10.1016/S0898-1221(99)00167-4.

[2]

L. Barreira and C. Valls, Center manifolds for nonuniformly partially hyperbolic diffeomorphisms, J. Math. Pures Appl. (9), 84 (2005), 1693-1715. doi: 10.1016/j.matpur.2005.07.005.

[3]

L. Barreira and C. Valls, Center manifolds for infinite delay, J. Differential Equations, 247 (2009), 1297-1310. doi: 10.1016/j.jde.2009.04.006.

[4]

L. Barreira and C. Valls, Robustness of nonuniform exponential trichotomies in Banach spaces, J. Math. Anal. Appl., 351 (2009), 373-381. doi: 10.1016/j.jmaa.2008.10.030.

[5]

L. Barreira and C. Valls, Lyapunov functions for trichotomies with growth rates, J. Differential Equations, 248 (2010), 151-183. doi: 10.1016/j.jde.2009.07.001.

[6]

L. Barreira and C. Valls, Nonuniform exponential dichotomies and admissibility, Discrete Contin. Dyn. Syst., 30 (2011), 39-53. doi: 10.3934/dcds.2011.30.39.

[7]

L. Barreira and C. Valls, Admissibility versus nonuniform exponential behavior for noninver-tible cocycles, Discrete Contin. Dyn. Syst., 33 (2013), 1297-1311.

[8]

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

[9]

S.-N. Chow and H. Leiva, Existence and roughness of the exponential dichotomy for linear skew-product semiflows in Banach spaces, J. Differential Equations, 120 (1995), 429-477. doi: 10.1006/jdeq.1995.1117.

[10]

Ju. L. Dalec'kiĭ and M. G. Kreĭn, Stability of Solutions of Differential Equations in Banach Spaces, Trans. Math. Monographs, Vol. 43, Amer. Math. Soc., Providence R.I., 1974.

[11]

S. Elaydi and O. Hájek, Some remarks on nonlinear dichotomy and trichotomy, in Non-linear Analysis and Applications (Arlington, Tex., 1986), Lect. Notes Pure Appl. Math., 109, Dekker, New York, (1987), 175-178.

[12]

S. Elaydi and O. Hájek, Exponential trichotomy of differential systems, J. Math. Anal. Appl., 129 (1988), 362-374. doi: 10.1016/0022-247X(88)90255-7.

[13]

S. Elaydi and O. Hájek, Exponential dichotomy and trichotomy of nonlinear differential equations, Differ. Integral Equ., 3 (1990), 1201-1224.

[14]

S. Elaydi and K. Janglajew, Dichotomy and trichotomy of difference equations, J. Differ. Equations Appl., 3 (1998), 417-448. doi: 10.1080/10236199708808113.

[15]

N. T. Huy and N. Van Minh, Exponential dichotomy of difference equations and applications to evolution equations on the half-line, Comput. Math. Appl., 42 (2001), 301-311. doi: 10.1016/S0898-1221(01)00155-9.

[16]

J. López-Fenner and M. Pinto, $(h,k)$-trichotomies and asymptotics of nonautonomous difference equations, Comp. Math. Appl., 33 (1997), 105-124. doi: 10.1016/S0898-1221(97)00080-1.

[17]

J. L. Massera and J. J. Schäffer, Linear Differential Equations and Function Spaces, Pure and Applied Mathematics, Vol. 21, Academic Press, New York-London, 1966.

[18]

M. Megan, A. L. Sasu and B. Sasu, Discrete admissibility and exponential dichotomy for evolution families, Discrete Contin. Dyn. Syst., 9 (2003), 383-397.

[19]

N. Van Minh and N. T. Huy, Characterizations of dichotomies of evolution equations on the half-line, J. Math. Anal. Appl., 261 (2001), 28-44. doi: 10.1006/jmaa.2001.7450.

[20]

O. Perron, Die Stabilitätsfrage bei Differentialgleischungen, Math. Z., 32 (1930), 703-728. doi: 10.1007/BF01194662.

[21]

K. J. Palmer, Exponential dichotomy and expansivity, Ann. Mat. Pura Appl. (4), 185 (2006), S171-S185. doi: 10.1007/s10231-004-0141-5.

[22]

V. A. Pliss and G. R. Sell, Robustness of the exponential dichotomy in infinite-dimensional dynamical systems, J. Dynam. Differential Equations, 11 (1999), 471-513. doi: 10.1023/A:1021913903923.

[23]

R. J. Sacker and G. R. Sell, Existence of dichotomies and invariant splittings for linear differential systems, III, J. Differential Equations, 22 (1976), 497-522. doi: 10.1016/0022-0396(76)90043-7.

[24]

B. Sasu and A. L. Sasu, Exponential dichotomy and $(l^p, l^q)$-admissibility on the half-line, J. Math. Anal. Appl., 316 (2006), 397-408. doi: 10.1016/j.jmaa.2005.04.047.

[25]

B. Sasu, Uniform dichotomy and exponential dichotomy of evolution families on the half-line, J. Math. Anal. Appl., 323 (2006), 1465-1478. doi: 10.1016/j.jmaa.2005.12.002.

[26]

B. Sasu and A. L. Sasu, Exponential trichotomy and p-admissibility for evolution families on the real line, Math. Z., 253 (2006), 515-536. doi: 10.1007/s00209-005-0920-8.

[27]

A. L. Sasu, Exponential dichotomy and dichotomy radius for difference equations, J. Math. Anal. Appl., 344 (2008), 906-920. doi: 10.1016/j.jmaa.2008.03.019.

[28]

A. L. Sasu and B. Sasu, Exponential trichotomy for variational difference equations, J. Difference Equ. Appl., 15 (2009), 693-718. doi: 10.1080/10236190802285118.

[29]

A. L. Sasu and B. Sasu, Input-output admissibility and exponential trichotomy of difference equations, J. Math. Anal. Appl., 380 (2011), 17-32. doi: 10.1016/j.jmaa.2011.02.045.

[30]

B. Sasu and A. L. Sasu, On the dichotomic behavior of discrete dynamical systems on the half-line, Discrete Contin. Dyn. Syst., 33 (2013), 3057-3084. doi: 10.3934/dcds.2013.33.3057.

[31]

X. Liu, Exponential trichotomy and homoclinic bifurcation with saddle-center equilibrium, Appl. Math. Lett., 23 (2010), 409-416. doi: 10.1016/j.aml.2009.11.008.

[32]

D. Zhu, Exponential trichotomy and heteroclinic bifurcations, Nonlinear Anal., 28 (1997), 547-557. doi: 10.1016/0362-546X(95)00164-Q.

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