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December 2018, 38(12): 6163-6193. doi: 10.3934/dcds.2018265

Principal Floquet subspaces and exponential separations of type Ⅱ with applications to random delay differential equations

1. 

Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, PL-50-370 Wrocław, Poland

2. 

Departamento de Matemática Aplicada, E.I. Industriales, Universidad de Valladolid, Paseo del Cauce 59, 47011 Valladolid, Spain

* Corresponding author: Sylvia Novo

Received  September 2017 Published  September 2018

Fund Project: The first author is supported by the NCN grant Maestro 2013/08/A/ST1/00275 and the last two authors are partly supported by MINECO/FEDER under project MTM2015-66330-P and EU Marie-Skłodowska-Curie ITN Critical Transitions in Complex Systems (H2020-MSCA-ITN-2014 643073 CRITICS)

This paper deals with the study of principal Lyapunov exponents, principal Floquet subspaces, and exponential separation for positive random linear dynamical systems in ordered Banach spaces. The main contribution lies in the introduction of a new type of exponential separation, called of type Ⅱ, important for its application to random differential equations with delay. Under weakened assumptions, the existence of an exponential separation of type Ⅱ in an abstract general setting is shown, and an illustration of its application to dynamical systems generated by scalar linear random delay differential equations with finite delay is given.

Citation: Janusz Mierczyński, Sylvia Novo, Rafael Obaya. Principal Floquet subspaces and exponential separations of type Ⅱ with applications to random delay differential equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6163-6193. doi: 10.3934/dcds.2018265
References:
[1]

Y. A. AbramovichC. D. Aliprantis and O. Burkinshaw, Positive operators on Kreǐn spaces, Acta Appl. Math., 27 (1992), 1-22. doi: 10.1007/BF00046631.

[2]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis. A Hitchhiker's Guide third edition, Springer, Berlin, 2006. doi: 10.1007/3-540-29587-9.

[3]

L. Arnold, Random Dynamical Systems, Springer Monogr. Math., Springer, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[4]

L. ArnoldV. M. Gundlach and L. Demetrius, Evolutionary formalism for products of positive random matrices, Ann. Appl. Probab., 4 (1994), 859-901. doi: 10.1214/aoap/1177004975.

[5]

J. A. CalzadaR. Obaya and A. M. Sanz, Continuous separation for monotone skew-product semiflows: From theoretical to numerical results, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 915-944. doi: 10.3934/dcdsb.2015.20.915.

[6]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.

[7]

T. S. Doan, Lyapunov exponents for random dynamical systems, Ph.D. dissertation, Technische Universität Dresden, 2009.

[8]

S. P. Eveson, Hilbert’s projective metric and the spectral properties of positive linear operators, Proc. London Math. Soc., (3) 70 (1995), 411–440. doi: 10.1112/plms/s3-70.2.411.

[9]

C. González-Tokman and A. Quas, A semi-invertible operator Oseledets theorem, Ergodic Theory Dynam. Systems, 34 (2014), 1230-1272. doi: 10.1017/etds.2012.189.

[10]

E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, American Mathematical Society Colloquium Publications, vol. 31. American Mathematical Society, Providence, R. I., 1957.

[11]

J. Húska, Harnack inequality and exponential separation for oblique derivative problems on Lipschitz domains, J. Differential Equations, 226 (2006), 541-557. doi: 10.1016/j.jde.2006.02.008.

[12]

J. Húska and P. Poláčik, The principal Floquet bundle and exponential separation for linear parabolic equations, J. Dynam. Differential Equations, 16 (2004), 347-375. doi: 10.1007/s10884-004-2784-8.

[13]

J. HúskaP. Poláčik and M. V. Safonov, Harnack inequality, exponential separation, and perturbations of principal Floquet bundles for linear parabolic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 711-739. doi: 10.1016/j.anihpc.2006.04.006.

[14]

R. JohnsonK. Palmer and G. R. Sell, Ergodic properties of linear dynamical systems, SIAM J. Math. Anal., 18 (1987), 1-33. doi: 10.1137/0518001.

[15]

U. Krengel, Ergodic Theorems, Walter de Gruyter, Berlin, 1985. doi: 10.1515/9783110844641.

[16]

Z. Lian and K. Lu, Lyapunov exponents and invariant manifolds for random dynamical systems on a Banach space Mem. Amer. Math. Soc. 206 (2010), vi+106 pp. doi: 10.1090/S0065-9266-10-00574-0.

[17]

R. Mañé, Ergodic Theory and Differentiable Dynamics, translated from the Portuguese by S. Levy, Ergeb. Math. Grenzgeb. (3), Springer, Berlin, 1987. doi: 10.1007/978-3-642-70335-5.

[18]

J. Mierczyński and W. Shen, Exponential separation and principal Lyapunov exponent/spec-trum for random/nonautonomous parabolic equations, J. Differential Equations, 191 (2003), 175-205. doi: 10.1016/S0022-0396(03)00016-0.

[19]

J. Mierczyński and W. Shen, Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2008. doi: 10.1201/9781584888963.

[20]

J. Mierczyński and W. Shen, Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. Ⅰ. General theory, Trans. Amer. Math. Soc., 365 (2013), 5329-5365. doi: 10.1090/S0002-9947-2013-05814-X.

[21]

J. Mierczyński and W. Shen, Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. Ⅱ. Finite-dimensional case, J. Math. Anal. Appl., 404 (2013), 438-458. doi: 10.1016/j.jmaa.2013.03.039.

[22]

J. Mierczyński and W. Shen, Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. Ⅲ. Parabolic equations and delay systems, J. Dynam. Differential Equations, 28 (2016), 1039-1079. doi: 10.1007/s10884-015-9436-z.

[23]

V. M. Millionščikov, Metric theory of linear systems of differential equations, Math. USSR-Sb., 77 (1968), 163-173.

[24]

S. NovoR. Obaya and A. M. Sanz, Topological dynamics for monotone skew-product semiflows with applications, J. Dynam. Differential Equations, 25 (2013), 1201-1231. doi: 10.1007/s10884-013-9337-y.

[25]

S. NovoR. Obaya and A. M. Sanz, Uniform persistence and upper Lyapunov exponents for monotone skew-product semiflows, Nonlinearity, 26 (2013), 2409-2440. doi: 10.1088/0951-7715/26/9/2409.

[26]

R. Obaya and A. M. Sanz, Uniform and strict persistence in monotone skew-product semiflows with applications to non-autonomous Nicholson systems, J. Differential Equations, 261 (2016), 4135-4163. doi: 10.1016/j.jde.2016.06.019.

[27]

R. Obaya and A. M. Sanz, Is uniform persistence a robust property in almost periodic models? A well-behaved family: almost periodic Nicholson sytems, Nonlinearity, 31 (2018), 388-413. doi: 10.1088/1361-6544/aa92e7.

[28]

V. I. Oseledets, A multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems (English. Russian original), Trans. Moscow Math. Soc., 19 (1968), 197–231; translation from Tr. Mosk. Mat. Obshch. 19 (1968), 179–210.

[29]

P. Poláčik and I. Tereščák, Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations, J. Dynamics Differential Equations, 5 (1993), 279-303. doi: 10.1007/BF01053163.

[30]

P. Poláčik and I. Tereščák, Erratum: Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations, J. Dynam. Diff. Eq., 5 (1993), 279–303; J. Dynamics Differential Equations, 6 (1994), 245–246. doi: 10.1007/s10884-006-9052-z.

[31]

M. S. Raghunathan, A proof of Oseledec's multiplicative ergodic theorem, Israel J. Math., 32 (1979), 356-362. doi: 10.1007/BF02760464.

[32]

D. Ruelle, Analycity properties of the characteristic exponents of random matrix products, Adv. in Math., 32 (1979), 68-80. doi: 10.1016/0001-8708(79)90029-X.

[33]

D. Ruelle, Ergodic theory of differentiable dynamical systems, Inst. Hautes Ètudes Sci. Publ. Math., 50 (1979), 27-58. doi: 10.1007/BF02684768.

[34]

D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math., (2) 115 (1982), 243–290. doi: 10.2307/1971392.

[35]

H. H. Schaefer, Topological Vector Spaces, Third printing corrected. Graduate Texts in Mathematics, Vol. 3. Springer-Verlag, New York-Berlin, 1971. doi: 10.1007/BF02684768.

[36]

W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc., 136 (1998), x+93 pp. doi: 10.1090/memo/0647.

[37]

P. Thieullen, Fibrés dynamiques asymptotiquement compacts. Exposants de Lyapounov. Entropie. Dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 49-97. doi: 10.1016/S0294-1449(16)30373-0.

show all references

References:
[1]

Y. A. AbramovichC. D. Aliprantis and O. Burkinshaw, Positive operators on Kreǐn spaces, Acta Appl. Math., 27 (1992), 1-22. doi: 10.1007/BF00046631.

[2]

C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis. A Hitchhiker's Guide third edition, Springer, Berlin, 2006. doi: 10.1007/3-540-29587-9.

[3]

L. Arnold, Random Dynamical Systems, Springer Monogr. Math., Springer, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[4]

L. ArnoldV. M. Gundlach and L. Demetrius, Evolutionary formalism for products of positive random matrices, Ann. Appl. Probab., 4 (1994), 859-901. doi: 10.1214/aoap/1177004975.

[5]

J. A. CalzadaR. Obaya and A. M. Sanz, Continuous separation for monotone skew-product semiflows: From theoretical to numerical results, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 915-944. doi: 10.3934/dcdsb.2015.20.915.

[6]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.

[7]

T. S. Doan, Lyapunov exponents for random dynamical systems, Ph.D. dissertation, Technische Universität Dresden, 2009.

[8]

S. P. Eveson, Hilbert’s projective metric and the spectral properties of positive linear operators, Proc. London Math. Soc., (3) 70 (1995), 411–440. doi: 10.1112/plms/s3-70.2.411.

[9]

C. González-Tokman and A. Quas, A semi-invertible operator Oseledets theorem, Ergodic Theory Dynam. Systems, 34 (2014), 1230-1272. doi: 10.1017/etds.2012.189.

[10]

E. Hille and R. S. Phillips, Functional Analysis and Semi-groups, American Mathematical Society Colloquium Publications, vol. 31. American Mathematical Society, Providence, R. I., 1957.

[11]

J. Húska, Harnack inequality and exponential separation for oblique derivative problems on Lipschitz domains, J. Differential Equations, 226 (2006), 541-557. doi: 10.1016/j.jde.2006.02.008.

[12]

J. Húska and P. Poláčik, The principal Floquet bundle and exponential separation for linear parabolic equations, J. Dynam. Differential Equations, 16 (2004), 347-375. doi: 10.1007/s10884-004-2784-8.

[13]

J. HúskaP. Poláčik and M. V. Safonov, Harnack inequality, exponential separation, and perturbations of principal Floquet bundles for linear parabolic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 711-739. doi: 10.1016/j.anihpc.2006.04.006.

[14]

R. JohnsonK. Palmer and G. R. Sell, Ergodic properties of linear dynamical systems, SIAM J. Math. Anal., 18 (1987), 1-33. doi: 10.1137/0518001.

[15]

U. Krengel, Ergodic Theorems, Walter de Gruyter, Berlin, 1985. doi: 10.1515/9783110844641.

[16]

Z. Lian and K. Lu, Lyapunov exponents and invariant manifolds for random dynamical systems on a Banach space Mem. Amer. Math. Soc. 206 (2010), vi+106 pp. doi: 10.1090/S0065-9266-10-00574-0.

[17]

R. Mañé, Ergodic Theory and Differentiable Dynamics, translated from the Portuguese by S. Levy, Ergeb. Math. Grenzgeb. (3), Springer, Berlin, 1987. doi: 10.1007/978-3-642-70335-5.

[18]

J. Mierczyński and W. Shen, Exponential separation and principal Lyapunov exponent/spec-trum for random/nonautonomous parabolic equations, J. Differential Equations, 191 (2003), 175-205. doi: 10.1016/S0022-0396(03)00016-0.

[19]

J. Mierczyński and W. Shen, Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2008. doi: 10.1201/9781584888963.

[20]

J. Mierczyński and W. Shen, Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. Ⅰ. General theory, Trans. Amer. Math. Soc., 365 (2013), 5329-5365. doi: 10.1090/S0002-9947-2013-05814-X.

[21]

J. Mierczyński and W. Shen, Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. Ⅱ. Finite-dimensional case, J. Math. Anal. Appl., 404 (2013), 438-458. doi: 10.1016/j.jmaa.2013.03.039.

[22]

J. Mierczyński and W. Shen, Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. Ⅲ. Parabolic equations and delay systems, J. Dynam. Differential Equations, 28 (2016), 1039-1079. doi: 10.1007/s10884-015-9436-z.

[23]

V. M. Millionščikov, Metric theory of linear systems of differential equations, Math. USSR-Sb., 77 (1968), 163-173.

[24]

S. NovoR. Obaya and A. M. Sanz, Topological dynamics for monotone skew-product semiflows with applications, J. Dynam. Differential Equations, 25 (2013), 1201-1231. doi: 10.1007/s10884-013-9337-y.

[25]

S. NovoR. Obaya and A. M. Sanz, Uniform persistence and upper Lyapunov exponents for monotone skew-product semiflows, Nonlinearity, 26 (2013), 2409-2440. doi: 10.1088/0951-7715/26/9/2409.

[26]

R. Obaya and A. M. Sanz, Uniform and strict persistence in monotone skew-product semiflows with applications to non-autonomous Nicholson systems, J. Differential Equations, 261 (2016), 4135-4163. doi: 10.1016/j.jde.2016.06.019.

[27]

R. Obaya and A. M. Sanz, Is uniform persistence a robust property in almost periodic models? A well-behaved family: almost periodic Nicholson sytems, Nonlinearity, 31 (2018), 388-413. doi: 10.1088/1361-6544/aa92e7.

[28]

V. I. Oseledets, A multiplicative ergodic theorem, Lyapunov characteristic numbers for dynamical systems (English. Russian original), Trans. Moscow Math. Soc., 19 (1968), 197–231; translation from Tr. Mosk. Mat. Obshch. 19 (1968), 179–210.

[29]

P. Poláčik and I. Tereščák, Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations, J. Dynamics Differential Equations, 5 (1993), 279-303. doi: 10.1007/BF01053163.

[30]

P. Poláčik and I. Tereščák, Erratum: Exponential separation and invariant bundles for maps in ordered Banach spaces with applications to parabolic equations, J. Dynam. Diff. Eq., 5 (1993), 279–303; J. Dynamics Differential Equations, 6 (1994), 245–246. doi: 10.1007/s10884-006-9052-z.

[31]

M. S. Raghunathan, A proof of Oseledec's multiplicative ergodic theorem, Israel J. Math., 32 (1979), 356-362. doi: 10.1007/BF02760464.

[32]

D. Ruelle, Analycity properties of the characteristic exponents of random matrix products, Adv. in Math., 32 (1979), 68-80. doi: 10.1016/0001-8708(79)90029-X.

[33]

D. Ruelle, Ergodic theory of differentiable dynamical systems, Inst. Hautes Ètudes Sci. Publ. Math., 50 (1979), 27-58. doi: 10.1007/BF02684768.

[34]

D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math., (2) 115 (1982), 243–290. doi: 10.2307/1971392.

[35]

H. H. Schaefer, Topological Vector Spaces, Third printing corrected. Graduate Texts in Mathematics, Vol. 3. Springer-Verlag, New York-Berlin, 1971. doi: 10.1007/BF02684768.

[36]

W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc., 136 (1998), x+93 pp. doi: 10.1090/memo/0647.

[37]

P. Thieullen, Fibrés dynamiques asymptotiquement compacts. Exposants de Lyapounov. Entropie. Dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 49-97. doi: 10.1016/S0294-1449(16)30373-0.

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