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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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[33]

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

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

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

[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.  Google Scholar

[33]

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

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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