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April  2020, 19(4): 2235-2255. doi: 10.3934/cpaa.2020098

Lyapunov exponents and Oseledets decomposition in random dynamical systems generated by systems of delay differential equations

Janusz Mierczyński 1,, , Sylvia Novo 2, and  Rafael Obaya 2,

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

 

Dedicated to Professor Tomás Caraballo on occasion of his Sixtieth Birthday

Received  September 2018 Revised  October 2019 Published  January 2020

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 MICIIN/FEDER under project RTI2018-096523-B-100 and EU Marie-Skłodowska-Curie ITN Critical Transitions in Complex Systems (H2020-MSCA-ITN-2014 643073 CRITICS).

Linear skew-product semidynamical systems generated by random systems of delay differential equations are considered, both on a space of continuous functions as well as on a space of $ p $-summable functions. The main result states that in both cases, the Lyapunov exponents are identical, and that the Oseledets decompositions are related by natural embeddings.

Keywords: Random dynamical systems, linear systems of delay differential equations, Lyapunov exponents, Oseledets decomposition.
Mathematics Subject Classification: Primary: 37H15, 37L55, 34K06.Secondary: 37A30, 60H25.
Citation: Janusz Mierczyński, Sylvia Novo, Rafael Obaya. Lyapunov exponents and Oseledets decomposition in random dynamical systems generated by systems of delay differential equations. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2235-2255. doi: 10.3934/cpaa.2020098
References:
[1]

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

[2]

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

[3]

J. A. Calzada, R. 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

[4]

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

[5]

M. C. Delfour and S. K. Mitter, Hereditary differential systems with constant delays. Ⅱ. A class of affine systems and the adjoint problem, J. Differential Equations, 18 (1975), 18-28.  doi: 10.1016/0022-0396(75)90078-9.  Google Scholar

[6]

J. Diestel and J. J. Uhl, Jr., Vector Measures, with a foreword by B. J. Pettis, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. doi: 10.1090/surv/015.  Google Scholar

[7]

T. S. Doan, Lyapunov Exponents for Random Dynamical Systems, Ph.D. dissertation, Technische Universität Dresden, 2009. Google Scholar

[8]

G. Froyland, S. Lloyd and A. Quas, A semi-invertible Oseledets theorem with applications to transfer operator cocycles, Discrete Contin. Dyn. Syst., 33 (2013), 3835–3860. doi: 10.3934/dcds.2013.33.3835.  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]

C. González-Tokman and A. Quas, A concise proof of the multiplicative ergodic theorem on Banach spaces, J. Modern Dynam., 9 (2015), 237-255.  doi: 10.3934/jmd.2015.9.237.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

J. Mierczyński, S. Novo and R. Obaya, Principal Floquet subspaces and exponential separations of type Ⅱ with applications to random delay differential equations, Discrete Contin. Dyn. Syst., 38 (2018), 6163-6193.  doi: 10.3934/dcds.2018265.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

J. A. Calzada, R. 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

[4]

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

[5]

M. C. Delfour and S. K. Mitter, Hereditary differential systems with constant delays. Ⅱ. A class of affine systems and the adjoint problem, J. Differential Equations, 18 (1975), 18-28.  doi: 10.1016/0022-0396(75)90078-9.  Google Scholar

[6]

J. Diestel and J. J. Uhl, Jr., Vector Measures, with a foreword by B. J. Pettis, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. doi: 10.1090/surv/015.  Google Scholar

[7]

T. S. Doan, Lyapunov Exponents for Random Dynamical Systems, Ph.D. dissertation, Technische Universität Dresden, 2009. Google Scholar

[8]

G. Froyland, S. Lloyd and A. Quas, A semi-invertible Oseledets theorem with applications to transfer operator cocycles, Discrete Contin. Dyn. Syst., 33 (2013), 3835–3860. doi: 10.3934/dcds.2013.33.3835.  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]

C. González-Tokman and A. Quas, A concise proof of the multiplicative ergodic theorem on Banach spaces, J. Modern Dynam., 9 (2015), 237-255.  doi: 10.3934/jmd.2015.9.237.  Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

J. Mierczyński, S. Novo and R. Obaya, Principal Floquet subspaces and exponential separations of type Ⅱ with applications to random delay differential equations, Discrete Contin. Dyn. Syst., 38 (2018), 6163-6193.  doi: 10.3934/dcds.2018265.  Google Scholar

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