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Attractors for semilinear wave equations with localized damping and external forces
Lyapunov exponents and Oseledets decomposition in random dynamical systems generated by systems of 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 |
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.
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. |
[2] |
L. Arnold, Random Dynamical Systems, Springer Monogr. Math., Springer, Berlin, 1998.
doi: 10.1007/978-3-662-12878-7. |
[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. |
[4] |
E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. |
[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. |
[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. |
[7] |
T. S. Doan, Lyapunov Exponents for Random Dynamical Systems, Ph.D. dissertation, Technische Universität Dresden, 2009. |
[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. |
[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] |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[2] |
L. Arnold, Random Dynamical Systems, Springer Monogr. Math., Springer, Berlin, 1998.
doi: 10.1007/978-3-662-12878-7. |
[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. |
[4] |
E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955. |
[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. |
[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. |
[7] |
T. S. Doan, Lyapunov Exponents for Random Dynamical Systems, Ph.D. dissertation, Technische Universität Dresden, 2009. |
[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. |
[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] |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
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