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