Two dynamical approaches to the notion of exponential separation for random systems of delay differential equations
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866917945116983296 |
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| author | Kryspin, Marek Mierczynski, Janusz Novo, Sylvia Obaya, Rafael |
| author_facet | Kryspin, Marek Mierczynski, Janusz Novo, Sylvia Obaya, Rafael |
| contents | This paper deals with the exponential separation of type II, an important concept for random systems of differential equations with delay, introduced in \JM\ et al.~\cite{MiNoOb1}. Two different approaches to its existence are presented. The state space $X$ will be a separable ordered Banach space with $\dim X\geq 2$, dual space $X^{*}$ and positive cone $X^+$ normal and reproducing. In both cases, appropriate cooperativity and irreducibility conditions are assumed to provide a family of generalized Floquet subspaces. If in addition $X^*$ is also separable, one obtains a exponential separation of type II. When this is not the case, but there is an Oseledets decomposition for the continuous semiflow, the same result holds.
Detailed examples are given for all the situations, including also a case where the cone is not normal. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_17990 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Two dynamical approaches to the notion of exponential separation for random systems of delay differential equations Kryspin, Marek Mierczynski, Janusz Novo, Sylvia Obaya, Rafael Dynamical Systems 37H15, 37L55, 34K06, 37A30, 37A40, 37C65, 60H25 This paper deals with the exponential separation of type II, an important concept for random systems of differential equations with delay, introduced in \JM\ et al.~\cite{MiNoOb1}. Two different approaches to its existence are presented. The state space $X$ will be a separable ordered Banach space with $\dim X\geq 2$, dual space $X^{*}$ and positive cone $X^+$ normal and reproducing. In both cases, appropriate cooperativity and irreducibility conditions are assumed to provide a family of generalized Floquet subspaces. If in addition $X^*$ is also separable, one obtains a exponential separation of type II. When this is not the case, but there is an Oseledets decomposition for the continuous semiflow, the same result holds. Detailed examples are given for all the situations, including also a case where the cone is not normal. |
| title | Two dynamical approaches to the notion of exponential separation for random systems of delay differential equations |
| topic | Dynamical Systems 37H15, 37L55, 34K06, 37A30, 37A40, 37C65, 60H25 |
| url | https://arxiv.org/abs/2305.17990 |