Relative Chaos for $C_0$-Semigroups Beyond Topological Notions
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866915792614850560 |
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| author | Nafia, El-Mehdi Ghazouani, Aziz El Omari, M'hamed El |
| author_facet | Nafia, El-Mehdi Ghazouani, Aziz El Omari, M'hamed El |
| contents | We investigate instability phenomena for linear evolution equations within the framework of $C_0$--semigroups on infinite--dimensional spaces. We show that Devaney chaos, being formulated in purely topological terms, may depend on the choice of topology and therefore fail to capture intrinsic dynamical behavior. To address this issue, we introduce a trajectory--based notion of relative chaos, defined with respect to a reference solution and measured in a fixed, physically meaningful norm. This criterion is independent of topological refinements and is shown to be strictly weaker than classical Devaney chaos. Its relevance is illustrated on boundary--driven reaction--diffusion--transport semigroups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_11009 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Relative Chaos for $C_0$-Semigroups Beyond Topological Notions Nafia, El-Mehdi Ghazouani, Aziz El Omari, M'hamed El Dynamical Systems 22E46, 53C35, 57S20 We investigate instability phenomena for linear evolution equations within the framework of $C_0$--semigroups on infinite--dimensional spaces. We show that Devaney chaos, being formulated in purely topological terms, may depend on the choice of topology and therefore fail to capture intrinsic dynamical behavior. To address this issue, we introduce a trajectory--based notion of relative chaos, defined with respect to a reference solution and measured in a fixed, physically meaningful norm. This criterion is independent of topological refinements and is shown to be strictly weaker than classical Devaney chaos. Its relevance is illustrated on boundary--driven reaction--diffusion--transport semigroups. |
| title | Relative Chaos for $C_0$-Semigroups Beyond Topological Notions |
| topic | Dynamical Systems 22E46, 53C35, 57S20 |
| url | https://arxiv.org/abs/2602.11009 |