Academic Journal
Streams, graphs and global attractors of dynamical systems on locally compact spaces
| Title: | Streams, graphs and global attractors of dynamical systems on locally compact spaces |
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| Authors: | De Leo, Roberto, Yorke, James A. |
| Source: | Discrete and Continuous Dynamical Systems. 47:308-340 |
| Publication Status: | Preprint |
| Publisher Information: | American Institute of Mathematical Sciences (AIMS), 2026. |
| Publication Year: | 2026 |
| Subject Terms: | FOS: Mathematics, FOS: Physical sciences, Dynamical Systems (math.DS), Mathematical Physics (math-ph), Mathematics - Dynamical Systems, Chaotic Dynamics (nlin.CD), Nonlinear Sciences - Chaotic Dynamics, Mathematical Physics |
| Description: | In a recent article, we introduced the concept of streams and graphs of a semiflow. An important related concept is the one of semiflow with {\em compact dynamics}, which we defined as a semiflow $F$ with a {\em compact global trapping region}. In this follow-up, we restrict to the important case where the phase space $X$ is locally compact and we move the focus on the concept of {\em global attractor}, a maximal compact set that attracts every compact subset of $X$. A semiflow $F$ can have many global trapping regions but, if it has a global attractor, this is unique. We modify here our original definition and we say that $F$ has compact dynamics if it has a global attractor $G$. We show that most of the qualitative properties of $F$ are inherited by the restriction $F_G$ of $F$ to $G$ and that, in case of Conley's chains stream of $F$, the qualitative behavior of $F$ and $F_G$ coincide. Moreover, if $F$ is a continuous-time semiflow, then its graph is identical to the graph of its time-1 map. Our main result is that, for each semiflow $F$ with compact dynamics over a locally compact space, the graphs of the prolongational relation of $F$ and of every stream of $F$ are connected if the global attractor is connected. 45 pages, 4 figures. arXiv admin note: text overlap with arXiv:2401.12327 |
| Document Type: | Article |
| ISSN: | 1553-5231 1078-0947 |
| DOI: | 10.3934/dcds.2025121 |
| DOI: | 10.48550/arxiv.2503.02262 |
| Access URL: | http://arxiv.org/abs/2503.02262 |
| Rights: | CC BY NC SA |
| Accession Number: | edsair.doi.dedup.....bbf3f917d2a6809404e8794a9b6dd771 |
| Database: | OpenAIRE |
| ISSN: | 15535231 10780947 |
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| DOI: | 10.3934/dcds.2025121 |