Streams, Graphs and Global Attractors of Dynamical Systems on Locally Compact Spaces
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| Format: | Preprint |
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2025
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| author | De Leo, Roberto Yorke, James A. |
| author_facet | De Leo, Roberto Yorke, James A. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_02262 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Streams, Graphs and Global Attractors of Dynamical Systems on Locally Compact Spaces De Leo, Roberto Yorke, James A. Dynamical Systems Mathematical Physics Chaotic Dynamics 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. |
| title | Streams, Graphs and Global Attractors of Dynamical Systems on Locally Compact Spaces |
| topic | Dynamical Systems Mathematical Physics Chaotic Dynamics |
| url | https://arxiv.org/abs/2503.02262 |