Streams, Graphs and Global Attractors of Dynamical Systems on Locally Compact Spaces

TL;DR

This work develops a unified topological framework for dynamical systems on locally compact spaces by introducing and relating global attractors, trapping regions, prolongational relations, streams, and various chain-based notions. The authors prove that a semiflow has compact dynamics precisely when a global attractor exists, and that many qualitative features of the semiflow are preserved under restriction to the attractor, with the time-1 map capturing the same chain-recurrent structure as the continuous-time flow. They establish conditions under which the global attractor is connected and show that connected attractors force connected prolongational graphs and streams, linking attractor topology to graph-based representations of dynamics. The paper further develops several notions of chains and Auslander-type streams, including the smallest stream and Sigma-chains, and demonstrates that, in the compact-dynamics setting, the attractor and chain-structure information can be reduced to analyses on $G_F$, often via fat trapping regions. Overall, the results provide a cohesive, morphism-friendly toolkit for analyzing global behavior of semiflows in locally compact spaces, with practical implications for reducing complex dynamics to tractable, attractor-centered representations.

Abstract

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.

Figures (4)

• Figure 1: Caption
• Figure 2: A semiflow with a non-invariant non-wandering set. The picture shows several orbits of a semiflow $F$ on the non compact space $X$ equal to the unbounded strip shown in figure where we identify points on the horizontal half-line $h$ passing through $A$ with points on the vertical segment $BC$ so that $A$ is identified with $C$ and points going to infinity on $h$ are identified with points going to $B$ on $BC$. Several orbits of $F$ are shown, each one painted in a different color. As the picture suggests, $\cap_{t\geq0}F^t(X)=\emptyset$, i.e. no subset of $X$ is $F$-invariant. The non-wandering set coincides with the blue orbit.
• Figure 3: A semiflow with a non-invariant non-wandering set.
• Figure 4: An example of semiflow $F$ where the prolongational graph of $F$ and that of the restriction of $F$ to its global attractor do not coincide.

Theorems & Definitions (144)

• Definition 2.1
• Definition 2.2: Orbits and limit sets
• Example 2.3
• Lemma 2.4
• Lemma 2.5
• proof
• Corollary 2.6
• proof
• Definition 2.7
• Example 2.8