2025/03/04 by De Leo, Roberto, Yorke, James A.
#Chaotic Dynamics (nlin.CD) #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2503.02262
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 FG of F to G and that, in case of Conley's chains stream of F, the qualitative behavior of F and FG 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.