Streams and Graphs of Dynamical Systems

TL;DR

The paper develops a unified, graph-based framework for qualitative dynamics by introducing streams—closed quasi-orders that extend the orbit relation—and by associating graphs that encode downstream relations among invariant sets. It generalizes Smale's spectral decomposition and Auslander's recurrence through the prolongational relation ${{\cal P}}_F$ and its nodes, and then extends these ideas to streams, which encompass chain-recurrence, strong-chain-recurrence, and their interplays with Lyapunov functions. Key contributions include connectivity results for prolongational graphs under compact dynamics, a detailed treatment of T-unimodal and logistic maps, and a demonstration that chain-based recurrence notions arise naturally as recurrent points of appropriate streams. The framework offers robust tools for analyzing invariant structures and their interrelations under perturbations, providing both theoretical insight and practical guidance for qualitative modeling of dynamical systems.

Abstract

While studying gradient dynamical systems (DSs), Morse introduced the idea of encoding the qualitative behavior of a DS into a graph. Smale later refined Morse's idea and extended it to Axiom-A diffeomorphisms on manifolds. In Smale's vision, nodes are indecomposable closed invariant subsets of the non-wandering set with a dense orbit and there is an edge from node N to node M if the unstable manifold of N intersects the stable manifold of M. Since then, the decomposition of the non-wandering set was studied in many other settings, while the edges component of Smale's construction has been often overlooked. In the same years, more sophisticated generalizations of the non-wandering set were elaborated first by Auslander in 60s, by Conley in 70s and later by Easton and other authors. In our language, each of these generalizations involves the introduction of a closed and transitive extension of the non-wandering relation, that is closed but not transitive. In the present article, we develop a theory that generalizes at the same time both these lines of research. We study the general properties of closed transitive relations ("streams") containing the space of orbits of a discrete- or continuous-time semi-flow and we argue that these relations play a central role in the qualitative study of DSs. All most studied concepts of recurrence currently in literature can be defined in terms of our streams. Finally, we show how to associate to each stream a graph encoding its qualitative properties. The current revision fixes some proof, adds some missing one and adds some example and clarification.

Figures (11)

• Figure 1: Two topologically inequivalent homoclinic saddle points on a surface. The prolongational relation for the corresponding flows is discussed in Example \ref{['ex: saddle point 1']}.
• 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: Two examples of bidimensional dynamical systems and their prolongational graphs. (Above) Phase-portrait of the physical pendulum on ${\mathbb{R}}$. In this case there are no attractors. Each periodic orbit (painted in blue) is a non-wandering node. Besides these nodes, the only other nodes are the boundaries $C_i$ of each disc foliated by concentric periodic orbits. For instance, $C_0$ is the disjoint union of the heteroclinic orbits $h_0$ and $g_0$ and the two saddle fixed points $s_0$ and $s_1$. Notice that this system, being Hamiltonian, has no attractor nor repellor and has a set of nodes with the power of continuum. The loops in the graph are due to the fact that each red node intersects its nearest neighborhoods. (Below) After replacing the periodic orbits with spirals spiralling outward, the fixed points $c_i$ become repelling and each interval of nodes above gets replaced by a repellor/attractor pair.
• Figure 5: A semi-flow with "purely non-wandering" points. (Left) In the picture we show the dynamics of the semi-flow $F$ in Example \ref{['ex:Klein']}, where $NW_F$ contains points that do not belong to the closure of the set of all limit points of $F$. The phase space $X$ is the Klein bottle, the arrows next to the sides show the way the opposite sides are glued. (Right) prolongational graph of $F$. The red saddle is the node of all points on the horizontal closed segment between $p_1$ and $p_2$. Each of the green fixed points is a node in itself. The point $c$ at the center of the green segment has a self-edge, each other point $x$ on it has an edge going to the point symmetric with respect to $c$ -- only two pairs of these edges are plotted.

Theorems & Definitions (208)

• Definition 2.0.1
• Definition 2.0.2
• Definition 2.1.1
• Proposition 2.1.2
• Proposition 2.1.3
• proof
• Definition 2.1.4
• Proposition 2.1.5
• Lemma 2.1.6
• proof