Infinite towers in the graphs of many dynamical systems

Chaotic attractors, chaotic saddles and periodic orbits are examples of chain-recurrent sets. Using arbitrary small controls, a trajectory starting from any point in a chain-recurrent set can be steered to any other in that set. The qualitative behavior of a dynamical system can be encapsulated in a graph. Its nodes are chainrecurrent sets. There is an edge from node A to node B if, using arbitrary small controls, a trajectory starting from any point of A can be steered to any point of B. We discuss physical systems that have infinitely many disjoint coexisting nodes. Such infinite collections can occur for many carefully chosen parameter values. The logistic map is such a system, as we show in a rigorous companion paper. To illustrate these very common phenomena, we compare the Lorenz system and the logistic map and we show how extremely similar their graph bifurcation diagrams are in some parameter ranges. Typically, bifurcation diagrams show how attractors change as a parameter is varied. We call ours graph bifurcation diagrams to reflect that not only attractors but also unstable periodic orbits and chaotic saddles can be shown. Only the most prominent ones can be shown. We argue that, as a parameter is varied in the Lorenz system, there are uncountablymany parameter values for which there are infinitely many nodes, and infinitely many of the nodes N-1, N-2, N-3, ... , N-infinity can be selected so that the graph has an edge from each node to every node with a node with a higher number. The final node N-infinity is an attractor.

Automated summary

Chaotic attractors, chaotic saddles, and periodic orbits are examples of chain-recurrent sets. The qualitative behavior of a dynamical system can be encapsulated in a graph, with nodes representing chain-recurrent sets. Physical systems can have infinitely many disjoint coexisting nodes, as seen in systems like the logistic map. Comparing the Lorenz system and the logistic map shows how similar their graph bifurcation diagrams are in certain parameter ranges.