Journal Article

Infinite towers in the graphs of many dynamical systems

Yorke, James A., De Leo, Roberto

2021

Overview

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 chain-recurrent 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 uncountably many parameter values for which there are infinitely many nodes, and infinitely many of the nodes N 1 , N 2 , N 3 , … , N ∞ 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 ∞ is an attractor.

• Publisher: Springer Netherlands,Springer Nature B.V
• Subject: Attractors (mathematics) / Automotive Engineering / Bifurcations / Chains / Chaos theory / Classical Mechanics / Control / Dynamical Systems / Engineering / Lorenz system / Mechanical Engineering / Nodes / Orbits / Original Paper / Parameters / Saddles / Trajectory control / Vibration