Necessity of multidirectional stretching for weakly dissipative chaos

Determine whether multidirectional stretching is a necessary condition for the existence of weakly dissipative chaotic attractors in three-dimensional continuous-time dynamical systems, specifically including the Lorenz–84 system. Establish rigorously whether weakly dissipative chaos can occur in the absence of multidirectional stretching in the Poincaré-section dynamics, or whether such multidirectional stretching is required to generate and maintain the characteristic thickness of these attractors.

Background

The paper analyzes the structure of the Lorenz–84 chaotic attractor, a prototypical weakly dissipative system whose attractor is thick and not adequately described by standard topological methods devised for strongly dissipative attractors. Using a color tracer mapping of the Poincaré section, the authors identify a prominent multidirectional stretching mechanism shaping the flow.

This multidirectional stretching is argued to be characteristic of weakly dissipative chaos and potentially essential to producing the observed thickness of the attractor. The authors explicitly formulate a conjecture that such multidirectional stretching is a necessary condition for weakly dissipative chaos, motivating a precise theoretical and/or computational test of necessity across three-dimensional flows, including the Lorenz–84 system.