Revisiting the Toda-Brumer-Duff criterion for order-chaos transition in dynamical systems

TL;DR

This paper critically revisits the Toda-Brumer-Duff criterion for chaos detection in dynamical systems, identifies its limitations, and proposes an improved version that correctly distinguishes between regular and chaotic behaviors.

Contribution

The authors analyze the shortcomings of the TBD criterion and develop an amended version that addresses key issues, improving its accuracy in chaos assessment.

Findings

The original TBD criterion can produce false positives.

Incorporating energy constraints improves the criterion's reliability.

The amended criterion correctly classifies dynamics in reference cases.

Abstract

TThe Toda-Brumer-Duff (TBD) is an analytical criterion for estimating the local exponential rate of divergence between nearby trajectories in dynamical systems, and it is employed as a test for assessing the existence of chaos therein. It is fairly simple, intuitive, and works well in several situations, hence gained quite a wide popularity, yet it is known to be not rigorous since predicts ``false positives'', i.e., flags as chaotic systems that are instead regular. We revisit here the TBD criterion in order to understand the causes of its failures, and pinpoint that the problem with it is due to two reasons: (a) the TBD criterion does not constrain the trajectories to lie on the same energy hypersurface; (b) it does not distinguish between the divergence of trajectories along or perpendicularly to the direction of the flow, the former being irrelevant for assessing the presence of chaos. We show how both points can be incorporated within the TBD framework, yielding an amended criterion which, when applied to some reference cases, interprets correctly the kind of dynamics observed.