  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.
