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

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.

Figures (5)

• Figure 1: $|({\bf w}_R)_1|$versus $\Lambda_R$ plots, at different energies for the Hènon-Heiles system. For each plot, a total of 3000 points were randomly sampled within the allowed phase space. The two plots with lower energy feature a cloud of points which aligns fairly close to the horizontal straight line $|({\bf w}_R)_1| = 1$, consistent with our hypothesis concerning regular motion. Conversely, the two plots at the higher energies, in the chaotic domain, feature an appreciable scatter of the points throughout the $|({\bf w}_R)_1| < 1$ region.
• Figure 2: The horizontal axis reports the real part of the complex-valued eigenvalue, $\Lambda_{I,r}$, when it is positive. The vertical axis reports the corresponding imaginary part, $\Lambda_{I,i}$. The straight line is $\Lambda_{I,i} = - \Lambda_{I,r}$. The data refer to the cases with the smallest and the largest energy out of those considered in Fig. \ref{['fig:1']}. All the points lie in the region $|\Lambda_{I,i}| > \Lambda_{I,r}$, supporting our view that the effective stretching of these trajectories is negligible.
• Figure 3: The figure is the counterpart of Fig. \ref{['fig:1']} for the anti-Hènon-Heiles system. The energy is $E = 0.05$, i.e, above the threshold for the onset of chaos according to the TBD criterion. As expected on the basis of our hypothesis, all points lie fairly close to the $|({\bf w}_R)_1| = 1$ curve.
• Figure 4: Poincarè plot for a trajectory with the potential (\ref{['Eq:11']}), obtained taking the intersections with the plane $y = 0$ and ${\dot y} <0$. The particle energy is $E = 0.5$ and $a = 2\times 10^{-4}$.
• Figure 5: The figure is the counterpart of Figs. \ref{['fig:1']},\ref{['fig:3']}. The energy is $E = 0.5$, and $a = 2\times 10^{-4}$. The points are scattered throughout the whole available interval, consistently with our guess about chaotic dynamics.