Revisiting the Toda-Brumer-Duff criterion for order-chaos transition in dynamical systems
Year: 2024
Authors: Sattin F., Salasnich L.
Autors Affiliation: Univ Padua, INFN, ENEA, Consorzio RFX,CNR,Acciaierie Venete SpA, Corso Stati Uniti 4, I-35127 Padua, Italy; Univ Padua, Dipartimento Fis & Astron Galileo Galilei & QTech, Via Marzolo 8, I-35131 Padua, Italy; Ist Nazl Fis Nucl, Sez Padova, Via Marzolo 8, I-35131 Padua, Italy; CNR, Unita Sesto Fiorentino, Ist Nazl Ott, via Carrara 1, I-50019 Florence, Italy.
Abstract: The 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.
Journal/Review: PHYSICA D-NONLINEAR PHENOMENA
Volume: 460 Pages from: 134067-1 to: 134067-6
More Information: Barbara Momo read carefully a draft of this work, making several useful suggestions. F.S. acknowledges fruitful discussions with D.F. Escande and I. Predebon. The authors wish to acknowledge two anony-mous referees whose comments greatly helped improve the paper. L.S. is partially supported by the European Quantum Flagship Project PASQuanS 2, by the European Union-NextGenerationEU founds: National Center for HPC, Big Data and Quantum Computing [Project No. CN00000013, CN1 Spoke 10: Quantum Computing] and PRIN Project Quantum Atomic Mixtures: Droplets, Topological Structures, and Vorticesof Ministero Universita e Ricerca, by the BIRD Project Ultracold atoms in curved geometriesof the University of Padova, Italy, by Iniziativa Specifica Quantumof INFN, Italy, and by the Project (Dipartimenti di Eccellenza) Quantum Frontiersof Ministero Universita e Ricerca.KeyWords: Toda-Brumer-Duff criterion; Hamiltonian systems; Chaos; Integrability; Tangent dynamicsDOI: 10.1016/j.physd.2024.134067Connecting to view paper tab on IsiWeb: Click here