Study of a dynamical system with a strange attractor and invariant tori

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journal

1 Scopus Citations
View graph of relations


  • Antonio Algaba
  • Manuel Merino
  • Bo-Wei Qin
  • Alejandro J. Rodríguez-Luis

Related Research Unit(s)


Original languageEnglish
Pages (from-to)1441-1449
Journal / PublicationPhysics Letters, Section A: General, Atomic and Solid State Physics
Issue number13
Online published5 Feb 2019
Publication statusPublished - 24 Apr 2019


A simple three-dimensional time-reversible system of ODEs with quadratic nonlinearities is considered in a recent paper by Sprott (2014). The author finds in this system, that has no equilibria, the coexistence of a strange attractor and invariant tori. The goal of this letter is to justify theoretically the existence of infinite invariant tori and chaotic attractors. For this purpose we embed the original system in a one-parameter family of reversible systems. This allows to demonstrate the presence of a Hopf-zero bifurcation that implies the birth of an elliptic periodic orbit. Thus, the application of the KAM theory guarantees the existence of an extremely complex dynamics with periodic, quasiperiodic and chaotic motions. Our theoretical study is complemented with some numerical results. Several bifurcation diagrams make clear the rich dynamics organized around a so-called noose bifurcation where, among other scenarios, cascades of period-doubling bifurcations also originate chaotic attractors. Moreover, a cross section and other numerical simulations are also presented to illustrate the KAM dynamics exhibited by this system.

Research Area(s)

  • Hopf-zero, Invariant tori, KAM theory, Reversible system, Strange attractor

Citation Format(s)