A newton method for locating invariant tori of maps

Harry Dankowicz, Gunjan Thakur

Research output: Contribution to journalArticlepeer-review


This paper presents an iterative method for locating invariant tori of maps. Specifically, through the introduction of a shift function on the torus corresponding to the projection of the dynamics on the torus onto toral parameters, a discrete system of equations may be formulated whose solution approximates the spatial location of the torus. As invariant tori of continuous flows may be considered invariant under the application of the flow for a fixed time, or may correspond to invariant tori (of one less dimension) of suitably introduced Poincaré maps, the methodology applies to discrete and continuous dynamical systems alike. Moreover, the insensitivity of the method to the local stability characteristics of the invariant torus as well as to the precise nature of the flow on the torus imply, for example, that the method can be employed for the continuation of unstable invariant tori on which the dynamics are attracted to a periodic trajectory as long as the torus is sufficiently smooth. The proposed methodology as well as reduced formulations that result from a priori knowledge about the invariant torus are illustrated through some sample dynamical systems.

Original languageEnglish (US)
Pages (from-to)1491-1503
Number of pages13
JournalInternational Journal of Bifurcation and Chaos
Issue number5
StatePublished - May 2006


  • Invariant tori
  • Newton's method
  • Parameter continuation

ASJC Scopus subject areas

  • Modeling and Simulation
  • Engineering (miscellaneous)
  • General
  • Applied Mathematics


Dive into the research topics of 'A newton method for locating invariant tori of maps'. Together they form a unique fingerprint.

Cite this