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Title: Three-dimensional tori and Arnold tongues

This study analyzes an Arnold resonance web, which includes complicated quasi-periodic bifurcations, by conducting a Lyapunov analysis for a coupled delayed logistic map. The map can exhibit a two-dimensional invariant torus (IT), which corresponds to a three-dimensional torus in vector fields. Numerous one-dimensional invariant closed curves (ICCs), which correspond to two-dimensional tori in vector fields, exist in a very complicated but reasonable manner inside an IT-generating region. Periodic solutions emerge at the intersections of two different thin ICC-generating regions, which we call ICC-Arnold tongues, because all three independent-frequency components of the IT become rational at the intersections. Additionally, we observe a significant bifurcation structure where conventional Arnold tongues transit to ICC-Arnold tongues through a Neimark-Sacker bifurcation in the neighborhood of a quasi-periodic Hopf bifurcation (or a quasi-periodic Neimark-Sacker bifurcation) boundary.
Authors:
 [1] ;  [2] ;  [3] ;  [4]
  1. Department of Mechanical and Intelligent Engineering, Utsunomiya University, Utsunomiya-shi 321-8585 (Japan)
  2. Organization for the Strategic Coordination of Research and Intellectual Property, Meiji University, Kawasaki-shi 214-8571 (Japan)
  3. Department of Electronics and Bioinformatics, Meiji University, Kawasaki-shi 214-8571 (Japan)
  4. Institute of Industrial Science, the University of Tokyo, Meguro-ku 153-8505 (Japan)
Publication Date:
OSTI Identifier:
22251013
Resource Type:
Journal Article
Resource Relation:
Journal Name: Chaos (Woodbury, N. Y.); Journal Volume: 24; Journal Issue: 1; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; BIFURCATION; DIAGRAMS; LYAPUNOV METHOD; MAPS; MATHEMATICAL SOLUTIONS; ONE-DIMENSIONAL CALCULATIONS; PERIODICITY; THREE-DIMENSIONAL CALCULATIONS; TORI; TWO-DIMENSIONAL CALCULATIONS; VECTOR FIELDS