Three-dimensional tori and Arnold tongues
- Department of Mechanical and Intelligent Engineering, Utsunomiya University, Utsunomiya-shi 321-8585 (Japan)
- Organization for the Strategic Coordination of Research and Intellectual Property, Meiji University, Kawasaki-shi 214-8571 (Japan)
- Department of Electronics and Bioinformatics, Meiji University, Kawasaki-shi 214-8571 (Japan)
- Institute of Industrial Science, the University of Tokyo, Meguro-ku 153-8505 (Japan)
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.
- OSTI ID:
- 22251013
- Journal Information:
- Chaos (Woodbury, N. Y.), Vol. 24, Issue 1; Other Information: (c) 2014 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); ISSN 1054-1500
- Country of Publication:
- United States
- Language:
- English
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