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.
Department of Mechanical and Intelligent Engineering, Utsunomiya University, Utsunomiya-shi 321-8585, Japan.
Munehisa
M
Sekikawa
Organization for the Strategic Coordination of Research and Intellectual Property, Meiji University, Kawasaki-shi 214-8571, Japan.
Naohiko
N
Inaba
Department of Electronics and Bioinformatics, Meiji University, Kawasaki-shi 214-8571, Japan.
Kyohei
K
Kamiyama
Institute of Industrial Science, the University of Tokyo, Meguro-ku 153-8505, Japan.
Kazuyuki
K
Aihara