Journal Article

Bifurcation scenarios for a 3D torus and torus-doubling

Naohikio Inaba, Munehisa Sekikawa, Yoshimasa Shinotsuka, Kyohei Kamiyama, Ken’ichi Fujimoto, Tetsuya Yoshinaga and Tetsuro Endo

in Progress of Theoretical and Experimental Physics

Published on behalf of The Physical Society of Japan

Volume 2014, issue 2
Published online February 2014 | e-ISSN: 2050-3911 | DOI:

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Bifurcation transitions between a 1D invariant closed curve (ICC), corresponding to a 2D torus in vector fields, and a 2D invariant torus (IT), corresponding to a 3D torus in vector fields, have been the subjects of intensive research in recent years. An existing hypothesis involves the bifurcation boundary between a region generating an ICC and a region generating an IT. It asserts that an IT would be generated from a stable fixed point as a consequence of two Hopf (or two Neimark–Sacker) bifurcations. We assume that this hypothesis may puzzle many researchers because it is difficult to assess its validity, although it seems to be a reasonable bifurcation scenario at first glance. To verify this hypothesis, we conduct a detailed Lyapunov analysis for a coupled delayed logistic map that can generate an IT, and indicate that this hypothesis does not hold according to numerical results. Furthermore, we show that a saddle-node bifurcation of unstable periodic points does not coincide with the bifurcation boundary between an ICC and an IT. In addition, the bifurcation boundaries of torus doubling do not coincide with a period-doubling bifurcation of unstable periodic points. To conclude, torus bifurcations have no relation with the bifurcations of unstable periodic points. Additionally, we exactly derive a quasi-periodic Hopf bifurcation boundary introducing a double Poincaré map.

Keywords: A30; A34

Journal Article.  3609 words.  Illustrated.

Subjects: Classical Mechanics