{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,24]],"date-time":"2026-04-24T02:56:34Z","timestamp":1776999394441,"version":"3.51.4"},"reference-count":26,"publisher":"World Scientific Pub Co Pte Lt","issue":"12","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2014,12]]},"abstract":"<jats:p>In continuous-time dynamical systems, a periodic orbit becomes a fixed point on a certain Poincar\u00e9 section. The eigenvalues of the Jacobian matrix at this fixed point determine the local stability of the periodic orbit. Analogously, a quasi-periodic orbit (2-torus) becomes an invariant closed curve (ICC) on a Poincar\u00e9 section. From the Lyapunov exponents of an ICC, we can determine the time average of the exponential divergence rate of the orbit, which corresponds to the eigenvalues of a fixed point. We denote the Lyapunov exponent with the smallest nonzero absolute value as the Dominant Lyapunov Exponent (DLE). A local bifurcation manifests as a crossing or touch of the DLE locus with zero. However, the type of bifurcation cannot be determined from the DLE. To overcome this problem, we define the Dominant Lyapunov Bundle (DLB), which corresponds to the dominant eigenvectors of a fixed point. We prove that the DLB of a 1-torus in a map can be classified into four types: A<jats:sup>+<\/jats:sup>(annulus and orientation preserving), A<jats:sup>-<\/jats:sup>(annulus and orientation reversing), M (M\u00f6bius band), and F (focus). The DLB of a 2-torus in a flow can be classified into three types: A<jats:sup>+<\/jats:sup>\u00d7 A<jats:sup>+<\/jats:sup>, A<jats:sup>-<\/jats:sup>\u00d7 M (equivalently M \u00d7 A<jats:sup>-<\/jats:sup>and M \u00d7 M), and F \u00d7 F. From the results, we conjecture the possible local bifurcations in both cases. For the 1-torus in a map, we conjecture that type A<jats:sup>+<\/jats:sup>and A<jats:sup>-<\/jats:sup>DLBs correspond to a saddle-node and period-doubling bifurcations, respectively, whereas a type M DLB denotes a double-covering bifurcation, and type F relates to a Neimark\u2013Sacker bifurcation. Similarly, for the 2-torus in a flow, we conjecture that type A<jats:sup>+<\/jats:sup>\u00d7 A<jats:sup>+<\/jats:sup>DLBs correspond to saddle-node bifurcations, type A<jats:sup>-<\/jats:sup>\u00d7 M DLBs to double-covering bifurcations, and type F \u00d7 F DLBs to the Neimark\u2013Sacker bifurcations. After introducing the mathematical concepts, we provide a DLB-calculating algorithm and illustrate all of the above bifurcations by examples.<\/jats:p>","DOI":"10.1142\/s0218127414300341","type":"journal-article","created":{"date-parts":[[2015,1,5]],"date-time":"2015-01-05T06:43:47Z","timestamp":1420440227000},"page":"1430034","source":"Crossref","is-referenced-by-count":22,"title":["Classification of Bifurcations of Quasi-Periodic Solutions Using Lyapunov Bundles"],"prefix":"10.1142","volume":"24","author":[{"given":"Kyohei","family":"Kamiyama","sequence":"first","affiliation":[{"name":"Department of Electronics and Bioinformatics, Meiji University, 1-1-1 Higashi-Mita, Tama-ku, Kawasaki-shi, Kanagawa 214-8571, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Motomasa","family":"Komuro","sequence":"additional","affiliation":[{"name":"Center for Fundamental Education, Teikyo University of Science, 2525 Yatsusawa, Uenohara-shi, Yamanashi 409-0193, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tetsuro","family":"Endo","sequence":"additional","affiliation":[{"name":"Department of Electronics and Bioinformatics, Meiji University, 1-1-1 Higashi-Mita, Tama-ku, Kawasaki-shi, Kanagawa 214-8571, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kazuyuki","family":"Aihara","sequence":"additional","affiliation":[{"name":"Institute of Industrial Science, The University of Tokyo, Komaba, Meguro-ku, Tokyo 153-8505, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2015,1,4]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1142\/S021812749500017X"},{"key":"rf2","first-page":"1250289-1","volume":"22","author":"Banerjee S.","year":"2012","journal-title":"Int. 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