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Double Hopf Bifurcation with Huygens Symmetry

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Title: Double Hopf Bifurcation with Huygens Symmetry
Author: Kitanov, Petko M.; Langford, William F.; Willms, Allan R.
Abstract: Double Hopf bifurcations have been studied prior to this work in the generic nonresonant case and in certain strongly resonant cases, including 1:1 resonance. In this paper, the case of symmetrically coupled identical oscillators, motivated by the classic problem of synchronization of Huygens' clocks, is studied using the codimension-three Elphick-Huygens equivariant normal form presented here. The focus is on the effect that the Huygens symmetry assumption has on the dynamic behavior of the system. Periodic solutions include the classical in-phase and anti-phase normal modes that are forced by the symmetry, as well as pairs of mixed mode phase-locked periodic solutions. The escapement paradox is explained. A theorem based on topological degree theory establishes the existence of quasiperiodic solutions in an invariant 3-torus that resembles a 2-torus slightly thickened to a solid toroidal shell, with the two principal radii of the 2-torus slowly modulated in time; that is, a \emph{toroidal breather}. Secondary bifurcations from the in-phase and anti-phase normal modes are explored, of codimension-one and -two, and it is shown that an Arnold tongue plays a fundamental role in the determination of whether secondary bifurcation gives birth to phase-locked periodic solutions or to quasiperiodic solutions. Detailed numerical analysis, using Matlab, extends the local bifurcation analysis to a more global picture that includes coexistence of multiple stable solutions and a "swallowtail" bifurcation of periodic solutions.
URI: http://hdl.handle.net/10214/6594
Date: 2013-01-24
Terms of Use: All items in the Atrium are protected by copyright with all rights reserved unless otherwise indicated.
Citation: SIAM J. App. Dyn. Syst. 12 (1) (2013) 126-174.


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