Geodesic
Center Manifold Reduction of the Hopf-Hopf Bifurcation in a Time Delay System
Christoffer Heckman1 ; Jakob Kotas2 ; Richard Rand3
1 Field of Theoretical & Applied Mechanics; Cornell University; Ithaca, NY USA
2 Department of Applied Mathematics; University of Washington; Seattle, WA USA
3 Department of Mathematics and Department of Mechanical & Aerospace Engineering; Cornell University; Ithaca, NY USA
ESAIM. Proceedings, Tome 39 (2013), pp. 57-65 Cet article a éte moissonné depuis la source EDP Sciences

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In this work, a differential delay equation (DDE) with a cubic nonlinearity is analyzed as two parameters are varied by means of a center manifold reduction. This reduction is applied directly to the case where the system undergoes a Hopf-Hopf bifurcation. This procedure replaces the original DDE with four first-order ODEs, an approximation valid in the neighborhood of the Hopf-Hopf bifurcation. Analysis of the resulting ODEs shows that two separate periodic motions (limit cycles) and an additional quasiperiodic motion are born out of the Hopf-Hopf bifurcation. The analytical results are shown to agree with numerical results obtained by applying the continuation software package DDE-BIFTOOL to the original DDE. This system has analogues in coupled microbubble oscillators.
DOI : 10.1051/proc/201339008

Christoffer Heckman 1 ; Jakob Kotas 2 ; Richard Rand 3

1 Field of Theoretical & Applied Mechanics; Cornell University; Ithaca, NY USA
2 Department of Applied Mathematics; University of Washington; Seattle, WA USA
3 Department of Mathematics and Department of Mechanical & Aerospace Engineering; Cornell University; Ithaca, NY USA 
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     author = {Christoffer Heckman and Jakob Kotas and Richard Rand},
     title = {Center {Manifold} {Reduction} of the {Hopf-Hopf} {Bifurcation} in a {Time} {Delay} {System}},
     journal = {ESAIM. Proceedings},
     pages = {57--65},
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     doi = {10.1051/proc/201339008},
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     url = {http://geodesic.mathdoc.fr/articles/10.1051/proc/201339008/}
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Christoffer Heckman; Jakob Kotas; Richard Rand. Center Manifold Reduction of the Hopf-Hopf Bifurcation in a Time Delay System. ESAIM. Proceedings, Tome 39 (2013), pp. 57-65. doi: 10.1051/proc/201339008

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