Center Manifold Reduction of the Hopf-Hopf Bifurcation in a Time Delay System
1 Field of Theoretical & Applied
Mechanics; Cornell University ; Ithaca, NY
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
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.
© EDP Sciences, SMAI 2013