Bifurcations of Periodic Solutions of Functional Differential Equations with Spatio-Temporal Symmetries
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We study bifurcations of periodic solutions with spatio-temporal symmetries of functional differential equations (FDEs). The two main results are: (1) a centre manifold reduction around a periodic solution of FDEs with spatio-temporal symmetries, and (2) symmetry-breaking bifurcations for symmetric rings of delay-coupled lasers. For the case of ODEs, symmetry-breaking bifurcations from periodic solutions has already been studied. We extend this result to the case of symmetric FDEs using a Centre Manifold Theorem for symmetric FDEs which reduces FDEs into ODEs on an integral manifold around a periodic solution. We show that the integral manifold is invariant under the spatio-temporal symmetries which guarantees that the symmetry structure of the system of FDEs is preserved by this reduction. We also consider a problem in rings of delay-coupled lasers modeled using the Lang-Kobayashi rate equations. We classify the symmetry of bifurcating branches of solutions from steady-state and Hopf bifurcations that occur in 3-laser systems. This involves finding isotropy subgroups of the symmetry group of the system, and then using the Equivariant Branching Lemma and the Equivariant Hopf Theorem. We then utilize this result to find the bifurcating branches of solutions in DDE-Biftool. Symmetry often causes eigenvalues to have multiplicity, and in some cases, this could lead DDE-Biftool to incorrectly predict the bifurcation points. We address this issue by developing a method of finding bifurcation points which can be used for the general case of n-laser systems with unidirectional and bidirectional coupling.