| Abstract: |
| The analysis of Hopf bifurcation of periodic solutions from an equilibrium state (both in non-equivariant and equivariant setting) has been done by many authors using different techniques. A natural counterpart of an equilibrium state (resp. periodic solution) in dynamical systems with continuous symmetries
is a relative equilibrium, i.e. an equilibrium modulo the group action (resp. relative periodic solution). M. Krupa proposed a method for analysis of the bifurcation of relative periodic solutions from a relative equilibrium for generic systems of smooth ODEs based on normal slice and center manifold reductions. Although, in principal, Krupa`s method can be applied to generic systems of smooth equivariant FDEs, we are not aware of any published work in this direction. On the other hand, if a system of ODEs/FDEs exhibits a lack of smoothness (systems with hysteresis), or lack of genericity (multiple/resonant eigenvalues crossing the imaginary axis non-transversally), absence of a flow (mixed FDEs), then the center manifold reduction is either impossible or meets serious difficulties. In my talk, I will show how to adapt the equivariant degree method for analysis of Hopf bifurcation of relative periodic solutions to FDEs with a lack of smoothness/genericity. As an application, a delay differential model of coupled identical passively mode-locked semiconductor lasers will be considered. |
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