| Abstract: |
| Second order differential equation $v``+g(v)=0$ satisfies the property: if $v(t)$ is a solution, then so is $v(-t)$. In the context of first order (time evolutionary) ODEs, this property, which is called {\em time-reversibility}, means that the system respects the involutionary symmetry connecting time-reversed evolution and forward-time evolution. Local bifurcations of periodic solutions to parametrized reversible systems of ODEs have been studied intensively by Buzzi, Lamb, Vanderbauwhede et al.
In FDE systems, {\em time-reversal symmetry} involves ``information traveling back in time``, which is paradoxical in common sense. Therefore, instead of time-reversible FDEs, one may consider (networks of) {\em space-reversible} FDEs. Motivating physical examples are related to stationary solutions to PDEs with non-local interaction: {\em mixed delay differential equations} (MDDEs) and {\em integro-differential equations} (IDEs).
In this talk, a $(\Gamma\times O(2))$-equivariant degree based method is proposed to study bifurcation of $2\pi$-periodic solutions in symmetric networks of reversible FDEs (the finite group $\Gamma$ reflects symmetries of coupling while $O(2)$ is associated with periodicity and reversibility). Such a bifurcation occurs when eigenvalues of linearization move along the imaginary axis (without change of stability of the trivial solution and possibly without $1:k$ resonance). In the case of $S_4$-symmetric networks of MDDEs and IDEs, we present exact computations of full equivariant bifurcation invariants. |
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