Monday, June 17, 2013

1306.3327 (Serhiy Yanchuk et al.)

Relative equailibria and relative periodic solutions in systems with
time-delay and $S^{1}$ symmetry

Serhiy Yanchuk, Jan Sieber
We study properties of basic solutions in systems with dime delays and $S^1$-symmetry. Such basic solutions are relative equilibria (CW solutions) and relative periodic solutions (MW solutions). It follows from the previous theory that the number of CW solutions grows generically linearly with time delay $\tau$. Here we show, in particular, that the number of relative periodic solutions grows generically as $\tau^2$ when delay increases. Thus, in such systems, the relative periodic solutions are more abundant than relative equilibria. The results are directly applicable to, e.g., Lang-Kobayashi model for the lasers with delayed feedback. We also study stability properties of the solutions for large delays.
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