Euler-Kirchhoff filaments are solutions of the static Kirchhoff equations for elastic rods with circular cross-sections. These equations are known to be formally equivalent to the Euler equations for spinning tops. This equivalence is used to provide a classification of the different shapes a filament can assume. Explicit formulas for the different possible configurations and specific results for interesting particular cases are given. In particular, conditions for which the filament has points of self-intersection, self-tangency, vanishing curvature or when it is closed or localized in space are provided. The average properties of generic filaments are also studied. They are shown to be equivalent to helical filaments on long length scales.

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International audienceWe unveil the bifurcations underlying polarization switching and injection locking in a vertical-cavity surface-emitting laser subject to optical injection. A Hopf bifurcation, not reported for conventional edgeemitting lasers, delimits the injection locking region and influences the polarization switching conditions. We furthermore theoretically show and experimentally observe periodic dynamics at the relaxation oscillation frequency in the noninjected mode together with wave-mixing dynamics in the injected mode. These dynamics precede the polarization switching leading to injection locking and are attributed to a torus bifurcation arising on a two-polarization mode solution

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