The purpose of this work is a systematic study of symmetric vortices for the Ginzburg-Landau model of superconductivity along a cylinder, with applied magnetic ÿeld parallel to its axis. The Ginzburg-Landau constant Ä of the material and the degree d of the vortex are ÿxed.
For any given parameters r (the radius of the cylinder) and h (the intensity of the applied magnetic ÿeld), one can ÿnd a symmetric vortex ( ; A) which satisÿes the boundary conditions |@B r =e id and curl(A) |@B r =he3. It is then shown that symmetric vortices form a family depending continuously on two real parameters and c which describe the behaviour at the center of the vortex. As the boundary conditions depend smoothly on those parameters, one can distinguish two main connected domains of vortices: the ÿrst one, deÿned by the boundary conditions |@B r =e id and |rA(r)| |@B r ¡ d, is a zone of stability where | | remains increasing; the second one, deÿned by the boundary conditions |@B r = e id and |rA(r)| |@B r ¿ d, is a zone where some instability can occur. Attention is focused on the smooth branch of vortices which separates those two domains: it is indexed by the parameter running from some L ¿ 0 to + ∞, and the limit as decreases to L corresponds to a Berger and Chen type vortex.