### Abstract

Self-consistent solutions of the nonlinear Ginzburg-Landau (GL) equations are investigated numerically for a superconducting (SC) cylinder, placed in an axial magnetic field, with a single vortex on the axis (m=1). Two modes, which show the original state of the cylinder, SC or normal (s 0 andn 0), are studied. The field increase (FI) and the field decrease (FD) regimes are studied. The critical fields destroying the SC state withm=1 are found in both regimes. It is shown that in a cylinder of radiusR and GL-parameter ϰ, there exist a number of solutions depending only on the radial co-ordinater corresponding to different states such as M,e, d, p,i, n, $$\bar n$$ ,n *, and the state diagram on the plane of the variables (ϰ,R) is described. The critical fields corresponding to intrastate transitions and the onset of hysteresis are obtained. The critical fieldH 0(R) dividing the paramagnetic and diamagnetic states of the cylinder withm=1 is determined. The limiting fields of supercooling or superheating of the normal state at which the restoration of the SC state occurs are established. It is shown, that (in both casesm=1,0) there exist two critical parameters, $$\kappa _0 = {1 \mathord{\left/ {\vphantom {1 {\sqrt 2 = 0.707}}} \right. \kern-\nulldelimiterspace} {\sqrt 2 = 0.707}}$$ and $$\kappa _0 = 0.93$$ , which divide bulk SC into three groups (with $$\kappa< \kappa _0 ,\kappa _0 \leqslant \kappa \leqslant \kappa _c $$ and $$\kappa > \kappa _c $$ ), in accordance with the behavior in a magnetic field. The parameters $$\kappa _0 $$ and $$\kappa _c $$ mark the boundary for the existence of a supercooled normal $$\bar n$$ -state in FD-regime and a superheated SC M-state in FI-regime respectively. It is shown, that the value $$\kappa _* = 0.417$$ , which was claimed in a number of papers as related to type-I superconductors, is illusory.