A closed orbit of a vector field becomes unstable when one or more of its Floquet multipliers leave the unit disc in the complex plane. Floquet multipliers occur either on the real axis or off the real axis in complex conjugate pairs, so in the simplest cases either one real or two nonreal Floquet multipliers pass across the unit circle. Our aim is to give a detailed description of the orbit structure in the latter case, provided that finitely many non-resonance and non-degeneracy conditions are satisfied. To analyze the orbit structure in this problem we construct invariant manifolds using the methods of Fenichel [2] and Sacker [9]. Then we construct invariant foliations near the invariant manifolds, using results of Fenichel [3]. These invariant foliations reduce our problem to studying maps and flows on low dimensional manifolds where the orbit structures are well known. At this time we give only a rough outline of results. Consider a family of vector fields depending on a real parameter p, with instability developing as p increases through zero. We transform the Poincare map of the closed orbit into a normal form, and isolate one significant coefficient, called p. Four cases must be considered, depending on the sign of p and the sign of Rep, the real part of ,8. Of these, the cases p > 0, Re ,8 > 0 and ,U 0, Rep 0 there is an invariant torus near the closed orbit (see Ref. [9]). The union of the closed orbit, the invariant torus, and the transit orbits joining the closed orbit and the invariant torus is an invariant manifold with invariant boundary, a solid torus. Every half orbit in a neighborhood of the solid torus either leaves a neighborhood of the solid torus or behaves asymptotically like a unique parametrized orbit in either the closed orbit or the two dimensional torus. In addition, there are foliations of codimension