Unstable and Stable Manifolds

Abstract

Poincare called the periodic orbits of maps the soul of the dynamics. The study of the stable and unstable manifolds of these orbits does indeed reveal a great deal about the dynamics. The unstable manifold of a fixed point p of a map F may be defined as the set of points q(0) that have a backward orbit coming from p, that is, a sequence of points q(i) with i = −1, −2, ∦, so that F(q(i−1)) = q(i) for which q(i) → p as i → ∞. The stable manifold of a periodic orbit may be defined for invertible processes as the unstable manifold for the inverted system. If a point p is a saddle fixed point of a map in the plane, then the stable and unstable manifolds are both curves that pass through p. The routines described below are not restricted to planar systems, but the manifold being computed must be a curve.