Integral manifolds of slow adaptation

Conditions are given under which an integral manifold ("slow manifold") exists for a nonlinear system representing a broad class of adaptive algorithms. The parameter update equation restricted to the manifold is an exact description of the slow adaptation process which can be approximately analyzed by averaging. Stability properties of the slow motion in the manifold are extended off the manifold by a set of conditions under which the manifold is shown to be attractive. This two-time-scale analysis provides a geometrical visualization of instability phenomena observed in adaptive systems and generalizes earlier local stability results.