Invariant surfaces of variable stability

This paper presents a brief description of the theory of invariant manifolds of variable stability in the context of their connection with the theory of solutions that are bounded on the whole axis. This approach allows various generalizations both to the case of increasing of the dimension of the invariant manifolds and to the case of multiple change of their stability. The sufficient conditions for the existence of an invariant manifold of variable stability are revealed. The continuity condition for the invariant manifold yields the analytic representation of the gluing function. The theoretical developments are illustrated by several examples. (paper)