  We consider the problem of asymptotic convergence to invariant sets in
interconnected nonlinear dynamic systems. Standard approaches often require
that the invariant sets be uniformly attracting. e.g. stable in the Lyapunov
sense. This, however, is neither a necessary requirement, nor is it always
useful. Systems may, for instance, be inherently unstable (e.g. intermittent,
itinerant, meta-stable) or the problem statement may include requirements that
cannot be satisfied with stable solutions. This is often the case in general
optimization problems and in nonlinear parameter identification or adaptation.
Conventional techniques for these cases rely either on detailed knowledge of
the system's vector-fields or require boundeness of its states. The presently
proposed method relies only on estimates of the input-output maps and
steady-state characteristics. The method requires the possibility of
representing the system as an interconnection of a stable, contracting, and an
unstable, exploratory part. We illustrate with examples how the method can be
applied to problems of analyzing the asymptotic behavior of locally unstable
systems as well as to problems of parameter identification and adaptation in
the presence of nonlinear parametrizations. The relation of our results to
conventional small-gain theorems is discussed.
