Bifurcation and Catastrophe††AMS (MOS) 1970 SUBJECT CLASSIFICATION: 58F99.‡‡Research supported by National Science Foundation (GP-7952X2) at the Institute for Advanced Study, Princeton, New Jersey.

This chapter presents a diffeomorphism F : R n → R n , F ( x ) = Ax + f ( x ), where f vanishes together with its first derivative at the origin. This chapter discusses exponential rate conditions for dynamical system. It presents a typical application of exponential rate conditions. A perturbation problem for invariant manifolds is stated in terms of exponential rates, and the study of these rates is the main step in solving the problem.