A GEOMETRIC APPROACH TO REGULAR PERTURBATION-THEORY WITH AN APPLICATION TO HYDRODYNAMICS

The Lyapunov-Schmidt reduction technique is used to prove a persistenc e theorem for fixed points of a parameterized family of maps. This the orem is specialized to give a method for detecting the existence of pe rsistent periodic solutions of perturbed systems of differential equat ions. In turn, this specialization is applied to prove the existence o f many hyperbolic periodic solutions of a steady state solution of Eul er's hydrodynamic partial differential equations. Incidentally, using recent results of S. Friedlander and M. M. Vishik, the existence of hy perbolic periodic orbits implies the steady state solutions of the Eul erian partial differential equation are hydrodynamically unstable. In addition, a class of the steady state solutions of Euler's equations a re shown to exhibit chaos.