C. Chicone, A GEOMETRIC APPROACH TO REGULAR PERTURBATION-THEORY WITH AN APPLICATION TO HYDRODYNAMICS, Transactions of the American Mathematical Society, 347(12), 1995, pp. 4559-4598
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.