HOMOCLINIC SOLUTIONS AND CHAOS IN ORDINARY DIFFERENTIAL-EQUATIONS WITH SINGULAR PERTURBATIONS

Ordinary differential equations are considered which contain a singula r perturbation. It is assumed that when the perturbation parameter is zero, the equation has a hyperbolic equilibrium and homoclinic solutio n. No restriction is placed on the dimension of the phase space or on the dimension of intersection of the stable and unstable manifolds. A bifurcation function is established which determines nonzero values of the perturbation parameter for which the homoclinic solution persists . It is further shown that when the vector field is periodic and a tra nsversality condition is satisfied, the homoclinic solution to the per turbed equation produces a transverse homoclinic orbit in the period m ap. The techniques used are those of exponential dichotomies, Lyapunov -Schmidt reduction and scales of Banach spaces. A much simplified vers ion of this latter theory is developed suitable for the present case. This work generalizes some recent results of Battelli and Palmer.