Persistent unstable equilibria and closed orbits of a singularly perturbed equation

with x E R a parameter and y E R. Let us suppose that g(x, 0) = 0 and that g(x, y) = 0 on a second curve C that passes through (b, 0) and is situated as in Fig. 1. Suppose in addition that (ag/ay)(x, 0) is negative for x < b and positive for x > b. Then the flow of (1) is in the direction of the arrows shown in Fig. 1. Thus the equilibrium y = 0 loses stability as x increases past b. The equilibria on C near (b, 0) are unstable for x < b and stable for x > b. Now let us assume that the parameter x is itself slowly varying: