Persistent Unstable Periodic Motions, II

We study the initial value problems of the two dimensional system: utsf u, t, I t.. 0.1. uts0su0 t1,Ii.qO e. where I t.sIiqet is a slowly varying parameter, f u, t, I. is analytic on all variables and periodic on t with period 2prv, and u0 t, I.are periodic solutions of the system ¨tsf ¨ , t , I. 0.2. where I is a constant parameter. We assume that the variational equations of 0.2., wtsf¨ u0 t, I., t, I.w, have the corresponding characteristic exponents l1 I.,l2 I.sl1 I. which move across the imaginary axis from the left half complex plane to the right half complex plane as I increases past Iy. We show that under the nonresonant conditions H4 and H5 that v / 2rn.