Persistent unstable periodic motions .1.

We study the initial value problems of the two dimensional system: (0.1) u(t) = f(u, t, I(t)) u/(t=0) = u(0)(t(1), I-i) + O(epsilon) where I(t) = I-i + epsilon t is a slowly varying parameter, f(u, t, I) is analytic on all variables and periodic on t with period 2 pi/omega, and u(0)(t, I) are periodic solutions of the system (0.2) upsilon(t) = f(upsilon, t, I) where I is a constant parameter. We assume that the variational equations of (0.2), w(t) = f(upsilon)(u(0)(t, I), t, I)w, have the corresponding characteristic exponents lambda(1)(I), lambda(2)(I)= lambda(1)(I) which move across the imaginary axis from the left half complex plane to the right half complex plane as I increases past I_. We show that under the nonresonant conditions H4 and H5 that omega not equal (2/n)\Im lambda(1)(I_)\ for n is an element of N (in Section 4), along with other generic conditions such as H1-H3 below, the separations of u(l(t)) from u(0)(t(1) + t, I(t)) do not occur at the critical point where I(t) = I_, rather, the bifurcations are substantially delayed until I(t) = Ii f Ei reaches I-q > I_ which is independent of epsilon, as epsilon --> 0(+). In other words, \u(t) - u(0)(t(1) + t, I(t))\ = O(epsilon) for t is an element of {t: I-i less than or equal to I(t) less than or equal to I-q} for some I-q = I-q(I-i) > I_ independent of epsilon, when epsilon --> 0(+). An exact formula for I-q is given for general situations. In near resonance cases, a sharp estimate of I-q is also derived. (C) 1996, Academic Press, Inc.