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 ana
lytic 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(up
silon, t, I) where I is a constant parameter. We assume that the varia
tional 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 separat
ions 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 del
ayed until I(t) = Ii f Ei reaches I-q > I_ which is independent of eps
ilon, 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_ independen
t 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.