Full text of DOI:10.1007/s003329910009

J. Nonlinear Sci. Vol. 10: pp. 197–221 (2000)
DOI: 10.1007/s003329910009
2000 Springer-Verlag New York Inc.
©
Slow Passage through a Saddle-Center Bifurcation
D. C. Diminnie1 and R. Haberman2
Department of Mathematics, Southern Methodist University, Dallas, TX 75275, USA
Received September 24, 1998; accepted August 9, 1999
Communicated by Stephen Wiggins
Summary. Slowly varying conservative one-degree of freedom Hamiltonian systems
are analyzed in the case of a saddle-center bifurcation. Away from unperturbed homo-
clinic orbits, strongly nonlinear oscillations are obtained using the method of averaging.
A long sequence of nearly homoclinic orbits is matched to the strongly nonlinear oscil-
lations before and after the slow passage. Usually solutions pass through the separatrix
associated with the double homoclinic orbit before the saddle-center bifurcation occurs.
For the case of slow passage through the special homoclinic orbit associated with a
saddle-center bifurcation, solutions consist of a large sequence of nearly saddle-center
homoclinic orbits connecting autonomous nonlinear saddle approaches, and the change
in the action is computed. However, if the energy at one specific saddle approach is
particularly small, then that one nonlinear saddle approach instead satisfies the nonau-
tonomous first Painleve´ transcendent, in which case the solution usually either passes
through the saddle-center homoclinic orbit in nearly the same way or makes a transition
backwards in time to small oscillations around the stable center (though it is possible
that the solution approaches the unstable saddle).
Key words. saddle-center bifurcation, passage through a separatrix, homoclinic orbits,
averaging, change in action, nonlinear saddle point, nonautonomous saddle approach,
first Painleve´ transcendent, generalized Riemann zeta function, slowly varying Hamil-
tonian systems
MSC numbers. 34C37, 34C35, 34C23, 58F14, 34C20, 58F05, 34C29, 34C15
PAC numbers. 0545-a, 0230Hq, 4520Jj, 4510Hj
1. Introduction
Due to center manifold reduction, typical bifurcation theory (saddle-node, transcritical,
and pitchfork) is one-dimensional (see [10], [11], and [12]). Even the amplitude for

198
D. C. Diminnie and R. Haberman
Hopf bifurcation is one-dimensional. Lebovitz and Schaar analyzed one-dimensional
transcritical [18] and pitchfork bifurcations [19] with the parameters slowly varying
dynamically (in time). Haberman [13] showed how local first-order nonautonomous dif-
ferential equations provided the transition between various slowly varying solutions and
described circumstances under which a delayed bifurcation exists. Neishtadt [23], [24]
proved that the unstable equilibrium may persist for slowly varying Hopf bifurcations.
Haberman [13] also described slow bifurcations in conservative Hamiltonian systems,
including pitchfork and transcritical, more recently studied by Maree´ [20]. In the case of
a saddle-center bifurcation, Haberman [13] showed how the first Painleve´ transcendent
provides a transition between small oscillations around the center and strongly nonlinear
nearly homoclinic oscillations, which were then (1979) analyzed only in a preliminary
fashion.
Slowly varying and weakly perturbed strongly nonlinear oscillations are analyzed
by the method of averaging (see [1] and [26]), which has a long history. In partic-
ular it is known that action is an adiabatic invariant for Hamiltonian systems. The
method of averaging is known to fail near an unperturbed homoclinic orbit (sepa-
ratrix). For slowly varying Hamiltonian systems, Cary, Escande, and Tennyson [6]
and Neishtadt [22] determined the O( ln ) change in the action in the slow pas-
sage through a separatrix by considering a large sequence of nearly homoclinic or-
bits. Henrard [14] extensively analyzed the result of Cary, Escande, and Tennyson [6],
and described many physical examples. Bourland and Haberman [2], [3], [4] analyzed
the slow passage through a separatrix for systems with a small dissipative perturba-
tion, obtaining an asymptotically accurate formula for the boundaries of the basins of
attraction, generalizing Cary and Skodje [7]. Recently, Vainshtein, Vasiliev, and Neish-
tadt [25], [27] analyzed three-dimensional dynamical systems corresponding to station-
ary incompressible fluid flows with periodic solutions, calculating the small change
in the adiabatic invariant due to the slow passage through a two-dimensional separa-
trix.
In this paper, the slow passage through a saddle-center bifurcation is studied. In
Section 2 we review the method of averaging for strongly nonlinear oscillations that
is valid away from the separatrix. In Section 3 we analyze nearly homoclinic orbits
that are shown to be relevant before the saddle-center bifurcation has actually occurred,
and in Section 4 we show how to use the averaged equations to obtain the boundary
of the basin of repulsion. In Section 5, we analyze the strongly nonlinear saddle point
and the corresponding saddle-center homoclinic orbit. The slow passage of a saddle-
center homoclinic orbit is shown in Section 6 to consist of a large sequence of nearly
saddle-center homoclinic orbits that are matched to the strongly nonlinear oscillations
before and after passage. The sequence of times associated with the nonlinear saddle
approaches are determined using the analytic continuation of the generalized Riemann
zeta function (see the Appendix), which enables us to obtain in Section 7 the relatively
large O( 5/6) change in action in slowly crossing the saddle-center homoclinic orbit.
In Section 8 we allow the energy to be particularly small at an approach to the saddle
point so that the first Painleve´ transcendent applies, in which case the solution usually
either passes through the saddle-center bifurcation or approaches the center, but it is
possible though unlikely that the solution approaches the saddle. An example is given
in Section 9.

Slow Passage through a Saddle-Center Bifurcation
199
xs(T)
Tc < Tsc
xs(Tsc) = 0
Tc = Tsc
Tc > Tsc
Fig. 1. Potential energy and phase portrait (fixed T) for saddle-center bifur-
cation.
2. Slowly Varying Strongly Nonlinear Oscillators
We analyze slowly varying conservative systems (a special case of slowly varying Hamil-
tonian systems)
d2x
dt2
= f (x, T ),
(2.1)
where T = t is the slow time scale and the slowly varying potential satisfies V (x, T ) =
R
x
f (x¯, T ) dx¯. Slowly varying equilibria (saddles and centers) are characterized by
xs (T )
f (x0(T ), T ) = 0, which form the leading-order term in a unique asymptotic solution of
(2.1). Here, we analyze a bounded saddle-center bifurcation so that we assume a saddle
and center only exist for T < Tsc = 0 and coalesce there as is shown in Figure 1. In
addition, we assume there is a second center which persists and, to be specific, stays to the
right throughout the bifurcation. The corresponding bifurcation diagram corresponding
to a saddle-center bifurcation is shown in Figure 2. It is important that the potential be
defined relative to a slowly varying saddle point when there is a saddle (T Tsc) so that
VT (xs, T ) = 0. At T = Tsc there is a strongly nonlinear saddle point which we assume
is located at x = 0. For T > Tsc it is less important how the potential is measured so we
assume it is zero at x = 0. The phase portraits with T fixed (see Fig. 1) have periodic
and homoclinic orbits that only approximate solutions to (2.1).

200
D. C. Diminnie and R. Haberman
x
T=Tsc
T
Fig. 2. Bifurcation diagram.
Away from the homoclinic orbits, solutions of (2.1) are slowly varying nonlinear
oscillators. We use the method of averaging [1], [3], [26] with energy-angle coordinates
1
dx
dt
E =
2 + V (x, T ),
(2.2)
(2.3)
2
Z
x
dx
=
√
,
2[E V (x, T )]1/2
ω(E, T )
xmin
where the xmin and xmax are the minimum and maximum of an oscillation and depend on
E and T and where the frequency of a strongly nonlinear oscillator satisfies ω(E, T ) =
R
xmax
xmin
dx
}
1. After a near identity transformation E = e + E1(e, , T ) +
√
{2
2[E V (x,T )]1/2
and
=
+
1(e, , T ) + , where e and represent the average energy and
phase, it can be shown that the leading-order averaged equations are particularly accurate
and may be used (with care) in describing the slow passage through homoclinic orbits
de
= ω(e, T )D(e, T ) + O( 2),
(2.4)
(2.5)
dT
d
= ω(e, T ) + O( 2),
dt
R
where D(e, T ) = 1/ω VT (x, T ) dt is the change in energy over one period ω1 . Action
0
R
dI
dT
1
ω
(I = p dq) is an adiabatic invariant (
= 0 since Ie = and IT
=
D), but we
prefer to use the energy. These averaged equations represent three different kinds of
nearly periodic solutions, those in the left (L) and right (R) wells and those solutions
outside (Out) the wells. The method of averaging is known to fail near the unperturbed
homoclinic orbits (e = 0).

Slow Passage through a Saddle-Center Bifurcation
201
3. Nearly Homoclinic Orbits (Tc < Tsc) for Slowly Varying
Saddle-Center Bifurcation
Neishtadt [21] showed that it takes an intermediate time to cross the separatrix, a long
time (O(ln )) in the fast time scale t but a short time (O( ln )) in the slow time scale
T . Specifically, the homoclinic orbit (separatrix) is crossed in a quasi-steady fashion with
the slow time T nearly fixed at Tc, where Tc is the time at which the averaged equations
(2.4) predict the averaged energy crosses the separatrix (e = 0). Near the unperturbed
homoclinic orbit, the solution consists of a large sequence of nearly homoclinic orbits.
The solution is approximated by a nearly homoclinic orbit between successive saddle
approaches. The energy change 1E from one saddle approach to the next is known to be
accurately approximated by the corresponding homoclinic Melnikov (Poincare´-Arnold)
R ∞
integral
∞ VT (x, Tc) dt, which depends on whether the solution x in the integrand
is approximately a left or right homoclinic orbit, which in this slow variation case also
depends on Tc
ML(Tc),
MR(Tc).
1E =
(3.1)
The Melnikov change in energy is known to equal the change in energy for periodic
solutions in the limit as the periodic solutions approach the homoclinic orbit, so that
D(0, T ) equals ML(T ), MR(T ), or ML(T )+MR(T ), depending on whether the periodic
solution is approximated by the left (E < 0), right (E < 0), or double homoclinic orbit
(E > 0).
In order to investigate the simplest dynamics (forwards and backwards in time) of a
saddle-center bifurcation, we will assume that each of the three Melnikov functions do
not change signs and that the dissipation function for periodic solutions D(E, T ) also
maintain the same signs as their corresponding Melnikov functions. There are two cases
of interest corresponding to a saddle-center bifurcation:
I. ML(Tc) > 0, MR(Tc) < 0, ML(Tc) + MR(Tc) < 0,
II. ML(Tc) > 0, MR(Tc) > 0, ML(Tc) + MR(Tc) > 0.
(3.2)
(3.3)
We study case I in this paper because case II corresponds to the “usual” dissipation
(backwards in time). The basic reason that ML(Tc) > 0 is that solutions initially (T =
Tinitial < Tsc = 0) in the left well have E < 0 and must pass through the homoclinic
orbit (E = 0) before the left well disappears (Tc < Tsc = 0). Thus, energy must
increase for all of these solutions, and consequently ML(Tc) > 0. A third case ML(Tc) >
0, MR(Tc) < 0, ML(Tc) + MR(Tc) > 0 cannot occur (assuming no sign changes) since
the disappearance of the left well implies that ML(Tc) → 0 as Tc → Tsc = 0. The
following more explicit demonstration of ML(Tc) > 0 may be of interest. The normal
form (rescaled for simplicity) of the saddle-center bifurcation near x = 0, T = Tsc = 0
√
(with the left well disappearing) is f
there) and center at x
to zero at the saddle point) is V
x2 +T , with saddle at x
+
T (since fx > 0
√
T (since fx < 0 there). The corresponding potential (set
x3
2
(
+ T x) 3 ( T )3/2. Here VT
x + ( T )1/2
,
3
and thus VT < 0 for x > ( T )1/2 and VT > 0 for x < ( T )1/2. This proves that
ML(T ) > 0 for T < Tsc = 0 (and small) since the left region only involves x small,

202
D. C. Diminnie and R. Haberman
e = E −εM /2−ε(M +M )/2
φ = −n
0
R
R
L
0
εM
R
φ = −n−1
e = E
0
Fig. 3. Two branches of stable manifold (backwards in time) of saddle, boundaries of the basins of
repulsion, exit saddle approach, and repelled orbits (dashed) from left and outer regions: Tc < Tsc
with MR(Tc) < 0, ML (Tc) > 0, MR(Tc) + ML (Tc) < 0.
but does not prove anything concerning MR(T ) since the right well is not restricted to x
small.
Orbits that correspond to the slow passage through the unperturbed homoclinic orbit
for fixed T = Tc(with Tc < Tsc = 0) are well-known and are shown in Figure 3. Since
ML(Tc) > 0, MR(Tc) < 0, ML(Tc) + MR(Tc) < 0, all solutions are captured into the
right well. Solutions that start in the left well (L) and those solutions that start in the top
region (Out) outside the well form thin alternating bands. The energy at the exit saddle
approach E0(last saddle approach with negative energy) ranges from MR(Tc) to 0:
MR(Tc) < E0 < 0.
(3.4)
The boundaries of the basins of repulsion are the two branches of the unstable man-
ifolds of the saddle point, and their energies (E0 = MR(Tc) and E0 = MR(Tc) +
ML(Tc) at the exit saddle approach can be computed using the Melnikov result (4.1).
Thus, if
MR(Tc) < E0 < MR(Tc)+ ML(Tc),
(3.5)

Slow Passage through a Saddle-Center Bifurcation
203
then solutions originated in the left well (L), and if
MR(Tc) + ML(Tc) < E0 < 0,
(3.6)
then solutions originated outside the wells (Out). The probabilities of release (opposite
to capture) are discussed in Neishtadt [21]
MR(Tc) + ML(Tc)
MR(Tc)
ML(Tc)
MR(Tc)
P(Out) =
and
P(L) =
.
(3.7)
These probabilities depend on the slow time Tc at which the slow variation equations
predict the average energy is zero.
When solutions are near the unperturbed homoclinic orbit, there is a well-known
large sequence of nearly homoclinic orbits, each surrounded by near saddle approaches
with energy Ei . The change in energy from one saddle approach to another can be
approximated by the appropriate Melnikov function. Solutions that go from the left to
the right have the sequence LLLLLLRRRRRR, and the energy at the ith saddle approach
satisfies
E0 + i ML(Tc),
i < 0,
0.
Ei =
(3.8)
E0 + i MR(Tc),
i
Solutions which go from outside the double homoclinic orbit to the right have the se-
quence LRLRLRLRLRLRRRRRRRRR, and the energy at the ith saddle approach is
similar to (3.8) but account of the even and odd saddle approaches must be made.
4. Use of the Averaged Equations in Separatrix Crossing (Tc < Tsc)
In the slow crossing of a separatrix, energy terms of O( ) are of particular importance.
The method of averaging fails as the separatrix is approached. To overcome this, Neish-
tadt [22] and Cary, Escande, and Tennyson [6] used the method of matched asymptotic
expansions to show that the sequence of nearly homoclinic orbits matches the method
of averaging before and after the slow passage through a separatrix. The difference be-
tween the energy E and the average energy e is O( ), and thus the O( ) correction to the
energy should be included. However, many authors including Bourland and Haberman
[2] showed that this O( ) energy difference vanishes at the minima and maxima of the
orbits, and thus the averaged energy may be used directly at the minima and maxima.
Since all solutions that pass through the separatrix fall into the right well with negative
energies, we consider final (instead of initial) conditions in the right well at T = Tf with
given energy e(Tf ) < 0 and phase 0
(Tf ) < 1 far enough from the separatrix so
that the averaging equations are valid there. The method of averaging (2.4) predicts that
the average energy e = 0 at T = Tc = tc where (2.5) predicts the phase to be c. The
separatrix is crossed with T fixed at Tc, and near the separatrix (e = 0) the averaged
equations can be approximated by quasi-steady equations
de
ω(e, Tc)D(0, Tc),
(4.1)
(4.2)
dT
d
ω(e, Tc)
,
dT

204
D. C. Diminnie and R. Haberman
where D(0, Tc) is the relevant Melnikov integral evaluated at Tc predicted by the slow
variation equations. Following Bourland and Haberman [3], [4], we note that near the
separatrix, the average energy equation behaves as an autonomous equation, and thus
the simple relationship follows:
de
= D(0, Tc).
(4.3)
d
Consequently, for two different phases along an orbit using the averaged equations near
the separatrix,
1e = D(0, Tc)1 .
c, where e = 0 and
(4.4)
The two phases we use are
=
=
n < 0 (it is important that
this is an integer) where e = E = E0. Since these calculations use averaging in the right
well, D(0, Tc) = MR(Tc), and it follows from (4.4) that
E0 =
MR(Tc)(n + c).
(4.5)
Using (3.5) and (3.6), we show that the number of orbits n equals the number predicted
by slow variation theory, n = [[ c]], the integral part of
c, and thus
mod
E0 = MR(Tc)(
)
< 0,
(4.6)
c
where (Q)mod is the modulus or fractional part of Q. Furthermore, it follows from (4.6)
using (3.5) and (3.6) that the specific repeller can be determined from initial conditions
using slow variation theory. If
MR(Tc) + ML(Tc)
mod
0 < (
)
<
,
(4.7)
c
MR(Tc)
then solutions originated outside the wells (Out), and if
MR(Tc) + ML(Tc)
mod
< (
)
< 1,
(4.8)
c
MR(Tc)
then solutions originated in the left well(L). The boundaries of the basins of repulsion
mod
mod
(unstable manifolds of the saddle point) correspond to (
)
= 0 and (
)
c
=
c
M (T )+M (T ) . In addition, this explains the probabilities (3.7). To obtain connection
R
c
L
c
MR(Tc)
formulas for the averaged energy corresponding to slow passage (backwards in time)
through the separatrix, we must also use slow variation theory for the nearly periodic
solutions in the left wells and outside the wells.
For the left well, the slow variation theory will predict e = 0 at T = TL where
=
L. We first determine
in a straightforward manner (TL is more difficult to
L
determine, and we will describe this later). Again, Neishtadt [21] showed that TL is
near enough to Tc so that (4.1)–(4.5) again apply. In particular, we employ the averaged
equation (4.4) with the following two phases along the orbit near the separatrix:
=
L
where e = 0 and
=
n
1 < 0 (again it is important that we choose this to be an
MR(Tc)
ML (Tc)+ MR(Tc)
integer) where e = E = E0
using the Melnikov idea for the
2
2

Slow Passage through a Saddle-Center Bifurcation
205
corresponding half orbits. Since these calculations are using averaging in the left well
here D(0, Tc) = ML(Tc), and thus from (4.4),
1
2
MR(Tc) ML(Tc)
ML(Tc)
mod
=
+
[(
)
c
1],
(4.9)
L
c
where (4.6) has been used. From (4.9) using (4.8), we can show that [[ L]] = [[
c
12 ]] with no restrictions on (
)
as should be required. If the method of averaging
mod
L
were valid throughout,
would equal c, so that (4.9) shows that one effect of the slow
L
passage (from the right well to the left well) through a homoclinic orbit is to introduce
a phase jump:
1
2
MR(Tc) ML(Tc)
ML(Tc)
mod
=
[(
)
c
1].
(4.10)
jump
A similar calculation can be applied to orbits that finish outside the wells. The method
of averaging predicts e = 0 at T = Tout where out . As for the left well, Tout will
be near enough to Tc for (4.1)–(4.5) to be valid with D(0, Tc) = ML(Tc) + MR(Tc).
Equation (4.4) is used along the orbit near the separatrix with
=
=
where e = 0
out
MR(Tc)
2
M (T )+ M (T ) . We
L
c
R
c
and
obtain
=
n
1 < 0 where, again e = E = E0
2
ML(Tc)
1
mod
=
(
)
,
(4.11)
out
c
c
ML(Tc) + MR(Tc)
2
where (4.6) has again been used. It follows from (4.11) using (4.7) that [[ out ]] =
1 ML (Tc)
2
[[
+
M (T )+M (T ) ]] with again no restrictions on its modulus. If the method of
c
L
c
R
averaging were validcthroughout,
would equal c. Thus, from (4.11) one effect of
out
passing slowly through a separatrix from the right well to oscillations outside the well
is to introduce the phase jump:
ML(Tc)
1
2
mod
=
(
)
c
.
(4.12)
jump
ML(Tc) + MR(Tc)
The complete connection formulas also require the times TL and Tout at which the
averaged energy is predicted to be zero by slow variation theory after the slow passage
through a homoclinic orbit. These times are comparatively difficult to calculate since they
involve matching the sequence of saddle times. We omit this here since these times have
been calculated in essentially equivalent situations by Cary, Escande, and Tennyson [6]
and Bourland and Haberman [2], [4]. However, in Section 6 we will calculate these times
by the same method for the related situation of the slow passage through a saddle-center
homoclinic orbit.
In the preceding analysis, we have assumed that the predicted separatrix crossing
time Tc is less that the saddle-center bifurcation time Tsc, in which case all the solutions
cross the separatrix before the saddle-center bifurcation has occurred. If Tc > Tsc, then
no separatrix crossing occurs, and the usual slow variation theory may be used without

206
D. C. Diminnie and R. Haberman
modification (for example action is conserved). An interesting situation arises if Tc = Tsc
(we will also consider what happens if Tc is near Tsc).
5. Saddle-Center Homoclinic Orbit and a Strongly Nonlinear Saddle Point
At T = Tsc, there is a one-branched homoclinic orbit connecting the strongly nonlinear
saddle point at x = 0 (where the energy is zero) to itself (see Fig. 1). The potential can
be approximated by a cubic at T = Tsc,
V (x, Tsc)
x3,
(5.1)
with 6
=
V
xxx (0, Tsc) = fxx (0, Tsc) > 0. The time-dependence of the “saddle-
center” homoclinic orbit is described by
Z
x
dx¯
V (x¯, Tsc)
√
=
|t ti |,
(5.2)
√
2
xmax
where ti is the center (symmetry) time of this solitary pulse at which x = xmax when
T = Tsc. This solitary pulse corresponds to the energy being a constant E = 0. However,
solutions take a long time to slowly pass through this kind of separatrix (as with the one
characterized by a linear saddle point). Thus, the solution will be approximated by a large
sequence of nearly solitary pulses of this kind which we parameterize by the symmetry
time ti . Although the energy is slowly varying, this homoclinic orbit will be sufficiently
accurate for our purposes, since higher order terms away from the saddle point will be
accounted for using the Melnikov integral for the change in the energy due to a slowly
varying potential. When the effect of the slow variation is included, the homoclinic orbit
is broken, and orbits spiral inwards as we have assumed MR(Tc) < 0 as also indicated
in Figure 4.
To obtain the asymptotic expansion of this saddle-center homoclinic orbit as t ti →
∞, we add and subtract the singularity using (5.1) and obtain
2
x
,
as
t
ti → ∞,
(5.3)
( t ti
c)2
q
R
xmax
0
2
xmax
1
1
√
√
where c =
+
[√
] dx. Note that the solitary pulse
2
x3
2
V (x,Tsc)
associated with the saddle-center bifurcation decays algebraically toward the strongly
nonlinear saddle point, unlike the usual solitary pulses associated with linear saddle
points which decay exponentially.
Solutions pass near the strongly nonlinear saddle point many times, which we char-
acterize by the energy Ei and the time of closest approach ti . We will define these more
precisely. In the neighborhood of the strongly nonlinear saddle (x = 0), the leading-order
differential equation is a sufficiently accurate approximation for our purposes,
d2x
dt2
= 3 x2,
(5.4)

Slow Passage through a Saddle-Center Bifurcation
207
E −εM
0
E
εM
R
0
R
0
φ = −n−1
φ = −n
Fig. 4. Stable and unstable manifolds of saddle at saddle-center bifurcation: Tc = Tsc with
MR(Tsc) < 0. The exit region is defined by MR(Tsc) < E0 < 0.
which satisfies conservation of energy
2
1
2
dx
dt
x3 = Ei ,
(5.5)
with phase portrait as shown in Figure 5. Ei = 0 corresponds to exactly a stable or
unstable manifold of the nonlinear saddle point, and (5.5) yields the simple result x =
E >0
E <0
i
i
t
E =0
t
i
i
i
Fig. 5. Nonlinear saddle point at saddle-
center bifurcation.

208
D. C. Diminnie and R. Haberman
2
2 . Of greater interest to us will be Ei = 0, in which case it is best to rescale (5.5)
(espke)cially since we will later make assumptions concerning the size of Ei )
(t
1/3
Ei
x =
x¯,
1/6
(5.6)
(5.7)
2
¯
t,
t
ti = Ei
¯
where t = ti (t = 0) corresponds to the “nearest” saddle approach (minimum where
dx
dt
= 0) or symmetry time for (5.5). This scaling shows that there are two versions of
(5.5) depending on whether the energy at the nonlinear saddle approach is negative or
positive:
2
1
2
dx¯
3
x¯ = 1,
(5.8)
¯
dt
where the upper sign in (5.8) corresponds to Ei > 0 and vice versa. We use this same
sign convention for many subsequent equations. The solution of (5.8) is
Z
x¯
1
dx
¯
√
= t .
(5.9)
(x3 1)1/2
2
1
Solutions of (5.5) develop a finite time singularity (x¯ → +∞) according to (5.9) at
Z
0(13 )0(16 )
∞
1
dx
1,
Ei < 0,
Ei > 0,
√
t
= √
=
√
(5.10)
(x3 1)1/2
3,
2
2 6
1
¯
¯
forwards and backwards in time at t = t and at t = t . The asymptotic expansion of
the singularity associated with the strongly nonlinear saddle point may be analyzed from
(5.9) but it is perhaps simpler to obtain the result directly from the differential equation
(5.4) or (5.5) once the singularity time is known (see (5.7) and (5.10)):
2
Ei
x
(t
t
sing)4 +
.
(5.11)
1/6t ,
2
(t tsing
)
28
2
Note that this double-pole singularity at t = tsing, where |tsing ti | = |Ei
|
will match to the algebraic decay (5.3) of the solitary pulse. By including the small
but necessary O(t
t
sing)4 term at the singularity, the differential equation is able to
communicate the value of the energy as we will describe.
We use a notation (see Fig. 6) where the saddle-center homoclinic orbit centered at
t = ti is preceded by the ith strongly nonlinear saddle approach ti and followed by the
i + 1st strongly nonlinear saddle approach ti+1. By matching this solitary pulse to the
both strongly nonlinear saddles, we have to leading-order
1/6 t + c,
1/6 t + c.
(5.12)
(5.13)
2
ti
ti = Ei
2
ti+1 ti = Ei+1

Slow Passage through a Saddle-Center Bifurcation
209
ti
ti*
ti+1
Ei
Ei+1
Fig. 6. Consecutive saddle-points surrounding a nearly saddle-
center homoclinic orbit.
By adding (5.12) and (5.13), we obtain a difference equation for successive saddle times
1/6 t + Ei+1
1/6 t + 2c.
(5.14)
2
2
ti+1 ti = Ei
If the energy were constant, then ti
ti represents the energy-dependent period. Thus,
1
the asymptotic expansion of the period P(E) of the periodic solution near the infinite
period homoclinic orbit for small E is
1
1/6 t + 2c,
(5.15)
2
= P(E) 2 E
ω(E)
so that (5.14) represents the average of the two periods associated with a periodic solution
of energies Ei and Ei+1. Here the energy is not constant, but can be determined by the
Melnikov energy for the right homoclinic orbit (in the limit as the saddle-node bifurcation
is approached):
Ei+1 Ei = MR(Tsc).
(5.16)
The exit region (see Fig. 4) MR(Tsc) < E0 < 0 is defined by the unique unstable
manifold of the strongly nonlinear saddle (the unique stable branch is less important).
The sequence of energies is simple but important,
Ei = E0 + MR(Tsc)i,
(5.17)
which is valid for positive and negative integers i. The solution consists of a sequence
of nearly saddle-center homoclinic orbits satisfying (5.17) which matches to the outside
orbits (backwards in time) and the right orbits (forwards in time).
From (5.11), this change in energy (5.16) corresponds to the change in the O((t
t
sing)4) term of the asymptotic expansion of the strongly nonlinear saddle. To find the
change in energy without using the Melnikov result (5.16), one calculates the O( )
perturbation to the solitary saddle-center homoclinic orbit and matches the resulting
outer solution to the leading-order solution in the neighborhood of the nonlinear saddle.
Ei
28
The O( ) outer solution must contribute a term A ( t ti
c)4. Thus,
= A
Ei+1
and
= A , so that
+
28
28
Ei+1 Ei = MR(Tsc) =
change in the O((t
t
sing)4) coefficient. (5.18)

210
D. C. Diminnie and R. Haberman
6. Slow Crossing of a Saddle-Center Homoclinic Orbit
Since the Melnikov function that accurately approximates the change in energy is the
same before and after the exit saddle approach, we show the method of averaging may
be used directly to minimums and maximums without a jump in phase. The method of
averaging (2.4) predicts the average energy e = 0 at T = Tc where (2.5) is used to
calculate the corresponding average phase
=
c, and for the passage to involve the
saddle-center homoclinic orbit the condition again is that Tc = Tsc = 0. The sequence of
strongly nonlinear saddle approaches (5.16) are valid. We again use (4.3) for the averaged
energy e(T ), here with D(0, Tc) = MR(Tsc) so that (4.4) becomes 1e = MR(Tsc)1
both before and after passage through the saddle-center homoclinic orbit. After passage
we choose e = 0 where
=
< 0 and e = E0 where
=
n, so that n = [[ c]]
c
and (4.6) follows:
mod
E0 = MR(Tsc)(
)
< 0,
(6.1)
c
so that we can determine E0 from final (or initial) conditions. Before passage, we choose
e = 0 at the unknown T = Tout where
=
and e = E0
MR(Tsc) where
out
=
n
1, and we immediately obtain
=
,
c
(6.2)
out
so that the method of multiple scales does not need to be modified in passing through
a saddle-center homoclinic orbit. In slowly crossing the saddle-center homoclinic orbit,
the solution obtained by the method of averaging is valid at minimums and maximums
near the unperturbed homoclinic orbit, though a more accurate description consists of
the large sequence of nearly homoclinic orbits described by (5.14) and (5.16). These
should be matched using the method of matched asymptotic expansions because we will
show the time predicted by the method of averaging is incorrect.
The sequence of approximately saddle-center homoclinic orbits is matched forwards
and backwards in time to the solution obtained by averaging. Earlier, we have matched
the energies. Here we match the saddle times. The times ti of the saddle approaches
associated with Tc = Tsc satisfy (5.14), which is a difference equation for the successive
saddle times
t
1/6
ti+1 ti = |E0 + MR(Tsc)i|
1/3
t
1/6
+ |E0 + MR(Tsc)(i + 1)|
+ 2c,
(6.3)
1/3
after using (5.17). We obtain from (6.3) the asymptotic expansion (top sign corresponds
to E > 0 and vice versa) as i →
∞
6
1/3
1/6
ti 2 | MR(Tsc)|
|i|5/6 t + 2ci +
,
(6.4)
5
where the size O( 1/6) of the constants
are important. These constants cannot be
determined from the form of the asymptotic expansion, but we will determine these
constants using an explicit solution of (6.3) involving t0 and the generalized Riemann

Slow Passage through a Saddle-Center Bifurcation
211
de
zeta function. The time-dependent averaged equations
= ω(e, T )D(e, T ) may be
approximated near the unperturbed saddle-center homodcTlinic orbit by (2|e
|
1/6t +
2
de
dT
2c) = MR(Tsc) using (5.15). Thus, at the saddle approaches,
6
1/3
MR(Tsc)(Ti
MR(Tsc)(Ti Tc), Ei < 0,
T
out ), Ei > 0,
2
|Ei |5/6 t + 2cEi =
(6.5)
5
where we have replaced e by Ei since the saddle approaches for the saddle-center
homoclinic orbit are minimums of the orbit. (Equation (6.5) explicitly shows the averaged
equations predict e = 0 in a finite time.) When (5.17) is substituted into (6.5), we see
that the saddle times given by (6.4) are compatible only if
2cE0
2cE0
Tc = Tsc
=
and
Tout
=
.
(6.6)
+
MR
MR
The jump in fast time is large
tout tc =
,
(6.7)
+
so that the method of averaging makes large errors in time since = O( 1/6), but the
jump in slow time is small, confirming that the passage of a saddle-center homoclinic
orbit is done in a quasi-steady fashion.
The constants in (6.4), the asymptotic expansion as i → ∞ of the saddle times
P
i
1
satisfying the difference equation (6.3), are obtained relatively easily from n=0(a +
1/6
n)
65 i5/6 + (16 , a), which is derived in the appendix using the analytic continuation
of the generalized Riemann zeta function, (16 , a):

1/6
1
E0
E0



2t
, 1
t
,
+


6
MR
MR
1/6
2
"
#
= t0 + MR(Tsc)
,
(6.8)
1/6

1
6
E0
E0




t
2
,
.
MR
MR
We recall that the energies Ei > 0 (corresponding to the first of (6.8)) for i < 0 and
Ei < 0 for i 0 (corresponding to the second of (6.8)). To derive (6.8), we have utilized
some elementary ideas such as (i + 1)5/6 i5/6, shifting sums, and letting j = i for
i < 0. From (6.8), we have an explicit expression for the jump in the fast time
tout tc
=
=
+
1
6
E0
1
6
E0
1/6
2
2
MR(Tsc)
t
,
+t
, 1
, (6.9)
+
MR
MR
where from (6.1),
E0
mod
0 <
= (
)
< 1.
(6.10)
c
MR(Tsc)
In Section 8, we show (6.9) is not valid near the boundaries where E0 = O( 6/5) or
E0
MR(Tsc) = O( 6/5).

212
D. C. Diminnie and R. Haberman
7. Change in Action for Slow Passage of a Saddle-Center Homoclinic Orbit
Action I (e(T ), T ) is an adiabatic invariant before and after the slow passage through a
saddle-center homoclinic orbit. Following Bourland and Haberman [4], the two different
constants can be expressed as I (e(T ), T ) = I (0, Tc) and I (e(T ), T ) = I (0, Tout ). Since
Tc = Tsc = Tout , but Tout Tc = O( 5/6), the jump in action when slowly crossing a
saddle-center bifurcation is O( 5/6) and is given by
1I = I (0, Tout
)
I (0, Tc) IT (0, Tsc)(Tout Tc) = MR(Tsc)(Tout Tc). (7.1)
Using (6.9), we obtain
2
1/3
1
6
E0
1
6
E0
5/6
5/6
1I =
|MR(Tsc)|
t
,
+ t
, 1
,
(7.2)
+
MR
MR
where (16 , a) is the analytic continuation of the generalized Riemann zeta function and
where we note (6.10). This change in action is slightly larger than the O( ln ) change
in action derived by Neishtadt [22] and Cary, Escande, and Tennyson [6] for the slow
passage through double-homoclinic orbits (with linear saddle points).
8. Nonautonomous Saddle-Point (the First Painleve´ Transcendent) and the Tran-
sitions to the Center and Saddle
For slowly varying conservative or Hamiltonian systems, the earlier parts of this paper
show that saddle-center bifurcations usually involve the connection of two strongly
nonlinear oscillations by a large sequence of nearly homoclinic orbits. There is another
family of somewhat different solutions in which there is a transition between strongly
nonlinear oscillations near the saddle-center homoclinic orbit and solutions near the
center or saddle involved in the saddle-center bifurcation.
We have described the leading-order nonlinear saddle point by the autonomous dif-
1
ferential equation (5.4) with fxx = 3 . However, a more accurate approximation to
the differential equation (2.1)2near the nonlinear saddle point x = 0 at T = Tsc would
be the nonautonomous equation (Ince [15]) known as the first Painleve´ transcendent,
d2x
dt2
1
2
=
fxx x2 + fT (T Tsc),
(8.1)
where T = t. Haberman [13] used the first Painleve´ transcendent to describe this kind
of transitional though he did not understand the details of the slow passage through a
separatrix at that time (1979). (If T is near Tsc the unfolding would yield an equivalent
d2x
dt2
1
2
equation
=
fxx x2 + x + fT (T Tsc) since it can be converted to (8.1) by shifting
x and T .) If T Tsc
x2 near the strongly nonlinear saddle, then the previous analysis
in the present paper for a saddle-center bifurcation is valid. Because the change of energy
due to Melnikov ideas is O( ), we may assume in our earlier analysis that the energy Ei
itself is O( ). From (5.5)–(5.7), this implies that x = O( 1/3) and t tsc = O( 1/6),
5/6
in which case T Tsc
x2 since
4/6. Thus, the strongly nonlinear saddle is the

Slow Passage through a Saddle-Center Bifurcation
213
leading-order equation almost all the time. However, if x is even smaller (closer to the
saddle), then it is possible for the first Painleve´ (8.1) to be valid. The first Painleve´ (8.1)
involves the balance of the three terms only if
2/5
x = O(
)
and
t
tsc = O( 1/5).
(8.2)
2/5
Here x is smaller than the scaling associated with the nonlinear saddle point (
1/3).
The first Painleve´ with scaling (8.2) still occurs at an intermediate time scale (slow from
the point of view of the fast time t, but fast from the point of view of the slow time
T ) since T Tsc = O( 4/5). The corresponding energy Ei defined by (5.5) is O( 6/5),
muchlessthanthe O( )energyassociatedwithourprevioussequenceofnonlinearsaddle
approaches. Since our sequence of saddle approaches will remain valid (backward in
time), this can only occur if one of the sequence of saddle approaches is near the boundary
of the basin of repulsion, i.e., either
6/5
E0 = O(
)
or
E0
MR(Tsc) = O( 6/5).
(8.3)
Philosophically, this can arise differently forward or backward in time. Forward in time,
as we will show, requires the transition to be near the coalescing saddle-center, while
backward in time puts constraints on the “final” energy such that the energy is almost
precisely zero at a saddle approach as described by (8.3).
Inanalyzingthepassagethroughvariouskindsofbifurcations, Haberman[13]showed
x
2
0T T
fx
that there are slowly varying solutions of (1.1), xsv
x0(T ) +
+
, whose
leading-order behavior satisfies 0 = f (x0, T ). To analyze solutions near a slowly varying
bifurcating center, we summarize Haberman [13] who used the method of multiple scales
with a slow time T and the usual fast phase associated with linear Liouville-Green type
asymptotics. When weakly nonlinear effects were included, the following solution was
obtained near the slowly varying center but away from the bifurcation point:
"
#
R
T
0
k1/2 dT
¯
x
xsv(T ) + 2A 1/2k 1/4 cos
+
1(T ) ,
(8.4)
where k = fx (xsv) = Vxx (xsv) and k1/2 is nearly the linear circular frequency. The
d
dT
A2
2
k
phase shift 1(T ) was shown to satisfy
1(T ) = 12k (3 fxxx
+
fx2x ). At the saddle-
1
center bifurcation fx (0, Tsc) = 0, so that f
fxx x2 + fT (T Tsc), where from now on
these derivatives are evaluated at (0, Tsc), in w2hich case fxx = 6 ; see (5.1). The center
2 fT
of the center-saddle bifurcation satisfies x0
(
)
1/2(Tsc T )1/2 so that k
fxx x0
as the saddle-center bifurcation is approached. Infxtxhe case of the slowly varying center
of the saddle-center bifurcation (at x = 0 at T = Tsc) as T → Tsc,
1/2
fT
x
(Tsc T )1/2
+
3
A2
2 fT
4(Tsc T )5/4
1/2 2A cos[ (12 fT )1/4
+
ln(Tsc T ) + ]
.
5
+
(8.5)
[12 fT (Tsc T )]1/8
(The stable manifold of the slowly varying saddle point can be obtained in a similar
way.) This outer expansion (8.5) fails when Tsc T = O( 4/5) with x = O( 2/5) giving

214
D. C. Diminnie and R. Haberman
rise to the scalings (8.2) of the first Painleve´. In particular we introduce the scaling of
the special nonautonomous nonlinear saddle point
fT2/5
(3 )3/5
2/5
x =
w,
(8.6)
(8.7)
1
4/5
T
Tsc
=
,
(3 fT )1/5
so that the normalized first Painleve´ arises
d2w
= w2 + .
(8.8)
2
d
The asymptotic expansions of the first Painleve´ are well known [15]. Using a Boutroux
transformation, the asymptotic expansion backwards in time of the first Painleve´ usually
either explodes or approaches the stable center, qualitatively similar to the phase portrait
of the left well in Figure 3. These solutions are separated by the stable manifold (back-
wards in time) of the saddle point. Solutions may oscillate around the center, approaching
the center backwards in time
"
#
√
√
A2p
24
2
4 2
1/2
5/4
w
(
)
+ Ap(
)
1/8 cos
(
)
ln( ) +
,
(8.9)
p
5
as
→
∞. This clearly matches to the oscillations around the slowly varying center
given by (8.5). All solutions of the first Painleve´ develop a finite time singularity as
→
forwards in time:
p
1
6
p
2
2
w
6(
)
(
)
p
(
p)3 + cp(
p)4 +
.
(8.10)
p
10
A solution of (8.8) can also have finite time singularities forward and backwards in
time. The solutions of the first Painleve´ provide connections so that (Ap, p) should be
determined by the two parameters associated with the singularity ( p, cp) and vice versa.
Explicit connection formulas now exist by isomonodromy methods [9], [16], [17]. The
asymptotic expansion of the singularity of the first Painleve´ can be seen to be slightly
different than the singularity of the autonomous nonlinear saddle point (5.11).
Haberman [13] showed that the finite time singularity of the first Painleve´ matches
into a nearly solitary pulse associated with the saddle-center bifurcation that is followed
by a sequence of such nearly homoclinic orbits. The singularity of the first Painleve´
matches to the algebraic decay (5.3) of the first saddle-center homoclinic orbit centered
at t = t0 (which we will determine in this way):
1/5
t0 tsc = (3 fT )
p + c.
(8.11)
It is most important that the order of magnitude of the time shift is O( 1/5) again an
intermediate time scale, this one associated with the nonautonomous nonlinear saddle
point. The shift of the center time is longer than the O( 1/6) time associated with the

Slow Passage through a Saddle-Center Bifurcation
215
“usual” strongly nonlinear saddle point since the solution passes even closer to the saddle
point.
E
28
We may think of the coefficient of (t
t
sing)4 as equaling
0 . Using (8.10) with
(8.6) and (8.7), we obtain
28
E0 =
( fT )6/5(3 )1/5cp.
(8.12)
It is important that the energy is small O( 6/5) where the first Painleve´ is valid. The
energy is determined from the important coefficient cp of the (
p)4 term in the
singularity (8.10) of the first Painleve´. The energy is not constant at a saddle approach
described by the first Painleve´ so that (8.12) only describes the correct coefficient of the
fourth power necessary to predict the energy at the next saddle approach,
28
E1 =
( fT )6/5(3 )1/5cp + MR(Tsc),
(8.13)
which will not be approximately zero. Thus, this and all subsequent saddle points will
be described by the usual nonlinear saddle for saddle-center bifurcation (not the first
Painleve´). In between these saddle approaches, the solution will be the nearly saddle-
center homoclinic orbit, so that the energies at the subsequent saddle approaches are
determined using Melnikov ideas,
28
Ei =
( fT )6/5(3 )1/5cp + MR(Tsc)i.
(8.14)
The times of the saddle approaches may be obtained as we did earlier from a sequence
of nearly saddle-center homoclinic orbits so that (5.14) may be used:
1/6 t + Ei+1
1/6 t + 2c.
(8.15)
2
2
ti+1 ti = Ei
Equation (8.14) is used for the energy.
These solutions near the saddle-center homoclinic orbit will match (as before) to
strongly nonlinear oscillations obtained by the method of averaging. We can use our
previously derived energy result, (6.1), with E0 given by (8.12):
28
mod
E0 =
( fT )6/5(3 )1/5cp = MR(Tsc)(
)
< 0.
(8.16)
c
mod
There is a thin band of solutions with (
)
= O( 1/5) corresponding to the first
c
Painleve´ if cp 0. If cp < 0, then E0 > 0, which means that E1 is close to
MR(Tsc)
as allowed (and E1, not E0, corresponds to the first Painleve´). Equation (8.16) can be
used forward in time, giving matching conditions that the strongly nonlinear oscillation
must satisfy. It is perhaps more natural to interpret (8.16) backwards in time. We can
obtain the first Painleve´ only for very special final conditions from which the energy is
predicted to be even smaller than usual; the energy needs to be O( 6/5) rather than O( ).
The saddle approach times ti are used to determine the singularity time from final
p
conditions. The sequence of saddle times satisfies (8.15) and starts from t1. However, the

216
D. C. Diminnie and R. Haberman
energy is much closer to zero (and the resulting period much larger) only for the saddle
point that satisfies the first Painleve´. Thus, in computing the saddle approach times, it
will be a valid approximation to replace E0 by 0 everywhere except i = 0. In this case,
(8.15) simplifies and its asymptotic expansion is determined using (A.6). We compare
the asymptotic expansion to (6.4) and obtain
1
6
1
2
1/6 t
, 1
.
(8.17)
2
= t1 2c + 2 MR(Tsc)
It should be noted that (16 , 1) = (61 ) is the usual Riemann zeta function.
is deter-
mined from final conditions (6.6), so that (8.17) shows how t1 is determined from final
conditions. Knowing t1 gives t from (5.13), so that is determined from (8.11) using
p
0
(5.13):
1
6
1/6 t
, 1 = tc tsc.
(8.18)
1/5
2
(3 fT )
p + 2 MR(Tsc)
Note that tc tsc is large while (tc tsc) = Tc Tsc is small. We would be justified
in keeping the smaller O( 1/6) term in (8.18) only if (5.13) and (8.17) are accurate to
o( 1/5).
As in Section 7, action is an adiabatic invariant before and after the slow passage
through the saddle-center bifurcation. When the solution, (8.4) and (8.9), is a linear
oscillation around the slowly varying equilibrium, the action is well-known to be ap-
√
2 fT
proximated by 4 A2 =
A2p. Thus, the change in action is
3
1I = 4 A2 I (0, Tc)
4
A2 I (0, Tsc) IT (0, Tsc)(Tc Tsc)
= 4 A2 I (0, Tsc) MR(Tsc)(Tsc Tc),
(8.19)
where I (0, Tsc) is the constant O(1) action associated with the homoclinic orbit of the
saddle-center bifurcation. Using (8.18) we obtain
1/5
1I = 4 A2 I (0, Tsc) + 4/5 MR(Tsc)(3 fT )
p
2
1
6
|MR(Tsc)|5/6 t
, 1 .
(8.20)
5/6
1/3
Since the solution passes much closer to the saddle point when the first Painleve´ is
valid, the change in action is O( 4/5) and larger than usual. The O( ) term will only be
asymptotically significant if the higher order corrections to the other terms are smaller.
Ap is determined from connection formulas associated with the first Painleve´ requiring
and cp (given by (8.16)). Equation (8.19) may be preferable since it shows how the
p
change in action is determined by final conditions (Tc).
9. Summary of Crossing Saddle-Center Homoclinic Orbit and an Example
We describe the different kinds of solutions that can occur backwards in time in the case
in which the averaged energy is predicted to be zero at the saddle-center bifurcation
time (Tc = Tsc). Solutions consist of strongly nonlinear oscillations that become a large

Slow Passage through a Saddle-Center Bifurcation
217
sequence of nearly saddle-center homoclinic orbits. After this large sequence,
I. Inmostcases, thesolutionpassesthroughthesaddle-centerhomoclinicorbit, reemerg-
ing as a strongly nonlinear oscillation resembling the saddle-center homoclinic orbits.
II. However, for a special class of final conditions in which the energy at a saddle
approach is predicted to be unusually small, one saddle approach is described by the
non-autonomous first Painleve´ transcendent. This first Painleve´ transcendent has a
finite time singularity forward in time that matches to the first of a series of nearly
saddle-center homoclinic orbits. In this case, three things can occur backwards in
time depending on the solutions of the first Painleve´ transcendent:
a. If the first Painleve´ approaches the stable center (backwards in time), this involves
the transition to small oscillations around the newly created stable center.
b. If the first Painleve´ approaches the unstable saddle point, this involves the highly
unlikely but still possible transition to the unstable saddle point (the two branches
of the stable manifold of the unstable saddle point).
c. The first Painleve´ may also have a finite time singularity backward in time, in
which case the solution becomes another nearly saddle-center homoclinic orbit,
and the solution will be very similar to case I.
1/5
The small band of nearly linear oscillations described by IIa has probability O(
)
occurring backwards in time. This band is bounded by IIb, the two branches of the
stable manifold of the saddle point, which separate IIa small oscillations from IIc, nearly
saddle-center homoclinic orbit. One branch of the stable manifold of the saddle point
consists of a large sequence of nearly saddle-center homoclinic orbits, while the other
branch is similar but preceded by a single passage around the slowly varying center.
As an example of a slowly varying saddle-center bifurcation, we briefly consider
d2x
dt2
=
(x 1 + t)(x2 + t).
(9.1)
The leading-order slowly varying equilibria are x = 1
saddle center bifurcation occurs at Tsc
t and x2
=
t so that a
=
tsc = 0, and the bifurcation diagram is as
5
shown in Figure 2. We choose = 10 so that there exists solutions very close to
the saddle-center homoclinic orbit, and we numerically integrate. We choose the final
energy such that Tc = Tsc = 0, and we vary the final phase. In Figure 7a, we show a
solution consisting of a sequence of nearly saddle-center homoclinic orbits having an
autonomoustransitionbetweenpositiveandnegativeenergies. Figure7bshowsasolution
that undergoes a nonautonomous (first Painleve´) transition between a sequence of nearly
saddle-center homoclinic orbits and small oscillations around the slowly varying center
√
x
t for t < 0. In Figures 7a and 7b, we also illustrate the projection of the
solutions in the two-dimensional (x, dx ) phase space, verifying that the solutions are
nearly saddle-center homoclinic orbitsd.t
Appendix: Analytic Continuation of the Generalized Riemann Zeta Function
The saddle times ti for nearly saddle-center homoclinic orbits (Tc = Tsc) can be obtained
by summing the difference equation (6.3). Since the sum is divergent, we are interested

218
D. C. Diminnie and R. Haberman
x 10−3
x 10−3
2
0
2
0
−2
−2
0.5
0.5
2
2
0
0
1
1
0
0
dx/dt
dx/dt
−0.5
−0.5
x
x
Fig. 7. Autonomous (a) transition between sequences of nearly saddle-center homoclinic or-
bits with positive and negative energies. Nonautonomous (b) transition (first Painleve´) be-
tween small oscillations around the center and a sequence of nearly saddle-center homoclinic
orbits.
instead in the asymptotic expansion of the saddle times as i → ∞. Similar convergent
sums can be expressed in terms of the generalized Riemann zeta function. We will follow
the well-known [8] ideas of analytic continuation as first described by Riemann and apply
it to the generalized Riemann zeta function [5]. We will show that the regularization of
our divergent sum equals the analytic continuation of the corresponding generalized
Riemann zeta function. The particular divergent sum of interest to us follows from (6.3)
1
Rm(a) = m (a + n)
(A.1)
X
1/6
n=0
with a > 0. The leading-order term in the asymptotic expansion for large m of (A.1) may
6
be obtained by elementary methods, Rm(a) 5 m5/6 + k, where we wish to determine
the more difficult constant k. An integral representation of the corresponding infinite
series is usually obtained by introducing an infinite geometric series with the gamma
R
function 0(61 ) = (a + n)1/6 ∞ x 5/6e (a+n)x dx requiring a + n > 0. Here, we must
instead use a finite geometric 0series
Z
∞
5/6
1
6
x
ax
0
Rm(a) =
e
x (1
e
mx ) dx.
(A.2)
1
e
0
The limit as m → ∞ of (A.2) does not exist because of the nonintegrable singularity
R
at x = 0, which we remove in the following way: 0(1 )Rm(a) = ∞ x 11/6(1
1
x
5/6
x
6
0

Slow Passage through a Saddle-Center Bifurcation
219
R
5/6
mx ) dx + ∞(e
x
11/6)(1
e
mx ) dx. The first integral can be easily eval-
x
1
ax
e
x
uated by letti0ng u = mx and integrating by parts, while the second integral has a limit
e
as m → ∞:
Z
∞
5/6
6
5
1
x
ax
11/6
Rm(a)
m5/6
+
e
x
dx.
(A.3)
0(16 )
x
1
e
0
We show that this regularized integral in (A.3) is related to the analytic continuation
of the generalized Riemann-zeta function. We introduce the function of the complex
az
1
e
z(1
variable z, F(z) =
) ( z)1/6
[
1 ], analytic in the cut plane to be described,
z
0( 16
e
)
z2
1
6
i
6
log( z)
1/6
arg(
where in the usual way ( z)1/6 = e
= |z|
e
z). We integrate along Rie-
mann’s contour C, starting along the real positive axis x from +∞, circling the origin in
the positive (counterclockwise) direction, and returning to +∞ along the positive real
axis. In this way the contribution around the singularity ( z)1/6 vanishes (since it is
1
R
R
5/6
less than a pole)
F(z) dz =
) (ei
∞(e
x
11/6) dx. Since
1
x
ax
/6
i
/6
e
)
x
0( 16
C
0
1 e
R
1
C ( z)1/6 dz = 0, because its antiderivative vanishes at the endpoints at infinity, it
z2
follows that
Z
6
5
1
1
( z)1/6
az
Rm(a)
m5/6
+
e
dz.
z
(A.4)
1
/6
i
/6
ei
e
z(1
e )
0(6 )
C
The analytic continuation of the generalized Riemann zeta function is defined [5]
by
Z
1
6
1
5
6
( z)1/6
az
, a
=
0
e
dz.
(A.5)
z
2 i
z(1
e )
C
From the reflection formula for gamma functions, 0(z)0(1 z) =
z , we have
sin
0(16 )0(56 ) =
. Thus,
sin( /6)
6
5
1
6
Rm(a)
m5/6
+
, a
.
(A.6)
The generalized Riemann zeta function is usually defined [5] and [8] by a convergent
series and its equivalent integral representation. The analytic continuation is introduced
to define the generalized Riemann zeta function when neither the infinite series nor its
usual integral representation exists. Here, we have derived the important formula (A.6)
6
that the regularized sum (limm→∞ Rm(a) 5 m5/6) equals the analytic continuation of
the divergent sum.

220
D. C. Diminnie and R. Haberman
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