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SIAM J. MATH. ANAL.
Vol. 22, No. 3, pp. 653-679, May 1991
(C) 1991 Society for Industrial and Applied Mathematics
006
THE BIFURCATIONS OF COUNTABLE CONNECTIONS FROM A
TWISTED HETEROCLINIC LOOP*
BO DENG?
Abstract. Codimension-two bifurcation phenomena associated with nondegenerate heteroclinic loops
are studied. The bifurcation curves of homoclinic orbits in the parameter space are characterized by the
twist structure of the heteroclinic loops at the bifurcation points. Among other things, it is shown that
heteroclinic orbits with any given winding number around a doubly twisted heteroclinic loop must bifurcate.
Applications of these bifurcation phenomena are also discussed.
Key words, twisted heteroclinic orbit, homoclinic orbit, periodic orbit, k-heteroclinic orbit, Sil’nikov’s
variables, exponential expansions, strong A-lemmas, entrance sets, exit sets, bifurcation equations
AMS(MOS) subject classifications. 34A34, 34C28, 34C99
1. Introduction. A heteroclinic loop takes place for a vector field F when there
exist two heteroclinic orbits z*(t) and z*(t), with z* connecting an equilibrium point
a l, to another one a2, and z2* connecting a2 to a l. To be precise,
z/*(t) -> ai
as --> -o
and
z/*(t) --> a
as --> +
for i,j= 1,2 and i#j. Figures 1.1 and 1.2 heuristically illustrate what could happen
to two structurally different loops when a planar vector field F is perturbed slightly.
In Fig. 1.1, either a homoclinic orbit or a periodic orbit would possibly bifurcate from
the loop, while in Fig. 1.2 a heteroclinic orbit winding around the original loop for
any finite times before reaching its destinations in both backward and forward evolutions would also be possible under perturbation. The very structure distinguishing
the second loop from the first one is that a given heteroclinic orbit arises from and
tends to the equilibria from different "sides" of the other heteroclinic orbit. The purpose
of this paper is to study the bifurcations of a generic two-parameter family of vector
fields in Ea, d _-> 2 which exhibit the above heteroclinic phenomena.
The first obvious generalization is to assume that the equilibria ai of the equation
:-- F(z)
(1.1)
FIG. 1.1
FIG. 1.2
Received by the editors April 26, 1988" accepted for publication (in revised form) March 23, 1990.
Mathematics and Statistics, University of Nebraska-Lincoln, Lincoln, Nebraska
? Department of
68588-0323.
653
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654
o
DENG
have the same dimension, m->_ l, for the stable manifolds W and the same unstable
1 and 2. Moreover,
dimension, n d m _-> l, for the unstable manifolds W’ for both
the ais are simple saddlepoints in the sense that
(1.2)
There exist principal eigenvalues Ai <0< for the linearization DzF(a) and
constants A <0</2 such that for any other eigenvalue , of DzF(ai) either
Reu<<Ai or Re,>/2i>/,for i=l and 2.
Not as a generalization but as a generic restriction to all the cases, we assume
that both equilibria are relatively contractive"
(1.3)
+/<0 for i=l and 2.
That is, the principal attraction of a dominates the principal repelling.
Concerning the structure on the intersection of the unstable manifold W’ of a
and the stable manifold W] of aj which must be nontransverse along the heteroclinic
orbit Fi := {z*(t)" 6 R}, we assume that they are in general position"
(1.4)
codim
where
and Tp W means the tangent space of a given manifold W at a base point p W. Also,
motivated by the strong h-lemma from Deng (1989), the following strong inclination
property, as another assumption, is also generic:
lim
t--)
(1.5)
lim
t-)Too
T,, W/ + Ta, WSi
Tz(t
T()= T,W’’+ T,W.
,
Here, W and W are the strong stable and strong unstable manifolds of a, respectively. See Fig. 1.3. If the vector field P is C then W and W are C as well (see,
instead (see
e.g., Shub (1987)). But, in general, W and W are proved to be C
Deng (1989)). Moreover, W and W are (m-1)- and (n-1)-dimensional, respectively, characterized by the fact that the limits
lim
(1.6)
lim
z(t)-a
0
for z(0)
-
WU\ W uu,
z(t)-a
,-+oollz(t)-all
exist and are equal to unit eigenvectors for the principal unstable eigenvalue and
principal stable eigenvalue, respectively. The strong inclination property is a generic
property provided F is C with r >-7. See Deng (1989) for the proof.
The last structural assumption reads
W/\ W’/") ("1 W\ Ws) for i,j= 1, 2 and j.
That is, by virtue of (1.6) this hypothesis says that the heteroclinic orbits arise from
(1.7)
F,
and tend to the equilibria along principal eigendirections. It is certainly a generic
condition.
The assumptions (1.4), (1.5), and (1.7) together are referred to as nondegeneracy.
They lead to our classifications of heteroclinic orbits into twisted and nontwisted as
follows.
THE BIFURCATIONS FROM A TWISTED HETEROCLINIC LOOP
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W
655
e-
e
W’’
e;
a2
W
(a)
(b)
(c)
FIG. 1.3. (a) a nontwisted loop, (b) a single twisted loop, (c) a double twisted loop.
Let
e-= lim (z*,(t)-a)/llz*,(t)-all,
(1.8)
e-=
lim
(z*(t)-a)/llz*i(t)-all.
By (1.6) and (1.7), they are unit principal eigenvectors. See Fig. 1.3. Choose pi G
and q F W]o sufficiently close
to the other equilibrium a. Let p z(0) and q z(T) for a large T 0. Because of
the strong inclination propey (1.5) and the principal asymptotic tangency (1.8),
choosing p and qj close enough to a and a, respectively, implies
F WTo sufficiently close to the equilibrium a
(1.9)
e[ Tp,
ef Tq,
d= Tp, +span (el),
Nd Tq +span (el).
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656
o
DEYG
Since Tz(,), 0 <= <- T, defines a homotopy from Tp, to Tqj, the following definition is
justified (see Fig. 1.3).
DEFNa’ION 1.1. Let Fi be a nondegenerate heteroclinic orbit connecting two
and ef point to opposite sides of Tp, and
simple saddles. Fi is said to be twisted if
it
is
nontwisted.
respectively.
Otherwise,
Tqj,
For the heteroclinic loop, cl (F1 (-.J F2) {a, a2} (-J 171 (-J F2, it is called double twisted
if both F and F2 are twisted, single twisted if and only if one of them is twisted, and
nontwisted if otherwise.
As the last assumption we assume
e
(1.10)
F’E a x2---, d is a generic C vector field with two parameters a
(a l, a2) E2, having a nondegenerate heteroclinic loop at a 0. Here, the
regularity r >_- 8.
By genericity we mean that our results will hold for a residual subset of C (d X 2, d
in the weak Whitney topology of Ck-convergence (see, e.g., Hirsch (1976)). To be
more precise, we include the following as equivalent conditions for (1.10):
(1.10a)
The continuation of Fi: Given the fact that the heteroclinic orbit is
a codimension 1 object by (1.4), we assume for every (a 1,0) there
exists a heteroclinic orbit F2(al) {z*2(t, al)" } from a2 to al such that
z2* is a C two-dimensional surface in the phase space. Similarly, z*(t, a2)
forms a C surface in d as (t, a2) takes all values from 2 and each of
the t-curves is a heteroclinic orbit from a to a2;
(1.10b) The transverse crossing of the stable and unstable manifolds along F:
d, (1, 0)
d(0,
lim--0
and
lim--0,
where dl(al, a2) denotes the continuously varying distance of W’;(a)fE and
W(a)Z with d(0,0)=0 and E is an arbitrarily chosen Poincar6 cross
section to F. A similar description applies for d2.
Note that since the Poincar6 mapping introduced between any pair of two cross sections
is diffeomorphism, the nonzero limiting property in (1.10b) above is independent of
the choice of the cross section Z.
Finally, to state our main theorems we need a few more terms. Let be a small
tubular neighborhood of the heteroclinic loop cl (F1 t_J F). A k-periodic (k-per) orbit
is a periodic orbit which is contained in OR and has winding number k in OR. Similarly,
the closure of a k-homoclinic (k-hom) orbit has winding number k in OR. Accordingly,
a k-heteroclinic (k-het) orbit F from a to a2 is such a heteroclinic orbit that cl (F t_J Fz)
has winding number k + 1. Similarly, we define a k-heteroclinic (k-het2) orbit from a
to al. Thus, F and F2 themselves are zero-heteroclinic orbits. Note that as long as R
is chosen small enough, the above definition is independent of any particular choice
of OR. Also, the terminology extends canonically to small perturbations of the vector
field F(., 0).
The first theorem, except for the directions of bifurcation and the k-heteroclinic
orbits, is taken from Chow, Deng, and Terman (1990).
THEOREM A. Suppose F is a generic two-parameter family of vector fields having a
nondegenerate heteroclinic loop connecting two relatively contractive and simple saddle
equilibria at a O, i.e., (1.2)-(1.5), (1.7), and (1.10a, b) are satisfied. Then there exists
a small tubular neighborhood OR of the heteroclinic loop cl (F kJ F2) and a neighborhood
THE BIFURCATIONS FROM A TWISTED HETEROCLINIC LOOP
657
of the bifurcation point a 0 in the parameter space such that up to a nonsingular and
differentiable change of the parameters, which leaves the axes invariant as sets, the
following is satisfied (cf. the bifurcation diagram Fig. 1.4):
(i) There exists a C r-7 curve a2 hOml (al) with al > 0 in f such that there exists
a homoclinic orbit to a in
if hOm l. Moreover, hom is asymptotically
if and only
+
and
the direction of the bifurcation is determined
the
as
to
a
a
0
tangent
positive 1-axis
by the twist of the heteroclinic orbit F2 as follows:
>0 if F is twisted,
hOml
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f
<0
otherwise.
to al for k >- 1 if F2 is not
twisted and there exists at least one 1-heteroclinic orbit from a to al on a C r-7 curve;
a= 1-het (al) for a >0 otherwise. Moreover, 1-hete is asymptotically tangent to the
0
a -axis as a
(iii) Analogous statements hold for homoclinic orbits to ae and k-heteroclinic orbits
from a to a2.
(iv) Let A={(al,a2):a2>hOml(al) if al>O or a>hom2(a2) if a2>O}. Then
there exists a periodic orbit in 71 if and only if a A.
(v) The homoclinic and periodic orbits do not coexist in all for a given parameter.
They are all unique and are 1-hom and 1-per orbits, respectively.
(ii) There does not exist any k-heteroclinic orbit from a2
+.
Our main result is as follows.
THEOREM B. In addition to the hypotheses of Theorem A, suppose the stable
manifolds of the equilibria a and a are all one-dimensional; then the following is satisfied
(cf. Fig. 1.4):
(i) If F2 is twisted but F1 is not, then the 1-heteroclinic orbit from a to al is the
Moreover,
unique k-heteroclinic orbit for all k >= 1 and a
.
1-het (a,) >hom, (al) for al > O.
(ii) If the loop cl (F U F) is double twisted, then there exist two sequences of C r-7
curves a2=k-het(a) with a>O and al=k-hetl (ae) with a>O in f, respectively,
satisfying
0 _--< k-het2 < (k + 1 )-het < hOml
all k >= 0 such that there exists a k-heteroclinic orbit from a to a if and only if
k-het2. Moreover, it is a unique heteroclinic orbit in 71 with respect to the parameter
and k-het is asymptotically tangent to the a -axis as a 0 +. Furthermore, the homoclinic
bifurcation curve hOml is inaccessible from below in the sense that for every al
for
a
-
k-het (a,) horn, (a,)
An analogous statement also holds for the k-hetl
-
as k- +.
curves.
Theorem A provides us with a useful clue to the twist of a given heteroclinic loop:
the two zero-heteroclinic continuation curves (which are the parameter axes in our
theorems) divide the neighborhood f into four sectors. The 1-homoclinic bifurcation
curves hom and hom lie in one sector for a double twisted loop, or in two adjacent
sectors for a single twisted loop, or in two opposite sectors for a nontwisted loop.
Keeping this fact in mind, let us examine the following bifurcation diagram Fig. 1.5
for traveling waves of the FitzHugh-Nagumo equation
vt=Vxx+f(v)-w, w,=e(v-yw), e,y_-->O,
where f(v)=-v+H(v-a) with H to be the Heaviside step function and O<a<1/2.
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658
loop.
o ozy
0
hom
(a)
(b)
FIG. 1.4. The bifurcation diagrams for (a) a nontwisted loop, (b) a single twisted loop, (c) a double twisted
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THE BIFURCATIONS FROM A TWISTED HETEROCLINIC LOOP
659
tl-t t/2-het,//"
-////--hom, /
(c)
FIG. 1.4.--continued
A traveling wave solution (v, w)(x, t) is a function (v, w)(z) of z=x+ct, c>=O.
Let u(z)= v’(z), then vc (v, u, w)(z) satisfies a first-order system of ODE
(1.11)
v’=u, u’=cu-f(v)+w,
w’=e-(v-yw).
c
For fixed 0 < a < 1/2 and 0 < e << 1, numerical as well as rigorous arguments from Rinzel
and Terman (1982) show that the front curve Or, on which there exits a front wave
connecting the rest steady state ff to the exitable state g as shown in Fig. 1.6, crosses
transversely the back curve 0B, on which there is a back wave from g to ft. Thus, at
the intersection point 0* there exists a front wave and a back wave traveling at the
same speed. This gives rise to a heteroclinic loop. Their numerical simulation also
shows that the impulse curve 0e, homoclinic to if, and the g’-impulse curve 0,
homoclinic to g, also bifurcate from the loop at 0". Note that their bifurcation directions
of asymptotic tangency are exactly opposite our bifurcation diagram Fig. 1.4 for
Theorems A and B. This is due to the fact that the steady states ff and g are relatively
contractive simple saddles having one-dimensional stable manifolds only for the time
reversed (z--z) system (1.11). What is most remarkable about these two curves is
that both of them lie in the same sector in the parameter space. In fact, this has been
rigorously proved (see (3.8) and (3.10) from Rinzel and Terman (1982)). Unfortunately,
however, we can only speculate that Theorem A suggests the double twist for the
heteroclinic loop. Indeed, we are facing a tantalizing dilemma here: either it is feasible
to check the transverse crossing condition (1.10b) and the double twist of the loop
due to the piecewise linearity of f but the vector field is not smooth enough, or it
becomes a fairly open problem to do so for a smooth vector field, e.g., the usual cubic
function f= v(v-1)(a-v). Nevertheless, the implication is interesting: for given
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660
o
DENG
OF
73
7
3’2
7
(a)
(b)
FIG. 1.5. (a) A heuristic bifurcation diagram produced from Rinzel and Terman (1982); (b) The conjectured complete diagram.
w
FIG. 1.6. The conjectured twist for the front-back wave loop.
parameters e, 0 < a < 1/2 and 0 < Y3 Y << 1 there would be infinitely many fronts traveling
at different speeds. The more "humps" a front were to carry the slower it would travel.
If the humps were "too many" (infinity) the traveling wave arising from the rest state
but would return to itself after a
would never be able to reach the exitable state
long excursion. Slowing down a little, it would become a traveling train, or periodic
orbit for the ODE. On the other hand, push y slightly to the right of Y3, the above
scenario would repeat for back waves and q-impulses.
By their numerical evidence on the stability of the primary front and back waves
with respect to the PDE, as well as other authors’ results on somewhat related stability
661
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THE BIFURCATIONS FROM A TWISTED HETEROCLINIC LOOP
problems, it has been demonstrated that the stability of a given impulse is closely
related to the direction at which the stable and unstable manifolds cross transversely
as the speed parameter c varies (see, e.g., Evans (1972), Jones (1984), Kokubu, Nishiura,
and Oka (1988)). Thus, we find the second implication is most interesting: there would
be infinitely many stable transition waves connecting two stable patterns.
However, all of these phenomena do not appear in the "chaotic" parameter region
discussed by Evans, Fenichel, and Feroe (1982) and Hastings (1982), where for a given
speed there are infinitely many impulses and traveling trains due to the Sil’nikov
saddle-focus homoclinic explosion for (1.11) (see also Sil’nikov (1967)), whereas there
would be a unique traveling front, or back, or impulse, or traveling train, except at
the bifurcation point 0* in our case. Indeed, as long as there are two bistable steady
states as shown in Fig. 1.6, the equilibria are not saddle-focus. Nevertheless, we would
probably not be too surprised by the enormous, stable, yet not "chaotical" transporting
capability that a nerve axion would inherit if our conjectures were true.
In contrast to our conjectures above, we will discuss the existence of nontwisted
heteroclinic orbits and thus the limited number of connections between two equilibrium
states for another type of reaction diffusion systems in 7. We will also discuss in that
section some ways newly discovered by other authors to check all the nondegenerate
and generic conditions (1.4), (1.5), (1.7), and (1.10a, b) for their examples to which
our theory is immediately applicable.
2. Preliminaries. This section is devoted to introducing the Sil’nikov variables for
a Poincar6 map around the loop.
Let 0< 6o be a small number and B(6o) --{Z: Z--Z ,’’’, z(d)), ]z(i (0} be the
6o-box of the origin. Let the coordinate be locally normalized near the equilibria
so that (x, y)=0 corresponds to z ai and the local stable, unstable manifolds are
given by the x-axis and y-axis in B(6o), respectively; i.e., Woc {y 0} B(8o) and
W’oc {x=O}B(6o). In addition, the directions of the first x-component x (1) and
and e-,
the first y-component y(1) are chosen to be the unit principal eigenvectors
respectively, as in (1.8). Let the points p and q from (1.9) in the definition of twist
(1)
be specifically given as Pi (xi, O) and q (0, y). We can assume xi
8o and Yi(1) 8o
e-
because of assumption (1.7) for the asymptotic tangency of F along the principal
eigendirections.
Let E and E’ be two small cross sections, or (d- 1)-dimensional boxes B(81)
and e-, respectively
with 0 81 80, centered at p and qi and perpendicular to
and
Poincar
sections
are
cross
provided 8o
(see Fig. 2.1). They
81 are sufficiently
be
of
whose
initial
the
subset
those
small. Let
trajectories in
points (Xo, yo)eE
This
at
the
time
exit cross section E’ at (xl, Yl)
corresponz(x0, Yo).
B(8o) first hit
dence gives rise to the local Poincar map I-I"
E’ by (Xo, Yo)-* (xl, Yl). Similarly,
by the continuous dependence on initial data and parameters we can define a global
Poincar map IIi E’ -* E. Here, without loss of generality Es is taken to be the domain
is a proper subset of E not containing any point from
of definition for I-Ii, whereas
the stable manifold Woc.
Let (, y)e a-1 and (x, )e a-1 be the normalized local coordinates on E and
E’ so that (0,0) corresponds to the center points Pi and qi, respectively. Indeed,
,y(")) (see
s=)-)i, and /=3-3i., where )---(X(2),
,X (m)) and )=(y(Z)
-()
the
is
snear
time
e
Sil’nikov
where
be
the
Let
principal
2.1).
a,
Fig.
i(a)
unstable eigenvalue for DzF(a, a). Then the Sil’nikov variable for the local map is
(s, :, 7) and the Sil’nikov domain is
e
r
r-
r
...,
A:= {(s,
,
o
662
DENG
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r
//"
q
constant
-r
.><> Q.:.-.y.r.._ r]
Y(/[
constant
e cross sections and te corresponding Poinear map en m
FIG. 2.1.
E2
ip2
const ant
1, n
2.
where So(a)= e -,(")o for some large but fixed to. Note that the dependence on a is
suppressed from rT, o’’, and ki.
It has been proved by Sil’nikov (1967) that for the initial point (Xo, Yo)e and
the end point (x,, y)e
with r-- r(Xo, Yo) time units apart, the initial Yo and the end
x, components are functions of the Sil’nikov variable:
yo := Y/(s,:, ’9, a) and Xl:=Xi(s,, "9, a).
Moreover, it has been observed by Deng (1989) that the maps p’A- o-, with (s, :, "9)with (s, :, ’9)- (X(s, :, "9, a), "9) are actually diffeo(, Y(s, sc, "9, a)) and p’" Ai
morphisms of class C r. Thus, p gives rise to a smooth change of variables for the
local Poincar6 map 1-I, which in turn is p’ under the new Sil’nikov variable. See
Fig. 2.2. More important, we have the following exponential expansion result.
PROPOSiTiON 2.1 (Deng (1988), (1989)). Let the strong stable manifold and the
strong unstable manifold also be normalized such that Ws= {x (1)= 0, y 0} and W
{x=0, y() 0} locally. Let vi(a) hi(a)/tx(a)-1 and (a) and i(a) be as in (1.2).
Then for
sufficiently small there exist C r-7 functions p(sc, a), ff(sc, "9, a),
Nil(S, "9, o), and R2(s, "9, a) over k such that
o-
’
r
,
X, s, "r], ce) qgi(, "r], ol s + + R
Y(s, sc, "9, a)= ,(, "9, a)s+ R,2(s,
(2.1a)
with q and
(2.1c)
No(s,
,
for all s, "9)
,
,
S,
"9,
"r],
ol),
a)
satisfying
(2.1b)
and Rij
,
,,
o
,
(i(, ’9, a)= emao+ o(([l / Inl /
IlJi "9, Ol e.6o + O((Iscl +1"91 + 8o)ao)
’9, ) satisfying
k
g ,1[= O(s l+v.+.’),
ID, R,11= O(s
IDe.)
,
) R,21 O(SI+"),
]O(e,,
A
,, 0 <--_ k <-_ r
]DsR,2[ O(sO’),
7, where
e,--(1, O,.. ",o)T m,
and
e,, =(1, O,..., 0) r
--F
),
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THE BIFURCATIONS FROM A TWISTED HETEROCLINIC LOOP
FIG. 2.2. The Sil’nikov change
of variables for the local map when rn
n
663
2.
and i>O is a constant not greater than rain {li(a)/txi(a), (li(OI.)--i(O))f ld(O)} for
all [a[ -< 60.
Equation (2.1a) is referred to as the exponential expansion, and i the expansion
coecient functions, and Rg, the remainders. From this proposition we immediately
have the following proposition.
PROPOSITION 2.2. For sufficiently small but fixed So, the domain
of the local map
contained
is
y)
the
on
sector
vertical
in
<
boundary point
Moreover,
every
H
for
(, y) (, (So, )), y) 6oSo/2. (See Fig. 2.2.)
It is also easy to see from (2.1c) that the functions X and
thus p and p,
can be C extended to s 0. From now on let us use the same notation for the extended
for the extended domain of A.
functions, but
Let us conclude this section with two lemmas which will be frequently used later.
LEMMA 2.3. Let the global map Hi be expressed as
P(x, a), y Qi(x, a),
under the new coordinates for
and Z. Let
.
,
__
,
,
,
DnP(O, O, O) 0
and M
D,Q(O, O, O) e
DnQ(O, O, O)
(a-2)(a-2)
where e (0, O, 0), or e. en both Mi and i are nonsingular for sufficiently small 60.
M
Proof Note first that all the column vectors of M except the middle one (0, e)
span the linear subspace Tp W7 E) + Tp W} E}), which has dimension d 2 by
the hypothesis (1.4) and the choices of E, which are transverse to the flow. On the
other hand, the remaining column vector is approximately parallel to the principal
unstable eigenvector e; by the exponential expansion property (2.1b). Thus, M
achieves its maximal rank by (1.9). Moreover, the strong inclination propey also
implies DnQ(0,0) is a diffeomorphism, thus the truncated square submatrix Mi
achieves its maximal rank d- 2 as well.
664
o
DENG
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LEMMA 2.4. Let Mi be the same as in Lemma 2.3 above with e
Ni
with f--(0, 1,’’’, O)W,
(2.2a)
(0, 0, 0)
and
LDnQi(O, 0, 0) f
(d-1)(d-1)
(0, 0,’’’, 1). Then there is a constant mo so that
lirn
6o->0
[det Mil> > 0
mo
0
and
(2.2b)
lim det N/= 0.
6o->0
o
-
0 and the strong inclination
Proof (2.2a) is true because of p(0, 0)/6-* e, as
property (1.5) and (1.9). Since (0,f) is contained in T,jW.u, (2.2b) is also true for the
[3
same reason.
3. Entrance and exit sets and their extensions. In this section we only consider the
heteroclinic connections from a2 to al. Analogous analysis and result can be immediately extended to the a to a2 connections. Again, the parameter ce is suppressed from
the text if no confusions arise.
Let Ex := Wloc fq Z denote the intersection of the local unstable manifold of as
with the exit cross section Z given in the previous section.
is referred to as the
initial exit set of W. It is obvious that there is a heteroclinic orbit (from a2 to a) if
and only if there is a solution of the initial exit set which also lies on the local stable
manifold of the other equilibrium a. Thus, we need to closely follow the images of
the initial exit set under those successive local and global Poincar6 maps. To be precise,
if we set the image of an empty set under a given map to be empty, then all the
following sets are well defined:
Ex
En2k := H2(Exk)
Ex/k:=Hi(En/k0p(A)),
i--1,2, k=l,2,...,
En k1: II(Ex 2k- ),
r
where p" Ais the Sil’nikov change of variables and p(Ai) is contained in the
domain
of Hi. En/k and Exk are referred to as the kth entrance set and the kth exit
set of W near a, respectively. They might be empty except for the initial exit set Ex2
and the first entrance set En near a. Nevertheless, we have the following.
PROPOSITION 3.1. There exists a (k-1)-heteroclinic orbit from a to al sufficiently
close to the loop F[’2 if and only if En,J-, Ex,J- for l<=j<-k-1, i-1,2, and
r
En k
Woc
.
By definition, for every point (:, y) Enlk
with z,J.
there exist (0, r/) Ex2 and
z o-
(,J., y), 1 -<_j -<_ k- 1 and i-- 1, 2 from the orbit of (0, r/) such that H21(0, r/)
H_ H2(zk-) (, y). Using the "pull back, we have a
z, H2 H(z) z,.
(s,
unique
r/i)A satisfying z i= Ps(r)- Thus, the following proposition is
valid.
’
,
PROPOSrrION 3.2. (3.2a) The kth entrance set to a, Enlk, is nonempty if and only
the
if following system of l (2k- 1)(d 1) equations has solutions for the l + n 1
unknown variables q, 1,
y with sr/=(s{, i, TJi) Ai satisfying the con<-_
and
straints si > 0 for all
2
1
<-j k- 1"
1,
,
n,(o, ) p()
1-I,(p’(’-’)) (, y);
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THE BIFURCATIONS FROM A TWISTED HETEROCLINIC LOOP
665
(3.2b) The kth exit set from a l, Ex k is nonempty if and only if after replacing (, y)
k
with k A the same statement of (3.2a) holds true"
by Pl(’l)
(3.2c) The kth entrance set to a2, En2k, is nonempty if and only if the following
system of 12 2k(d- 1) equations has solutions for the 12 4-n- 1 unknown variables
/,
with
i (s ji,
rl
rlk :, Y
srll,
Ai satisfying the constraints s > 0 for all
1, 2 and 1 <-j < k:
I’[21(0, y) p,(’)
nl(p’()) (, y);
(3.2d) The kth exit set from a2, Ex2k
is nonempty if and only if after replacing (, y)
by p(k) with k2 A2 the same statement of (3.2c) holds true.
Solving equations (3.2a-d) is more difficult with the constraints s > 0 than without
them when those maps p and p’ are considered as the extended maps on the extended
domain zi introduced in the previous section. Let us now study the extended equations
and leave the consideration of the constraints to the next section.
Note that each system of the extended equations (3.2a-d) might locally define an
(n 1)-dimensional manifold near the origin of the/-dimensional Euclidean space NI,
where
li + n- 1. In fact, we have the following.
LEMMA 3.3. Suppose the equilibria are simple saddle, relatively contractive, and the
heteroclinic loop is nondegenerate. Then there exists a small constant 6 > 0 independent
of k such that in the 6-box B(6) of the origin in Nl each of the systems of the extended
equations (3.2a-d) defines an (n 1)-dimensional manifold J/[ in B (6) which contains
the origin and can be written as the graph of a C vector-valued function of the last
n 1 components, i.e., either 33 (y(2,..., y() or rh, where
11 + n 1, or 1 + n 1,
accordingly.
Proof. We prove the lemma by the implicit function theorem for equation (3.2a)
only since the other cases are identical. By using the notation II 0 (Pi, Q) p (i, Y)
and p’ (X, r/i) from the last section, we see that solving equation (3.2a) is equivalent
to solving the zero of the following equation:
r,
a,(t’, 3)
o,
where
-l+P(O, )
YI + Q2(O,
--2 d- PI(X1, ’1)
(?,g)=
-
-Y2+Q!(X, hi)
+ P.(x"k-1
-y + Q2(X2k-1
X= X(s,
,
Y=
k-1
k-1
q2
k2-1 k2-1
k-1
y(,
Y(s, so{, r/{). It is obvious that the existence of the
rl{) and
heteroclinic loop FlU F2 (at a =0!) implies (0, 0)=0. A simple calculation yields
that the ll x ll square Jacobian matrix 0/0ff at (sr, fi) (0, 0) has the following diagonal
property
det _--7 (0, O)
Idet diag (M2, M1 ,’’’, Ml, M=)I,
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666
o
DENG
where the diagonal blocks have the forms of Lemma 2.3 with all the blocks Mi except
the last M2 taking e=-Oj(0, 0, 0). Therefore, the Jacobian matrix is nonsingular.
Hence, by the implicit function theorem there exists a 3 > 0 such that (’, 33) 0 defines
in B(3) an (n- 1)-dimensional manifold
(containing the origin at a 0) which is
the graph of a C function of the variable 33. Moreover, because of the diagonal block
structure it is not difficult to see that 3 can be chosen to be independent of the number
13
of the equations 11 (2k- 1)(d 1).
Let P be the canonical projection from E=Itl-d-lE d-1 onto the last d-1
components. Then Lemma 3.3 implies that the projection PJ// of the manifold J/ is
also the graph of a C function of the last n- 1 coordinates. Therefore, we have the
following.
DEFINITION 3.4. n/k:=P and lxk:=p’(P) are called the extended kth
entrance set and the extended kth exit set (of W) near ai, respectively, according to
whether is taken to be the manifold defined by the extended equations (3.2a-d) for
the entrance sets or the exit sets in Lemma 3.3.
Since all the extended entrance and exit sets exist in some small but fixed 3-box
B(3) of the center points on the entrance and exit cross sections, respectively, we can
of the heteroclinic loop cl (F1 F2) such
easily construct a tubular neighborhood
that the intersections of with the entrance and exit cross sections (E and E’) are
exactly those 3-boxes. Thus, we only need to consider the real entrance and exit sets
(of W) in B(3) and rename Eft k :-" Eft k B(3) for simplicity of notation, where/ n
or x. Now we are read,y to compare these sets with their extensions. Because the kth
extended entrance set Enk near al is a graph over the last n- 1 coordinates fi on E
and the nonemptiness of its intersection with the local stable manifold W of a forces
33 0, we have proved the following.
COROLLARY 3.5. If a (k-1)-heteroclinic orbit from a2 to al exists in 11, then it
must be unique (for the corresponding parameter).
k
COROLLARY 3.6. Let
=graph G, H) with
G(fi) and yl)= H(fi). If
dim W dim W 1, IH(33)1 <_- 3oSo/4 and the derivative Ion(fi)l <-- 1/2for all Ifil < 3, then
nk O tri f if and only if 0< H(0), where 3o and So are as in Proposition 2.2.
Proof Since dim W dim W 1, G(y) 30, the x-component of the center
of the entrance section E T. Since the boundary point (30, Yo)=(30, Y(so, O))6Otr
satisfies >- 3oSo/2 > max HI by Proposition 2.2 and our assumption, the two boundary points (30, Yo) and (30, 0) must be in different sides of k if 0< H(0). The path
connectedness of implies there must be a point (30, Y(sl, 30, 0)) lnk. This shows
0< H(0) is sufficient. To show that it also necessary, suppose it false, i.e., H(0)_-<0.
Since nk graph (H), then 1)= n(0)-<_0. Thus, yl)-yol<=lDHIl[<-1fi[/2 by our
assumption. Let Yl 6 n/kf-)tr7 # ; then Ifill<y 1 holds true by Proposition 2.2. It
[-1
follows that Ifil-yo<y])-yo)<-ll/2 and 0-< 13311/2<y01), a contradiction.
Our main result of this section is as follows.
THEOREM 3.7. Let lnk graph G, H) with
G() and y 1 H(). If the derivak
<
tire of H satisfies DHI
k, where
for all k, i= 1, 2 and Eft is nonempty, then E k
=
n
:
yo
n
o-
yo
.t
ni or Xi.
Proof. Suppose it is false; then there exists a first E k such that E k k. We
claim first fl x. If fl ni then there exists a point Po n/k_ Enk. Hence, there exists
< 0, where
and the
a point (, 3)
with Po P(, 3), and (, 3) has at least one s j=
projection P are as in Definition 3.4. To be precise, say s =< 0.
Let us first note the following: Denote
/3 k according to whether it is obtained
by the extended equations for the kth entrance set when/3 n or the exit set when
/3 xi. Now it is not difficult to see that if p /3 k then the point q, whose components
since q satisfies
consist of the first (2j 1)(d 1) + n 1 components of p, belongs to
x
667
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THE BIFURCATIONS FROM A TWISTED HETEROCLINIC LOOP
the first (2j-1)(d- 1) equations (cf. Proposition 3.2). Similarly, if q is obtained by
keeping the first 2j(d- 1)+ n- 1 components of p, then q is in gx.
Now, resume our assumption s-<0 and let qo be such a truncated point of Po
which belongs to gx. Then P(qo) /x
since _-< 0. This contradicts our assumption for
Hence, the claim holds true.
Since we have
it follows that there exist
(ss, r/s) zi with
fq
for the
and p(sr2)
j 1, 2, s, =< 0, and s2 > 0 such that p(srl)
same reason as above on the truncated point q. Since
is path connected, being a
graph over the path connected set [931 < 6, there exists a
(So, sCo, r/o) 6 ,i with So 0.
Hence, P(’o) (sCo, Y(0, so, r/o)) (sCo, 0) 0r f’] Wloc. That is, (sCo, 0)
graph (G, H). This implies ly)l _-<
I1--< 11/2 by our assumption for all (s y)
fq
It follows that
since by Proposition 2.2 ]93] < y() for all (sc, y)
This
[3
contradicts p,*.(’2) En f) c ln f) r,*..
Ex
n.
Ex x,
s
n- Enn
sro
n
o-
,r
IDHI
,En o,
.
nn.
4. Bifurcation equations. From now on, we shall spell out the parameter explicitly
wherever it is necessary. In this section we consider the constraints s{ > 0 in terms of
their sign changes with the parameter, in particular the sign changes of y(1)= H(0).
Here (G, H) gives the graph of the extended entrance set. For this reason, we consider
the following equations"
1-I2(0, r/, c)= (, y),
II21(0, r/, a)--P(’l, a),
II21(P(2 a), ce)= (, y),
172,(p(sr, c), a)= p(’,, a),
II12(P(, ce), c)= (sc, y),
17112(P(’1, ce), a)= p(sr2, a),
(4.1a)
(4.1b)
(4.1c)
(4.1d)
(4.1c’)
(4.1d’)
which introduce every new s{ or y(1) into our recursive construction of the entrance
and exit sets in the last section, where
(&, sc, r/).
Note that each of the systems above defines a system of d-1 equations with
l=(d- 1) + (n- 1)+2 unknown variables for (4.1a-b) or/2- 2(d- 1)+2 variables for
(4.1c-d’), including the parameters a and a2. Thus, presumably, each of them defines
an (l (d 1))-dimensional manifold in R6 accordingly. Indeed, we have the following
lemma.
LEMMA 4.1. Suppose the conditions of Theorem A are satisfied. Then there exists a
small constant 6 > 0 such that in the &box B(6) of the origin in t, each of the extended
equations (4.1 a-d) defines an (l (d 1)) -dimensional differential manifold in B (6)
which can be written as the graph of a C function of the last n- 1)+ 2 variables (, a)
or r/, a) when li l or the first m and the last n 1) + 2 variables si, i, a) or
(&, r/, a) when li 12, accordingly. Moreover, up to only one nonsingular and differentiable change of the parameter for all the equations (4.1a-d’) considered, the following
bifurcation equations are satisfied for solutions to (4.1a-d’) with the corresponding
,
,
alphabetical order:
(4.2a)
(4.2b)
(4.2c)
(4.2d)
(4.2c’)
m,y (1)+
l
+ m,y (")= a+
Og2 -II-
mly (I)+
q-"
m.y (")= a2 + ’r2(a )s +u2 + 0(([/:721 -iI- [$21 <)Is=l l/t"2 + 13312),
)S12 +v2 -’1- O((l2l -’[- ]S2I ff2)lS2] 1+"2 + (In,I + ISll
t/0/lY (’>+’’" +tfi,Y (" O1-71(O)S1l+Vl /O((I,I/Isll 1 )IS, 1+ "’/11 2)
S1
2 "It- T2(
668
BO DENG
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(4.2d’)
where vi, f’i are as in Proposition 2.1. Here mi mi(t) are differentiable functions of a
satisfying 1/(26o) < ml <2/60, mi/ml o(1) as 6o-O fori # 1 and the analogousproperties also hold for ri. Moreover, the scalar functions, ’i % (a), called twist functions, are
nonzero, differentiable, and satisfy
(4.3)
’i(0)
<0
>0
if F
is twisted,
otherwise.
Furthermore, the change of the parameters leaves the parameter axes, as sets, invariant,
but may reverse their directions.
The basic framework for the proof of this lemma, in particular, the derivation of
the bifurcation equations through a modified Lyapunov-Schmidt reduction, has much
in common with the spirit of Chow, Deng, and Terman (1990). Thus, we will prove
it in the Appendix with necessary modifications given to the twist terms ’(a)s +’ and
the order estimates on the higher-order terms.
The following corollaries concern the conditions of Corollary 3.6 and Theorem
3.7 when the parameter is taken into consideration.
COROLLARY 4.2. Let n/k= graph (G(., a), H(., a)) with
G(;, a) and y(=
H(, a). Then 6o and 6 can be chosen sufficiently small but fixed such that IH(, a)l <=
6oSo/4 and IDH(f, a)[<=1/2 for all ]1, ]al<6 and k>=l, where D is the differentiation
operator in y.
Proof Using (4.2c) or (4.2c’), we have
=
[max
IDHI t.[maxil
lY<’>[ <=
and
(6)]/m1()
+ 0(6%)]/m1(o)=o(1)+26o0(6"),
lmi(a)’6 /
Imi(a)l
(1)6 /
where Vo min {/21, /’*2, 1, 2}" Choosing 6o and 6 so small but fixed implies the desired
estimates.
COROLLARY 4.3. 6 can be chosen small so that if ln/k= graph (G(., a), H(., a))
crosses the stable manifold Wloc f’l E, then it does so transversely in aj in the sense that
0H(0, a)/0aj>0 for I1<, In other words, if H(O, a)=0 then for the fixed ith
component o a i, H(O, ce > 0 if and only if aj > o0
Proof. Using (4.2c) or (4.2c’) again, we have ml(a)H(0, a)=% + o(11/o) with
2}. Thus
Vo min { v, v2,
,
( [ Om----A
0H(0, c)/oc_-> 1
H(O, a)
< by an appropriately chosen small
COROLLARY 4.4. The first entrance set Enll to al intersects the domain
of the
local map nonempty if and only if 02 > O.
Proof By (4.2b), s=2/o(InlllSll/lsl/,), implying s>0 if and only
for
cr
if
a>0.
5. Proof of Theorem A. As we mentioned earlier, Theorem A has been proved in
Chow, Deng, and Terman (1990), except for the directions of homoclinic bifurcations
and the k-heteroclinic orbits. Thus, we are going to outline the proof from that paper
and provide the necessary details for the other part of the proof.
669
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THE BIFURCATIONS FROM A TWISTED HETEROCLINIC LOOP
Consider the Poincar6 map II1 rI21 II2 I-[2 II from a subset of 0" into the
entrance section E near al. Using the Sil’nikov changes of variables, we can similarly
reduce the problem of finding periodic points of II1 into solving a system of equations
for the unknown Sil’nikov variables with the constraints si > 0. The conditions of the
nondegenerate heteroclinic loop and the relative contraction for the simple saddle
equilibria imply that the extended system has a unique solution parametrized by a by
the implicit function theorem. This uniqueness allows us to consider the simple
homoclinic, periodic orbits only. Thus, by the implicit function theorem, we solve
(’*, ’2*)(a)= (s*, :*, 7*, s*, :*, /*)(a) as the solution for the extended equations
I-I2(p’(’l, a), a)= p(’2, a) and Hl(p’(’, a), a)= p(sr, a), where ri (s, :, /i)
into the bifurcation equation (4.2d’)
with I’*(a)[ O([a[). Now, substituting sr* and
and (4.2d), we have s*2= a +O(la[ +o) and s*=a+O(la[+). It follows that the
map a- (s*, s2*) is a diffeomorphism. Thus, the sector A for the periodic orbits is
given by s* > 0 and s2* > 0 and the curve hom for 1-homoclinic orbits from a to a
is given as a piece of the boundary 0A with s*- 0 but s2* > 0. Substituting s* 0 and
s2* > 0 into (4.2d’) and (4.2d), again we have s*2 a + O([a[)[s*[ +o, implying al >0,
’*
and
o
+ ()*’+ + o([ [)[*1 ’+,
az=hom (a)=[-Zz(a)+O(lal)]]s*2[ +2. Therefore,
(.)
the bifurcation direcimplying
tions in (i) hold true because of (4.3). Finally, to complete the proof, we only need to
prove (ii).
Let us consider the 3(d- 1) equations 1-I2(0, rt, a)= p(sr, a),
II,(p’(’, a), a) p(’2, a), and 1-I2,(p’(’2, a))
including the two parameters al and
following 3(d 1) 1 equations first:
a2,
(:, 0) for 3(d-1)+l variables,
and assume
F2 is twisted first. We
solve the
-
-, + P2(0, ’q, a)
--rl+ Q2(0, ’q,a)
P(X1, r/l, a)
=0,
(, )= --2
+
Y2 QI(X,, "rl,,
(5.2)
02
with
where
"-
(Q(),
-
-+ n(x, n, )
, Q(f)), and solve the leftover equation Q(I)(X2,
(r/, s, :,
, s, :,
r/,
r/:, a)=0 later,
:). The existence of the heteroclinic loop implies
(0, 0)=0 and a simple calculation shows the Jacobian square matrix satisfies
det
(0, 0)
[det diag (M2, M,, M)[,
where Mi and M are the same as in Lemma 2.3 with e being (0, 0) in M. Hence,
it follows from Lemma 2.3 that the Jacobian 0/0sr(0, 0) is nonsingular and sr can be
solved as a C function sr* of a satisfying ’*(0)=0 by the implicit function theorem.
Substituting sr=sr*(a) into the remaining equation Q(I)(X*, r/*, a)=0, we find it
equivalent to solving a from
(5.3)
s*=P2(X2*
O= Q:(X*
’0* a),
n*
a).
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670
o
Dn6
Notice that this equation has the form of the connecting equation (4.2c) and the
corresponding bifurcation equation (4.2c) of Lemma 4.1 applies Thus, it is equivalent
to
Since s2*,
o(lal), this equation always has a unique solution a2: 1-het (al) for
every a by the implicit function theorem Since F2 is twisted, then r:(a)< 0. This
implies 1-het>0. Also 1-het= O(lall’/). To see if the constraint s*(a)>0 and
s*(a) > 0 are satisfied at a 1-het we need to consider the other two bifurcation
equations (4.2b) and (4.2d’) corresponding to the first two connections:
*=
(5.5a)
S*l o=+ o(In* lls* l+ls* l/,),
From (5.5a) and a2 1-het (a)> 0 it is obvious to see that s* >0 is automatically
satisfied. Moreover, s* O(a2)= O([a11+2). Substituting this order for s* into (5.5b)
yields s2* > 0 if and only if a > 0 since a is the leading term in the right-hand side
when s* and a are of order O(la,l+). Let 1-het2 := 1-het I,>o be the desired curve
To show the nonexistence of k-heteroclinic orbits for k>-1 under the nontwist
assumption for F2, let us solve a system of equations similar to (5.2). Analogously, it
is equivalent to solving those a from the following bifurcation equations so that
j*
si (a) > 0 for all i,j:
1"
Ol1-1"- ’,Is,1"
S2
(5.6)
0
=
/
1+
’+ o((l’,*l+lsl*ll)lsll*ll+l+(lnJ*l+ls’*lOls*l)
=ls*l /= / o((1*1 / Is*l =)lsg*l /=).
Since r > 0, the last equation implies a2 < 0. This forces
This completes the proof.
s1" < 0 from the first equation.
6. Proof of Theorem B. To prove Theorem B we need the following three lemmas.
When dim W 1, two given graphs
graph (/-/) over 33 on the entrance set
E are denoted by E =< E if H(33) <- H2(33), or E1 < E2 if H(33) < H2(33) for all common
y. A point p=(6o, y)E is said to satisfy p-<_ (<)E graph (H) if y()-< (<)H(33). In
what follows, let
be the corresponding first entrance set of
W’ocf3 E’ and
W to a. Their definitions are analogous to Ex2 and End, respectively. Now we have
Lemma 6.1 (cf. Fig. 6.1).
to ai
LEMMA 6.1. If dim W1--dim W 1 and all the entrance sets Fni and
up to a given number k >-j are nonempty graphs over the fi-axis, then
E
Fx:
Fn
En
(6.1a)
If F:
(6.1b)
If both F1
Proof.
is twisted but
F1 is not, then
En < En/< Fnl < Enl for 3 <=j <- k;
Fe are twisted, then
Fn < Enk < En ki-1 <
and
< En < En.
The proof is based on the following two simple observations: (1) The range
o-’ of the local map Hi contains all the exit sets and lies to one side of the corresponding
initial exit graph (FXl or
Ex).
The images of the exit sets under the global map
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THE BIFURCATIONS FROM A TWISTED HETEROCLINIC LOOP
671
lie below (above), i.e., <(>), the corresponding first entrance graph to the other
equilibrium aj if the connection Fi from ai to aj is twisted (nontwisted); (2) For a
given pair of entrance graphs near a given equilibrium ai ordered by <, the ordering
(<) for the consecutive entrance graphs near the other equilibrium aj is (not) to be
reversed if the connection F from a to aj is (not) twisted.
To show (6.1(a)), we have Exc G, thus En+l <
for all j_-> 1 by (1). By (2),
the single twist and En{ < End1 imply En2 < En] +1 for all j -> 2. Since
< En1/2 for j => 1
+1
< Fn2 by (2) (see Fig. 6.1(a)). To show (6.1(b)), we have
by (1), we have
<
+1
+
for all j>_- 1 by (1). The twist of F2 and
and
<
< Fnl imply Fnl <
2
for all j>_-0 by (2). The double twists and En < En imply En] < En2 by (2). Last, a
+1
for all j-> 1 (see Fig. 6.1(b)).
<
simple inductive argument shows
LEMMA 6.2. If dim W dim W 1 and [a2[ < al, then the kth exit set Exk from
a is nonempty if all the previous k- exit sets
from a are nonempty and the kth
extended entrance set
near a has nonempty intersection with the domain tr of the
local map H
exit set Ex2k-1 from a2 is nonempty. Since
Proof We first claim that the (k-1)st
kExk-, the (k-1)st entrance set En2 to a2 is nonempty. By [DH(.,a)[<-1/2 of
Corollary 4.2 and Theorem 3.7, we have Enk-l=
=graph (G(., a), H(., a)).
Using the bifurcation equation (4.2c’) we have
En
En
En
En
Fn
En
En
En
Ex
.nk
lnzk-
H(O, G) [Gl + O({o[l+%)]/ml(o) > O,
(a)
,(b)
FIG. 6.1. (a) F is twisted but
F
is not twisted;
(b) Both
F
and F are twisted.
En Fn
En
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672
BO DENG
..
since al is dominating by our assumption o > 121. It follows from
of Corollary 4.2 and Corollary 3.6 that
Hence,
n2-1 o-
Ex2-I II2(En2-
IH(Y, c)l -< 6oSo/4
r) #
En
The claim is proved. Obviously, this claim implies the kth entrance set
to al is
by Corollary 4.2 and Theorem 3.7. Thus, the
nonempty and equal to its extension
13
condition nfq # implies the kth exit set
near al is nonempty.
Combining Corollary 3.6, Theorem 3.7, and Corollaries 4.2, 4.3 above we have
the following important result.
LEMMA 6.3. If dim W dim W 1 and Pi Wiloc(a)f"l Ei ,nki at some a
o
o
nifqr if and
a
(a o1, a) with k >= 1, then for the fixed ith component ai ai,
is nonempty then the same statement
where
< Moreover, if
only if a >
holds true for En/.
Proof Let o graph (H(., a)). Corollary 4.3 implies H(0, a) > 0 for ai a if
and only if aj > aj.
Since Corollary 4.2 implies the conditions
< and ]HI _-< oSo/4
of Corollary 3.6, we conclude from Corollary 3.6 that
f3 r7 # for a a if and
By the condition DHI < of Corollary 4.2 and Theorem 3.7, we have
only if
r
a,
n
Ex
En
n
n
=> 7.
IDHI
.
En/ lni
Proof of Theorem B. (i)
Corollary 4.2 implies the conditions of Corollary 3.6. The
Hence, by
nonemptiness of the kth entrance set En k with k >= 3 implies En2 f3 o-1
Corollary 3.6, 0<H(0), where En2=graph(H), implying the center point
Wloc(a) f3E lies below the second entrance set En2. It follows from (6.1a) that the
existence of k-heteroclinic orbits from a2 to al is impossible for k-> 2. This together
with Theorem A(ii) proves the uniqueness of the 1-heteroclinic orbit from a2 to
Let 1-het2 be as in Theorem A. To show 1-het2 (a)> homl (a) for al>0, we
notice that Fn11 is nonempty in the spirit of Corollary 4.4. By (6.1a) we have Enl2 < Fn
En 1. It follows that with a2 increasing, the homoclinic connection Pl Fnll takes place
before the single heteroclinic connection Pl Enl2 does by the transverse crossing
property of Corollary 4.3 and Lemma 6.3.
(ii) First we zoom in the region of the parameter where (k 1)-heteroclinic orbits
(from a2 to al) can take place. We first claim there exists a (k-1)-heteroclinic orbit
only if 0<=ce2<hOml (tel) with al>0, where homl is the bifurcation curve for the
1-homoclinic orbit at a l. If a < 0 then En2 <
< p2 for all k >= 1 by Corollary 4.4
for k >_- 1.
and the twist of F1. Thus, Enk C? o-2
by Corollary 3.6. Hence, tnlk+l
If a2> hOml (al) then Pl < Fnl by Corollary 4.3 By Lemma 6 1 Pl < Fnl < Enlk for all
k_-> 1. This proves our claim.
The existence of these heteroclinic bifurcation curves (k-1)-het2 in the region
0--<aa<homl (al) is an immediate consequence of the following claim: in 0_-<a2 <k
homl (al) there exists a unique a2 (k- 1)-het2 (al) for every k _-> 1 such that Pl < Enl
if and only if a2> (k- 1)-het2 (al). We proceed by induction. When k 1, it is trivial
by the existence of the primary heteroclinic orbits and the transverse crossing property
of Corollary 4.4. Suppose the claim holds true for k- 1. Then, by Lemma 6.3 we have
for 0/2 < (k 2)-het2 (al); hence, (k 1)-heteroclinic orbits do not exist.
En -1 f-I
for (k-2)-het2 (al) % a2hom
Again, by Lemma 6.2 and 6.3 we have Enk
cel. On the other hand, Corollary 4.4 implies Fn2 f’l or2 # G5 and thus Fn] # G5 for these
parameters. The transverse crossing properties of Lemma 6.3 and Corollary 3.6 imply
It follows from the conFn l<pl for a2<homl (al). Lemma 6.1 implies Fnl <
tinuity of En] on the parameter that when 0<a-(k-2)-het2(al)<< 1, Enk-1 is
sufficiently close to the stable manifold W, and Ex k-ll is close to FXl Therefore,
holds. Since Pl Fnl < Enlk at a2 hOml (al), there must be an
Fnl <
Fn
r
En.
.
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THE BIFURCATIONS FROM A TWISTED HETEROCLINIC LOOP
673
such that Pl E Enlk at (a, ak-). Because Lemma 6.3 implies this crossing is transverse,
k-1
Let ak- := (k- 1)-bet2 (a). This completes the claim.
Pl < Enkl if and only if a2> a
Furthermore, by the property of transverse crossing, the function (k-1)-bet2 is also
differentiable by the implicit function theorem.
Finally, to show the inaccessibility of the homoclinic bifurcation curve hom from
below, we suppose to the contrary that (k- 1)-het2 (a l)- <hOml (eel) for some a
for all k. Let limk_, En k E Then E ->_ p > Fn at
Then, at a o (a a )
2 En k
0
a Thus, Pl E because otherwise
< Enk <p for sufficiently large k for En kl
would be sufficiently close to the stable manifold W and tnk/l would be empty.
Therefore, Pl < E But, in this case, by moving a2 down a little, i.e., 0< a-a2 << 1,
p < E=< Enk would still hold true for all k and a =(a a2). Thus, there would exist
a k such that a2<(k-1)-het2(al). This would imply Enk<p by our second claim
[3
above. This is a contradiction.
ce
o,
.
Fn
.
.
,
7. Remarks. (a) It seems that the bifurcation equations (5.6) are solvable for
and z2 are all negative. Incidentally, by neglecting
the higher-order terms, the truncated recursive formulas do give rise to a monotone
increasing set s_i* for i=l,...,k-1 with se1" =a+ra1+,.’>0 and amonotone decreasing set si* for i= 1,..., k with slk* := a+ z2[sk-I*I+2=0. Furthermore, the (k1)-het curve is then defined by the recursive formulas. Unfortunately, this argument
fails when those fuzzy error terms are taken into consideration. A similar situation of
our losing control over the full system appears in the homoclinic bifurcations with
resonant principal eigenvalues (i.e., h(0)-/Zl(0) in our notation) studied by Chow,
Deng, and Fiedler (1990). This is the reason we impose the condition dim Ws= m 1
and approach our problem topologically by considering the entrance and exit sets and
their extensions. We feel that this restriction may not be merely technical, since without
it the position of Fnl relative to Enk may behave in an unpredictable way.
(b) It can be easily seen from the bifurcation equation (5.1) that the asymptotic
tangency of the homoclinic curve homi is completely determined by the sign hj +/xj
for #j. That is, homi is asymptotically tangent to the ai-axis at a -0 for a relatively
contractive aj, or the a-axis for a relatively repelling ay, or tangent to none of them
for an a with principal resonant eigenvalues, i.e., hj +/x =0 at a =0. In all cases,
however, k-heteroclinic orbits are expected to bifurcate at a twisted heteroclinic loop.
In particular, as in the homoclinic doubling bifurcation for a twisted homoclinic orbit
(see, e.g., Chow, Deng, and Fiedler (1990)), a double homoclinic bifurcation will also
probably take place at a single twisted loop with the resonant eigenvalues, i.e.,
(1 + q)(1 + ,)- 1 at ce-0, or in the case where the heteroclinic loop is degenerate
(see Yanagida (1986)).
Relaxing the equal dimensionality dim W dim W assumption will also lead to
countable k-heteroclinic connections which, in contrast to Theorem B, take place in
an open set of the parameter space. This was observed by Deng (1989). Also, as the
principal eigenvalue (either stable or unstable) becomes a pair of complex, the system
itself at a --0 becomes rather chaotically complicated (see Tresser (1984) for the case
where dim W dim W, and Bykov (1980) where dim W dim W).
(c) The heteroclinic loop gives us another new bifurcation point which an oriented
homoclinic path can hit globally in the parameter space (cf. Fig. 7.1). The orientation
of a given homoclinic path is determined by the nonzero orbit index of the periodic
orbits nearby. See Mallet-Paret and Yorke (1982), Fiedler (1985), and Chow, Deng,
and Fiedler (1990) for more details on the orientation relative to the orbit index. Let
us suppose the homoclinic paths hom and hom_ are oriented as shown in the figure.
s j* (a)> 0 if the twist functions
’
674
o
DENG
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k-het,
/ hom
i
The twisted heteroclinic
FIG. 7.1.
is the orbit index.
loop_O-heq
If it is assumed to be
k-het
O-het
then the region A is on the right
of homi
curves.
W
-yw+O=O
=f(v)
FIG. 7.2
Following hom,, it will hit and terminate at the heteroclinic loop bifurcation point
a 0. But, right at this point, the homoclinic orbit to a, trades itself to a homoclinic
orbit to a2 and the curve hom2 arises from a 0 as if it is the continuation of hom.
For this reason, we may also call our heteroclinic loop bifurcation the homoclinic
trading bifurcation. However, if we "follow" (actually we do not know how at this
moment) a heteroclinic path in the double twisted case, we will find doubly infinite
heteroclinic trading partners at the homoclinic trading place a =0. Certainly, this
immediately complicates any "global heteroclinic path following" attempt. But it also
gives us one more hope that a global homoclinic path following result seems on its
way (see a detailed discussion from Chow, Deng, and Fiedler (1990)).
(d) While writing these remarks, I received a preprint by Kokubu, Nishiura, and
Oka (1988). I found that our notion of twisted heteroclinic loop has been propagating
faster than I could finish writing this paper. Their work demonstrates that the nondegenerate conditions (1.4), (1.5), (1.7), and (1.10a, b) are verifiable. Indeed, motivated
by the idea for the Mel’nikov function and the method of singular limit eigenvalue
problem developed by Nishiura and Fujii (1987) and Nishiura (1989), they derive not
only an analogous function to detect the transverse crossing of the stable and unstable
manifolds (also see, e.g., Kokubu (1988)), but also a computable twist function to
detect the strong inclination property and the twist of a given heteroclinic orbit at the
same time for the system of ODE for the traveling waves, (v, w)(x, t) (v, w)(x + ct).
The reaction diffusion system they consider is as follows:
ev, e2 v + f v w,
(7.1)
Wxx -Jl- I) llW -Jl" O,
where f =-v3+ v. Starting at a standing front wave and a standing back wave which
forms a heteroclinic loop (i.e., at c 0), they manage to obtain the local codimensionthree bifurcation unfoldings with c, O, y being the relevant parameters and globally
W
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THE BIFURCATIONS FROM A TWISTED HETEROCLINIC LOOP
675
extend the local bifurcation diagram. Among the most interesting is the nontwistedness
of all the resulting heteroclinic loops involved due to various symmetries exhibited by
the system. According to Theorem A, this implies that there are no multiple heteroclinic
connections other than the persistence of the zero-heteroclinic orbits from the loops.
They also show that some of the finite connections are actually unstable.
Perhaps some comparisons between their system and the FitzHugh-Nagumo
equation considered in the Introduction are worthwhile. First of all they model systems
of different worlds--chemical reactions, predator and prey populations for the former
while nerve impulses for the latter. Theoretically speaking, however, they are the same
system but at different values of the diffusion parameter for the w dynamics. Indeed,
if we move the origin to the left equilibrium state in Fig. 7.2, the parameter 0 is the
same as the parameter a in the FitzHugh-Nagumo equation (1.11). Rescaling the time
and the space variables in (7.1) yields
wt 6Wx + e(v yw),
VXX +f(v) w,
where we renamed := z/e and e := ez. Thus, it is the same FitzHugh-Nagumo equation
except for a large diffusion coefficient 6 for w. Since both systems have the same
symmetries, I think the appearance of the second diffusion simply "untwists" the
twisting structure somewhere. Thus, it is natural to ask whether it happens at some
6o> 0 or just at 0 =0. Indeed, there are two types of bifurcations involved. When
6o 0 the system of the ODE is singularly perturbed. When > 0, however, to untwist
a heteroclinic loop a heteroclinic orbit must be degenerate in general and, in particular,
it must violate the strong inclination property. None of these bifurcations problems has ever been fully investigated. Nevertheless, the idea developed in this paper
offers more hope for solving the bifurcation of twists than the problem of singular
perturbation.
V
o
Appendix.
Proof of Lemma 4.1. The proof for the first half part of the lemma is identical to
the proof of Lemma 3.3 in 3. Thus, we omit it here. Using the notation from 2, we
write equations (4.1a-4.1d) in the following equivalent forms:
(L z, )=0,
where
with the subindex
a, b, c, d in correspondence with the alphabets in
the equation numbers. Here
(r/2, :1) and z varies with equations as follows.
(I)a(, Z, 0)--
"
Q2(O,
r/,
-
z--yl,
, g,
, )(C, z, )= (e(x_,
,(’, z,
(o,
= (Sl,
1)
0(x, n, )-y
a(, z, a)
,
Q(X2,
,
a)- Y,
Y(s,
r12,
z
z
,
(s, :, y),
(s,
,
s, rl),
where X X2(s,
rl, a) and Y
rl, a). Delete the rnth component of
and let =((a),
(m-),(m+),
(a-)7". We solve c=0 first by the
implicit function theorem for r=sr*(z, a) and then solve the reduced remaining
equation (") (’*, z, a) 0 later.
Because of the existence of the heteroclinic loop we have (0, 0, 0) 0. Moreover,
the square Jacobian 0/0sr(0, 0, 0)- M is nonsingular by Lemma 2.3. (Note that this
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676
BO DENG
also implies 002/0T/(0 0, 0) nonsingular.) Hence, by the implicit function theorem
there exists a differentiable function
of Izl, [a[ < 6 satisfying st*(0, 0) 0 and [sr*[ < 6
such that (’, z, a)=0 if and only if
sr*(z, a). To solve the reduced equation
(m)(sr*, z, )=0, we need some important facts about ’*. Since usually are not
the same for different indexes
a, b, e, or d, we denote it by ’*, accordingly. Since
and
do not depend on the first y-component yl),
and
are functions
of )1 and a only. Moreover, since X2(0, 2, r/, a)=0 and Y(0, s, r/l, a)=0 by the
0
property (2.1a) of exponential expansions, it is easy to see that when equations
for all are restricted to sl =0, s2=0, and =0 (whichever applies) they are all
reduced to the same equations as follows"
’*
a
’=
c
sr.
’*
’*
r/, )--1 =0,
(r/2, ,).
Q2(0, r/2, a) 0,
Thus, the solution depends only on a. Therefore, the following functions of restrictions
"
(A.0)
"
sr*[;,=o, ’*[s,=o, sr*]s2=0,;,=o, and ’*[,=s2=o are in fact equal to the same function of a,
say (u, v)(a), which is the solution to (A.0). Note also that [’*- (u, v)r[ O([;l[ +[s2[)
or O([sl[ + [s2[), accordingly.
As another preparation, we need the following procedures one way or another.
Expand (P, Qz)(X, rt, a) at (x, rt)= (0, u),
(P)(x,
( P2) +
a)( )
x
8(P2, Q2)
(O, u,
Q2,
Q
o(x, n)
"q u
where (P2, O2)= (P2, Q2)(0, u, a). Expand q(s r/, a) at (:, r/)= (0, u) and q,(s
at (s r/) (v, 0), respectively:
(A.2)
X(s2, 2, T]2, og) (2as+v2-[- O[(l2[[-lT]2- ul)ls211+v2[-]s211+v2+’2],
Y(s1, 1, T/l, )-- ,aS1 + O[(ISl- vl / Ir/ll)lXll / ISll’+v’],
(A.3)
where o2 o(O, u, a) and /la (V, O, 1). Let
(A.1)
rt,
a)
,
,
,
r/,
a)
oP (o, u,
oQ2
(0, u, a)
(d-1)x(d --2)
and
P (o, u, a) -I
M2(o
IOn
(O, u, )
0
--
a and Lemma 2.3 we may assume that, without loss
of generality, Lz(a) has the maximal rank d-2 which is attained by the submatrix
//(a). Note that this also implies 0O2/0r/(0, u, a) nonsingular. Also, up to rn-1
permutations we may still call
Then, by the continuity of u on
(= (’
and we have
4,
.-(.-1
]det M2(a)[ > mo6o for [a] < 6 by Lemma 2.3 and Lemma 2.4.
677
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THE BIFURCATIONS FROM A TWISTED HETEROCLINIC LOOP
(0,.
Now we are ready to solve (’)(’*, z, a)=O. Since
(m), 0," ", O)(Sr*, Z, a). This implies
.,
det [L:(a), O(sr*, z, a)]
(’*, z, a)-=O, (’*, z, a)=
(-1) d-’)+" det /l:(a)om)( *, z, a).
Hence, (I)(’(sr*, z, a)=0 is equivalent to
det [L2(a), O(’*, z, a)] =0.
(A.4)
Since the simplifications for these equations are all identical we will only treat two
a and
c here, with emphasis on how the nonsingular change of
typical cases
parameters and the functions mi and ’i are obtained for all the bifurcation equations.
When
a, substituting (x, r/) (0, r/z*) into the Taylor expansion (A.1) and using
[r/- U] 0(I;11) we have
(*,z,)=
Q
+
0(P, Q)
Or/
(0, u,,)(n2m-u)+O(I.,12)
yl
Since the second and the fourth terms all belong to the range of L2(a), they will
disappear in (A.4). This implies
det
[L2(a), (y0)]=det[L2 (P2
2).
Dividing this equation by det M2(a) and expressing the left-hand side in terms of a
homogeneous linear combination in y]i) we have
m,(a)yl(1)qt-
m,(a)y] ") c2(a,
where
mi
[ ( ) ] /det
O
det L2( a )’
M2( a
and
(A.5)
c2
det L2(a),
Q2
det M2(a),
and ei En has zero components except for the ith component of 1. This has the form
of (4.2a). Let us show that the functions rni satisfy the required properties and postpone
the discussion of c2 until later.
Since (0, q,j(0, 0))/60-*(0, el) as 60-*0 by the exponential expansion property
(2.1b), det M2(a) is approximately the product of 6o and the numerator for ml as
6o-* 0. Hence, for small but fixed 60 we have 1/26o < ml < 2/60. Also, it follows from
Lemma 2.4 that
mi_det[L2(a),|O][/det[L2(a),|ol[r[\]/[[\’} =o(1)
ml
k
\
/ J/
Before we check the properties for
k
2,
\ el /
as
60-*0.
let us first obtain the bifurcation equation
(4.2d)o Substitute (x, r/)=(X*, r/*) with X*z=X(s, 2, r/*, a) into (A.1) and use
IX2*[ O([s2[ 1+"2) and Jr/z* u[ O([sl] + Is2[). Then substitute the exponential expansion
(A.2) for X*. Finally, substitute the obtained (A.1) and (A.3) with sol :* into the
678
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function
Bo OENG
d
(’*, Z, a). We have
,,
* z, o
Q
+
+
o
o(P2Q2)
(o, u, o
(o,
,
s
, )(n*- u)- (0)
+ 0(([2[+[$2[2)[$2[1+v2+([1[+[$1[)[$1[).
Similarly, the third and fouh terms belong to the range of Lz(a ); hence, they disappear
in equation (A.4). This yields
(
+det
[ L()’O(P’Ox
Q)
(0, u,
)] s+
+ (the same form of higher order).
o
we obtain the desired form
Dividing both sides by det M() det L(), l)],
s c+ rs + (the same form of higher order),
where the Nnction c of is the same as (A.5), and
l+p
"=det[ L()’O(P’Ox (O,u,)p/detM().
Q)
Now we show c(, 0)=0 and Oca/O(O, 0) 0 and (4.3). Recall that Q
Thus, from (A.5) we obtain
det 0Q/0(0, u, )
det M(
C2(ffl, if2) (__l)(m_l)
0.
2
Because oQ/o(O, u, ) is nonsingular it suces to show n(l=0 when =( 0)
and 0.(/0 0 at
(0, 0) by the product rule of differentiation. It is trivial to
check (=0 at =(1,0) because of the existence of the primary heteroclinic
connections from a to a on the -axis. Also, since (P, Q)r=(p, Q)r(0 u,
is on the unstable manifold W() for (0, u)e Woc()
by (1.10b) for the
and Q 0 we have
distance between W and W on
,
0 < d2(l, 2) <
()
](P2, Q2)-(, 0)] ]2.
min
(#,0)
ilo
(1)
This implies Q2
at
(0, 2) has a constant sign for 2 > 0, say >0, since 0 < d2(O,
by our assumptions. Therefore,
d(0, 2)
()
2
2
0< for=(0,2).
)/
> 0 at
Passing the limit 2 0+ above, we have 02.2
(0, 0) by (1.10b). This
completes the proof for c.
To show (4.3), we notice first that u(0)=0 and the set of all the column vectors
and
of M2(0) forms a base for
TZ
ox
o(x,
n
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THE BIFURCATIONS FROM A TWISTED HETEROCLINIC LOOP
679
Project this vector onto the one-dimensional linear space span {(0, $1o)r}, which is
complementary to the range of L2(0); namely, span {Tp,(W’f’lE), Tp,(WocVIE)}.
We obtain O(P2, Q2)/ox(O, O, 0)O2o "72(0, qlO) r + h with h range L2(0). Hence,
det [L.(O), O(P2, Q2)/ox(O, O, O)q2o]
and -72 r2(0) follows. Of course, z2(0)> 0 if and
[-I
Definition 1.1. This completes the proof.
"72 det M2(O)
only if 12 is not
twisted by our
Acknowledgments. The author is indebted to the reviewers for their many useful
suggestions. He also has benefited from many conversations with S. N. Chow, J. K.
Hale, and J. Mallet-Paret. Special thanks go to D. Terman, who corrected the author’s
misunderstanding of his work with J. Rinzel, which was the key motivation for beginning
this work.
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