BIFURCATION OF A HOMOCLINIC ORBIT WITH A SADDLE-NODE EQUILIBRIUM. (Submitted by: R.D. Nussbaum)

Differential and Integral Equations, Volume 3, Number 3, May 1990, pp. 435-466.

BIFURCATION OF A HOMOCLINIC ORBIT WITH A SADDLE-NODE EQUILIBRIUM

SHUI-NEE CHowt

Center for Dynamical Systems and Nonlinear Studies, School of Mathematics Georgia Institute of Technology, Atlanta, Georgia 30332 USA

XIAO-BIAO LIN

Department of Mathematics, North Carolina State University, Raleigh, NC 27695 USA

(Submitted by: R.D. Nussbaum)

1. Introduction. Bifurcation of periodic orbits from an orbit f0 homoclinic to a hyperbolic equilibrium point was extensively studied by Silnikov in [12, 13, 14].

In this paper, we study the codimension two unfolding of an orbit f0 homoclinic to a saddle-node equilibrium in IRd. We allow the eigenvalues of the equilibrium to have both positive and negative real parts, except for a simple eigenvalue

>.

0 = 0.

Several new methods are employed. The bifurcation diagram is obtained by the bifurcation function which is derived by the method of Liapunov-Schmidt decom- position. The idea of exponential trichotomy (see Hale and Lin [7]) is employed to study the linearized equation around f0 . We show that the linear operator in- duced by the linearization around f0 is Fredholm, which is essential for applying the Liapunov-Schmidt method. We use the method of cross sections and Poincare mappings to study the bifurcation of the periodic orbits from f0 . This requires the extension of the domain of the Poincare mapping to a whole neighborhood of fo since flows starting on a cross section near

r

0 do not always return. Although ex- tended mapping is not very nice as a point mapping, submanifolds of certain type, called the u-slices, behave nicely under the extended mapping. Roughly speaking, they are stretched in the unstable directions and compressed in the center and sta- ble directions. Thus, a u-slice under the Poincare mapping can be found to be fixed and hence bifurcation of periodic orbits is proved.

Lukyanov [10] has considered the situation that we discuss here in the special case where the equilibrium point has a simple zero eigenvalue and all others have negative real parts. As pointed out by Schecter [11], the bifurcation diagram of Lukyanov is incorrect. Using the Melnikov function, Schecter [11] gave a correct bifurcation diagram for differential equations in IR2 • Vegter [17], using the theory of singularities of maps, has studied a related problem in IR2 .

Our main results are presented in §2. Basic facts concerning exponential tri- chotomies and Fredholm properties of the linearized system around fo are given in §3. In §4, bifurcations near

r

⁰ are classified geometrically in terms of center Received May 5, 1989, in revised September 5, 1989.

tpartially supported by DARPA and NSF #DMS8912289.

AMS Subject Classifications: 34C35, 34C45, 58F12, 58F14.

An International Journal for Theory & Applications

manifold, center stable and unstable manifold and fibrations of these manifolds.

This is based on many previous works on the center (center stable and unstable) manifold. See for example van Gils and Vanderbauwhede [6] and Chow and Lu [4].

In §5, bifurcation functions for the existence of homoclinic and heteroclinic orbits are given which clarify the geometrical conditions in §4. The novelty here is the generalization of classical methods to the case of a nonhyperbolic equilibrium. The most difficult part of the paper is in §6 in which bifurcation of periodic orbits is studied. Although the basic idea of reducing the problem to the study of certain maps comes from the work of Silnikov [11, 12, 13], several extensions are needed.

Lemma 6.1 is essential for the bifurcation analysis and presents a local analysis using Silnikov's method of reparametrization for a nonhyperbolic equilibrium.

We received Deng's preprint [18] shortly before submitting this paper. In his work, Deng obtained the bifurcation diagram for homoclinic solution interacting with saddle-node, transcritical or pitchfork equilibrium. However, as noted in [18], his method is different from ours.

2. Main results. Consider the following ordinary differential equation in IRd :

x(t) = f(x, J.L), X E IRd (2.1)

where f: IRd x M-+ IRd, f..L

=

(J.L1 , J.Lz) EM and M is a neighborhood of f..l

=

0 in IR2 .

Let D be the differentiation operator and a( A) be the spectrum of a matrix A. Let (a, J.L) E IRd x M be an equilibrium of (2.1); i.e., f(a, J.L) = 0. Let W"(a; J.L), Wu(a, J.L), W⁸⁸(a; J.L) and wuu(a, J.L) be the stable, unstable, strong stable and strong unstable manifolds of x =.a in [Rd at f..L E M for equation (2.1). Let Wz~c(a; J.L) be the local stable manifold of the equilibrium x =a at parameter f..L for equation (2.1). Other local invariant manifolds will be denoted in a similar way.

Assume the f(O, 0)

=

0. We rewrite equation (2.1) as an autonomous equation in

[Rd X M:

{ x ⁼

^f(x,J.L)

fJ,

=

0. (2.2)

Local center, local center stable and center unstable manifolds at (x, J.L)

=

(0, 0) of equation (2.2) are well-defined (see §7 Appendix) and denoted by Wz~c' Wz~c'

and W1^~':': respectively. We note that these local invariant manifolds Wz~c' Wz~c' and Wz~':': <;: IRd x Mare not uniquely defined. If IJ.LI is sufficiently small (we allow f..l

=

0), then define

Wz~c(f..l)

=

{x E IRd

I

^{(x, J.L) E} Wz~c}·

Since fJ,

=

0, Wz~c(f..l) is a local invariant manifold near x

=

0 for equation (2.1).

Similarly, we define Wz~c(p,) and Wz';,'~(p,). Let T(t; p,) be the flow generated by equation (2.1). Given any local center manifold W1^~c(f..l), we define

wc(J.L)

= U

T(t, J.L)Wt~c(f..L).

tER

Similarly, we define wcs(J.L) and wcu(J.L). For the existence and smoothness of these invariant manifolds, we refer the readers to Carr [1], Chow and Hale [2], Chow

BIFURCATION OF A HOMOCLINIC ORBIT 437

and Lu [3, 4), Chow, Lin and Lu [5], Hirsch, Pugh and Shub [9] and van Gils and Vanderbauwhede

[16].

Consider the following hypotheses:

(i) f E Ck, k ~ 4. Dp.f(x, J.L) is uniformly bounded in Rn x M.

(ii) f(O,O) = 0 and a(Dxf(O,O)) =iA-n,···

,.Ll,>.o,>.1,··· ,>.m},

with Re

>--n :::; · · · :::;

Re

>--1 <

>.o

=

0

<

Re

>.1 :::; · · · :::;

Re

>.m,

n

+

m

+

1

=

d, (iii) >.0 = 0 is simple with right and left eigenvectors er and et,

(iv) for J.l

=

^0, (2.1) has a homoclinic orbit fo: x

=

q(t) such that q(t) E W1^~c(O) as t .- -oo and q(t) E W1^~~(0; 0) as t .- +oo,

(v) q(t)/lq(t)l.-er as t .- -oo and e,_ · er

>

0, (vi) e,_ · Dxxf(O, 0)( er, er)

>

0,

(vii) e,_ · D,..d(O, 0)

>

0, e,_ · D,..d(O, 0) = 0,

(viii) wcs(o) and wcu(o) intersect transversely along

ro.

By Theorem A in the Appendix (§7), even though local center stable or unstable manifolds are not unique, assumption (viii) is meaningful and is independent of the choice of local center stable (or unstable) manifolds.

By (ii), dim W1^~c(J.L) = 1 for all small J.l· By (vi) and (vii), we have a generic saddle-node bifurcation of x = 0 at J.l = (0, 0). In fact, by choosing a suitable coordinate system for J.l (see Lemma 4.2 in §4) we may assume that for Ill

<

0, there exist two equilibria on W1^~c(J.L); one is a source S(J.L) in W1^~c(J.L), another is a node N(J.L) in W1^~c(J.L). S(J.L) and N(J.L) are hyperbolic equilibria in Rd. For Ill = 0, S(J.L) and N(J.L) collapse and form a saddle-node SN(J.L) on W1^~c(J.L). For J.L1

>

0, there are no equilibria on W1^~c(J.L). The solutions on W1^~c(O; J.L) are denoted by Ql(t,J.L), Q2(t,J.L) and Qa(t,J.L) (see in Figure 1.). Note that if J.ll

>

0, then there exists a unique solution Ql(t,J.L) in W1^~c(J.L).

Figure 1.

For a homoclinic solution q(t, J.L) of equation (2.1), we say that the orbit f(J.L) of q(t, J.L) is tangent to wc(J.L) (or W⁸(a; J.L), or wu(a; J.L)) at t

=

±oo if the tangent space of r(J.L) approaches the tangent space of wc(J.L) (or W⁸(a; J.L), or wu(a; J.L)) as t .- ±oo. Our main result is the following:

Theorem 2.1. There exists a C¹ function G(J.L) (see (5.7) in §5) such that if D,..2G(O)

>

0 (explicit formula for Dp.2G(O) can be found in (5.10)) then there exists C¹ change of variables in the parameter space M such that bifurcation of phase flows of (2.1) in a neighborhood of x = 0 is completely determined by the

Ji..z -7

6. 2 7.

-6 2 - . J.L,

2 -5

8. _9.

Figure 2. Unfolding of saddle-node homoclinic.

signs of f..ll, f..l 2 and a C¹ curve /-ll = ~(f..l2) which is quadratically tangent at 1-12 axis and is contained in the half plane {!-L

I

f..ll :::; 0}. The bifurcation diagram is depicted in (f..l1 , f..l 2)-space (Figure 2) and in phase space (Figures 2.1-2.9). There are 9 cases:

Case 1. Figure 2.1: /-ll

=

/-l2

=

0. There exists a saddle-node equilibrium and a lwmoclinic orbit f0 such that f 0 E W1^~c(O) as t-+ -oo and E W1^~~(0) as t-+ +oo.

Case 2. Figure 2.2: /-ll

>

0. There are no equilibria and there exists a unique periodic orbit Til' near fo.

Case 3. Figure 2.3: /-ll = 0, f..l 2

<

0. There exists a saddle-node equilibrium SN(f..l) and a homoclinic orbit r(f..l) near f0 such that f(f..l) is tangent to W1^~c(f..l) at t

=

-oo and r(/-1) E W1^~8c(f..l) as t-+ +oo

BIFURCATION OF A HOMOCLINIC ORBIT 439 Case 4. Figure 2.4: ~(p,2⁾

<

p,1

<

0, p,2

<

0. There are (a) two equilibria, a saddle S(p,), a node N(p,) in W1^~c(p,); (b) a (local) heteroclinic orbit fzoc(M) s;;; W1^~c(M) which is the orbit of q3(t, p,) in Fig. 1 that goes from S(p,) to N(p,); (c) a (global) heteroclinic orbit f(p,) close to f0 that goes from S(p,) to N(p,) and r(fL) is tangent to the orbit ofq1(t,p,) in W1^~c(M) at t = -oo and f(p,) is tangent to the orbit of q2(t, p,) in W1^~c(M) at t

=

+oo.

Case 5. Figure 2.5: p,1 = ~(p,2)

<

0, M2

<

0. This case is topologically equivalent to Case 4 and r(p,) E W1^~~(N(p,); p,) as t-+ +oo.

Case 6. Figure 2.6: p,1

<

^~(p,2⁾

<

0. This case is topologically equivalent to Case 4 and f(p,) is tangent to the orbit of q3(t, p,) in W1^~c(M) at t = +oo.

Case 7. Figure 2.7: p,1 = ~(p,2⁾

<

0, 1-t2

>

0. This case is the same as Case 4 except that the global heteroclinic orbit becomes a homoclinic orbit f(p,) which is tangent to W1^~c(M) at t = -oo and is in W1^~~(S(p,); p,) as t -+ +oo.

Case 8. Figure 2.8: ~(M2)

<

Ml

<

0, P,2

>

0. There exist S(p,), N(p,) and fzoc(M) as in Case 4; however, there is no homoclinic or heteroclinic orbit f(p,) close to f0 .

There is a periodic orbit TI(p,) near f0 .

Case 9. Figure 2.9: p,1 = 0, 1-t2

>

0. There exists a saddle-node equilibrium SN(p,) on W1^~c(p,). There is no heteroclinic or homoclinic orbit, but there is a periodic orbit TI(p,) near f0 .

In all the cases, one and only one periodic, homoclinic or heteroclinic orbit can bifurcate from f0 . The periodic orbit TI(p,), if it exists, is hyperbolic with dim Wu(TI(p,))

=

dim wcu(O), where wu(TI(p,)) is the unstable manifold of the peridic orbit TI(p,).

Remark 2.2. For simplicity, our main result is stated in a two dimensional param- eter space. Let

j

be a Ck vector field in a neighborhood of

f.

We can show that there exist C¹ nonlinear functionals'p,i(J) and ~-t2(f) such that Dp,i(f) and Dp,2(f) are linearly independent and the bifurcation diagram is completely determined by Mi and M2·

3. Exponential trichotomies and Fredholm operators. Consider a linear nonautonomous system

i;(t)- A(t)x(t)

=

^h(t) (3.1)

and its associated homogeneous system

x(t)- A(t)x(t) = 0. (3.2)

Assume that A(t) is a continuous n x n matrix function oft E IR. Let X(t, s) be a fundamental matrix of equation (3.2). Assume that X(t, t) = I, the identity matrix, for all t E IR. Note that X(t, r)X(r, s) = X(t, s), for all t, r, s E IR. We say that X ( t, s) (or equation 3.2)) has an exponential trichotomy on an interval J s;;; IR, is there exist constants a

<

v- c

<

v

+

c

<

(3 and K

>

0 and continuous projections Pu(t), P8(t) and Pc(t) satisfying I= Pc(t)

+

Pu(t)

+

P8(t), for all t E J and

X(t, s)P;(s)

=

P;(t)X(t, s), fori= c, u, s (3.3)

and

t,

s E J such that for all

t,

s E J

IX(t, s)Ps(s)i ~ Ke'Jt(t-s), t ~ s IX(s, t)Pu(t)l ~ Ke-f3(t-s), t ~ s IX(t, s)Pc(s)i ~ Ke(v+E)(t-s), t ~ s

IX(s, t)Pc(t)l ~ Ke(-v+E)(t-s), t ~ s.

The adjoint system of (3.2) is

y(t)

+

A*(t)y(t) = 0

(3.4)

(3.5) which has a fundamental matrix Y(t,s) = [X(t,s)*J-¹ = X(s,t)*. If X(t,s) has an exponential trichotomy on J, then Y(t, s) also has an exponential trichotomy on J with projections being P;;,(t), P;(t) and P;(t), t E J. Properties similar to (3.3) and (3.4) may be obtained by simply taking adjoints in (3.3) and (3.4).

Lemma 3.1. Let a ERn be an equilibrium of a nonlinear equation

:i:(t) = f(x(t)), (3.6)

and q(t) be a solution of (3.6). Suppose that q(t) --.a as t--. +oo (or t--. -oo);

then the linearized equation

:i:(t)-Dxf(q(t))x(t) = 0 (3. 7)

has an exponential trichotomy on [0, +oo) (or ( -oo, OJ). Moreover, let 'f/

>

0 be so small that if A E a{Dxf(a)} and -ry ~ ReA ~ ry, then ReA

=

0. Then one can choose v = 0, a

<

^-ry

<

^-E

<

^+E

<

'f/

< f3

in (3.4), provided that the constant K

>

0 is sufficiently large. Furthermore, dimPi(t)

=

dimWi(a), where i

=

u,c,s.

Proof: See Hale and Lin [7] (Lemma 4.3). 1

Let 11 and 12 be two real constants. Let C⁰(11,12 ) denote the set of continuous functions x : Ill --. IRn such that lx(t)le-l'•t, t ~ 0 and lx(t)IC~'²t, t ~ 0 are finite.

The set of such functions form a Banach space C0 (r1 , 12 ) with the norm

Let Ck(/1 , 12 ) be the Banach space of all the Ck functions x(t) such that Dix(·) E C0 (r1, 12), i = 0, 1, ... k, with the norm

k

lxlck(l'.m) =

L

IDixlco(/'.,,.2)·

i=O

Suppose a and bERn are two equilibria for equation (3.6) (possibly a= b), and q(t) is a solution of (3.6) with q(t) --. a as t --. -oo and q(t) --. b as t --> +oo. Let

1

>

0 be so small that if A E a{Dxf(a)} U a{Dxf(b)} and -1 ~ReA~/, then

ReA = 0. Then we have the following lemma.

BIFURCATION OF A HOMOCLINIC ORBIT 441 Lemma 3.2. Let :F: C¹(±-y, ±-y) ---+ C⁰(±-y, ±-y) be a linear operator defined by :F(y) = h, where y E C¹(±-y, ±-y) and hE C⁰(±-y, ±-y) are related by the following equation:

y(t)- Dxf(q(t))y(t) = h(t), (3.8)

and "± -y" means we may choose either +'Y or --y. Then :F is a Fredholm operator, and index of :F is determined by the following relations:

( 1) If :F : C ¹ ('Y, 'Y) ---+ C⁰ ('Y, 'Y), then Ind :F = dim wu (a)- dim wu (b);

(2) If :F: C¹('Y, --y)---+ C⁰('Y, --y), then Ind:F =dim wu(a)- dim wcu(b);

(3) If :F: C¹(--y,-y)---+ C⁰(--y,-y), then Ind :F =dim wcu(a)- dim wu(b);

(4) If :F: C¹(--y, --y)---+ C0 ( --y, --y), then Ind :F =dim wcu(a)- dim wcu(b).

The range of :F is determined by:

R(:F) = { h :

f~:

'¢(t)* h(t)dt = 0, for all solutions '!f;(t)

of the adjoint equation of (3.8) and'¢ E C⁰(±-y, ±-y)},

where the sign of 'Y for'¢ E C⁰(±-y, ±-y) is chosen to be the opposite of the sign of 'Y in the range of :F ~ C⁰(±-y, ±-y).

Proof: See Hale and Lin [7] (Lemma 4.5). 1

Remark. If hE R(:F) ~ C⁰('Y,--y) and'¢ E C⁰(--y,-y) is a solution of the adjoint equation of (3.8), then'¢ E C0 ( --y

+

b, 'Y-b) for some b

>

0 and

J

^{-oo +oo} '¢(t)* h(t)dt

converges. The same remark holds for all other choices of ±-y.

4. Center manifolds, fiberations and heteroclinic orbits. Consider equa- tion (2.2):

{ x

⁼ f(x,fL)

jJ, = 0. ( 4.1)

From the conditions on the eigenvalues of Dxf(O, 0), there is a linear change of coordinates such that x = (y, w), y E IR, w E Rm+n, and ( 4.1) becomes

{

il

= h(y, w, 1-L)

w =

Aw

+

g(y,w,{L) /1,=0

(4.2)

where his real-valued, w = (u,v), u E IRm, v E IRn, A= diag(A1,A2), A¹ is an m x m matrix and A2 is ann x n matrix, Recr(A1) > 0, Recr(A 2)

<

0, g = (g1,g2), h(O, 0, 0)

=

^0, g(O, 0, 0)

=

^0, D(y,w)h(O, 0, 0)

=

^0, D(y,w)g(O, 0, 0)

=

^0.

By assumption (ii), a(At)

=

{.At.··· , .Am} and cr(A2)

= P-n, · · · ,

..\_1}. Hypoth- esis (vi) implies that 8²h(O, 0, O)f8y² > 0, and (vii) implies that 8h(O, 0, 0)/8p,1 > 0, and 8h(0,0,0)/81-L2 = 0.

We are interested in the flow of ( 4.2) near /k

=

0 and x = 0. By a Ck change of coordinates, we may assume in a small neighborhood of the origin x

=

^{0 that}

Wi~:C,(~L) ={xI x = (y, w), u = 0}, Wz';;~(!k) ={xI x = (y, w), v = 0}, and

wl~,(!k) ={xI X= (y, w), u = 0, v = 0}.

We now perform another change of variable to facilitate further investigation.

It is known that W1^~c(~L)

c

Wz'~;'c(~L), and W1^~c(~L) is invariantly fibered by Ck submanifolds W(y,o)(!k) which passes through x

=

(y, 0) E Wt'~c(!k)· We shall use

w~ (lL) to denote the fibers. Any two points X1 and Xz E w~ (!k) satisfy the property that

IT(t; ~L)x1- T(t: ~L)xzl :S: Ce't as t __. +oo,

where T(t; !k) is the flow generated by equation (2.1), supReO'(A2 )

= ,\_

1

< r <

0 and

r

can be chosen to be as close, but not equal, to sup Re 0'( A2 ) as we want by allowing the constant C to be large. Note that points in W1^~c(~L) are not contracted as fast. We may write in a small neighborhood of x = 0 :

w~o (~L)

= {

(y, 0, v)

I

y

=

Yo

+

¢(yo, v, /k)}.

It is known that ¢(yo,V,/k) is ck- 1 in the variables (yo,V,/k)· Since ¢(yo,O,/k) = 0 and the nonlinear terms g and h in equation (4.2) are small together with all the derivatives up to order k- 1, by using the fibers as coordinates on W1^~8c(~L) we can make the fibers flat. Consequently, h(y,O,v,/k)

=

h(y,O,O,/k). Similarly, wcu(lt) is invariantly fibered by w;(/k). Using fibers as coordinates, we have h(y, u, 0, /k) =

h(y, 0, 0, !k)· Thus, we have the following.

Lemma 4.1. There exists a small neighborhood of x

=

0 and a ck- 1 change of coordinates such that in the new coordinate system and in that neighborhood we have for equation (4.2)

Wt~⁸c(!k) ={xI X= (y, w), u = 0},

WD~,(!k) ={xI x = (y, w), v = 0}, and J,V1^~JC(!k) ={xI x = (y, w), u = 0, v = 0}.

Moreover, W1^~c(~L) is invariantly fibered by

wcu(p,) is invariantly fihered by

w;o(~L)

=

{(y,w)

IY

=Yo, v

=

0},

and the nonlinear terms in equation (4.2) satisfy: h(y, u, 0, tt)

=

h(y, 0, v, ~L)

=

h(y, 0, 0, IL), Y1(y. 0. V.tt) ⁼ 0 and gz(y, ^U, 0, /k)

=

0.

Proof: Although the results presented here are not new, see Hirsch, Pugh and Shub [9] for mappings. and Takens (1971), Fenichel (1974) for flows. We give a

BIFURCATION OF A HOMOCLINIC ORBIT 443 proof for the ck- 1 foliation on W1^{~•c(Jt) by}

w;o

^(JL) in Theorem B of Appendix B of this paper for completeness. The idea of the proof is from Chow, Lin and Lu [5] on a more general case. I

We now look for equilibria of ( 4.2) in a small neighborhood of x = 0. This is equivalent to solving

h(y, 0, 0, Jt)

=

0.

Here we have an elementary saddle-node bifurcation in IR¹ and the following is well known.

Lemma 4.2. There is a ck- 1 change of coordinates in the parameter space such that in the new coordinate system, (4.2) has a saddle-node equilibrium SN(Jt) for JL1 ⁼ 0; two equilibria S(Jt) and N(Jt) for JL1

<

0, N(Jt) is an attractor and S(JL) is a repeller on W1^~c(Jt); no equilibria for JL1

>

0.

Proof: See Chow and Hale [2]. I

We note that S(Jt) (or N(JL)) is not defined for JL1

>

0 and is not smooth at JL1 = 0. We now introduce another coordinate system to obtain smooth dependence on parameters. Let JL1 = -8r and J.l2

=

82 in the half space JL1 ::; 0. Then there is a ck-J function r: M---+ IRl+m+n such that 8r1(0, O)j881

>

0 and r2(81, 82) = 0, where r = ( r-1, r2), r1 E IR and r2 E IRm+n. Moreover,

N( )

=

N( 8 8 )

= {

r(81,82) ^if 81

<

0 Jt 1' 2 r(-81,82) if 81

>

0, S ( )

=

S ( 8 8 )

= {

r ( 81, 82) ^if 81

>

0

Jt 1' 2 r(-81,82) if 81

<

0, SN(Jt)

=

SN(81, 82)

=

r(O, 82).

The specific fibers w;(JL), JL1 ::; 0, passing through (y, 0) = N(Jt), S(Jt) or SN(Jt) will be very useful for determining the bifurcation diagram of (4.2). In the (81,82 )

coordinates, these fibers are

w:(

6 6 _{1' 2} )(M) and

w:(-{j

_{1' 2} 6 )(JL).

II Ill Ill II Ill

Figure 3.

Figure 3 shows the flows on W1^~c(Jt) for JL1 :S 0. The horizontal direction cor- responds toy in the center manifold. Note that S(81,82) is always on the right of N(81, 82) ^{in W}1^~c(Jt). The local center stable manifold W{~,"c(Jt) is divided into three regions, I, II and III: region I consists of all the points that are to the left

of

w:(

61 ,62)((h,82) and

w:(_

61 ,02)(81,82); region III consists of all the points that are to the right of w:(o _{l1 2} 6 )(81,82) and ws(-' , _{T Ul,U2} )(81,82); region II consists of all the points that are in between

w:(o

1.D2/81,82) and

w:(_

61 ,02l(81,82). For the case 81 = 0, N(81 , 82) and 5(81 , 82) collapse into SN(O, 82) and II disappears while I and III persist.

Consider the homoclinic orbit f0 . By assumption (iv), in a neighborhood of 0, f0

has two connected parts, f

0

^and

rt.

^f

0

^{~ W}1^{~c(O) as t-+ -oo and}

rt

^{~ W}1^~8c(O; 0) as t -+ +oo. Let ~ be a transverse cross section to

rt.

Assume that U C ~ is a small ( d-1) dimensional disc centered at

rt n

~. We have that

rt n

^U is a unique isolated point if U is sufficiently small. Under assumption (viii) in §2, wcu(o) and wcs (0) intersect transversely along f0 . Thus, for l11l sufficiently small, U intersects wc•(Jl) transversely and the intersection is an n-dimensional submanifold. Also, for each fiber W~(Jl), U n W~(Jl) is an (n- I)-dimensional submanifold. We have a similar situation for U

n

wcu(J.L). Assumption (viii) and the smooth dependence of W1^~~(J.L) and W1^~8c(Jl) on 11 imply that for 1111 sufficiently small {U

n

wcu(Jl)} and {U n wcs(J.L)} intersect transversely and the intersection is a unique isolated point

which is Ck in Jl. Let the first component of E(J.L) be E(J.Lh which is also Ck in Jl.

Let r1 be the first component of r( 81 , 82 ) as before. This gives the following.

Lemma 4.3. If 111-:::; 0 and -111

=

8r, 81

2

0, then E(J.L) E region I if E(J.Lh

<

r1(-81,82)

E(J.L) E region II ifr1(-81,82)

<

E(J.Lh

<

r1(81,82) E(J.L) E region III if E(J.Lh

>

r1(81, 82)

E(J.L) E W1~~(N(J.L);Jl) if E(J.Lh = r1(-81,82) E(J.L) E W1~~(S(J.L);J.L) if E(J.Lh = r1(81,82).

Similar results hold for 81 -:::; 0.

Corollary 4.4. For equation (4.2), in a neighborhood off0 ,

(a) there exists a homoclinic orbit f(J.L) ~ W1^~~ (S(J.L); Jl) as t-+ +oo and f(J.L) ~ W1~c(S(J.L); Jl) as t-+ -oo if and only if 111

<

0 and E(J.Lh = r1 (81, 82), 81

>

0;

(b) there exists a heteroclinic orbit f(J.L) ~ W1^{~~ (N(J.L);} J1) as t -+ +oo and f(J.L) ~ W1~c(S(J.L); Jl) as t-+ -oo if and only if 111

<

0, E(J.Lh = r 1( -81, 82), 81

>

0;

(c) there exists a heteroclinic orbit r(J.L) ~ W1~c(N(J.L); Jl) which is tangent to the orbit of q2(t, Jl) in region I (see Figs. 1 and 3) at t = +oo and f(J.L) ~ W1~c(S(J.L); Jl) as t-+ -oo if and only if 111

<

0, E(J.Lh

<

r1(-81,82), 81

>

0;

(d) there exists a heteroclinic orbit f(J.L) ~ W1~c(N(J.L); J.L) which is tangent to the orbit of q3(t, Jl) in region II (see Figs. 1 and 3) at t

=

+oo and f(J.L) ~ Wt':,c(S(J.L); J.L) as t-+ -oo if and only if111 < 0, r1(-81,82) < E(J.Lh < r1(81,82), 81 > 0;

(e) there exists a homoclinic orbit to S N (Jl), which is tangent to the orbit of

q2 ( t, Jl) in region I (see Figs. 1 and 3) at t = +oo and is tangent to the orbit of

q1 ( t, Jl) in region III (see Figs. 1 and 3) at t = -oo if and only if 81 = 0 and E(J.Lh < r1(0,82).

Proof: We only need to see that r(J.L) ~ W1~c(S(J.L); Jl) as t ^-+ -oo. This can be shown by continuous dependence. I

BIFURCATION OF A HOMOCLINIC ORBIT 445 Corollary 4.4 allows us to relate the bifurcation diagram with the sign of the ck- 3

functions of p,. In §5, we shall show that the conditions in Lemma 4.3 and Corollary 4.4 can be described by an integral along f 0 , known as the bifurcation function.

5. Liapunov-Schmidt reduction and bifurcation functions. In this section we consider only J.ll :::; 0. Let

bf

= - J-11, b2 = Jl2. Thus (J.Ll, J-12) = (

-bf,

b2).

Let E(p,) be the point defined in §4. We will find conditions on J.l under which E(p,) E

w:(o,,o

2)(J.L),

w:(-o,,o

2)(J.L), region I, region II, or region III. First, we will find J.l for which E(p) E

w:(o,,o

2)(J.L). Assume that 'Y

>

0 is a constant, 'Y

<

min{Re.>. 1, -ReLI} (see assumption (ii) in §2). It is clear that in this case, there exists a heteroclinic (or homoclinic) solution x(t) of (2.1), Jx(t)- r(b1, b2)J :::;

ce-'

¹

as t -+ +oo ( C is some fixed constant depending on 'Y), and the orbit of x( t), denoted by f(p,) is tangent to that of q1 ( t, Jl) as t -+ -oo, where Jl = (

-bf,

b2).

Let B1(t) be a smooth real-valued function satisfying B1(t) = 0 fort

2:

-1 and B1(t) = 1 fort:::; -2. Let B2(t) = B1( -t). The solution x(t) can be written as

x(t) = q(t) + el(t)(ql(t, p,)-q(t)) + e2(t)r-(b) + z(t) (5.1) where Jl = (

-bf,

b2) and r(b) = r(b1, b2). Note that q1(t, p,) = q1(t, (

-b?,

b2)) is smooth in b for t in any compact subset of [fit We shall assume that q1 ( t, 0) = q( t).

Observe that the center manifold is not unique so this can always be achieved by choosing a center manifold to contain q(t). After a phase shift in x(t), we may assume that Jz(t)J :::; Ce-,[t[ as t-+ ±oo, for some constant C

2:

1. This suggests that we search for a solution x(t) in the form (5.1) with z E C 1('Y, -'"'().Substituting (5.1) into (2.1), the equation for z(t) becomes

z(t)-Dxf(q(t), O)z(t) = N(z, b)(t) (5.2) with

N(z, b)(t) = f(q(t)

+

B1(t)(q1(t, p)- q(t))

+

B2(t)r(b)

+

z(t), b)

- j(q(t), 0)- at 8 (B1(t)(q1(t, p,)-q(t))

+

B2(t)r(b))-Dxf(q(t), O)z(t).

It is not hard to see that for small z E C⁰('Y, -'"'(), N(z, b) E C⁰('Y, -1) and IN(z, b) leo(/,-!)= 0

(lzl~oh,-l) ⁺

^Jbl).

Moreover, N : C⁰('Y, -1) -+ C⁰('Y, -'"'() is Ck and is continuous with respect to 8 E M. By Lemma 3.2, (5.2) can be written as

:Fz

=

N(z, 8) (5.3)

where F: C¹('Y, -1) -+ C⁰('Y, -1) is Fredholm with Fredholm index IndF = -1.

From assumption (viii), x = q(t) is the only solution of (3.6) which approaches zero as t -+ ±oo. However, q(t) -+ 0 is slower than

e'

^{1 as} t -+ -oo. Therefore, the null space of :F is {0} and codim R(:F) = 1. In other words, there exists, up to a scalar factor, a unique nontrivial solution 'lj;(t) for the adjoint equation

x(t)

+

[Dxf(q(t), O)]*x(t)

=

0 with 'lj; E C 1(-"(,"f). We actually know more about 'lj;(t).

(5.4)

Lemma 5.1. (i) 11/l(t)- etl ---+ 0 as t---+ +oo, where et is the left eigenvector of the eigenvalue

>.

0 (assumption (iii))

(ii) 11/l(t)l:::; Ce-rt as t---+ -oo where Cis a constant.

Proof: Let A(t)

=

Dxf(q(t),O) and A(oo)

=

Dxf(O,O). We note that A(t) ---+

A(oo) as t---+ ±oo. However, A(t) ---+ A(oo) exponentially as t---+ oo while A(t) ---+

A( oo) at a slower rate which is not exponential as t ---+ -oo. Let :F be the linear operator in Lemma 3.2. Since we will consider :F as operators on different spaces, we will use subscripts to indicate the differences. Consider :F1 : C1 ( -"(, -"() ---+

C0 ( -"(,-"(),we have Ind:F1 = 0. Since dimN(:FI) = 1 (N(:Fl) is the null space of :F1 ); i.e., N(:F1 ) is spanned by {q(t)}. We have codimR(:FI) = 1. Therefore, there exists a unique solution (up to scalar factor) 1/l(t) of (5.4) such that 1/1 E C⁰('Y, "f).

This proves (ii) of Lemma 5.1. It remains to show (i). Next consider :F2: C¹('Y, "f)---+

C⁰('Y, "f). Ind:F2 = 0. Since dimN(:F2) = 0, we have codim R(:F2 ) = 0. Thus, 1/l(t) E C⁰('Y, 'Y)\C0 ( -"(, -"(); i.e., 11/l(t)l :::; Ce-rt as t ---+ +oo (slowly growing) but 1/1( t) does not approach 0 like e--rt as t ---+ +oo.

Write (5.4) as i:(t) = B(t)x(t), B(t) = -A*(t). Since IB(t)- B(oo)l ---+ 0 expo- nentially. we have

!

⁰⁰ ltlhiB(t)l dt

<

⁰⁰

for any integer h. From a general result given in Theorem 10.13.2 of Hartman [8], to every nontrivial solution x(t) of (5.4), there exists a nontrivial solution y(t) of y(t) = B(oc)y(t) satisfying

lx(t)- y(t)l = o(ly(t)l)

as t---+ +oo. Let y(t) be the solution of y(t)

=

B(oo)y(t) with 11/l(t)- f;(t)l = o(lfJ(t)l).

y(t) is slowly growing or bounded, but not decaying like e--rt as t---+ +oo. From the properties of linear autonomous equations, it is clear that y(t) ---+ Get as t ---+ +oo, for some c =I= 0, and we may set c = 1. I

Let Q be a projection onto R(:F). Equation (5.3) is equivalent to the following:

{ :Fz-QN(z,8)=0

(I- Q)N(z, 8) = 0. (5.6)

Using the uniform contraction mapping theorem, we obtain a continuous function i(8) E C⁰('Y, -"() such that i(O) = 0 and i(8) solves the first equation in (5.6) for small z and 8. Substituting into the second equation in (5.6), we obtain in a neighborhood of b

=

0 a continuous function G(8) and the equation:

G(8)

= i:

^{1/l(t)* N(i(8), 8) (t) dt}

⁼

^{0 (5.7)}

is equivalent to equation (5.6). We shall call G(8) the Melnikov function and G(8) = 0 the bifurcation equation for the existence of a heteroclinic (or homoclinic) solution of (2.1).

BIFURQATION OF A HOMOCLINIC ORBIT 447

We now indicate how to show that G(c5) is in fact C 1 in c5. First, it is not hard to see that there is a small ~:

>

0 such that z( c5) E C⁰ ( 'Y

+

~:, - 7 -~:) and z( c5) is continuous in c5 in the new norm. We then showN: C⁰('Y

+

E, -"(-E) X M--+ C⁰('Y, -"() is C¹ with respect to (z, c5) jointly. The difficulty here is that q1 (t, J.L) is not smooth in c51 in the uniform norm for t E R-. However, observe that for t ~ -2,

f(q(t) + BI(QI(t, J.L)- q(t)) + B2(t)r(8) + z(t), c5)

{)

- {)t (BI(t)(qi(t, J.L)- q(t))- f(q(t), 0)

= f(qi(t, J.L)

+

z(t), c5)- :t Q1(t, J.L) =

fo

^{1 fx(QI(t, J.L)}

⁺

vz(t), c5)z(t) dv.

Observe that

I t

₆q1(t, J.L)ie.t ^--+ 0 as t--+ -oo for any ^E

>

0. Based on this, it is not hard to show that N : C⁰('Y

+

E, - " ( -t:) X M --+ C⁰('Y, -"() is C1.

We are now able to show z(c5) E C⁰('Y, -7) and N(z(c5), c5) E C⁰('Y, -7) are both C1 in c5. The idea here is similar to the use of scaled Banach spaces in proving the smoothness of center manifolds (see [18]). Hence,

Since 1/;(t) is the solution of the adjoint equation, integrating by parts, we obtain for any t 1

>

0 :

[ttll

^{1/J(t)* [ Dxf(q( t), 0)}

{)q~~

J.L) - :t [{)q1t8 0)]] dt

= - [

1/J(t)"

{)q~8

⁰⁾

[tl '

^(5.8)

where q(t, J.L) = q(t, J.L(c5)) = B1(t)q1(t, J.L)

+

B2(t)r(8). Note that q(t, J.L) = q!(t,J.L) for t

<

-2. Hence, 8q(t,J.L(0))/8c5 satisfies the following equation fort ~ -r, where r

>

0 is sufficiently large:

w

^{= Dxf(qi(t,} J.L(O)), O)w.

Since q1(t, J.L(O)) ⁼ (y(t), 0, 0), it follows from Gronwall's inequality that

8ij(ta~(O))

^{= O(e-L(t)t), as t--+ -oo.}

This implies

lim ?J;(t)*[8q(t, J.L(0))/8c5] = 0.

t~-oo (5.9a)

Next, consider t

>

2. We have that q(t,J.L) = r(c5) and 8q(t,J.L(0))/8c5--+ 8r(0)/8c5 as t --+ +oo. By Lemma 5.1,

lim ?J;(t)*[8q(t, J.L(0))/8c5]

=

1/J( +oo)* {)r{)(c50) .

t-+oo (5.9b)

It follows from (5.8) and (5.9) that by letting h -+ +oo we have

Therefore,

and the second integral converges. Since

*t

¹8=₀

=

0, we have

here we use the facts that ⁸;~~)

=

Cer, C

>

0, 1/1( +oo)

=

eg and eg · er

>

0. Hence, 8G(O) ₀

~<-

We assume the following generic hypothesis on G : 8G(O)

Joo

^*

~ = -oo 1/1 (t)D11-2J(q(t), 0) dt

>

0. (5.10)

We have shown that the existence of a heteroclinic (homoclinic) trajectory connect- ing S(8) to r(8) and tangent to

we

as t ^-+ -oo, tangent to

w:(

8 ) as t ^-+ +oo is equivalent to G(8)

=

0, which defines a codimension one hypersurface; i.e., curve, in ( 81 , 82 )-space. The existence of the homoclinic orbit with a saddle-node equilibrium is governed by the system of equations

which defines two curves intersecting transversely at 8

=

0 in (81,82)-space.

In §4 we have shown that the fibers

wz

of wcs (f-l) are well-ordered and we can speak of the right or left side of a fiber. We have shown that E(f-l) E

w:(

8)(8) is equivalent to G(8)

=

0. Thus, we know that E(f-l) is to the right of

w:(

8)(8) if one of the conditions G(8)

>

0 or G(8)

<

0 is satisfied. Similarly, E(f-l) is to the left of ^{w:(.5) (} 8) if G( 8)

<

0 or G( 8)

>

0 is satisfied. Let us fix 82

=

0 and increase 81 slightly from 81 = 0. Thus, r(8) moves to the right, like 0(81 ) and the perturbation of wcu(J.L) is like 0(8?). Therefore, we expect to have E(J.L) moved to the left of

w:(

8)(8) for 8

=

(81 , 0) 0 < 81 < t. Observe that 8G(0)/881

<

0. We conclude that E(J.L) is on the left of

w:(

8) ( 8) if and only if G( 8)

<

0, and E(f-l) is on the right of

w:(

8)(8) if and only if G(8)

>

0. These facts can be proved rigorously, but we shall not render it here.

We know that for every f11

<

0, there are two equilibria r(81 , 82) and r( -81, 82) with 8?

=

-J.L1 . So it is obvious that the condition for E(f-l) E

w:(-

8,,82/8) is that

BIFURCATION OF A HOMOCLINIC ORBIT 449 G( -61 , 62 ) = 0. The graph defined by this equation is a submanifold of codimension one in (61,62)-space and is a reflection of the graph of G(61,62) = 0 with respect to 61 = 0. The neighborhood of (61,62) = 0 is divided by the two submanifold G(61 , 62) ^{= 0 and G(} -61, 62) = 0 into four parts:

Figure 4.

For a description of Regions I, II, and III see Figure 3 in §4.

The following is the main result of this section.

Lemma 5.2. Assume that D~"2^G(O)

>

0. Then there exists a C 1 curve t-t 1 = ~(t-t2)

with ~(0)

=

0, ~(·) :::; 0 and~ is quadratically tangential to t-t2-axis at t-t

=

0. In a neighborhood o£0, if t-t 1

<

~(t-t2), E(t-t) E Region II, i£0;::: t-t1

>

~(t-t2) and t-t2

<

0, then E(t-t) E Region I; ifO;::: t-t 1

>

~(t-t2), and t-t2

>

0, then E(t-t) E Region III. If t-t 1

=

~(t-t2), and t-t2

>

0, then E(t-t) E W 55 (S(t-t), f..t). If f..tl

=

C(t-t2) and t-t2

<

0, then E(t-t) E W 55 (N(t-t), f..t).

Proof: Consider the equation G(61 , 62 ) = 0. Since G(O, 0) = 0 and ⁸~}~)

<

0, we can solve 61 as a function of 62; i.e., 61 = A(62). Moreover, since ⁸~}

2

^°)

^>

^{0, DA(O)}

ⁱ⁼

0. Note that t-t1 = -6? and and t-t2 = 62, the desired curve is t-t 1 = -(A(t-t2)) 2. The rest of the proof follows from Figure 4.

6. Periodic orbits. In this section we show that if t-t 1

>

0 or if t-t 1 :::; ^{0 and} E(t-t) E Region Ill, then there is a unique periodic orbit II~" bifurcating from f0 .

Moreover, II~" is hyperbolic with dim wu(II~") =dim wcu(o).

We will use the same notation as in §4. We assume that E > 0 is sufficiently small and the assumptions on the smallness of E will be described later. We assume that

for the study of the local flow near x = 0, a truncation has been made to equation (4.1) so that the vector field

f

has compact support in the ball {x: lxl ~ 2c} and IDfl ~ L(r:) with L(r:) -+ 0 as E -+ 0. Moreover, assume that a ^{c k - l} change of variable has been made so that Lemma 4.1 is valid. Let t0

>

0 be a sufficiently large constant. Any constant independent oft: and t0 will be denoted by C1, C2, · · · . We use C1(t:),C2(t),···, to denote constants that depend onE and approach 0 as

E -+ 0. Let o:, o: 1 and (3 be constants satisfying 0

<

(3

< -

o:

<

min { Re )q, - Re .L

d,

and o:

<

o:1

<

-3/3/2

<

-(3

<

0.

Consider equation (4.2). As in Silnikov [12], we consider a two-point boundary value problem in studying the flow near the equilibrium x = 0. The following lemma is crucial in this section and is a genralization of Silnikov's reparametrization theorem to a nonhyperbolic equlibrium.

Lemma 6.1. There exists f!

>

0 and Eo

>

0 such that if E

<

^t:0 , then for each vo E !Rn, u1 E !Rm, Y1

>

0 and to

>

0, with Ivai ~ i! and lu1l ~ f there exists a unique solution (y(t),u(t),v(t)) for (4.2), 0 ~ t ~to with y(to) = Y1, u(to) = u1 and v(O) = v0 . Denote the solution by

y(t) = y*(t,to,Yl,ul,Vo,f..L) +yc(t,to,yl,f..L) u(t)

=

u*(t, to, y1, u1, vo, f..L),

v(t)

=

v*(t, to, Y1, u1, vo, f..L)

where Yc(t, to, Y1, f..L) is the solution on Wc(f..L), with Yc(to, to, Y1, f..L) = Yl· We have the following estimates:

ly*(t)l ~ Cl(i!)eadoei3(to-t) lu*(t)l ~ Cl(f)eadto-t) lv*(t)l ~ C1(f!)e"''t,

ify1

>

lr(b)l.

Proof: Consider the following integral equation on 0 ~ t ~ to :

(6.1)

y(t) = i t [h(yc

+

y, u, v, f..L)(s)- h(yc, 0, 0, f..L)(s)] ds, (6.2)

to

u(t)

=

cA(t-to )ul

+

i t eA(t-s) 91 (Yc

+

y, u, v, f..L)( s) ds, (6.3)

to

Observe that

BIFURCATION OF A HOMOCLINIC ORBIT

jh(yc, u, v, p,)- h(yc, 0, 0, p,)j ~ C3lullvl, lhyl ~ L(E), lg1ul ~ L(E), lg2vl ~ L(E),

jeA(t-s)l ~ C4e<>(to-t), leBtl ~ C4e"'t.

Let L(f) ~ (3/2,

c3

^{° Cl(f.)/(3 ~} 1/2, L(f)C4

<

Ia- all/2 and 2C4f = Cl(f.).

The right-hand side of the integral equation (6.2)-(6.4) defines a mapping 1i: (y, u, v) __, (y, u, v).

451

(6.4)

If ly(t)l ~ Cl(f.)e"''toe,B(to-tl, lu(t)l ~ Cl(f)e"'dto-t), and lv(t)l ~ C1(f.)e"''t, then we have

IY(t)l

~

^{lto (}

~IYI ⁺

^{C31ullv1) ds}

~

^{lto {}

~Cl(f.)ea,toe,B(to-s) +

C3[CI(f.Wea,to} ds

<

Cl(f.) ea,toe,B(to-t)

+

C3[CI(f)j2 ea,toe,B(to-t)

- 2 f3

!u(t)l

~

C4lu1lea(to-t)

+

lto C4e<>(s-t)L(E)Iu(s)lds

<

C - a(t0-t)

+

L(E)CI(f.)C4ea1(t0-t)

<

C (-) ai(t0-t)

_ 4Ee I _a- I _ 1 f e

a1

Let X be the Banach spaces of continuous functions (y, u, v) in 0 ~ t ~ t0 with norm

ll(y,u,v)llx =max

sup jy(t)e-n,toe-,B(to-tlj, o::; t::;to

sup ju(t)e-n,(to-tll, o::; t::;to

sup lv(t)e-a'tl o::;t::;to

Thus, 1{: X--> X maps a C1(E")-ball in X into itself provided that

Ju

1

J:::;

E",

Jv

0

J:::;

E"

and E

>

0 is small. Furthermore, 1{ is a uniform contraction provided that t

>

0 and

E

>

0 are small. Therefore, equation (6.2)-(6.4) admits a unique solution (y*, u*, v*) in the C1(E")-ball in X.

The estimates for

J¥t-J, J

⁸

..ft·J,

and /⁸

at·l

come directly from (4.2). Except for the estimates involving partials with respect to t0 in ( 6.1), all the other estimates can be proved similarly. Thus, we will only prove the estimates for /

* /, /

^~~:

^I,

^and

I~~~

I·

We start with the following integral equation

ay i t ( ay au av)

-a (t) = hy-a

+

hu-a

+

hv-a (s)ds,

to to to to to

aau (t) ⁼ eA(t-to) ( -Au1- 9l(y, u, v, Jl)(to))

to

t

^{A(t-s) ( ay au av )}

+}toe 9ly ato

+

9lu ato

+

9lv ato (s)ds,

av _

t

^{B(t-s) ( ay au av)}

ato - Jo e 92y ato

+

92u ato

+

92v ato (s)ds.

Since

lhul :::; C5lvl, lhvl :::; C5lul, IY1yl + IY1vl :::; C5lul

and

IY2yl + IY2ul :::; C5lvl,

we have

I :t: (t)l :::; {

1t (

C5Cl(E")e"'(t-•)e<>l•e<>Jtoei3Cto-s) + C5Cl(E)e"'(t-s)e<>lseaJ(to-s) +C5L(t)ea(t-s)e<>is)ds}·ll(ay, au, av)ll .

ato ato at0 x If

_2C_s_C-cl-'(-'l)_+_C,-s_L-'-( E-'-) < ~

Ia- a1l -

2' then

l ay(t)l<~ll(ay

au av)ll e"'ltoei3(to-t) ato - 2 ato ' ato ' ato x '

l au(t)I<Ce"'l<to-tl+~ll(ay

au av)ll e"'J(to-tl

ato - ⁵ 2 ato ' ato ' ato X '

BIFURCATION OF A HOMOCLINIC ORBIT 453

Thus,

II (

~, g~

,

^g~

) II

x ::; C ^5. This gives the desired estimates for

I

~ I, I ~~: I, and

l_7ft; av•

I

0

We now prove the last estimate in (6.1). Observe that 8ycf8t0 satisfies the following initial value problem:

{

.:!_

0Yc (t) = hy. 0Yc (t)

dt 8to 8to

~~~(to) ⁼

^-h(y, 0, 0, f-1),

where -h(y, 0, 0, f-1)

=

e1

>

0 is a constant. Since lhyl ::; /3/2, we have

Thus,

(8y*

+

^Yc) (t)

<

^{- e} e-~(t0 -t)

+

C ea,toef3(to-t).

8to - ^{1 5}

Let t0 be so large that

We have that

I

We note that although Lemma 6.1 is proved for the truncated system, it is valid for the original system, if Y1, £ and lf-11 are small, and - f

<

Yo

<

Y1 in the case f-11

>

0.

Let y* (0, t0 , y1, u 1, v0 , f-1)

+

Yc(O, to, y, f-1) = Yo· Estimate (6.1) allows us to invert to as a function of (y0 , y1, u 1, v0 , f-1). In the case of f-11 ::; 0, such inversion is possible only ify0 ::; y1 and Yo is in the range ofy*(O)+Yc(O). If f-11::; 0 and Yo< Y1 and Yo is not in the range of y* (0)

+

Yc(O), we set t0 =

+oo.

By a shift in they direction, continuous with respect to f-1, we assume that: (i) for (-11 ::; 0, S(f-1) = {(y,u,v) = (0,0,0)}:

(ii) for (-11 = 0, SN(f-1) = {(y, u, v) = (0, 0, 0)}; (iii) for (-11

>

0, the solution of

~~ (y, 0, 0, f-1)

=

0 is y

=

0. We have the following:

Lemma 6.2. Assume that E > 0, i' > 0 are suflicicntly small and the assumptions of Lemma 6.1 are true. For fixed Y1

>

0, u1, Vo and f-1, Yo= y*(O,to,Y1,u1,vo,f.1)

+

Yc(O, t0 , y1, f-1) is strictly decreasing as to ----+

+oo.

Moreover, for f-11 ::::; 0 we have y0 ----+ 0 as t0 ----+

+oo.

On the other hand, for (-1₁

>

0, there exists

t

0 such that limto--->fo Yo = -c Moreover, if f-11

>

⁰ and Yo, f-1 are allowed to change, with IYol ::; <"3

and lf-11 ::; f3, then t0 = to(Yo, f-1) ----+

+oo

as f3 ----+ 0.

Proof: The monotonicity is obvious from (6.1). From the property of the fiow on the center manifold, it is clear that if (-11 ::::; 0 then limto--->oo Yc(O, to, y1, f-1)

=

0. Fur- thermore, limto--->+oo y*(O, to, Y1, u1, Vo, f-1) = 0 from Lemma 6.1. Thus, limto--->oo Yo=

0, provided (-11 ::::; 0.

To show the last assertion, suppose it is false. Then there exist sequences { t~},

{y~}, {t:~}, j = 1, 2, · · ·, and h

>

0 such that

IYbl

~ t:~,

tb

^{~ t1 and t:~ ----. 0 as}

j ----. oo. Let JL1 = 0. Let E be so small that C1 (E) ^~ yi/2. It can be shown that if fy*(O,to,··· )f ~ C1(f)e(<>¹+,6)t,, then Yc(O,to,y1,JLI)

2:

y1e-{3tt/²

>

0. We have that y0 = y*(O)

+

Yc(O)

2:

li,fe-f3ti/2 . Thus, y0 is bounded below by a positive constant for JL1 = 0. By the uniform continuity of y0 with respect to to and JL, Yo is bounded below by a positive constant for 0

<

^JL1

<

t:4 where t:4

>

0 is a small constant. This contradicts

IYbl

~ t:~----. 0. I

As in §4, for JL = 0, the homoclinic orbit fo has two connected components

rt

and f

0

in a small neighborhood of x = 0. Consider a cross section ~1 =

{xI

x = (y,u,v), y = di} for some 0

<

d1

<

t:. Thus, f_

n

^~1 = M1 = (d1,0,0). Also consider a cross section ~0 = {xfx = (y,u,v), v = (v 1,··· ,vn), vn =do} for some 0

<

do

<

c Hence, ~0

n r

+ = M0 = (0, 0, tJ), where tJ = (0, · · · , do). In a 6;-neighborhood of Mi E ~i the flow of (4.2) is transverse to ~;, where i = 0, 1, and 60 and

th

can be chosen independently of JL if JL and f are sufficiently small.

By following the trajectories of (4.2), a diffeomorphism T1(JL) : ~1 ----. ~0 can be defined with the domain being a D1-neighborhood of M 1 in ~1. Also, by following the local flow a diffeomorphism To(JL) : ~0 ^{----. ~}1 can be defined in the domain a0

which is in a Do-neighborhood of M0 in ~0^. In the above definitions for To and T1

we assume that Do and D1 are sufficiently small. Observe that not all the points in the Do-neighborhood will return to ~1 when JL1 ~ 0 even if Do is very small.

y

u

Figure 5.

The mapping 1u(Jl) : a0 C ~0 ^{----. ~}1 can easily be described by the parametrized version (see [10]). Set 1'o = (v~, ... ,v

0)

^{= (v0,v}

0),

^{where v0 = (v~, ...}

,vg-

1 ). If (yo, uo, vo) E ao, then =do and fvof <Do. Let 0

<

d1

<

t:, [u1[

<

D1 ~ E, Do::::; E,