On the existence of periodic solutions in the limited explodator model for the Belousov-Zhabotinskii reaction

then A contains the

KAMKE’STHEOREM[7]. Suppose that (2.1) is cooperative in Q. Let R, y: [0, + co) -+ R be CL maps. If X(t), y(t) are the solutions of (2.1) and x(O) < p(O), then n(t) 4 70)

for all t E [0, + 00).

(2.2)

LEMMA2.2 [4-61. Suppose that (2.1) is an irreducible cooperative system. Let 4, Y E Q and x < p, R # jj, and the orbits through 2, y be in R. Let O(X) and o(y) be compact and nonempty. Define E c Sz to be the set of equilibria of (2.1). Then exactly one of the following cases holds: (a) ~(9 X MY)), (b) O(X) = o(y) c E. By lemma 2.2 and time reversal, it is easy to prove the following. COROLLARY 2.1. Suppose that (2.1) is an irreducible competitive system in a p-convex open set Q. Let r be a nonconstant periodic orbit of (2.1). Then no two points of the unstable manifold M”(T) of r are related. 3. RESULTS It is easy to see that /Rt is invariant for (1.2), and that (1.2) has a unique equilibrium (x0, yo, zo), where x0 = D = B/P, y. = 1 + p/D, z. = (D/A)(l + p/D). Clearly (x, y, z) = (Pe’ - p, 0,O) is a solution of (1.2). THEOREM3.1. Let AD > 1. Suppose that the Jacobian matrix of (1.2) at (x0 ,yo, to) has two conjugate characteristic roots with positive real parts. Then all nonconstant orbits on the

1361

Limited Explodator model

unstable manifold of (x,, , y,, , z,,) of (1.2) approach the same periodic orbit of (1.2), the Floquet multipliers of which have absolute values less than or equal to one. In order to prove theorem 3.1, we need several lemmas (lemmas 3.1-3.8). By the transformation: X=

x-x,,

Y=y

Z=z--0,

-Yo,

(3.1)

(1.2) can be written as

X - (B + D)Y + (1 + P)AZ - XY, (Z=

(3.2)

(1+;)X+DY-AZ+XY.

Let .-D,Y.-(l+$),Z+l+$)].

(X,Y,Z)ER~IX

Q is a positively invariant set for (3.2) and (0,0,O) is the unique equilibrium of (3.2) in Q. (X, Y, Z) = (pe’ - ,D - x0, -y,, -zo) is a solution of (3.2). Now we consider (3.2) restricted to Q. The Jacobian matrix of (3.2) at (0,0,O) is -- P

-D

0

-(B + D)

(1 + P)A

D

D

the characteristic

9

-A

equation of which is &I)

=

A3 + a,12 +a,1 + a0 = 0,

(3.3)

wherea2=A+B+D+~/D~O,a,=-D+~/D(A+B),ao=AB(1+~/D)~O.ao>O

implies that (3.3) has a negative characteristic Let

root -1,(1,

> 0).

and Qs=((X,Y,Z)~QIXzzO,

Y(:O,ZrO).

From the assumption that (3.3) has another two characteristic roots with positive real parts and the fact that q$-p/D) > 0 and 9(-A) > 0, one can prove the following. LEMMA 3.1.

The stable manifold of (0,0,O)

of (3.2) points into Q, U Qs .

I362

B.

TANG

Introduce the following surfaces: s,: y=

-ii

D

& 1

s2:z=(1 + /I)‘4 K &:Z=i K

P

w,tn

l+$

p$!+(x,Y),

X+(B+D)Y+XY >

3

l+’ D x + DY + XY >

1

2 (pj(X, Y).

Note that k = 0 on S,, Y = 0 on S, and Z = 0 on S,, respectively. parts: Q = S, U SC U S;, where

S, divides Q into three

&+ = 1(X, Y, Z) E Q 1 Y > w&W &- = lW, I’, Z) E Q 1 Y < w,(W. S2 divides Q into three parts: Q = S2 U SC U S;,

where

S; = 1(X, Y, Z) E Q 1Z > (02(X, Y)l, S,- = (W, Y, -3 E Q 1Z < v,zV, Y)l.

And S3 divides Q into three parts: Q = S3 U ST U S;, where %+ = ](X, Y, Z) E Q t Z > P&C

YN,

&- = (W, Y, Z) E Q 1Z < PO’,

YN.

Clearly, k < 0 for (X, Y, Z) E &+,

X > 0 for (X, Y, Z) E S;,

Y > 0 for (X, Y, Z) E S,‘,

Y < 0 for (X, Y, Z) E S;,

i < 0 for (X, Y, Z) E SC,

i > 0 for (X, Y, Z) E S;.

Now consider the vector field of (3.2) on S,, S, and SJ. On S,, k = 0, n, 2 d/dt[ Y - v/i(X)] = I’. If Y > 0 and n, > 0, then the direction vectors of (3.2) on S, point into ST. If Y < 0 and n, < 0, then the direction vectors of (3.2) on S, point into S;. On S2, Y = 0, n, & d/dt[Z - (p2(X, Y)] = (Y(X + 0) + (PA + &D)X)/((l + p)A)(Y + + D)) B V/~(X) and 1 + p/D). Now Y + 1 + c(/D > 0, so if Y > -(PA + p/D&7/(X n2 > 0, then the direction vectors of (3.2) point into S:, and if Y < 02(X) and n2 < 0 then the direction vectors of (3.2) on S2 point into SC. Similarly, on S, , n3 4 d/dr [Z - q&X, Y)]. If Y > [p - b/0)( l/(D + S))]X P VIJX) and n3 > 0 then the direction vectors of (3.2) on S3 point into S:. If Y < wJX) and n3 < 0 then the direction vectors of (3.2) on S, point into S-J. Consider S,: Y = w2(X) and S5: Y = v3(X). The surfaces S; (i = 1, . . . . 5) divide Q into twenty-four parts, fifteen of which form a positively invariant set for (3.2). Consider the open two-dimensional regions (I-VIII) shown in Fig. 2. Denote by “d” (down) the region in Q where Z 5 min((02(X, Y), p3(X, Y)], by “b” (between) the region in Q where minlrp&K Yh u13W, Y) 5 Z 5 maxb2W,

Y), v3W, YN,

1363

Limited Explodator model

by “u” (up) the region in Q where Z 2 maxMX,

Y), (PAX, Y).

Use these letters and the numbers in Fig. 2 as indices to denote the corresponding example,

sets of Q. For

Qtb = ((Xv Y, Z) E QI 93(X) I I’-( WAX), 93(X, Y) 5 Z -( 92(X, YN. LEMMA

3.2.

Consider the set Q* = Up=, Qi, where

Q, = Qtb u Qrrb U Qw u Qrrcr = i(X, Y, 0 E Q 1Z 5 92(X, I’), Y 5 90).

X 5 01,

Q2 = Qrrrd U Qrvd U Qrru

or 492(X,

n

5

z

5

e(X,

0

y

5

w2Gn

x

2

01,

Q3 = Qrvb = ((XV Y, Z) E Q t 92(X, Y) 5 Z 5 93(X, Y), ‘44X) s Y 5 Vi(X), X 1 01, Q4 = Qvb u Qvrb U Qv,, U Qv~u = ((X, Y, Z) E Q 1Z 2 92(X, Y), Y 2 W,(X), X 2 01, Qs = Qvnu U Qvmu = 1(X, Y, Z) E Q 1Z 2 92(X, Y), Y 2 W,(X), X 5 01, Qa = Qvrrrb = {(X, Y, Z) E Q

)93(x,

y)

I

z

5

92(x,

y),

W,(x)

I

y

5

v/z(x),

x

5

0).

Then (i) Q* is a positively invariant set of (3.2); (ii) every orbit of (3.2) not on the stable manifold of (0, 0,O) of (3.2), which starts from a point outside Q*, must enter Q* in finite time. Proof. (i) We are going to prove that on the boundary aQ* of Q*, the orbits of (3.2), except (X, Y, Z) = (pe’ - p - x0, -ye, -z,), point into Q*. It is easy to see that i3Q* = LJ,!:, II;, where

D, = ((X, Y, Z) E Q 1Z = 9,(X, Y), X 5 0, Y 5 W,(X)) D2=

(X,Y,Z)EQ~Z=-;

Y 5 w,(X)

,

D3=l(X,Y,Z)~QIX=-D,Z~92(-D,Y)), Da = ((X, Y, Z) E Q 1Z I 93(X, Y), Y = v/,(X), X = 01, D,=((X,Y,Z)EQIX>-D,

Y=O,Z=O),

136-l

B.

w,(X),

X 1 01,

Y, Z) E QI Z 5 (POW, Yh

Y = w,(X),

X 2 01,

Y, Z) E Q 1Z = v,Z(X, Y),

Y 2 u/,(X),

X 2 01,

= 1(X, Y, Z) E Q 1Z 1 rp,W, Y),

Y = w,(X),

X 2 OJ,

D,,

= (W,

Y, Z) E Q 1Z = p2(X,

Yh

Y L w2(X),

X 5 01,

4,

= f(X,

Y, Z) E Q 1 Z 2 lo2W, Yh

Y = wl(X),

X 5 01,

D,,

= i(X,

Y, Z) E Q 11p3(X, Y) 5 Z 5 92(X, Y),

D,,

= ((X, Y, Z) E Q 1Z = 9,(X,

4

=

(W,

D,

= l(X,

D,

= M,

4

I’,

Z)

E

QI Z = 93(X,

TANG

Y),

Y I

Y), w,(X)

I

Y = wz(X),

X I 01,

Y 5 J/~(X), X I 01.

Also, it is easy to check the following properties. (1’) w,(X) 5 w&X) 5 MX), 9JX, Y) 5 9AX. Y) for X 5 0, e(X)

2 w,(X)

(2’) On S,, if if On S,. if if On S3, if if (3’) On S,: Y

Z > Z < Y> Y< Y> Y<

2 WAX), 9dX, Y) 2 (PAX, Y) for X 2 0; 9,(X, Y), the orbits of (3.2) point into ST. 9*(X, Y), the orbits of (3.2) point into S;; w2(X), the orbits of (3.2) point into S:,

v2(X), the orbits of (3.2) point into S;; V/~(X), the orbits of (3.2) point into S,+, v3(X), the orbits of (3.2) point into S;. = ty2(X), let S,- = ((X, Y, Z) E Q 1 Y < w2W)L

then when 9JX, Y) I Z I 9*(X, Y), X I 0 (i.e. i 5 0, I’ I 0, k 4 0), d/dt[ Y - w2(X)] = I’ + (PA + pc/D)(D/((X + D)‘)k I 0. This implies that the orbits of (3.2) which start from (X, Y, Z) E D,, point into S;. From the above properties, (i) follows. To prove (ii), consider Q - Q* = U,!l, Qi, where

QCI= Qru U QII~ = f(X, Y, Z) E Q

1Z > 92(X, Y),

Y < w,(X),

X < 01,

1Z > 93(X, Y),

Y < u/l(X),

X > 01,

z < 92(x, y),

y > WI(~),

x > 01,

z

y

x

QIO= Qmu U QIV~ = fW, Y, Z) E Q QI 1

=

Qw

U Qw

= 1(X, Y,Z)

E 91

Q,2 = QVIM U Qvm, =NXYJhQi

<

93(X,

Y),

>

w*Wh

<

01,

Q13 = Qvm = f(X,

Y,Z)

E QI

92(x,

y)

<

z

<

93(x,

y),

y

>

v/2(.0,

x

<

01.

1365

Limited Explodator model

In Q9. X > 0, Y > 0, Z < 0. If an orbit I- = (X(t), Y(t), Z(t)) remains in Q9 for large t, then because of X < 0, Z > -A/D( 1 + p/D), there exist X0 5 0, and Z,, L - A/D(l + p/D) such that lim X(f) = X0, and lim Z(f) = Z,. Since X(t) > 0, 0 I X(t) + D < X0 + D. If Y(t) f-+O ,-+o= is unbounded, then Y(f) -+ +oo as t -+ fco, so that i(f) = (1 + ,D/D)X - AZ + (D + X)Y -+ +a as f -+ +m, a contradiction. Hence Y(f) is bounded. Then there exists a Y0 such that lim Y(f) = YO. So (X,, , Y,, Z,) is an equilibrium of (3.2). By the uniqueness of the equilibrium ,-+m of (3.2), (X0, Ye, Z,) = (0,0, 0),again a contradiction. Similarly, orbits of (3.2) not on the stable manifold of the equilibrium (0, 0,O) of (3.2) do not remain in Q,,, U Q,, U Qlz U Q,, for large f. The above conclusions along with the proof of(i) impliy (ii). n LEMMA 3.3.

Suppose AD > 1. Then every orbit of (3.2) in Q* advances in the following manner:

QI~Q~~Q~-*Q~~Qs-‘Q~~QI. Proof.

We first prove that no orbit of (3.2) in Q* remains in Qi (i = I,& . . . ,6) for large f. By lemma 3.1, no orbit of (3.2) in Q* (except the equilibrium (0, 0,O)) stays on the stable manifold of the equilibrium (0,0,O). In Qr, X > 0, Y < 0. If an orbit I = (X(f), Y(f), Z(f)) remains in Qr for large f, then from X < 0, Y > - (1 + p/D), it follows that there exist X, I 0 and Y, L - (1 + p/D) such that lim X(f) = X0, and lim Y(f) = Y,. Also there exists a sequence f-+m I-+m (f,n: m =

such that

1,2, . . . . lim f,,, = +a) m-+m

lim Y(t,J = 0, i.e. m-+4, -

(

I + 5

as m + +oo.

>

X(1,) - (B + D)Y(f,,,) + (1 + PM.%,)

- X(f,,,) Y(f,“) ---*0,

Hence Z(f,) + Z, = (oz(X,, YO) as m -+ +a. Now consider the orbit I, = with initial point (X,(O), Y,(O), Z,(O)) = (X,, Y,, Z,). Then for every T E R, X,(T) = X(r, X0) = X(T, lim X(f)) = lim X(t + r) = X0, Y,(r) = lim Y(t + r) = Y,. This ,-+m t-+0;, I-+m means I, = (X0, YO,Z,(f)). By 0 = Y, = - (1 + p/D)X,, - (B + D)Y, + (1 + &lZ,(f) X,, YO, we have Z,(f) = Z, = (oz(X, , Ye). Hence (X0, Y,, Z,) is an equilibrium. By the uniqueness of the equilibrium of (3.2), (X0, Y,, Z,,) = (0,0, 0), i.e. (0,0,O) is a limit point of I-, a contradiction. In QrrIb, X > 0, Y > 0, Z > 0. Since AP > 1, there exists Xs = (AP + p)/(A/? - 1) > 0 such that Y = v2(Xs) = 0. So m QIIIb, X < X5, Y < 0, Z < l/A(l + ,4D)Xs. This implies that no orbit of (3.2) remains in QnIb for large f. It is easy to check that on the common boundary of Qmb and QIIId U QrVb9the direction vectors of (3.2) point into QIIId U QtVd. In QrrId U QIVd, k > 0, Y < 0, i > 0. If an orbit I = (X(f), Y(t), Z(f)) remains in QIrrd U QrVd for large f, then Y > - (1 + p/D) implies that there exists Y, L - (I + p/D) such that lim Y(f) = Y,. Since p(f) > 0, 0 > Y(f) > Y, for f L 0, so that -(Y(f) + flu/D) < I-+m - (YO+ p/D)5 1. If X(t) is bounded, there exists an X0 such that lim X(f) = X0. Now there I-+(X,(f),

Yl(f), Z,(f))

1366

B. TANG

t-D,-

(I +

$-

L-g (I++_)) Fig. 1.

is a sequence (t, : m = 1.2, . . . ; lim t, = +m) such that lim Y(t,,J = 0, which implies that ?Tl-++m m-+a, there exists a 2, such that lim Z(l,) = Z,,. Since Z(t) > 0, lim Z(f) = Z,. But the uniquem-+m [-++m ness of the equilibrium implies that (X(t), Y(t), Z(t)) + (X0, YO,Z,) = (0, 0, 0) as t --+ +oo, a contradiction. If X(j) and Z(f) are unbounded, then X(l) ---t +a~, Z(t) + +co as t + +co. Since 2(r)=-(Y(t)+$X(r)-DY(I)5X(O-DY,,

X(t)lc,e’+DY,.

Now Y(t) = PAZ(r) - BY(t) - Z(f) < 0, so Z(r) > PAZ(t) - BY(t), Z(t)zc,eaA’ + Y(O)//Ql, where ct , c2 are positive constants (because of X(t) -+ +oo, Z(t) + +a0 as t * fco). Hence x(t)

- (B + D)Y(t)

I

1 (

D,-II

+ (1 + P)AZ(t)

+;I)

I

Fig. 2.

- X(t)Y(t)

1367

Limited Explodator model

where cj is a constant. Then /3A > 1 implies that p(t) + +m as t + +a, a contradiction. If X(t) is unbounded and Z(f) is bounded, then there exists a Z,, > -D/A(l + p/D) such that X(t) * +oo, Z(t) 4 Z, as t + +co. Now P + i = &AZ - DY), (Y + i’) = p(Ai - Dp) > 0, so L + i is monotonely increasing. Now lim (AZ(f) - DY(t)) = AZ, - DY,. If AZ, t-+m

DY, > 0,

then there are E,, > 0 and G > 0 such that p(AZ, - DY,) > y(t) + i(t) > PC,, > 0 for t 2 T,, which implies Y(t) + Z(t) + +a~ as t ---) +a~. Since Y(t) < 0, we have Z(f) + +a, as t -* +co, which contradicts the fact that Z(t) is bounded. If AZ,, - DY, < 0, y(t) + i(t) < P(AZ, - DY,) < 0, which implies Y(t) + Z(f) --t --a0 as t * +oo. But Y(t) and Z(t) are all bounded, also a contradiction. Hence lim (AZ(f) - BY(t)) = AZ, - DY, = 0. Now there I’+m exists a sequence (t, : m = 1,2, . . . ; lim t,,, = +Q)) such that lim i(t,) = 0, i.e. m-+00 m-+cO

lim

!?I-+0

Y&)

+ 1+

5

>

X(t,)

+ DY(t,)

- AZ&)

= 0.

lim X(f,) = +oo, lim Y(?,,J = Y, and lim Z(t,) = Z,,, so lim Y(t,) = - (1 + p/D) = !?I-+= WI-+CO m-+C0 !Pl-++m Y,. Therefore Z, = (D/A)Y, = - D/A(l + p/D), which contradicts the fact that to > -D/A(l + j/D). A similar argument shows that there is no orbit of (3.2) which stays in Uf= 3 Qi for large t. In summary, we have proved that no orbit of (3.2) remains in Qi (i = 1, . . . ,6) for large t. It is easy to check that on the common boundary of Qi and Qi+, (i = 1, . . ., 6; Q, 4 Q,), the direction vectors of (3.2) point into Qi+l from Qi. n

By lemmas 3.2 and 3.3, one obtains the manner of advance for orbits of (3.2) as follows.

BY means of the transformation

u = x,

u = Y,

w=-z,

(3.4)

1368

B. TANG

we rewrite system (3.2) as

ri = -

5u

- Dv - uv =

f(u,

v),

u - (B + D)v - (1 + /3)/l w - uu = g(u, u, w),

ti = -

(3.5)

[ti=-(1

+$)u-Dv-Aw-uv=h(u,u,w).

Under the transformation (i = 1, *.., 13), respectively.

(0

(3.5), Q, Q* and Qi (i = 1,2, . .., 13) become Q’, Q** and Q; It is easy to check that

f” 5 0; g,, 5 0, g, 5 0, g, < 0, h, < 0, h, I 0, h, < 0, I

in Q’,

i.e. system (3.5) is an irreducible competitive system; (ii)’ (0,0,O) is a unique equilibrium of (3.5) in Q’; (iii)’ Q** is a positively invariant set for (3.5). LEMMA 3.4. Let Q*** = IJfi, Q,!. Then there is no nonconstant approaches the equilibrium (0, 0,O) in negative time. Proof.

This follows from the proofs of lemmas 3.2 and 3.3.

orbit of (3.5) in Q*** which

W

Lemma 3.4 implies the following lemma. LEMMA 3.5. The global two-dimensional in Int(Q**) (except (0, 0,O)).

unstable manifold U’ of (0,0,O) of (3.5) is contained

Represent the local two-dimensional unstable manifold W by fi = &(O, r;l), [8], where (a, 0, ti)r = M(u, v, w)r are the coordinates of the stable and unstable manifolds of the linear part J’ of (3.5) at (0, 0, 0), and M is a change-of-basis matrix. Also, 6 is C2 and the unstable manifold W is tangent to the unstable manifold of (ti, 0, G)r = J’(fi, 0, G)r at (0,0,O). This implies (&, &.)l~O.O~= (0,O). LEMMA 3.6. W is represented 4, c 0, 4, c 0 near (0,O). Proof.

locally near (0, 0,O) by a unique C2 function u = +(u, w), and

Similar to the proof of lemma 1 in (91. n

The local unstable manifold W can be extended via the flow of (3.5) to the global unstable manifold U’ which is the union of all orbits of (3.5) with (0,0,O) as an o-limit set (lo]. By lemma 3.6, it is easy to prove the following. LEMMA 3.7. U’ contains no related points.

Limited Explodator

1369

model

Let S=

(u,O,w)ER3I--DIf4~0,

- (1 +1B)a(l

++

+(1

+;)I,

[ then S

C

Q; and S divides Q; into two regions: S- = ((u, u, w) E IR31u I 0) n Q;, SC = ((u, u, w) E lR’ ) u 1 0) f~ Q;.

On S, the orbits of (3.5) point into S- from S+. By lemma 3.3, one can define the Poincare map P on S: define P(u, u, w) be the point at which an orbit of (3.5) starting at (u, u, w) E S next meets S. Then P is continuous on s - ((0, 0, 0)). Let L = U’ fl S. Then L is homeomorphic ((0, 0, 0)) c Int(Q**).

LEMMA 3.8.

to [0, r) for some r < +oo and L -

Proof. Because two eigenvalues of the Jacobian matrix of (3.5) at (0, 0,O) are imaginary, the orbits on U’ spiral back to (0, 0,O) as t + -co. Hence the inverse P,-’ of Pr., the restriction of P to L, is defined everywhere on L. Since By lemma 3.7, L = UieJ, Li, where Li are disjoint connected components. (0,0,O) E L, without loss of generality, assume that (0,0,O) E L,. If L has other components, which differ from L1, then there exists a 4 E Lj # L,, with j E J,. Since lim PIem = m-+oD

(0,0,O) E L,, so

for sufficient large m PLm(q) E L,. Now PFm(L1) = L, and Pi”’ is a one-to-one map, hence Pi”(Lj) = L,, i.e. Lj = L,, a contradiction. Therefore, L has only one connected component. By lemma 3.7, L is monotone. Since L - ((0,0,O)) C Int(Q**), L is homeomorphic to [0, r) for some r < +oo. H Proof of theorem 3.1. By lemma 3.8, L has an end point pe = (u,, 0, w,) E Int(S), where

Int(S) =

i

(u, 0, w) E R3 I -D

< u c 0, - (1 +lD)a(l

+$),<

w<;(l

+$)I.

Now we prove that the orbit &(p,) is periodic and every nonconstant orbit &(q) of (3.5) starting from q E L approaches this periodic orbit in positive time. Consider the orbit 4,(q) with q = (q”‘, 0, qC3’)E L. Let P(q) = (P,(q), 0, P,(q)). We claim that P,(q) < q(l) and P,(q) > q (3). If not, then by lemma 3.7, q(l) < P,(q), qC3’> P,(q) (see Fig. 3, where S is bounded by the dotted lines). Since lim Pi”‘(q) = (0,0, 0),and P,(q) c 0, m-+=

P,(q) > 0, the uu-plane projection of the orbit&(q)

must cross itself at some point q1 (see Fig. 4, where 4, p(q) are the uu-plane projections of the corresponding points q, P(q) in Fig. 3, respectively). The orbit projection cannot cross the negative u-axis in the positive u-direction. The two points of&(q) which projects onto q, are related. This contradicts lemma 3.7. By reiterating P, one obtains a sequence (Pm(q): m = 1,2, . . .] with P;“+‘(q) < P?(q), P?+‘(q) > P,m(q). Now P?+‘(q) > u,, PT+’ (q) c w, and P is continuous, so lim P”(q) m-+=

exists. In view of the arbitrariness

of q, the decreasing of P;“(q) and the increasing of P;“(q),

B. TANG

1370

0

0 IS

L----e

--_-----_I

i I ’

D $,+$)

1 w

Fig. 3.

lemma 3.8 implies that

lim P(g) = (u,, 0, w,), so that P(u,, 0, w,) = lim P’““(q) = /?I-+m-++m (u,, 0, w,). Hence the orbit &(p,) is periodic. Denote this periodic orbit by I,. The arbitrariness of q also implies that every orbit with some point on L approaches I,. Now we prove by contradiction that the Floquet multipliers of r, have absolute value less than or equal to one. Suppose that I, has at least one Floquet multiplier with absolute value greater than one. According to lemma 2.1, there exists a k-dimensional (k 2 2) C’ unstable manifold M”(IJ of I,, and the dimension of the manifold M”(lJ fl Int(S) is at least one. According to corollary 2.1, there exist a q E M”(T,,) n Int(S) and a q E L such that q and 4 are related. Now a(q) = r,, CX(Q)= [(O, 0, 0)). so there exists certain point 4r(pe) E I, such that it is related to (0,0,O) by Kamke’s theorem, but that is impossible. By the transformations (3.1) and (3.4), theorem 3.1 is proved. n Remark. (1) Lemmas 3.1,3.2 and 3.3 imply the existence of periodic orbits of (3.5).

(2) By means of the transformation: (X, Y, Z) y (-Z, X, Y), (3.2) becomes a new system which obeys most of the assumptions in [9], except that the positive invariant set may not be bounded and h, varies in sign.

Fig. 4.

Limited

Explodator

1371

model

In view of the equivalence of systems (1.2) and (3.5), we only consider (3.5) under the coordinate system (u, u, w) in the following discussion. THEOREM 3.2.

The periodic orbits of (3.5) are unrelated.

Proof. Let Ii, r, be two different periodic orbits of (3.5). If I’i and I, are not unrelated, then there exist two points q E I-,, 4 E r, such that q and 4 are related. Without loss of generality, assume that q < 4. Then by lemma 2.2, I, < I,. But by lemma 3.3, there exist (u,, 0, w,) E Int(S) (l I, and (u,, 0, w,) E Int(S_) n I,, where Int(S_) = ((u, 0, w) E R’ 1u 2 0, w I - l/((l + &I)(1 + j/D)u). Clearly, (u,, 0, w,) and (uz, 0, w2) are unrelated, a contradiction. n

In order to study further the relative position and the number of the periodic orbits of (3.5), suppose that (3.5) satisfies the following condition. (H) All periodic orbits of (3.5) are hyperbolic. In the following suppose that the conditions of theorem 3.1 and (H) hold. Let P be the Poincare map as in theorem 3.1. Suppose that I is a periodic orbit of (3.5). Let qr = (Q, 0, We) = r fl Int(S). Define ZN(qr) = UU, 0, w) E Int(S) I u > h-, w < WI-I, EX(qr)

= ((u, 0, w) e

Int(S) I 24< f+, w > w,].

Suppose that r is another periodic orbit of (3.5) and qr = (uy, 0, WY) = r t7 Int(S). Then by theorem 3.2, qr E ZN(qr) or qr E EX(q,-). Let I,, be the periodic orbit in theorem 3.1 and qro = r,, n Int(S). LEMMA 3.9.

The number of periodic orbits of (3.5) is finite.

Proof.

Since all periodic orbits of (3.5) are hyperbolic, they are isolated [ll]. Suppose that the number of periodic orbits of (3.5) is infinite. Choose a sequence of periodic orbits (Tm: 171= 1,2, . . . ). Denote I, rl Int(S) by qm, then q,,, is a fixed point of P: P(q,) = qm * (4,“: m = 1,2, . ..) is a bounded sequence, so there is a convergent subsequence, without loss of generality, assume that lim qm = q. According to the proof of theorem 3.l,ZN(u,) m-+X7 contains no fixed points of P, so q # 0. By the continuity of P, P(q) =

lim P(q,) =

!?I-+-

lim qm = q.

m-+C0

Hence the orbit through q is periodic, but it is not isolated. This is a contradiction. Definition 3.1. ZN(q,-) n EX(r)

Two different periodic orbits I, r are called neighboring if EX(q,) contains no fixed point of P.

n

fl ZN(qr)

or

Let I be a periodic orbit of (3.5) and N(T) = ((u, u, w) E Int(Q’) 1(u, u, w) is not related to any point of I], IN(T) = ((u, u, w) E Int(Q’) I( u, u, w) is not related to any point of I and there

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exist pr = (pi, pt, p’,) E r and qr = (q;, qt, 4;) E r such that u E (pk. qi) or u E (Sk, pr) and u, w have the same property], EX(T) = N(T) - IN(T). Remark.

In the definition of IN(T), pr and qr may change for different (u, u, w).

It is easy to prove the following lemma. LEMMA3.10. Let G = G, U Gz be open and G, n Gz = r$. Then G, and Gz are open. Clearly EX(T) fl IN(T) = 6. Since Iv(r) = ((I- + ii?:)’ U (r - IR:)‘) f7 Int(Q’) is open (where A’ is the complement of A, A + K & (a + b 1a E A, b E Kj), ZN(r) and Z%(T) are open. Since N(T) is positively invariant for (3.9, IN(T) and EX(T) are positively invariant for (3.5). LEMMA3.11. Let r and r be neighboring totically stable simultaneously.

periodic orbits. Then r and T can not be asymp-

Proof. By contradiction. Let r and T be two neighboring periodic orbits of (3.5) and suppose that they are asymptotically stable. Without loss of generality, suppose qr E EX(qr). Then qr E ZN(qr) and ZN(qr) fl EX(qr) contains no fixed point of P. ZN(qr) fl EX(qr) is a nonempty open set with respect to S. qr E ZN(qr) implies qr E IN(r), so r C IN(T). Hence EX(T) fl IN(T) is nonempty and open. Define AS(r) = (p E Int(Q’) 1b,(p) + r as t -, +oo). Since r and T are asymptotically stable, AS(T) and AS(T) are open. Since r c AS(T), AS(r) n EX(T) fl ZN(T) is a nonempty open set. If AS(T) fl KY(T) 0 ZN(f) = EX(lY) f7 ZN(r), then by T c a(EX(T) rl IN(f)), there exists a p E T such that p E d(AS(T)), which contradicts the asymptotic stability of r. So AS(r) n EX(r) n IN(r) c EX(r) n IN(f), then there exists ap E &4S(T)) n EX(T) fl IN(T). By the positive invariance of EX(T) and IN(T), w(p) E Cl(EX(T) fl IN(r)) (where CL(A) is the closure of A). According to theorem C in [4], o(p) is a periodic orbit of (3.5). By theorem 3.2, o(p) c EX(T) n ZZV(T). Since EX(T) n IN(f) fl Int(S) c EX(qr) n ZZV(qr) and q,@, = 0(p) n M(S) E m(r) n IN(T) n In@), q,@, E EX(qr) tl ZN(qr). This contradicts the fact that EX(qr) n ZN(qr) contains no fixed point of P. H

LEMMA3.12. Let r be an unstable periodic orbit of (3.5). Denote L(T) = M”(T) tl Int(S), where Mu(r) = l(u, u, w) E Q’ I a ( u, u, w) = I-) is the unstable manifold of IY. Then L(T) is homeomorphic to (rt,rJ for --CQ< rl < r, < +oo. Proof. (3.5) can be changed into a cooperative system by time reversal, so by (H) and theorem 2.4 in [5], M”(T) is a two-dimensional manifold. M’(T) intersects S transversely. Off the straight line: w = - ~~(0, u), the vector field of (3.5) is transverse to S, so L(T) is a one-dimensional C’ manifold [ 121. By corollary 2.1, L(T) is the union of curves which cannot intersect: L(T) = Uj EJ, Lk for some index set J, . For any p E M”(T), cr(p) = r, so 4,(p) spirals into r in negative time. Hence P-’ is defined on L(r). The remaining proof is similar to the proof of lemma 3.8. 1

Limited

Explodator

model

1373

0

q J,

”

P(9)

qr

w

Fig. 5

LEMMA3.13. Suppose that I is an unstable periodic orbit of (3.5). Let err and ezr be two end points of L(T) = M”(T) n Int(S). Then (+,(eir) 1t E (- m, +a)) (i = 1,2) are two asymptotically stable periodic orbits of (3.5). By corollary 2.1, L(T) = L,(I) U L,(T), where t,(f) n L,(T) = qr, L,(T) - qr = Suppose elr E ZZV(q,),e,, E EX(qr). By lemma 3.12, L,(T) is homeomorphic to [r, r2) and L*(r) is homeomorphic to (or, r] . e,, is an end point of L,(T) and e,, B L,(T). e2r is an end point of L,(T) and e2r $ L2(r). The following proof is made for err. A similar argument also holds of ezr. For any q E L,(T), corollary 2.1 implies that q E IN(T), so 9,(q) E IN(T) for all t E [0, +oo). Hence P(q) E L,(T). Let q = (4”‘. 0, qc3’), P(q) = (P,(q), 0, P,(q)). An argument similar to the proof of theorem 3.1 show that P,(q) > q(l), P,(q) < q (3). Reiterating the Poincart map P, one obtains that lim P”(q) = e,,, so the orbit through err is a periodic orbit w(err). By the arbitrariness of q, Proof.

M”(T) fl ZN(q,) and &(I’) - qr = M”(T) n EX(q,).

m-+m

every orbit with some point on L,(T) is asymptotic to o(e,r). If w(err) is unstable, then the unstable manifold M”(o(err)) of w(e,r) meets Int(S) in a l-dimensional manifold M”(w(err)) fl Int(S). According to corollary 2.1 and lemma 3.12, there are two pointsp E M”(o(err)) n Int(S) andp E L,(T) such thatp andp are related. Without loss of generality, assume that p
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there must be the fixed point qr,, of P after finitely many times such that EX(qr,,,) contains no fixed point of P. It is easy to prove that the periodic orbit r,,, through qr, is asymptotically stable. Since the periodic orbits alternate between asymptotic stability and unstability, and r, and r,,, are asymptotically stable, M = 2p for some positive integer p. From the above it follows that the fixed points {O)U {qrj:j = 0, 1, . . ., 2~) of P are on the invariant curve of P:

where L(0) is the curve L in the proof of theorem 3.1, i.e. L(0) = U’ fl Int(S), and L(T2j_ 1) = fl Int(S), for j = 1, . . . , p. Divide Int(S) into two parts: EX(qn,) and Int(S) - EX(q,,,). Then EX(qrzP) contains no fixed points of P. If there exists a fixed point q of P such that q E Int(S) - EX(qrzp), then by the above proof, there is a 4 E L^such that Q is related to q. Without loss of generality, suppose 4 < q, then by lemma 2.2, a((~) -C cu(q). But a(q), CY(Q) are periodic orbits of (3.5). So this contradicts theorem 3.2. Therefore all fixed points of P are on t. Define

M”(T~j_,)

Then E is a connected invariant surface of (3.5) and the equilibrium and periodic orbits of (3.5) are all on 1. By corollary 2.1 and lemma 3.7, ,Ycontains no related points. Hence, the number of periodic orbits of (3.5) equals to m + 1 (= 2p + I), and is odd. 1 Acknowledgements-1 ments and help.

am much indebted to Professor Bingxi Li and Mr Jiancheng Yuan for their many encourage-

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A new skeleton mechanism for the halate driven

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