0387986995Basic TheoryC

290 IX. AUTONOMOUS SYSTEMS

and

(IX.3.22)

Let Q(t) be the 3 x 3 matrix whose column vectors are (at, qo)), fi(t), and

,fi(t), i.e., Q(t) = [ f (p1t, rlo)) 6(t) q"3(t)! . The transformation w = Q(t)v changes the linear system

Using these notations, write (IX.3.22) in the form

with K(r) > 0 and limK(r) = 0 for lull + lull < r and Ivll + 1v21 < r. Observe r-O

that the two multipliers of the linear system

are also multipliers of system (S2). Therefore, using system (IX.3.23), Theorem IX-3-4 can be proven. In general, we obtain more precise information concerning the behavior of solutions in a neighborhood of a periodic solution in this way.

The materials of this section are also found in 1CL, Chapter 13, §2, pp. 321-3271.

292 IX. AUTONOMOUS SYSTEMS

Step 2. Since tj E C+(r)l ), there exists a sequence {tk : k = 1,2.... } of real

there exist two distinct real numbers rl and T2 such that it = p(rl, ill) = (r2, ill) and hence p(t, rl = p(t, p(ri , ili )) = p1t, p(r2i ill )). This, in turn, implies that p(t + T1, ill) = p(t + r2i ill) for t > 0. Therefore, the orbit C(ill) is periodic in t of period Irl - r21. Furthermore, C(ill) C C+(rjo). Note that there is no stationary point on C(qj ).

Step 3. Since C+(ilo) is connected, it follows that distance(C(ijl), C+(ilo)-C(ni)) = 0. Hence, if C(ijl) # L+ (10), there exists a sequence { k E C+(i o) : k = 1, 2.... }

there exists a transversal t such that E e and lk E t (k = 1, 2.... ). Note that l; E C(ill) C C+(ilo). Then, k = t c C(ill) (k = 1, 2, ... ). This is a contradiction.

Thus, it is concluded that C+(rlo) = C(iji).

Now, we complete the proof of Theorem IX-4-1 as follows.

Proof of Theorem IX-4-1-

If C+ (, ) does not contain any stationary points, then (i) follows (cf. Lemma

IX-4-4). If C+(no) consists of stationary points only, we obtain (ii), since C+(ilo) is connected. If f-+ (Q contains stationary and nonstationary points, then

any point it E C+(ilo) does not contain nonstationary points (cf. Lemma IX-4-4). This is true also for t < 0. Hence, (iii) follows.

Observation IX-4-5. In cases (i) and (iii), the set 1R2 - C+(ilo) is not connected.

Furthermore, if an orbit C(i)) is contained in C+(i o), two sides of the curve p1t, tl") belong to two different connected components of R2 - C+(t ). In fact, if we consider a simple Jordan curve C which intersects with the orbit C(tl at n transversally, then the curve p(t,,o) intersects with C in a neighborhood of two distinct points on C infinitely many times (cf. Figure 5).

Theorem IX-4-6. If C(ijo) n C+(ilo) 0 0, then C+(ijo) = C(rjo) and either no is a stationary point or the orbit C(ilo) is periodic.

Proof

In this case, C(ilo) C C+(ilo). If C+ (8o) contains nonstationary points, the orbit

C(i o) consists of nonstationary points. Choose a transversal a at ilo. Then, it can be shown that C(ilo) is periodic in a way similar to the proof of Lemma IX-4-4, since r1o E C+(tlo).

cycle (cf. Figures 6-1 and 6-2).

The materials of this section are also found in [CL, Chapter 16, §§1 and 2, pp.

391-3981 and (Har2, Chapter VII, §§4 and 5, pp. 151-1581. In [Hart[, the PoincareBendixson Theorem was proved without the uniqueness of solutions of initial-value problems.

IX-5. Indices of Jordan curves

In this section, we explain applications of index of a Jordan curve in the plane to the study of solutions of an autonomous system in 1R2. Consider again an R'-valued function f (y-) of y E 7,k2 whose entries are continuously differentiable on the entire

y-plane R. Denote also by pit, J) the unique solution of the initial-value problem

dg

= f (y1, yi(0)

dt

To begin with, let us introduce the concept of indices of Jordan curves. Let C be a Jordan curve y = il(s) (0 < s < 1) with the counterclockwise orientation (cf.

Figure 7). Assume that &(s)) # 0 for 0 < s < 1. Set u(s) = f ('l(s)) (0 <

If(s))1

s < 1). Then, il(s) is the unit vector in the direction of f'(il(s)). There exists a real-valued continuous function 0(s) defined on the interval 0 < s < I such that

a (s) _ Isin0(s)I'

Definition IX-5-1. The index of the Jordan curve C with respect to the vector

field f (y) is given by I f{C) = 0(1) - 8(0) .

2r,

This definition is independent of the choice of a parameterization rj(s) of C and the function 0(s). Let us denote by D the domain bounded by C (cf. Figure 7).

Observation IX-5-2. If the domain D is divided into two domains DI and D2 by a simple curve £, then 1 f{C) = l f{",) + If-(8D2) if f (rl) 0 on £ U C, where 8D? (j = 1, 2) denote the boundaries of domains DI and D2, respectively, as the portions of I f{8Di) and I j{d Dt) along £ canceled each other (cf. Figure 8).

Observation IX-5-3. If All) A 0 on C U D, then I f{C) = 0. To show this, divide

V into sufficiently small subdomains, use the fact that the vector field 19(s) has no change on a small subdomain, and apply Observation IX-5-2.

Observation IX-5-4. Assume that a point d E V is a stationary point, i.e., f(d) = 0. Assume also that f(,) 96 0 on C U V except at a. Then, If(C) depends only on aa. Therefore, we define the index of an isolated stationary point d with respect to f by Ilea) = I f(OV), where V is a neighborhood of as such that there are no stationary points in V other than a.

Observation IX-5-5. If D contains only a finite number of stationary points

N

dl, d2, ... , dN, then I j<C) = FI J(a"k). Proof of this result is similar to that of

k=1

Observation IX-5-3.

Observation IX-5-6. If f (y) # 0 on C and if f(,7(s)) is tangent to C at every point i(s) (0 < s < 1) of C, then I1{C) = 1 (cf. (CL, Chapter 16, §4, Theorem

4.31).

Proof.

It is evident that the index of C with respect to f and the index of C with respect

to the tangent vector to C are the same, i.e.,

If4C) = hn)(C)-

Assume without any loss of generalities that

# ( 0 )=# ( I ) ) = 0 and i (s) > 0 f o r 0 < s < 1,

where rig (s) (j = 1, 2) are the entries of the vector r"(s) (cf. Figure 9).

For 0<r<s<1,set

Then, v(T, s) is continuous in (T, s) for 0 <_ r < s < 1. Hence, studying how 16(0, s) changes from s = 0 to s = 1 and how i6(T,1) changes from T = 0 to T = 1, we obtain Ij.(.)(C) = 1. 13

Observation IX-5-10. Let d be an isolated stationary point. Assume that a neighborhood V of d is divided into a finite number of sectorial regions S1, S2,... , SN by a finite number of orbits C(rji ), C(il3),... , C(iv) in such a way that

(a)rli, Tt2, - , nN E 8V,

(Q)either p1r,rjk) E V fort > 0 and tends to 99 as t -+ +oo, or p5r,ilk) E V for t < 0 and tends to d as t - -cc (cf. Figures 13-1 and 13-2).

Let us assume also that each sectorial region Sk satisfies one of the following four conditions:

(I)if it E Sk and {il - d] is sufficiently small, then C(r)) C Sk and pit, q-) tends to d as Itl +oo (cf. Figure 14),

(II)if i) E Sk and Jil - dl is sufficiently small, then pit, i)) E Sk only on a finite t-interval ak < t < Yk (cf. Figure 15),

(111) if it E Sk and ail - dl is sufficiently small, then p1t,i) tends to dd in Sk as t +oc, but plt, q") does not tend to dd in Sk as t -+ -oo (cf. Figure 16-1),

(IV) if ij E Sk and Jrj - al is sufficiently small, then p1t,71) tends to ad in Sk as t -. -oc, but plt, q+) does not tend to dd in Sk as t - +oo (cf. Figure 16-2).

The sectorial region Sk is said to be elliptic (respectively hyperbolic) in case (I)

(respectively (II)). In cases (III) and (IV), the sectorial region Sk is said to be parabolic. Let us denote by E (respectively H) the total number of elliptic sectorial regions (respectively hyperbolic sectorial regions) among St, ... , SN. Then, it is known that

E--H+2

(IX.5.3)

2

This result is similar to formula (1X.5.2) of Observation IX-5-9. For a proof in detail, see (Har2, Chapter VII, §9, pp. 166-172).