0387986995Basic TheoryC

298 IX. AUTONOMOUS SYSTEMS

= aP f (rte, where A is a real variable and p is an integer. Set y = r os0 I. Then,

where

F1(0) = f 1(cos 8, sin o) cos 0 + f2(cos 0, sin 0) sin 0,

Sl F2(0) = -f1(cos0,sin6)sin0 + f2(cos0,sin0)cos0

(cf. Exercise VIII-3). If a ray to is defined by F2(0) = 0, then to is an orbit of the

system dff = f (r))). Thus, the entire y"-plane can be divided into sectorial regions

by those orbits QB determined by equation F2(0) = 0. For example, in the case of system (IX.5.5), we obtain Fl (0) = - sin 0 and F2(0) = cos 0. Observe that

Hence, E = 2 and H = 0 (cf. Figure 19). Therefore, I f-{0) = 2 (cf. Example

IX-5-15).

The materials of this section are also found in [CL, Chapter 16, §§4 and 5, pp.

398-402] and [Har2, Chapter VII, §§2 and 3, pp. 146-151, and §§8 and 9, pp.

161-1741. For a geometric and topological treatment of indices, see (Mil.

EXERCISES IX

IX-1. For each of the following three systems, using the given function V (x, y), show that all orbits are bounded as t -. +oo .

Examples.

(a)Every orbit in Iy21 < 1 of the system with fl(yl,y2) = y2 and f2(yi,y2) _ yl(y2 - 1) is periodic.

(b)The stationary point (0,0) is a stable spiral point if fl(yl,y2) = y2(2 - sin(yly2)) and f2(yj,y2) = 2yi(y2 - 1).

(c)The stationary point (0, 0) is an unstable spiral point if ff(yi, y2) = y2(2 +

sin(yiy2)) and f2(yi,y2) = 2y, (y22 - 1).

Verify these statements by using the function V(yl, y2) = yi - ln(1 - y2).

IX-4. Let us consider a system

where f, (y-) and f2(y-) are continuously differentiable with respect toy in a domain

Do C R2. Assume that

(i)system (S2) has a periodic orbit p-(t, ik) of period 1 which is contained in the domain Do,

(ii) RA t' v) 6,

Show that the periodic orbit p(t, jo) is orbitally asymptotically stable as t -' +00.

Hint. This is called Poincnre's criterion. Look at

dzV 49f

dt =

Then, I det Y(1)I < 1 for the fundamental matrix solution Y(t) of this system suc that Y(0) = 12 (cf. (4) of Remark IV-2-7). Hence, an eigenvalue p of Y(1) must satisfy the condition dpi < 1. (The other eigenvalue of Y(1) is 1.) Now, use Theorem

IX-3-4.

IX-5. Assume that two functions f (x) and g(x) are continuously differentiable in x E R. Assume also that the differential equation dt2 + f (x) drt + g(x) = 0 has

a nontrivial periodic solution x(t) of period I such that f f (x(t))dt > 0. Show

0

where

(1)the entries of the 1R2-valued function f (y-) is continuously differentiable on the entire y-plane,

find the total number E of elliptic sectorial regions, the total number H of hyper- bolic sectorial regions, and the total number of parabolic sectorial regions in the

where the entries of the l 3-valued functionf is continuously differentiable on the entire y-space R3. Assume that (S3) has a periodic orbit pit,ip) of period 1 such that f (r'(t, o)) 0 0. Assume also that the first variation of system (S3) with respect to the solution p(t,' o), i.e.,

d9 Of

(r(t,io))u,

dt

has three multipliers p1 = 1, p2i and p3 satisfying the condition: 1P21 < 1 and jp3j > 1, respectively. Construct the general orbits pit, r) of (53) such that distance(At,17),

C(s'7o)) tends to0ast-+too.

IX-9. Show that the differential equation d-t2 + (x + 3)(x + 2) drt + x(x + 1) = 0 does not have nontrivial periodic solutions.

Hint. Two stationary points are a node (0,0) and a saddle (-1,0). Furthermore,

for the function V (x, y) = x2 + y2,

(3)find the set S = {(x): dVdt = 01,

(4)examine if S is an invariant set,

(5)find the phase-portrait of orbits.

IX-11. For the system

dx f(x,y,x) = - x(1 - x2 _ y2)2 + 3xz + y, dt =

dt = 9(x,y,z) = - y(1 - x2 - y2)2 + 3yz - x,

dz

h(x,y,z) = - z - 3(x2 + y2),

dt =

(1)find all critical points and determine if they are asymptotically stable,

(2)find

for the function V(x, y, z) = x2 + y21+ z2,

(3)find thesetS=((x,y,z): dV =0},

(4)find the maximal invariant set M in S,

(5)find the phase portrait of orbits.

IX-12. Find the a phase portrait of orbits of the system

IX-13. Let f (x, y) and g(x, y) be real-valued, continuous, and continuously differentiable functions of two real variables (x, y) in an open, connected, and sim-

ply connected set D in the (x, y)-plane such that 8f (x, y) + Lf (x, y) j4 0 for all

(x, y) E D. Show that the system dxt = f (x, y), d = g(x, y) does not have any nontrivial periodic orbit that is contained entirely in D.

IX-14. Let p(z) be a polynomial in a complex variable z and deg p(z) > 0. Set x = 3t(z] and y = Q [z]. Verify the following statements.

(i) In the neighborhood of each of stationary points of system

there are no hyperbolic sectors. Also, system (A) does not have any isolated nontrivial periodic orbit.

(ii) In the neighborhood of each of stationary points of system

(B) dt = 3R[p(z)], L = -`3'[p(x)],

there are no elliptic sectors. Also, system (B) does not have any nontrivial periodic orbits.

IX-15. Find the phase portrait of orbits of the system

that it has exactly five stationary points and all of them are centers.

IX-17. Consider a system

where the entries of the R'-valued function f are analytic with respect toy in a domain Do C R'. Assume that

(i)system (S2) has a periodic orbit pit, ijo) of period 1 which is contained in the domain Do,

(ii)AP-Tt,no)) 0

(iii)for any open subset V of Do which contains the periodic orbits p(t, i"p), there

exists an open subset U of V which also contains p'(t, o) such that for any point i in U, the orbits p'(t, i of (S4) is contained in V and periodic in t.

Show that if U is sufficiently small, for any point i in U, we can fix a positive period of p1t, y) so that is bounded and analytic with respect to i in any simply

connected bounded open subset of U.

Hint. Apply the following observation.

Observation. Let Do be a connected, simply connected, open, and bounded set in

. Suppose that, for any pointy E Do, there exists a j such that T3 [yl = y, where j may depends on Y.

Then, there exists a jo such that Tjo [yj = y for all y E Do.

Proof.

Set

Then,

(1) E. is closed in Do,

+00

(2) Do = U E,,

3=1

(3) Do is of the second category in the sense of Baire.

Hence, for some jo, the set Ego contains a nonempty open subset (cf. Baire's Theorem). Since Tea is analytic, we obtain Tao [ 7 = y" for all y" E Do. O

For Baire's Theorem, see, for example, [Bar, pp. 91-921.

X-1. Two-point boundary-value problems

In this section, first as an application of Theorems 111-2-4 and 111-2-5 (cf. [Kn]), we prove the following theorem concerning a boundary-value problem

Theorem X-1-1. If the function F(t, yt, y2) is continuous and bounded on a region 11 = {(t, Y1, y2) : a < t < b, lyt I < +oo, Iy2I < +oo}, then problem (X.1.1) has a solution (or solutions).

Proof.

For any positive number K, the set Aa = {(a, or, y2) : Iy21 < K} is a compact and connected subset of ft We shall show that A0 satisfies Assumptions 1 and 2 of §111-2 for every positive number K. In fact, writing the second-order equation (X.1.1) as a system

y1(t) = y1(a) + J y2(s)ds, a

y2(t) = y2(a) + JF(s,yi(s),y2(s))ds` .

Hence, if (a, y1(a), Y2 (a)) E A0, we obtain

Jy2(t)I < K + M(b - a),

ly1(t)I < lal + [K + M(b - a)](b - a),

where I F(t, yi, Y2)1 < Al on Q. Therefore, Ao satisfies Assumptions 1 and 2 of

§111-2. Thus, Theorem 111-2-5 implies that SS is also compact and connected for every c on the interval Te = {t : a < t < b}.

We shall prove that if K > 0 is sufficiently large, the set Sb contains two points

(771,772) and (t;t, (2) such that

yI(b)I can be made as large as we wish by choosing Iy2(a)I sufficiently large. Thus, there are two points (nt, n2) and ((1, (2) in Sb such that (X.1.3) is satisfied.

Since the set Sb is compact and connected, there must be a point (Q, () in the set Sb. This implies the existence of a solution of problem (X.1.1). 0

Lemma X-1-4. Let x(t, to,rl) be the solution of the initial-value problem

where a < to < b, (to, l;) E Ao. Then, for any given positive number M, there exists a positive number a(M) such that l x'(t, to, l;, il) j < a(M) if

choose a(M) > 0 for any given positive number M in such a way that a(M) > 111 and

where x(T) = x(r, to, t ,17). Then, since x'(r) > 0 for ri < r < r2, it follows that

This contradicts the choice of L by (X.1.9). Therefore, Lenuna X-1-4 is true for to < r < t. We can treat the case t < r < to similarly, since if we change t by -t, the differential equation x" = f (t, x, x') becomes x" = f (-t, x, -x'). 0

Lemma X-1-5. Set

7 = rim {a(ba)I wi(a)+e, -w2' (a)-eI,

where e is an arbitrarily fixed positive number. Let also x(t, c) be the solution to the initial-value problem