308 |
X. THE SECOND-ORDER DIFFERENTIAL EQUATION |
Then,
(1)two curves x = x(t, c) and x = w2(t) meet for some t on the interval a < t < b if c>'Y;
(2)two curves x = x(t, c) and x = wl (t) meet for some ton the interval a < t < b if c < -ry.
Proof.
For part (1), by virtue of Lemma X-1-4, x'(r,c) > La if (r,x(t,c)) E Ao for
a < r < t. The proof of part (2) is similar.
Proof of Theorem X-1-3.
Now, let us complete the proof of Theorem X-1-3. The main point is that when two curves x = x(t,c) and x = w2(t) or two curves x = x(t,c) and x = w, (t) meet, they cut through each other. So look at Figure 1.
(b, B)
(a. A)
FIGURE 1.
Example X-1-6. Theorem X-1-3 applies to the boundary-value problem
d2x |
_ |
(X.1.10) |
x(0) = A, x(1) = B |
dt2 |
Ax' |
if A is a positive number. In fact, assume that 0(u) is a suitable positive constant. If wi(t) = sinh(ft) -a and w2(t) = sinh(ft)+$ with two positive numbers a and
,3 such that -a < A < j3 and sinh(/) - a < B < sinh(VrA_) +,Q, all requirements of Theorem X-1-3 are satisfied.
If A is negative, Theorem X-1-3 does not apply to problem (X.1.10). Details are left to the reader as an exercise.
2. APPLICATIONS OF THE LIAPOUNOFF FUNCTIONS |
309 |
X-2. Applications of the Liapounoff functions
In this section, using the results of §IX-2, we explain the behavior of orbits of a system
d |
[yj __ |
Y2 |
(X.2.1) |
y[-h(yi)y2 - 9(yi) |
dt |
l)y2J |
- 9(y1) |
JLet us assume that |
as t +oo. Set y = I yl and f (yl = [ h(yi |
h(x), g(x), and dd(x) are continuous with respect to x on the entire real line R.
Also, we denote by p(t,r) the solution of (X.2.1) satisfying the initial condition y(0) = n
First set V (y) = + G(yl ), where G(x) = fox g(s)ds. Then,
2Y2
09 = [9(yi ), Y21, |
8y'- f (y-) = -h(yi) |
y22. |
|
|
Set also, S = { g : R Y) = 0 y. Then, U E S if and only if either h(yl) = 0 or
Y2 = 0.
Observation X-2-1. Denote by M the set of all stationary points of system
(X.2.1), i.e., M = {9: g(yl) = 0, y2 = 01. Then, M is the largest invariant set in
S if the following three conditions are satisfied:
(1)h(x) > 0 for -oo < x < +oc,
(2)h(x) has only isolated zeros on the entire real line IR,
(3)g(x) has only isolated zeros on the entire real line R.
The proof of this result is left to the reader as an exercise (cf. Figure 2, where
0, |
0 and |
0). |
|
By using Theorem IX-2-2, we conclude that |
lim p(t, y) = 17 E M if conditions |
|
|
|
t-+00 |
(1), (2), and (3) are satisfied and if the solution p(t,g) is bounded for t _> 0. Note that G+(rt) is a connected subset of M.
In Observation X-2-1, the boundedness of the solution p(t, i) for t _> 0 was assumed. In the following three observations, we explore the boundedness of all solutions of (X.2.1). Set
G(x) = o g(s)ds |
and |
H(x) = fX h(s)ds. |
Jo |
|
o |
Observation X-2-2. Every solution p(t, q) of (X.2.1) is bounded for t > 0 if
(i) h(x) > 0 for -oc < x < +oo and (ii) |
lim G(x) = +oo. |
1x1 |
+00 |
This is a simple consequence of Theorem IX-2-3. In fact, lim V (Y-) = +oo.
310 X. THE SECOND-ORDER DIFFERENTIAL EQUATION
Observation X-2-3. Every solution pit, |
of (X.2.1) is bounded for t > 0, if |
(i) h(x) > 0 for -oo < x < +oc, (ii) urn |
IH(x)l = +oo, and (iii) xg(x) > 0 |
I=t-+o0 |
for -oc < x < +oo. |
|
Proof |
|
Change system (X.2.1) to |
|
(X.2.2) |
d |
[zl] |
Z2 - H(z1) |
dt |
|
-g(z1) |
|
22 |
by the transformation
Y1 = Z1, Y2 = z2 - H(zi).
Denote by qqt,() the solution of (X.2.2) such that z(0) _ . Set V, (z-) = zz2 +
G(z1). Then,
|
L9 V, |
|
8z" ,P = -g(z1)H(zl) |
|
ay - [g(zl), |
|
|
|
z21, |
|
|
Note that g(x)H(x) > 0 for -oo < x < +oo and that |
|
|
dt |
V1(ggt,C)) |
- z(g-(t,C)) F(glt,S)) S 0 |
for t > 0. |
|
|
|
|
|
Hence, setting q"(t, S) = |
z1(t, |
and V, = c > 0, we obtain |
|
2[z2{t,S)]2 < |
c |
for |
t>0, |
since G(x) > 0 for -oo < x < +oo. Therefore, Figure 3 clearly shows that q1 t, C)
is bounded for t > 0. This implies that all solutions of (X.2.2) are also bounded for
t>0.
2. APPLICATIONS OF THE LIAPOUNOFF FUNCTIONS |
311 |
Observation X-2-4. Every solution p(t, >)1 of (X.2.2) is bounded for t > 0 if |
|
(i) = lim H(x) = +oo, (ii) |
lim H(x) = -oo, (iii) g(x) > a for x > as > 0, |
+00 |
= --Cc |
|
and (iv) g(x) < -a for x < -ao < 0, where a and ao are some positive numbers.
Proof.
In Observation X-2-3, the Liapounof function
Vi(z) = |
G(zi) = 2[y2 + H(yl)]2 + G(yi) |
was used. Now, let us modify Vl to a form
V2(yl = 2[y2 + H(yi) - k(yi)J2 + G(yl)
Then,
OV2 |
_ [[y2 + H(yi) - k(yi )J {h(yi) - dyyt )1 + 9(yi ), |
Y2 + H(1h) - k(yi) I |
|
|
|
|
1 |
09 |
|
|
|
|
and |
a z |
(Y'){y22 |
J |
J |
|
+ [H(yi) - k(yi)J y2} |
|
|
dy, |
|
|
- 9(yi) [H(yi) - k(yl)J
Using (i) and (ii), three positive numbers M, a, and c can be chosen so that
|
|
fa [H(x) |
- cJ > M |
for x > a > ao, |
|
|
-a [H(x) + c] > M |
for |
x < -a < -ao. |
|
Also choose a function k(x) so that |
|
|
|
(1) |
k(x) = |
{ c |
for |
x > 2a, |
|
-C |
for |
x!5 -2a, |
|
|
|
|
|
(II} |
Jk(x)J < c |
and |
dk(x) |
> 0 |
for - on < x - +oo, |
|
|
|
|
dx |
|
|
and
dk(x) > m > 0 for JxJ < a dx
for some positive number m (cf. Figure 4).
k
k=c
x
k=-c
312 X. THE SECOND-ORDER DIFFERENTIAL EQUATION
|
If a positive number b is chosen sufficiently large, |
|
|
|
|
|
OV2 .f |
0 |
for |
lyil ? 2a, |
or |
1y21 > b. |
|
In fact, |
|
|
|
|
|
|
|
|
(A) -g(yl) [H(bi) - k(yl)) |
|
- a IH(yl) - cI < -M < 0 |
for yl > a, |
|
a IH(y1) + cj < -M < 0 |
for yl < -a |
|
|
|
|
|
and |
|
|
|
|
|
|
|
|
|
(B) |
Y2 + IH(yj) - k(yi )11/2 ? 1 |
for |
1yi 15 2a, 1y21 > b |
|
if b > 0 is sufficiently large. Therefore, |
|
|
|
|
|
(i) |
av2 ' f = -g(yi) IH(yi) - k(yi)1 < 0 for |
Iyi I |
>- 2a, |
11/21 < +00, |
|
(u) |
f < -- |
(y1 |
< 0 |
for |
a < 1yi I G 2a, |
1y21 ? b, |
|
|
09 |
|
|
|
|
|
|
|
|
and |
|
|
|
|
|
|
|
|
|
(iii) |
OV2 f < - m {y22 + IH(yi) - k(yi)1 y2} |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
g(yt)IH(yi) - k(yi)I < 0 |
forlyil < a, |
1y21 ? b |
if b > 0 is sufficiently large.
Since lim G(x) = +oo, Theorem IX-2-3 implies that every solution of (X.2.2)
1XI +00
is bounded.
Example X-2-5. For the van der Pol equation
2 + e(x2 - 1) dt + x
= 0,
where a is a positive number, h(x) = e(x2 - 1) and g(r) = x. Hence,
G(x) = 2x2 |
and H(x) = e (3x3 - x) |
Therefore, conditions (i), (ii), (iii), and (iv) of Observation X-2-4 are satisfied. This implies that every solution of the van der Pol equation
|
|
X.2.3 |
d |
yl |
y2 |
|
( |
dt |
y2 |
-E(Y - 1)y2 - yi, |
|
) |
is bounded for t > 0. System (X.2.3) has only one stationary point 6. It is easy to see that 0 is an unstable stationary point as t -+ +oo. Therefore, using the
Poincaare-Bendixson Theorem (cf. Theorem IX-4-1), we conclude that there exists at least one limit cycle. In §X-3, it will be shown that system (X.2.3) has exactly one periodic solution.
-aye < 0, where f (y =
3. EXISTENCE AND UNIQUENESS OF PERIODIC ORBITS |
313 |
Example X-2-6. For a given positive number a, the system
(X.2.4) |
[yi |
_ |
y2 |
d Y2 J |
|
[ -aye - sin (yl ) |
|
|
satisfies three conditions (1), (2), and (3) of Observation X-2-1. But, (X.2.4) does
not satisfy conditions of Observations X-2-2, X-2-3, and X-2-4. Therefore, in order to prove the boundedness of solutions of .(X 2.4), we must use some other methods.
In fact, using the Liapounoff[ijfunction V (y) = -yZ - cos (yl ), we obtain ev. f =
WY=
2Since V (y-)
t +oo
exists for every solution p-(t, ii) of (X.2.4). Now, observe that
(1)We must have a < 1. Otherwise we would have y2(t,,)2 > 2(a+cos(yi(t,rte)) for t > 0. This implies that dt V (pl t, r7-')) < -2a(o - 1) < 0. This contradicts
(X.2.5).
(2)If -1 < a < 1, the solution p(t, 7) must stay in one of connected components of the set {y" : V(y) < a + e < 1} for large positive t. Those connected components are bounded sets.
(3)In case a = 1, we can show the boundedness of p(t, t) by investigating the
behavior of solutions of (X.2.4) on the boundary of the set {g: V(y-) < 1).
X-3. Existence and uniqueness of periodic orbits
In this section, we prove the following theorem (cf. (CL, p. 402, Problem 51).
Theorem X-3-1. Assume that
(i) two real-valued functions h(x) and g(x), and dg(x) are continuous for -oo <
X < +00,
(ii)g(-x) = -g(x) and h(-x) = h(x) for -oo < x < +oo,
(iii)g(x) > 0 for x > 0,
(iv)h(0) < 0,
(v)H(x) = f h(s)ds has only one positive zero at x = a,
0
(vi) h(x) > 0 for x > a,
(vii) H(x) tends to +oo as x -- +oo.
Then, the system
(X.3.1)
d [Yyj
dt [-h(yi)y22- 9(yi)
has exactly one nontrivial periodic orbit and all the other orbits (except for the
stationary point 0) tend asymptotically to this periodic orbit as t |
+oo. |
314 X. THE SECOND-ORDER DIFFERENTIAL EQUATION
Change system (X.3.1) to
d |
zi) |
(X.3.2) |
Lz2- |
dt Lz2J |
by the transformation |
|
|
Y2J |
Lz2 |
-zH(zi), |
|
Setting |
|
|
|
|
|
|
V(z) = |
|
+ G(z1), |
|
where |
|
2 |
|
|
|
G(x) = |
J0z g(s)ds |
and |
z = |
[z2] |
look at the way in which the function V(z) changes along an orbit of (X.3.2). For example,
dt V (zI = 9(zl)[z2 - H(zi)] - z29(z1) = -g(z1)H(z1)
along an orbit of (X.3.2). Hence,
dz2 |
|
g(zl) |
dzi |
|
z2 - H(zi) |
dzi |
z2 |
- H(zl)' |
dz2 |
_ |
9(z1) |
dV |
9(z1)H(z1) |
dV |
H(zi) |
dz1 |
z2 |
- H(z1)' |
dz2 |
|
|
|
along an orbit of (X.3.2). |
|
|
|
|
|
|
|
z1(t' a)be |
the orbit of (X.3.2) such that z(0, a) |
Observation 1. Let z(t, a) = IZ24,a)] |
[0]. Then, V (i(0, a)) = 2 a2. There exists exactly one positive number ao such
that
[
z1(ao, ao) = and z2 (t, ao) >0 for 0:5t < oo
0 ]
for some positive number co, where a is the unique positive zero of H(x) given in condition (v) (cf. Figure 5).
Observation 2. Since 0 when z2 = H(z1), there exist two positive numbers
r(a) and Q(a)such that
|
z(r(a), a) |
0 |
and |
0 < zi (t, a) :5a |
for 0< t < r(a) |
|
-Q(a) |
|
|
|
|
|
|
|
if 0 < a < ao. Also, since H(x) < 0 for 0 < x < a, we obtain |
|
dtV(z'(t,a)) = -g(zi(t,a))H(zi(t,a)) > 0 |
for |
0 < t < r(a) |
3. EXISTENCE AND UNIQUENESS OF PERIODIC ORBITS |
315 |
except for a = ao and t = oo. Therefore, |
|
|
(A) V(z"(r(a),a)) - V(z(0,a)) > 0 |
for |
0 < a < a0 (cf. Figure 6). |
(0.ao) |
|
|
zI = 0
Observation 3. If ao < a, there exists a positive number r0(a) such that
|
|
|
|
J z1(ro(a), a) = a, |
0 < z1(t, a) < a |
for 0 < t < ro(a), |
10 < z2(t,a) - H(zl(t,a)) |
for |
0 < t < ro(a) |
(cf. Figure 7). In particular ro(ao) = Co (cf. Observation 1).
If the variable t is restricted to the interval 0 < t < 7-0 (a), the quantity z2(t, a) can be regarded as a function of z1(t,a), i.e.,
z2(t,a) = Z(zl(t,a),a),
where Z(x, a) is a continuous function of (x, a) for 0 < x < a and a > a0, and continuously differentiable for 0 < x < a and a > ao except for x = a and a = a0.
Furthermore, Z(x, al) < Z(x, a2) for 0:5 x:5 a if a0 < a1 < 02 (cf. Figure 7).
Set 2(x, a) = I Z(z, a) J . |
Then, |
|
|
|
, d |
V(Z(x,a)) |
- |
g(x)H(x) |
0 |
for |
0<x<a |
dx |
|
Z(x,a) - H(x) > |
|
|
and, hence, |
|
|
|
|
|
|
|
3jV(Z(x,a1)) |
> ±V(i(x,a2)) |
|
for 0 < x < a |
if 00 < 01 < 02. Thus, we obtain
(I)V(zro(al),al)) - V(z(O,al)) > V(E(ro(a2),a2)) - V(E(O,a2)) > 0
316 |
X. THE SECOND-ORDER DIFFERENTIAL EQUATION |
Observation 4. If a0 < a, there exists a positive number ri(a) such that ri(a) > ro(a), zl (r1(a), a) = a, and zi(t, a) >a for ro(a) <t <7-1 (a) (cf. Figure 8).
(0. a) |
(a, z2(rp(a), a)) |
O,a |
(a. z2(r0(a), a)) |
|
|
|
|
I |
( ) |
z2=H(z0 |
|
z2=H(zt) |
(O.a0) |
|
(O.ao) |
|
|
z2=0 |
|
=0 |
zi =0 |
zi=a |
Z, =0 |
|
FIGURE 7. |
FIGURE 8. |
Note that z2(ri(a),a) < 0 and that H(zi(t,a)) > 0 for ro(a) < t < 7-1(a) since zi (t, a) > a on this t-interval. Regarding zi (t, a) as a function of z2(t, a) for z2(r1(a), a) < z2 < z2(ro(a), a), we obtain
(II) 0 > V(zi(ri(ai),ai))-V(zlro(ai),ai)) > V(z(ri(a2),a2))-V(z(ro(a2),a2))
for ao < al < a2 in a way similar to Observation 3 (cf. Figure 8).
Observation 5. If ao < a, there exists a positive number r(a) such that r(a) >
71(a), z1(r(a), a) = 0, and 0 < zi (t, a) < a for r1(a) < t < r(a) (cf. Figure 9).
Note that z2(t, a) < H(zi (t, a)) < 0 for r2 (a) < t < r(a). Again, regarding Z2 (t, a) as a function of zi (t, a) in the same way as in Observation 3, we can derive
(III) V(z(r(ai),al))-V(z-'(ri(ai),ai)) > V(i(r(a2),a2))-V4flri(a2),a2)) > 0
if a0 < aI < a2 (cf. Figure 9).
Observation 6. Thus, by adding (I), (II), and (III), we obtain
(B)V(z(r(aI),at)) - V(z(0,a1)) > V(zr(a2),a2)) - V(z0,a2)) > 0
if a0 < al < a2. This implies that the function G(a) defined by
g(a) = V((r(a),a)) - V(z(0,0 = Zz2(r(a),a)2 - 2a2
is strictly decreasing for a > ao as a -+ +oo. Also, 9(a) > 0 for 0 < a 5 a0 (cf.
(A)).
Observation 7. Since
dV |
= 0 |
uniformly |
for |
0 < z1 < a, |
lim |
|
jz21-+oo dzi |
|
|
|
|
z1lim00 a = +oo |
uniformly |
for |
- oo < z2 < +oo, |
3. EXISTENCE AND UNIQUENESS OF PERIODIC ORBITS |
317 |
it follows that
a+oolim (V(i(ro(a),a)) - V(z(0,0J = 0, a+oolim (V(z(rl(a),a)) - V(z(ro(a),a))J = -oo,
lim JV(z(r(a),a)) - V(i(ri(a),a))J = 0.
a-.+00
Therefore,
(C) |
|
lim |
Cg(a) = -oo. |
|
|
|
a-'+00 |
|
|
Thus, we conclude that g has exactly one positive zero a+, i.e., |
|
|
|
|
- 2a2 |
> 0, |
0<a<a+, |
(X.3.3) |
9(a) |
Zz2(r(a), a)2 |
= 0, |
a = a+, |
|
|
|
|
< 0, |
a>a+ |
From (X.3.3) and symmetric properties (ii) of the functions h(x) and g(x), we conclude that a(t, a+) is the only periodic orbit, and all the other orbits tends to z"(t, a+) asymptotically since
Iz2(r(a), a)I > a |
if 0<0<a+, |
Iz2(r(a),a)I < Of |
if a> a+. |
Thus, we complete the proof of Theorem X-3-1. 0
Remark X-3-2. Condition (iv) of Theorem X-3-1 can be replaced by the following condition:
(iv') there exists a positive number 6 such that H(x) < 0 for 0 < x < S.
