(cf. Figure 21).
The general solution of (VIII.8.2) is
yI(t) = e-tr(7f)sin (L(7 2a-2t
y2(0 = e-tr(771 cos (!2e_2t
where y(0) and r(fl) are given by (VIII.8.5) and (VIII.8.6), respectively. Every orbit of (VIII.8.2) goes around the point (0, 0) only a finite number of times. Figure 22 shows that the point (0, 0) is a stable proper node as t -+ +oo.
Remark VIII-8-3. If condition (iii) of the assumption given above is relaxed, the point (0, 0) may become a stable spiral point (cf. Exercise VIII-10).
Proof of Theorem VIII-8-1.
We prove Theorem VIII-8-1 in two steps. To start with, using the polar coordinates, write system (VIII.8.1) in the form
|
|
dr |
= Ar + gI (y) cos(9) + g2(y) sin(g), |
(VIII.8.7) |
J |
dt |
|
dO |
|
|
r |
= -g1(y) sin(g) + g2(y COs(9) |
|
t |
(cf. (VIII.7.4)). By virtue of the assumption given above, there exists a positive number ro such that
(VIII.8.8) |
2Ar < Ar + gi (y-)cos(9) + g2(y) sin(g) < 2r < 0 |
for0<r<ro.
Step 1. We show first that the point (0, 0) is a stable node. To do this, let
(r(t), 9(t)) be a solution of (VIII.8.7) satisfying the initial condition
(VIII.8.9) |
r(0) = p, 9(0) = w, |
where p is a positive number and w is a real number. This solution exists for t > 0. Furthermore, estimate (VIII.8.8) implies that the function r(t) satisfies
8. PERTURBATIONS OF A PROPER NODE |
269 |
the following two conditions: (a) r(t) is strictly decreasing as t -+ +oo and (b)
lim r(t) = 0. Therefore, we obtain (c) 0 < r(t) <_ p for t >_ 0. t-. +00
Since r(t) is strictly decreasing for t _> 0, the variable t can be regarded as a function of r, i.e., t = t(r), where t(p) = 0 and lim t(r) = +oo. Use r as the
independent variable. Then, (VIII.8.7) becomes
( V1II . 8. 10 ) |
d9 |
F(r, 0) _ |
-gi (y-)sin(g) + g2(Y)cas(e) |
|
Tr |
|
r[Ar + g1(y) cos(0) + g2(y) sin(0)J |
Let 0 = 4D(r) = 0(t(r)) (0 < r < ro) be the solution of (VIII.8.10) satisfying the initial condition (b(p) = w. Then,
fi(r) = w + JPr F(s, I(s))ds (0 < r < ro).
Estimate (VIII.8.8) implies that
2Ar2 I - gt (y sin(0) + 92(Y) cos(0) I |
F(r, 0) < ar2 (I g1(y)I + Ig2(y1I) |
for 0 < r < ro.
to
JP F(s, $(s))ds =
r(t)
F(s, (s))ds
P
0
Now, by virtue of condition (iii), the improper integral
lim r F(s, 4(s))ds exists. Observe that 0(t) = 4(r(t)) = w + r-.0 JP
and that lim r(t) = 0. |
Therefore, |
lim 0(t) |
= w + |
t--+oo |
|
t-'+oo |
|
Thus, we conclude that the point (0,0) is a stable node of
JP
(VIII.8.1) as t +oc.
Step 2. In this step, we prove that the point (0,0) is a proper node. To do this, for a given real number c, find a solution (r(t), 0(t)) of (VIII.8.7) such that
rim 0(t) = c and |
lim |
|
r(t) = 0. |
Selecting a sequence {pm : m = 1, 2, ... } of |
t-+oo |
t -+oo |
|
|
|
|
positive numbers such that 0 < p,,, < ro and |
lim pm = 0, define a sequence of |
functions {cm(r) : in = 1, 2.... } by the initial-value problems |
d0 |
= F(r, 0), |
|
0(Pm) = c |
(m = 1, 2.... ), |
WT |
|
|
|
|
|
|
respectively. Those functions are defined for 0 < r < ro, and |
|
|
|
c + / r |
F(s, 4m(s))ds |
(m = 1, 2, ... ). |
|
|
|
|
P |
|
|
Set |
|
|
|
|
|
|
Cm = |
|
lim |
4)m(r) = c + jP1o. F(s, Om(s))ds. |
|
|
- +«; |
|
|
|
270 VIII. STABILITY
Then, 4b,,,(0) = c,,,, |
lira c,,, = c, and |
|
|
m=+oo |
|
(r) = cm + J r F(s, 4,,,(s))ds |
(m = 1,2.... ) |
|
0 |
|
for 0 < r < ro. |
Since the sequence {4D m(r) : m = 1, 2.... } is bounded and |
equicontinuous on the interval 0 < r < ro, assume without any loss of generality
that |
lim |
fi(r) exists uniformly on the interval 0 < r < ro (otherwise |
|
M +00 |
|
|
|
choose a subsequence). It can be shown easily that |
|
|
|
fi(r) = c + |
fF(s,(s))ds |
(0 r ro). |
Therefore, 7(r) is a solution of (VIII.8.10) such that rim 1(r) = c.
Define r(t) by the initial-value problem
dr = )r + |
92(17(r))sm(4'(r)), |
r(0) = ro, |
dt |
|
|
where g(r) = r [c?1]. The function r(t) is defined for t > 0, is decreasing,
and tends to 0 as t -4 +oo. Set 0(t) = $(r(t)). Then, it is easily shown that
(r(t), B(t)) is a solution of (VIII.8.7) such that |
limo 0(t) = 1i mt(r) = c. |
0 |
t |
+0 |
|
The materials of this section are also found in (CL, §3 of Chapter 151.
VIII-9. Perturbation of a spiral point
In this section, we show that (0,0) is a stable spiral point of (VIII.7.1) as t -, +oo
if (0, 0) is a stable spiral point of the linear system |
= Ay". The main result is |
the following theorem.
Theorem VIII-9-1. If
(i) two eigenvalues of A are not real and their real parts are negative,
(ii)the entries of the L2-valued function ff(y-) is continuous in a neighborhood of the point (0,0),
(iii)lira MI = 6,
g-alyl
then the point (0, 0) is a stable spiral point of (VIII. 7.1) as t +oo.
Proof. |
|
Assume that |
|
(VIII.9.1) |
all = a22 = a < 0 and a12 = -a21 = b 0 0. |
10. PERTURBATIONS OF A CENTER |
271 |
Then, from system (VIII.7.4), it follows that
dr
ar + 91(yjcos(9) + 92(ylsin(g),
dt
r t = -br - 91(y-) sin(g) + g2(Y)cos(0).
Therefore, a positive number ro can be fouind so that r(t) < r(0) exp 1at J for t > 0
if 0 < r(0) < ro. This, in turn, implies that |
lim r(t) = 0 and that dt < - if |
|
|
t -+oo |
|
|
2 |
b > 0, while de > - 2 if b < 0, whenever 0 < r(0) < ro. Therefore, lim00(t) = -oo |
if b > 0 and |
lim 0(t) = +oo if b < 0, whenever 0 < r(0) < ro. Thus, we conclude |
t |
too |
- |
that (0, 0) is a stable spiral point of system (VIII.7.1) as t -+ +oo. 0
The materials of this section are also found in fCL, §3 of Chapter 151-
VIII-10. Perturbation of a center
We still consider a system (VIII.7.1) under the assumption that the entries of the RI-valued function §(y) is continuous in a neighborhood of 0 and satisfies condition (VIII.7.2). In this section, we show that if the 2 x 2 matrix A has two purely
imaginary eigenvalues Al = ib and A2 = -ib, where b is a nonzero real number, then the point (0,0) is either a center or a spiral point of (VIAssume
without any loss of generality that the matrix A has the form [°b b J |
and system |
(VIII.7.4) becomes |
|
|
|
|
|
|
dr |
_ |
|
|
|
(VIII. 10 . 1) |
dt |
= 91Mcos (0) + 92(y-)sin (0) , |
|
dB |
= -b + |
1 |
|
|
|
(B) + 92(y')cos (B)). |
|
dt |
|
r |
|
|
Observe that system (VI I I.10.1) can be written in the form |
|
(VIII.10.2) |
dr = |
|
91(y')cos (0) + 92(y-)sin (0) |
|
|
|
|
|
|
d8
-b + r (-91(yjsin (0) + g2(y-')cos (0))
If a positive number p is sufficiently small, the solution r(0, p) of (VIII.10.2) satis- fying the initial condition r(0, p) = p exists for 0 < 0 < 27r.
Case 1. If there exists a sequence (pm : rn = 1, 2, ...) of positive numbers such that
lim p,,, = 0 |
and r(2xr, Pm) = Pm |
(m. = 1, 2, ... ), |
m-.+00 |
|
|
then the point (0,0) is a center of (VIII.7.1) in the sense of Bendixson (Bent, p.
Case 2. Assume that b < 0. If there exists a positive number po such that
r(2ir, p) > p (respectively < p)
whenever 0 < p < po, then the point (0, 0) is an unstable (respectively stable) spiral point as t - +oo (cf. Figures 24-1 and 24-2).
FIGURE 23. |
FIGURE 24-1. |
FIGURE 24-2. |
Example VIII-10-1. The point (0,0) is a center of the system
d |
[ yuj + |
yi +y2 I |
|
dt [y2J |
yy1J |
since drd8 = 0.
Example VIII-10-2. The point (0, 0) is an unstable spiral point of the system
|
d |
|
= [ Y2 |
I |
yI |
|
|
|
|
|
-Yi J |
+ ` y 2 + 2 |
|
|
|
dt |
l Y2 l |
Y2 |
Y2 |
|
|
|
|
|
|
as t - -oc, since dr |
= r2, which has the solution r = |
1 |
r(0) = 1 and |
r +oo when t |
dt |
|
|
|
|
ro - t |
ro |
ro. (Note: t < ro.) |
|
|
|
|
Example VIII-10-3. For the system |
|
|
|
|
dt |
= - y, |
d y = x + x2 - xy + cry2, |
|
the point (0, 0) is (a) a stable spiral point if a < -1, (b) a center if a = -1, and
(c) an unstable spiral point if a > -1.
Proof.
Use the polar coordinates (r, 0) for (x, y) to write the given system in the form
(1+rcos8(cos20-cos0sin8+asin28)) |
= r2sin8(cos28-cos0sin8+asin20). |
10. PERTURBATIONS OF A CENTER |
273 |
Setting
too
r(9, c) _ E r,,, (9)c"', where r(0, c) = c,
m=1
and comparing r(2r, c) and c for sufficiently small positive c, it can be shown that
r1(t) = 1, |
r2(2r) = 0, |
r3(2r) = (1 + Or |
|
|
4 |
Thus, r(27r, c) < c if a < -1. Therefore, (a) follows. Similarly, r(2r, c) > c if a > -1. Therefore, (c) follows.
In the case when a |
x2 - xy - y2 = (x - +2 ) (x - (1-2 ) . Set |
W = - (1 |
Then. a = (1 |
Therefore, changing x and y by u = |
X - (1 + ,/)y and v = x - (1 - )y, the given system is changed to |
2 |
|
2 |
|
|
|
du |
Wv(1 + u), |
dv = ---u(1 + v). |
|
|
dt |
|
|
|
Hence, on any solution curve, the function |
|
|
W2(v |
|
|
t2W2v2 |
+ |
- ln(1 + v)) + (u - ln(1 + v)) = u2 |
is independent of t. This implies that in the neighborhood of (0, 0), orbits are closed curves. This shows that (0, 0) is a center. 0
Example VIII-10-4. The point (0,0) is a center of the system
dy = bAy' +9(y), |
y= y'J |
dt |
|
y2 |
if (1) b is a nonzero real number, (2) A = 101 |
1J, (3) the entries of the R2- |
valued function"g"(yj is continuous and continuously differentiable in a neighborhood
of 0, (4) lim = 6, and (5) there exists a function M(y) such that M is positive g-6 1Y1
valued, continuous, and continuously differentiable in a neighborhood of 0 and that
8(Mf1)(y) + 8(M 2)(y..) = 0 in a neighborhood of 0, where f, (y-) and f2 (y-) are
the entries of the vector MY + §(y).
Proof
The point (0, 0) is either a center or a spiral point. Look at the system
dt =bAy + 9(y) |
and |
7t- = M(y- |
|
|
Using s as the independent variable, change the given system to ds = M(y)(bAy+
g"(y)). Upon applying Theorem VIII-1-7, we conclude that (0, 0) is not asymptoti- cally stable as t - ±oo. Hence, (0, 0) is a center.
The materials of this section are also found in [CL, §4 of Chapter 15] and [Sai,
§21 of Chapter 3, pp. 89-1001.
EXERCISES VIII
VIII-I. For each of the following five matrices A, find a phase portrait of orbits of the system d:i = AY.
5 |
1 |
I 14 1]' [ 11 31 ' [ 15 11]' [1 |
5 |
VIII-2. Find all nontrivial solutions (x, y) = (d(t), P (t)). if any, of the system
dt |
= |
-xy2, |
_ -x4y(1 + y) |
|
|
satisfying the condition |
lim (th(t),tb(t)) = (0,0). |
c-.+00 |
|
VIII-3. Let f (x, y) and g(x, y) be continuously differentiable functions of (x, y) such that
(a) f (0, 0) = 0 and g(0, 0) = 0,
(b) (f(x,y),g(x,y)) # (0,0) if (x,y) 0 (0,0),
(c)f and g are homogeneous of degree m in (x, y), i.e., f (rx, ry) = rm f (x, y) and g(rx, ry) = rmg(x, y). Let (r, 8) be polar coordinates of (x, y), i.e., x = r cos 8 and y = r sin 0. Set F(8) = f (cos 0, sin 8) and G(O) = g(cos 8, sin 8).
(I)Show that
|
|
dr |
= rm(F(8) cos 8 + G(8) sin 8), |
|
(S) |
dt |
|
|
|
|
|
d8 |
= |
rm-1(-F(8)sin8 + G(O)cosO). |
|
|
dt |
|
|
|
|
(II)Using system (S), discuss the stability property of the trivial solution of each of the following three systems:
4ii = x2 - |
y |
2 |
dt = 2xy; |
(i) |
|
|
|
= x3 (x2 + y2) - 2x(x2 + y2)2, |
= -y3(x2 + y2); |
(iii) |
d = x4 - 6x2y2 + y4, |
dy = 4x3y - 4xy3. |
VIII-4. Let J be the 2n x 2n matrix defined by (IV.5.2) and let H be a real constant
2n x 2n symmetric matrix. Show that the trivial solution of the Hamiltonian system
a = JHy is not asymptotically stable as t +oo.
V11I-5. Show that the trivial solution of the system
dy'
d = Ay" + 9(t, y-)
is asymptotically stable as t +oo if the following conditions are satisfied:
(i)A is a real constant n x it matrix,
(ii)the real part of every eigenvalue of A is negative,
(iii)the entries of the IRI-valued function g(t, y-) are continuous in the region
0(ro) = To x D(ro) = {(t, yl : 0 < t < +oo, Iyj < ro) for some ro,
(iv) g(t, yj satisfies the estimate
19(t, y) _< eo(t, y')Iyj |
for (t, y) E o(ro), |
where Eo(t, y) is continuous, and positive, andfun Eo(t, y) = 0 in L(ro). |
|
+Ii7 -.o |
VIII-6. Show that the point (0, 0) is a stable improper node of the system |
dy = Ay + Ng + 9(y), |
y = Iyil, |
where
(1)A is a negative number,
(2)N is a real constant nilpotent 2 x 2 matrix and N # 0,
(3)the 1R2-valued function y'(yj is continuous in a neighborhood of
(4)19(y-) 11-- cly-1I+" for some positive numbers c and v in a neighborhood of
Hint. Suppose N = 0 |
CA |
2 . If we set yj = r cos 0 and y2 = r sin 6, we obtain |
0 |
0 |
d = r IA _ |
EA sin2 cos 0 + 9i (y-)cos 9 + 92(y-)sin O |
|
, |
where 9"(y'i = {9i].
92(V
lim r(t) = 0 for t > 0.
t+0 -
Hence, if r(0) is small, r(t) is bounded by r(0) and
This implies that if 1#(0)I is small, g(t) is bounded
and lim y(t) = 0'. Look at
t-+o
y(t) = eAtetN {(o) + |
-sN9(y($))d$] |
276 VIII. STABILITY
If we set y(t) = e' e'Nu, we obtain
|
d(t) = y-(0) + |
r e-aae-aN9(g(s))ds. |
|
|
|
0 |
|
Choose e > 0 so that 1 - e > 1 |
Then, condition (4) implies that |
|
|
+ |
+00 |
|
6(+ |
= 9(0) + |
e-aae_a' |
|
0 |
|
|
|
exists.
Now,
(1) if d(+oo) = 0, then d(t) = 0 identically, since, in this case,
|
d(t) = |
f |
e-X$e-ajV9(y(s))ds; |
|
|
|
+a |
|
|
(2) if t7(+oo) = |
0 0, then t |
line-a`y"(t) = i"(+00); |
|
(3) if iZ(+oc) = |
RZJ with 2 # 0, then |
limo ( e _) |
0 2 |
|
L
Hence, the point (0, 0) is a stable improper node of the given system.
VIII-7. Determine whether the point (0,0) is a center or a spiral point of the system
dt = -y, |
dt = 2x + r3 - x2(2 - x)y. |
|
VIII-8. Show that the point (0, 0) is the center of the system
dt = y + xy3 - y7, |
_ -x + xy2 - yg. |
Hint. This system does not change even if (t, y) is replaced by (-t, -y).
VIII-9. For the system
dt = 3y + 5x(x2 + y2), |
dty = 27x + 5y(x2 + y2), |
find an approximation for the orbit(s) approaching (0,0) as t - +oo. VIII-IO. Show that the point (0, 0) is a stable spiral point of the system
d |
[y,] |
- |
[yj + |
1 |
-- |
[ y2 |
dt |
y2 |
yIn yi + y2 |
yi |
Hint. If we set yl = r cos 0 and y2 = r sin 0, the given system becomes
VIII-11. Suppose that a solution ¢(t) of a system
dy _
dt
AJ + §(y-)
satisfies the condition
lim 4(t) = 0
t-r
for a positive number r. Show that if
(1)A is an n x n constant matrix,
(2)the n-dimensional vector W(if) is continuous in a neighborhood of 0,
(3)lien 9(Y) = 0,
V-6 !y1
then ¢(t) = 6 for all values oft.
(Note that the uniqueness of solutions of initial-value problems is not assumed.)
VIII-12. Assume that
(i)AI.... .. 1n are complex numbers that are in the interior of a half-plane in the complex A-plane whose boundary contains A = 0,
(ii) there are no relations \, = plat + P2 1\2 + |
+ pnan for j = 1, ... , n and |
non-negative integers p,,... , pn such that pi + - - + pn >_ 2, |
(iii) f(y' = |
yam' ft, is a convergent power series in ff E Un with coefficients |
fpEC'. |
jp1>2 |
|
|
|
|
|
Show that there exists a convergent power series ((u) _ |
ul'Qp with coefficients |
|
|
Inl>2 |
_ |
dg
Qp E Cn such that the transformation if = u" + Q(ur) changes the system - _
A#+ f (y-) to j = Au, where A = diag[.1t, A2,4, ... , A ].
VIII-13. Show that the trivial solution of the system
dx - -st(x +X22) + x2ezl+ 2, |
dr |
-22(22 + 22) _ Xlez'+ 2 |
|
|
dt |
|
|
is aymptotically stable as t |
+oo. Sketch a phase portrait of the orbits of this |
system. |
|
|
|
|
VIII-14. In Observation VIII-6-2, it is stated that if one of the |
is not real, |
then y" is a spiral point. Verify this statement.
VIII-15. Discuss the stability of the trivial solution of the system
j1 = x21 |
722 = xi(I - x1) |
as t - +00.
