0387986995Basic TheoryC

228 VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

Observation in Case 2. Set

Bo (f) =

	(VII-6-9)	B1(t, E, iZ) _ -P(t, E)`1	u	8P	(t, E1,
				8e
			I<j<m

where µ E C' and the entries of µ are u1, P2, ... , µm. Then, differential equation (VII.6.6) can be written in the form

(VII.6.10)

dt = {Bo(h(t)) + B,

Notice that the entries of the matrix B, (t, E, µ) are periodic in t of period tv and that B1 (t, E, 6) = 0. Furthermore, any two distinct eigenvalues of Bo (6) do not

27n	IE + lµI < r}.
differ by integral multiples of -. Set D(r) =

In Case 2, the following lemma is used.

Lemma VII-6-4. There exist n x n matrices P, (t, e, {'t) and HI (E, µ) such that

(:)the entrees of P1 (t, E, ui) are continuous in (t, E, u') E R x D(r) and analytic in (E, )i) E D(r) for each fixed t E Ii£, where r is a suitable positive number,

(i:) P, (t + w, F,)!) = P, (t, e, it) for (t, F, tt+) E 1 x D{r

(iii)P1(t, E, 6) = O for (t, i) E R x 0(r),

(iv)the entries of H1 (E, u') are analytic in (E, Et) E D(r) and H1 (E, 6) = 0 for

E E O(r),

(v) at P, (t, e, µ') exists for (t, E, p) E R x D(r) and is given by

19 Pi(t,e,u) ={Bo(e) + B1(t,e,µ)) (In + Pi(t,E,ls)}

(VII.6.11)

+ P1 (t, i,#)) { Bo(E) + Hi (E, µ) }

for (t, E, µ) E IR x D(r-), where I, is the n x it identity matrix.

Remark VII-6-5. Equation (V1I.6.11) can be simplified as

	(VII.6.12)	a P1(t,Ej) = Bo(E)P1(t,E,i) - Pi(t,E., )Bo(Ej
		+ {B1(t,E,N)P:(t,E,Ji) -	+B1(t,,rA-)} -Hl(E,u-)-

Proof of Lemma VII-6-4.

M

Given p = (pi,... ,p,,,), where the p, are non-negative integers, denote EIpjI

=1

and 4' um by Ipl and 91, respectively. Set

Pi (t, E, u') _	gP1,p(t, E),	B, (t, e,
ipl>1	y	$pl?1

Ip1?1

6. AN APPLICATION OF THE FLOQUET THEOREM

229

Then, equation (VII.6.12) is equivalent to

8

(VII.6.13)	p = B0(E)P1,p		- P1,pBo(E) + Q1,p(t, E) -
where
(VII.6.14) Q1,p(t,E') =		E	(B1,ptP1,p2 - Pi,p2Hl,ps) + B1,p.
		P1+p2=p, 0p1IJ{r21>_1)
Hence,
(VII.6.15)	Pi,p(t,e-) =		{c( + 1 exp[-sBo(0

x 1 Q1,p(s, ) - H1,$,(e)) exP[sBo(E))ds} exp[-t B0(Z)1,

where C(e) and H1,p(e) are n x n matrices to be determined by the condition that

P1,p(t,e) is periodic in t of period w, i.e.,

exp[wBoQ-)1C(E) - C(i) exp[wBo(E)1

(VII.6.16)	- exp[wBo(E)1	exp[-sBo(e)1H1,p(i) exp[sBo(i)1ds
		J 0
	_ - exp[wBo(E)1	exp[-sBo(e)1Ql.p(s, e) exp[sBo(t1ds.

0

It is not difficult to see that condition (VII.6.16) determines the matrices C(ep) and Hl,p(ej. Then, the matrix P1,p(t, E) is determined by (VII.6.15). The convergence of power series P1 and H1 can be shown by using suitable majorant series. 0

In the same way as the proof of Theorem VII-6-3, the following theorem can be proven.

Theorem VII-6-6. The transformation

(VII.6.17)	it = {In + Pi (t,h(t),h'(t)))v"
changes differential equation (VII.6.10) to

dv = f H(i(t)) + H1(h(t), h'(t))
dt	l
(VII.6.18)	- [In + P1(t, h(t), hh'(t)),'1	I hj(t) aP1(t, h(t), h'(t})
		l

+ h (t) a_ (t, h{t), h'(t))J j v,

230	VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

where P1(t, E, µ) and H1(E, µ) are those two matrices given in Lemma VII-6-4.

Observe that under assumption (VII.6.3), we obtain

In +	P1(t,h(t),h'(t))j_1
f0+00	8Ej

(VII.6.19)

+ hJ (t) a. (t, h(t), h'(t))J dt < +oo.

Observe also that the matrix H(h(t))+H1(h(t), h'(t)) does not contain any periodic terms. Furthermore, H(6) + H1(6, 6) = H(0). It is clear that the derivative of

H(h(t)) is not necessarily absolutely integrable over to < t < +oo. Therefore, in order to apply the argument of MI-5, the matrix H(h(t)) must be examined more closely in each application. Details are left to the reader for further observation.

EXERCISES VII

VII-l. Find the Liapounoff's type number of each of the following four functions f (t) at t = +00:

(1) exp [t2sin (1L1)	]'	(2) exp Lfo	sint(dt](3) inin(exp[3t],exp[5t]sin67rt]),

(4) the solution of the initial-value problem y"-y'-6y = eat, y(O) = 1, y'(0) = 4.

VII-2. Find a normal fundamental set of four linearly independent solutions of

the system dy = Ay" on the interval 0 < t < +oo, where

252	498	4134	698
A = -234	-465	-3885	-656
15	30	252	42
-10	-20	-166	-25

Hint. See Example IV-1-18.

VII-3. Let 4i(x) be a fundamental matrix solution of the system

	log(l+x)	e=	expx2
=	x3	1+ 1+ x	sin x
	exp(e')	cosx	arctanx

on the interval 0 < x < +oo. Find Liapunoff's type number of det 4i(x) at x = +oo.

EXERCISES VII

231

VII-4. Assuming that the entries of an n x n matrix A(t) are convergent power

series in t-1, calculate lim p log I (t) for a nontrivial solution ¢(t) of the differential

equation t = A(t)9 at t = +oo.

Hint. Set t = e".

VII-5. Let dY = xP-'A(x)i be a system of linear differential equations such that

y E C" is an unknown quantity, p is a positive integer, the entries of an n x n matrix

A(x) are convergent power series in x-1, and lim A(x) is not nilpotent. Show that

there exists a solution g(x) of this system such that	lim	ln(l y-(Teie)j)	is a positive
number for some real number 6.	r-.+oo	rp

Hint. Set x = re!e for a fixed 0 and t = rp. Then, the given system becomes
d#	e+'0			e' PO	e'
	= -A(re's)y'. Note that			lim -A(re'B) _ -A(oo). The eigenvalues
	P	ei_,		r-.+oo P	P
	ei>'		)Iof
	A(oo) are	p '	where the a, are the eigenvalues of A(oo). Now, apply
	P
Corollary VII-3-7.

VII-6. For each of two matrices A(t) given below, find a unitary matrix U(t) analytic on (-oo, oo) for each matrix such that U-' (t)A(t)U(t) is diagonal or uppertriangular.

r	Ot	11	1	t	0
(a) A(t) =		0J, (b) A(t) =	t	1 + 2t	0
L	i		0	0	1+t2

Hint. See [GH].

VII-7. Show that if a function p(t) is continuous on the interval 0 < t < +oo and

lim t-Pp(t) = I for some positive integer p, the differential equation d2t +p(t)y =

t +oo

0 has two linearly independent solutions rlf(t) such that

77.k(t) =	P(t)-114(1 +o(1))exp [±ij t	)ds(,
77, (t) = P(t)114(+1 +o(1))exp [f iJ t		JJ
		d4] ,
	to

as t +oo, where to is a sufficiently large positive number and o(1) denotes a quantity which tends to zero as t - +oo.

Hint. Use an argument similar to Example VII-4-8.

m-1

VII-8. Let Q(x) = x' + E ahxh and P(x, e) _ ,Tn+l + Q(x), where in is a

h=o

positive integer, x is an independent variable, and

am-1) are complex

232	VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS
parameters. Set A(x, f) = {(0)		11.	Also, let Ro, po, and ao be arbitrary but
fixed positive numbers. Suppose that 0 < po < 2. Show that the system
	d:V
	= A(x,f)y,		y = 17121

has two solutions y, (x, f) (j = 1, 2) satisfying the following conditions:

(a)the entries of y, (x, f) are holomorphic with respect to (x, f, ao,... , am_ 1) in the domain

D={(x,f,ao,...,am_,): rEC, 0<Ifl<ao, Iargfl<Po,

laol + ... + Iam-I I < Ro},

(b) y', (x, f) (j = 1, 2) possess the asymptotic representations

exp	111 +o(1)JP(x,f)
	P(t,f)1/2dtl
L fro	J L it - o(1)]P(x,f)1/4

11 +o(1)] P(x,E)-1/4 exp l - Jro P(t,E) 1/2dt, f -l + o(1)]P(x, f)'/4,

respectively as x - +oc on the positive real line in the x-plane, where xo is a positive number depending on (Ra.po.ao) and o(1) denotes a quantity which tends to 0 as x ---. +oo on the positive real line uniformly with respect to (f, ao,... , am-I) for IEI < ao, I arg EI < po, and Iaol + ... + lam-1l < Ro.

Hint. Use a method similar to that for Exercise V1I-7. See, also, [Nful and [Si13,

Chapter 3].

VII-9. Let A1(t)...... (t) be n continuous functions of t on the interval Zo =

{tER:0<t<+oo}such that R.eja1(t)-aJ+1(t))>1(j=1,...,n-1) on

Also, let A(t) be an n x n matrix such that the entries are continuous in t on Zo

and that

sup(1 + p - t)	1 1 p I A(s) l ds - 0 as t -+ +oo.
p>t	1

Show that there exist a non-negative number to and an n x n matrix T(t) such that

(1) the derivative	(t) exists and the entries of T(x) and . (x) are continuous
E
in t on the interval 2 = It : to < t < +oo},
(2) t limo T(t) = 0,
(3) transformation	(I + T(x)) z changes the system
	dt =	(diag[A1(t),A2(t),...	+T(t))y
to
d			... , An(t) +
dt = diag(A1(t) + bl (t), A2(t) + b2(t),

where y E C, i E C, n functions b1(t),... ,b,,(t) are complex-valued and continuous in t on T, and lim b,(t) = 0 (j = 1, ... , n).

t +00

EXERCISES VII

233

VII-10. Show that if R(t) is a real-valued and continuous function on Zo = {t E It :

	+00
0 < t < +oc} satisfying the condition J	JR(t)j < +oo, the differential equation
0
d2y + (1 + R(t))y = 0 has a solution q5(t) such that		lim (O (t) - sint) = 0. Also,
dt2		c-+OQ	lim n = 7r.
show that 0(t) has infinitely many positive zeros an such that
			n+00 n
VII-11. Show that every solution of a differetial equation			tz + R(t)y = 0 has

at most a finite number of zeros on the interval Zo = {t E Lea : 0 <_ t < +oo} if R(t) is a real-valued and continuous function on Zo satisfying the condition

+a,

tIR(t)j < +oc.

L

+00

Remark. See (CL, Problem 28 on p. 1031. For the case when f tjR(t)j = +oo,

o

f+a

but J IR(t)l < +oo, see Example VI-1-i.

0

VII-12. Suppose that u(t) is a real-valued, continuous, and bounded function of t

	r+00
on the interval 0 < t < +oo. Also, assume that J		ju(t)jdt < +oc. Show that if
X is an eigenvalue of the eigenvalue problem	0
d'y	f+00
7t2 + u(t)y = Ay, y(O) = 0,	J0	y(t)2dt < +oo,

then 0 < A < sup Ja(i)l. o<:<+c*

VII-13. Let A(t) be an n x it matrix whose entries are real-valued, continuous, and periodic of period 1 on R. Show that there exist two n x n matrices P(t, c) and

B(c) such that

(a)the entries of P(t, c) and B(c) are power series in c which are uniformly convergent for -oc < t < +oo and small JEJ,

(b)P(t, 0) = I, (-oo < t < +oo), where In is then x n identity matrix,

(c)the entries of P(t, c) is periodic in t of period 1,

(d)P(t, c) and B(E) satisfy the equation

eP

8t = cIA(t)P(t, e) - P(t, c)B(c)j.

VII-14. Apply Lemma VII-6-2 to the following system:

dy	= [nL + (a + 8cos(2t))(K - L)Jg,	g _
dt		[Y2J

where n is a positive integer, the two quantities a and,3 are real parameters, and

K =	f0	and L = [ 01	0]
	U]

234	VII. ASYMPTOTIC BEHAVIOR OF LINEAR SYSTEMS

VII-15. Using Theorem VII-6-3, find the asymptotic behavior of solutions of the differential equation

2 + {1 + h(t) sin(at)} rl	= 0,	h(t) =	ln(2 + t)

as t -+ +oo, where a is a real parameter.

CHAPTER VIII

STABILITY

In the previous chapter, we explained the behavior of solutions of linear systems as t -. +oo. In this chapter, we look into similar problems for nonlinear systems.

To start with, in §VIII-1, we introduce the concepts of stability and asymptotic stability of a given particular solution as t +oo. We illustrate those concepts with simple examples. Reducing the given solution to the trivial solution by a simple transformation, we concentrate our explanation on the stability property of the trivial solution. It is well known that the trivial solution is asymptotically stable as t +oo if real parts of eigenvalues of the leading matrix of the given system are all negative. This basic result is given as Theorem VIII-2-1 in §VIII-2. The case when some of those real parts are not negative is treated in §VIII-3. In particular, we discuss the stable and unstable manifolds. In §VIII-4, we look into the structure of stable manifolds more closely for analytic differential equations. First we change a given system by an analytic transformation to a simple standard form. By virtue of such a simplification, we can construct the stable manifold in a simple analytic form. This idea is applied to analytic systems in IR2 in §VIII-6. In

§§VIII-7-VIII-10, using the polar coordinates, we explain continuous perturbations of linear systems in J2. In §VIII-5, we summarize some known facts concerning linear systems with constant coefficients in JR2. The topics discussed in this chapter are also found in [CL, pp. 371-388], [Har2, pp. 160-161, 220-227], and [SC, pp. 49-96]. The materials in §§VIII-4 and VIII-6 are also found in [Du], [Huk5], and

[Si2].

VIII-1. Basic definitions

Let us consider a system of differential equations

	(VIII.1.1)	dy	t=
		d

where y E lR" and the R'-valued function At, M is continuous on a region

A(ro) = Zo x D (ro) = {(t, y1 E J"+I : 0 < t < +oo, 1171 < ro}.

Also, assume that a solution mo(t) of (VIII.1.1) is defined on the entire interval 10 and that (t, mo(t)) E A(ro) on Yo. The main topic in this chapter is the behavior of solutions of the initial-value problem

(VIII.1.2)	dt =	f (t, o'

as t +oo. To start, we introduce the concept of stability.

235

236	VIII. STABILITY

Definition VIII-1-1. The solution do(t) is said to be stable as t - +oo if, for any given positive number c, there exists another positive number b(c) such that whenever I&(0) - 7 < 5(c), every solution 4(t) of initial-value problem (VIII.1.2) exists on the entire interval Zo and satisfies the condition

(VIII.1.3)

I$o(t) - (t)I < E

on 10.

Remark VIII-1-2. Suppose that the initial-value problem

(VIII.1.2.r)

dt = At' y),

y(r) = it

has the unique solution y = (t, r, i) if (r, rl) E 0(ro). Then, qs(t, r, 771 is continuous with respect to (t, r, '). Therefore, for any r E Zo and any given positive numbers T and f, there exists a positive number p(r,T,e) such that whenever 100(r) - )1 < p(r, T, c), the solution 0(t, r, 71) exists on the interval 0 < t < T and I0o(t) -

¢(t, r, Y-7)1 < E on the interval 0 < t < T (cf. §11-1). This implies that if the solution

0o(t) is stable as t +oo, then for any r E Zo and any positive number c, there exists another positive number b(r, c) such that whenever loo (,r) - n7 < b(r, E), the solution (t, r, rt) exists on the entire interval 10 and (t, r, J) satisfies the condition

IV'0(t) - (¢(t, r, T-D I < E on I.

We also introduce the concept of asymptotic stability.

Definition VIII-1-3. The solution do(t) is said to be asymptotically stable as t-4 +00 if

(i)the solution do(t) is stable as t - +oo,

(ii)there exists a positive number r such that whenever I o(0) - i7 < r, every

solution fi(t) of initial-value problem (VIII. 1.2) satisfies the condition

(VIII.1.4)

lim I&(t) - (t)I = 0.

Remark VIII-1-4. Set g= E+ ¢0(t). Then, system (VIII.1.1) is changed to

(VIII.1.5)

dt = At' z + do(t)) - f'(t, do(t)).

Hence, the study of the solution o(t) of (VIII.1.1) is reduced to that of the trivial solution z1(t) = 0 of (VIII.1.5). Thus, the solution .o(t) of (VIII.1.1) is stable

(respectively asymptotically stable) as t -+ +oo if and only if the trivial solution of

(VIII.1.5) is stable (respectively asymptotically stable) as t -- +co. In the following sections, we shall study stability and asymptotic stability of the trivial solution.

The following three examples illustrate stability and asymptotic stability.

Example VIII-1-5. The first example is the system given by

(VIII.1.6)

ddtl = -yi,

dt = (yl - y2)y2

1. BASIC DEFINITIONS

237

To find the general solution of (VIII.1.6), solve the first-order equation

(VIII.1.7)	dye	= -y2 + .
	dyi	y1

The transformation u = b2 changes this equation to a first-order linear differential

du	2	c - 2'I	vt a-2n
equation - = 2u - -. Thus, we obtain u(yl) = ezvi			dq . Since
7Y1	yl		rl

u = yz > 0, the quantity c must be positive. It is evident that y1(t) = y e-t satisfies the first equation of system (VIII.1.6) with the initial value yt(0) = y.

Therefore,

(VIII.1.8)	yl(t) = -( e-t,			1
				112(t) = ±
				u(yi(t))
is a solution of (VIII.1.6). Observe that
	y<yl(t)<0	if y<0 and 0<y1(t)<y if y>0
for t > 0.	ry, (t)-2		dt) < 0 for t > 0. Thus, y2(t)2 <		e-2Yt(t)
	Therefore, J.	77			C

e2171

= e21',1e2'ry2(0)2 for t > 0. This proves that the trivial solution of system

c
(VIII.1.6) is stable as t	+oc. Furthermore, since
fvt(t) a-2q			fYi a-2n
lim J	-dt7 = lim		di = -oo,
t-+o0	r)	Y1 -0	11

the trivial solution of system (VIII.1.6) is asymptotically stable as t -- +oo. This result is shown also by Figure 1.

Y2

FIGURE 1.

Example VIII-1-6. The second example is a second-order differential equation

(VIII.1.9)

+ g(t7) = 0,

d-t2