0387986995Basic TheoryC

Thus, we conclude that

T f (x(t))dt > 0.

J

This implies that every periodic orbit is orbitally asymptotically stable. Hence, there exists at most one periodic orbit.

X-8. For a nonzero real number c and a real-valued, continuous, and periodic function f (t) of period T which is defined on -oo < t < +00, find a unique solution 0(t, E) of the differential equation

dt = -y + f (t)

E

which is periodic of period T. Also, find the uniform limit of 0(t, c) on the interval

-oo<t<+ooasE-+0.

X-9. Assuming that X(t) satisfies the condition

on an interval 0 < t < t, where a is a positive constant, and that x(t, A) is the unique solution to the initial-value problem

where e is a positive parameter, or is a real number, and ,8 is a positive number,

find lim y(t, e) for any fixed t > 0.

f 0+

Hint. Look at E dt = c - y2. Then, c = Q + a2 > a2. Hence, y2 increases to c

very quickly. Then, use Theorem X-7-1 [Na6], or find the solution explicitly (cf.

Exercise I-1).

X-11. Find, if any, solution(s) y(t, c) of the boundary-value problem

4dt2 + 2ydty = 0, y(0) = A, y(1) = B,

in the following six cases: (1) 0 < B < A, (2) B = A, (3) B > IAA, (4) B = -A > 0,

(5) CBI < -A, and (6) B < 0 < A, assuming that e is a positive parameter. Also, find lim y(t, e) for 0 < t < 1 in each of the six cases.

e 0+

Hint. Use explicit solutions together with the Nagumo Theorems (Theorems X- 1-3 and X-7-1) on boundary-value and singular perturbation problems. See also

Exercises X-10 and X-12, and [How].

X-12. Let f (x, y, t, e) be a real-valued funcion of four real variables (x, y, t, e).

Assume that

(i) 0(t) is a real-valued, continuous, twice-continuously differentiable function on the interval Zo = {t : 0 < t < 1) and satisfies the conditions 0 =

f (¢(t), 4'(t), t, 0) and 4(1) = B on Zo, where B is a given real number,

(ii) the function f (x, y, t, e) and its partial derivatives with respect to (x, y) are continuous in (x, y, t, e) on a region R = {(x, y, t, e) : Ix - ¢(t)I < rl, IyI <

+00, t E Zo, 0 < E < r2}, where r1 and r2 are positive nmbers,

(iii)If (4(t), 4'(t), t, e) I < Ke for t E Zo, where K is a positive number,

(iv)there exists a positive number µ such that Of (x, y, t, e) < -p on R,

(v) there is a positive-valued and continuous function '(s) defined on the interval

R,

(vi) A is a given real number.

Then, there exists a positive number co such that for each positive f not greater than co, there exists a solution x(t, E) of the boundary-value problem

such that

Ix(t,E) - di(t)I < IA - h(0)led=te-ft/` +C2e on Zo,

where c1 i c2, c3, c4, and cs are positive numbers.

Hint. Cf. Theorem X-1-3. See also [How].

X-13. Let a(t), b(t), and f (t) be real-valued, continuous, and continuously differ- entiable functions of t which are defined on the interval 0 < t < 1. Also, let 4o(t)

For a positive number e, denote by y(t, e) the unique solution of the initial-value problem

d3y + d-t2 + a(t) dy + b(t)y = f (t),

3

y(0) ='7o(E), y'(0) = 171 W, y"(0) = 12(E)-

Show that ly(t, e) -00(t)I+ Iy'(t, e) -40 (t)I +Iy"(t, e) -40 (t) I tends to zero uniformly

e-0+.

X-14. Let Y E lR", y E Rm, t E R, and e E R. Also, let ft f, y, t, e) and g(x", y, t, e) be respectively R'-valued and 1R'"-valued functions of (x, y, t, e). Assume that

(i)an R"-valued function fi(t) and an R'-valued function t%i(t) are continuous and continuously differentiable on an interval Zo = {t : a < t < b} and satisfy

the system of equations

dx" = f (Y, i, t, 0), 0 = g(x, y, t, 0) on Zo, dt

(ii)f (Y, y, t, e), g(x', y, t, e), and their partial derivatives with respect to (x, y-) are continuous on a region R = {(:e, y, t, e) : I x - m(t) I < rl, Iy - j(t) I < r2, t E lo, 0 < e < r3 }, where rl, r2, and r3 are positive numbers,

(iii) all eigenvalues of the matrix (1(t), ti(t), t, 0) are less than a negative num-

ber M on .70.

Show that the initial-value problem

has a unique solution (2(t, e), y(t, f)) for t E Zo and 0 < e < r4 if e + IC(e) -

(a)I + Ir(e) - t'(a)I is sufficiently small on 0 < e < r4i where r4 is a positive number. Also, show that Ii(t, e) - ¢(t)I + I y(t, e) - tj,(t)I - 0 uniformly on Zo as e + IC(E) - (a)I + I#(E) - (a)I - 0.

Hint. See [LeL].

X-15. Let x E lR", y E R"', t E R, and e E R. Also, let t, e) and g(i, y, t, e) be respectively R"-valued and R'-valued functions of (i, y", t, e) which are periodic in t of period T. Assume that

(i)an R"-valued function fi(t) and an R"'-valued function t%'(t) are continuous, continuously differentiable, and periodic of period T on the interval Zo = it :

-00 < t < +oo} and satisfy the system of equations d = f (:F, y", t, 0), 0 =

g"(x, y, t, 0) on Za,

(ii)f(i, y", t, e), g(i, y", t, e), and their partial derivatives with respect to (x, y') are continuous on a region R = {(i, y, t, e) : Ix - m(t)I <- ri, I y - 7G(t)I < r2, t E Zo, IEI < r3}, where rl, r2, and r3 are positive numbers,

(iii)there exists a real m x m matrix P(t) such that

(iiia) the entries of P(t) and P(t)-1 are real-valued, continuous, continuously differentiable, and periodic of period T[Baton Zo,

m1 x ml matrix with eigenvaules havingnegative real parts on 10i while B2(t) is a real (m - ml) x (m - ml) matrix with eigenvalues having positive real parts on Zo,

(iv) the system

CHAPTER XI

ASYMPTOTIC EXPANSIONS

In §§V-1 and V-2, we defined formal solutions of a system of analytic dif- ferential equations. Formal solutions are not necessarily convergent. For ex- ample, as we mentioned it in Remark V-1-4, the divergent formal power series

has an actual solution f (x) = el/=J t-1e-hIldt for x > 0. Integrating by parts, we

0

el/=10,7 tNe-l/tdt = xN+2 - (N + 2)el/= rstN+1e-l/tdt < xN+2, we conclude that

M=0

an asymptotic representation of an actual solution by means of a formal solution. In this chapter, we explain the asymptotic expansions of functions in the sense of Poincare and in the sense of the Gevrey asymptotics. In the Poincare asymptotics,

flat functions are characterized by the condition lim E f) = 0 for all positive inte- xm

gers m, whereas in the Gevrey asymptotic, flat functions are characterized by the condition If (x) I exp(cIxI-k) < M as x -i 0, where c, k, and M are some positive numbers. Generally speaking, the Poincare asymptotics is too general for the study of ordinary differential equations. A motivation of the Gevrey asymptotics is also given by the Maillet Theorem (cf. Theorem V-1-5). In §XI-1, we summarize the basic properties of asymptotic expansions of functions in the sense of Poincare. The Gevrey asymptotics is explained in §§XI-2-XI-5.

For more information concerning the Poincare asymptotics, see, for example,

[Wasl]. The Gevrey asymptotics was originally introduced in [Wat] and further developed in [Nell. To understand the materials concerning the Gevrey asymptotics of this chapter, [Ram 1], (Ram 21, [Ram3], [Si17, Appendices], [Si18], and (Si19] are helpful.

XI-1. Asymptotic expansions in the sense of Poincare

In this section, we explain the asymptotic expansions in the sense of Poincare.

Let x = a be a point on the extended complex x-plane. Consider a formal power series

342

1. ASYMPTOTIC EXPANSIONS IN THE SENSE OF POINCAR$ 343

Let D be a sector in the x-plane with vertex at x = a and Do be a neighborhood of x = a in D. Assume that f (x) is defined and continuous in Do.

Definition XI-1-1. The formal series (XI.1.1) is said to bean asymptotic (series) expansion of f (x) as x -k a in V if for every non-negative integer N, there exists a constant KN such that

for all x in Do.

Such an asymptotic relation is denoted by

(XI.1.3) f(x) p(x) as x a in D.

This definition of an asymptotic expansion of a function was originally given by H.

Poincare [Poi2J.

Before we explain some basic properties of asymptotic expansions, it is worthwhile to make the following remarks.

Remark XI-1-2. The vertex x = a can be x = oo. In that case, the asymptotic

on

m=0

Remark XI-1-3. Assume that f (x, t) is a function defined and continuous in (x, t) for x in D and t a parameter in a domain H in the t-plane. A formal power series

N

F c,,,(t)(x-a)'", where c,(t) is a function oft, is said to bean asymptotic (series)

M=0

expansion of f (x, t) as x 0 in V if for every non-negative integer N, there exists a function KN(t), independent of x, such that

for all x in Do. If, moreover, KN(t) are independent of t, then the asymptotic expansion is said to be uniform with respect to t.

Remark XI-1-4. If f (x) is holomorphic (i.e., analytic and single-valued) in a neighborhood of x = a, by virtue of the Taylor's Remainder Theorem, f (x) admits its Taylor's series expansion as its asymptotic expansion in any sector with vertex at x = a.

Theorem XI-1-5. For a continuous function f (x), there is at most one asymptotic expansion as x -+ a in a sector with vertex at x = a.

Proof

Assume that there are two asymptotic expansions of f (x) at x = a

M=0

Then, for every non-negative integer N, there exist two constants KN and LN such that

m=0

for all x in a neighborhood Do of x = a in D. For N = 0, we have Ieo - 7ol <

(Ko + Lo)lx - al for x in Do. Let x a in D; then, we obtain co = 70

Now, assume that Ck = 7k for k = 0, 1,... , N - 1. Then, from (XI.1.5) and

(XI.1.6), it follows that ICN -7NIIx-aIN < (KN+LN)Is-alN+l for x in Do. Let

x -+ a in D. Then, cN = 7N. Thus, cm = 7m is true for every non-negative integer

m. O

Theorem XI-1-6. Assume that f (x) and g(x) are two continuous functions such that

(XI.1.7)

m=0 h+k=m

Proof.

N

Fix a non-negative integer N and put f (x) _ > cm(x - a)'+Ei(x)(x-a)N+i

M=0

N

and g(x) _ > 'ym(x - a)m + E2(x)(x - a)N+1 Then, there exists two constants

m=0

KN and LN such that

1. ASYMPTOTIC EXPANSIONS IN THE SENSE OF POINCAR$ 345

for x in Do. Therefore,

N

(XI.1.11) If (x) ± 9(x)] - E I(Cm ± 1'.](x - a)m < (KN + LN)I x - aIN+1

M=0

for x in Do. Thus, (XI.1.8) holds. Also, it is easy to see that there exists a positive constant k such that

for x in Do. Thus, (XI.1.9) holds.

Theorem XI-1-7. Suppose that f (x) is a function holomorphic in 0 < Ix - al < 8 and admits an asymptotic expansion (XL1.4) with V = (0 < Ix - at < b}. Then, the asymptotic series actually converges to f (x) in 0 < Ix - at < S.

Proof.

Since ralim f (x) = co, x = a is a removable singularity. Thus, f (x) is holomorphic

in Ix - al < b. Therefore, the asymptotic series agrees with the Taylor's series (cf. Remark XI-1-4 and Theorem XI-1-5). Thus, the asymptotic series converges in Ix - aI < 8.

For a vector z E cm with entries (z1,z2,... zm) and p = (p1,p2,... Pm) with non-negative integers p, (j = 1, 2,... , m), we define I6oI = P1 + p2 + + p,,, z' = zr z2 ... zm , and Iz7 = max{Iz1I, IZ21, ... , Ixml}.

Theorem XI-1-8. Suppose that F(x, z) is a function with power series expansion

00

F(x, ) = F.(x)zY, which converges uniformly forx E D, I zl < bo, where Fp(x)

Ip1=o

00

is continuous in D and admits an asymptotic expansion Fp(x) c 1: Fpk(x - a)k

k=0

as x -, a E D. Define the formal power series in (x,

k=0

If f (x) is a continuous Cm-valued function with entrywise asymptotic expansions

00

f (x) ^-- > f (x - a)' __ p"(x) as x -' a in D,

1=0