0387986995Basic TheoryC

258 VIII. STABILITY

changes (VIII.6.3) to another system

where f2(() is a polynomial in C with real coefficients which has one of the following two forms:

where co is a nonzero real number and tno is a positive integer.

Observation VIII-6-3. Let us look into Case I. Assume that system (VIII.6.3)

is

Note that transformation (VIII.6.5) does not change (VIII.6.3.1). Using a trans- formation of form (VIII.6.5), change h(ul, u2) so that

Since (u1,u2) = (ce`t,de-'t) is a solution of system (VIII.6.3.1) with a complex arbitrary constant c, the formal series

cue-,t)

cell + h(eett,

ce-u + h(ce-tt cett)

is a formal solution of system (VIII.6.2) which depends on two real arbitrary constants.

Let

be the solution of system (VIII.6.2) satisfying the initial condition

are power series in { and which are convergent uniformly on any fixed bounded interval on the real t line.

Using (VIII.6.6), we obtain

of system (VIII.6.2) are identical. The first of these two is convergent; hence, the second is also convergent. This finishes Case I.

Observation VIII-6-4. Let us look into Case II. Assume that system (VIII.6.3) has the form

Note that we have (VIII.6.4).

Since (ul,u2) = (ce't(1+co(ce)'"°) ee-u(1+c0(ct)'"0)) is a solution of system

(VIII.6.3.2) with a complex arbitrary constant c, the formal series

ce't(I+co(ce)'"0) + h(ceit(l+co(ct)"'O),

ee-'t(1+c0(ct)_0))

(FS-2) v = f (u') =

h(ce-tt(1+co(ce)'"° ), celt(1+co(ct)"O) )

ee-'t(1+c0(ce)'"0) +

is a formal solution of system (VIII.6.2) which depends on two real arbitrary constants.

Again, as in Observation VIII-6-3, let (S) be the solution of system (VIII.6.2) satisfying the initial condition (C). Set

ee-tt(1+c0(ct)'"0) + h(ee-it(t+co(ct)'"O),ceut(l+co(et)"O))

as formal power series in (c, e). This series has a formal period

Solving (VIII.6.7) with respect to (c, e), we obtain two power series in

(VIII.6.9)

Using (VIII.6.9) as equations for P({,£), it can be shown that the formal power

is convergent. (Here, some details are left to the reader as an exercise.) On the other hand,

T(ct)

since the quantity hP+1,P are real (cf. (VIII.6.4)). This proves that C(Cr/ ) is conver-

C(l, 7)

gent. Hence, c(l;, ) and 4) are convergent. Thus, the proof of the convergence of A g) in Case II is completed.

Thus, it was proved that if all of the coefficients w,,, on the right-hand side of system (VIII.6.3) are real, the point (0,0) is a center of system (VIII.6.1). The general solution of (VIII.6.1) can be constructed by using various transformations of (VII.6.1) which bring the system to either (VIII.6.3.1) or (VIII.6.3.2). Periods of solutions in t are independent of each solution in Case I, but depend on each

solution in Case H.

7. PERTURBATIONS OF AN IMPROPER NODE AND A SADDLE PT. 261

Remark VIII-6-5. The argument given above does not apply to the case when the right-hand side of (VIII.6.1) is of C(O°) but not analytic. A counterexample is given by

where h(r2) = e-1jr2 sin I r I. Using the polar coordinates (r,O), this system is changed to

dt = rh(r2) and d

and periodic solutions are given by r2 = 1 with any integers nt. This is an

MR

example of centers in the sense of Bendixson (cf. [Ben2, p. 26] and §VIII-10).

For more information concerning analytic systems in JR2, see [Huk5].

VIII-7. Perturbations of an improper node and a saddle point

In the previous section, the structure of solutions of an analytic system in JR2 was showen by means of the method with power series. Hereafter, in this chapter, we consider a system

under the assumption that

(i)A is a real constant 2 x 2 matrix,

(ii)the entries of the R2-valued function §(y) are continuous in g E JR2 near 0' and satisfies the condition

In this section, it is also assumed that the initial-value problem

has one and only one solution as long as g belongs to a sufficiently small neighborhood of 0.

Remark VIII-7-1. In the case, when the entries of an 1R2-valued function f (Z-) are continuously differentiable with respect to the entries z1 and z2 of i in a neigh-

borhood of i = 0 and f (0) = 0, write the differential equation dt = f(E) in form

(VIII.7.1) by setting

Using thepolar coordinates (r, 0) in the g-plane, write the vector g in the form y = r [c?}. Then, system (VIII.7.1) is written in terms of (r, B) as follows:

d=r[a, i cos2(B) + a22 sin2(9) + (a12 + a21) sin(g) cos(9)]

+ 91(-1 COs(0) + 92() sin(g),

(VIII.7.4)

In this section, we consider the case when the two eigenvalues Ai and A2 of the matrix A are real, distinct, and at least one of them is negative, i.e., A2 > Al and Ai < 0. Throughout this section, assume that all = Al, a22 = A2, and a12 = a21 = 0. Then, in terms of the polar coordinates (r, 0), system (VIII.7.1) can be writen in the form

(VIII.7.5)

Observation VIII-7-2. The point (0,0) is not a proper node of (VIII.7.1). In

0 and e(t) --+ w as t - +oo or -oo, then d8 --* (A2 - AI) sin w cos w as t -+ +oo or -oo. Hence,

sin w cos w = 0. This implies that w = -7r, - 7r, 0, or

2

Observation VIII-7-3. For any given positive number e > 0, there exists another positive number p(e) > 0 such that

if 0 < r < p(e) (cf. Figure 13).

Observation VIII-7-4. If 0 > A2 > Al, there exists a positive number ro > 0

such that dt < Air for 0 < r < ro.

7. PERTURBATIONS OF AN IMPROPER NODE AND A SADDLE PT. 263

Observation VIII-7-5. In the case when 1\2 > 0 > AI, find a real number wo

such that 0 < wo < 2 and tanwo = -1. Then, for any given positive number

2

E > 0, there exists another positive number p(E) > 0 such that

if 0 < r < p(E) (cf. Figure 14).

Observation VIII-7-6. If two positive numbers f and p are sufficiently small,

Observations VIII-7-2 and VIII-7-3 show that the behavior of orbits of (VIII.7.1) in the sectorial region S = {(r cos 0, r sin 0) : $6 < E, 0 < r < p} looks like Figure

15.

Denote by fl the set of all real numbers w such that

(i)IwI 5 E,

(ii)the orbit (r(t), 8(t)) of (VIII.7.1) such that r(0) = p and 0(0) = w leaves the sectorial region S through the boundary 0 = -E.

Set a = sup(w : w E 0). Then, ja$ < E. Furthermore, the orbit (r(t), B(t)) of

(VIII.7.1) such that r(0) = p and 0(0) = a satisfies the condition lim r(t) = 0

and Iim 6(t) = 0. This can be shown by using the continuity of orbits with+00 respect c-.+oo

to initial data (cf. Figures 16-1 and 16-2). Here, use was made of the uniqueness of solutions of initial-value problem (VIII.7.3).

7. PERTURBATIONS OF AN IMPROPER NODE AND A SADDLE PT. 265

(I +E)Yj .V

(I +e)Yi

Observation VIII-7-10. If we assume some condition on smoothness of §(y-),

there is only one orbit of (VIII.7. 1) that tends to the point (0, 0) in the direction

9 = 0 as t - +oc.

Theorem VIII-7-11. Assume that all = A1, a22 = A2, and a12 = a2I = 0, and that

(1)A2 > \1 and Al < 0,

(2)g(j) is continuous in a neighborhood of

(3)lim " = 0,

y-.o lye

(4) a9(y exists and is continuous in a neighborhood of 6.

2

Then, there exists a unique orbit y(t) _ [01(1)1 such that

02(t)

01(t) > 0 for t > 0,

Proof.

The existence of such orbit is seen in previous observations. To show the uniqueness of such orbit, rewrite (VIII.7.1) in the form

'\

266 VIII. STABILITY

G(yi, u) = -

92(yiu) - a 91(yl U

1 + -19i(y1iZ) aiyi

ac

It is easy to show that lim 5:u (y1, u) = 0 uniformly on Jul < 1. Hence, there exists v --o

a non-negative valued function K(yj) such that

!G(yi, u) - G(y1, v)! < K(yi )lu - vl

whenever Jul < 1, !v! < 1 and !y1! is sufficiently small. Furthermore, slim K(yt)

=0.

Let u = 01(yi) and u = ip2(yi) be two solutions of (VIII.7.9) such that

(1)ti i (yi) and .'2(y1) are defined for 0 < yi < r) for some sufficiently small positive number 'I,

Set '(yi) = 01 (y1) - s (yi ). By virtue of uniqueness, assume without any loss of generality that y(yi) > 0 for 0 < yi < ri. Then, for a sufficiently small positive r?,

0 < y1 < 17. Therefore, ;1(yi) = 0 for 0 < y1 < r).

The materials of this section are also found in (CL, §§5 and 6 of Chapter 151.

VIII-8. Perturbations of a proper node

In this section, we consider a system of the form

(VIII.8.1) dt = All + where y" E R2, under the assumptions that

(i)A is a negative number,

(ii)the entries of the R2-valued function g(y} are continuous in a neighborhood

of 0,

(iii)1§(y -)l < K!y"l1+i in a neighborhood of 6, where K is a nor-negative number and v is a positive number.

The main result is the following theorem.