On the existence of Ck solutions to Briot-Bouquet systems

JOURNAL

OF DIFFERENTIAL

EQUATIONS

3, 282-285

(1967)

On the Existence of CR Solutions Briot-Bouquet Systems*

to

RICHARD J. VENTI Sandia

Laboratory,

Albuquerque,

New

Mexico

Received June 28, 1966

1. Consider

INTRODUCTION

the real system x(dyldx)

= 4

+ Y(% Y),

(1)

where x is a scalar, y, Y are vectors, A is a constant matrix, YE CK (1 < K < co) in a neighborhood of (x, y) = 0, and Y(x, r) = o(l X, y I) I denotes the Euclidean norm). By a solution of (1) as Ix,Yl+O (I we shall mean a real vector-valued function 4 = 4(x) which vanishes at x = 0 and is continuously differentiable and satisfies (1) in a neighborhood of x = 0. This note gives sufficient conditions for the existence of CK solutions of (1); the main result is: THEOREM 1. Let the eigenvaluesof A with real parts greater than or equal to 1 be denotedby A, ,..., A, and put A, = 1. Under the above hypotheses,if R(hi) # K

(j = l,..., n)

11 C mdi # 4

(j = l,..., n)

(2)

and, in caseK > 2, i-0

for all setsof nonnegativeintegersm, ,..., m, satisfying

2<

i

mi Q

K,

i=O

then there exists a CK solutionof Eq. (1). * This

work

was supported

by the

United

282

States

Atomic

Energy

Commission.

CKSOLUTIONS

TO BRIOT-BOUQUET SYSTEMS

283

When y is a scalar, (1) reduces to the classicalBriot-Bouquet equation, which has been treated by a number of authors; an extensive discussion and bibliography concerning this equation can be found in [2], Chap. III. Conditions (2) and (3) can probably be weakened, but they allow us to apply, in a simple manner, certain resultson invariant manifolds and normal forms. In general, some condition on the hj is obviously necessary,at least that Ai # 2,..., K. This is seen by substituting finite Taylor seriesfor the componentsof y in (1) and comparing coefficients. If fl is in the real Jordan form and the nonlinear part of (1) corresponding to X, ,..., h, is o(l x,y 1”) as 1x, y 1-+ 0, then Condition (3) may be dropped; in fact it is precisely to obtain this circumstance that (3) is assumed.It should be noted that even when the nonlinear term in (1) is o(l x, y 1”) as j x, y I + 0, (1) may fail to have a CK solution when (2) is violated; (see [3], Section 5).

2.

AN EQUIVALENT

PROBLEM

Considerthe system &=--x,

j=-Ay-Y(x,y)

(--co
< co),

(4)

the dot denotes the time derivative, and x, y, (1 are as in (1). A curve S in (x, y)-space is called an invariant curve of (4) if every solution of (4) which is on S at t = 0 remainson S for all t > 0. We note that a Cl function 4 = 4(x), satisfyingC(O) = 0, is a solution of (1) if and only if for someE > 0 s = {(x9 4(x)) : I x I -=c4

(5)

is an invariant curve of (4). Hence Theorem 1 is equivalent to: THEOREM2. Under the hypotheses of Theorem 1 there exists an invariant curve S of (4) having the form (5), where 4 is of class CK and 4(O) = 0. Proof of Theorem 2. We may assumethat A is in the real Jordan form, and by changing notation we rewrite (4) as

k = -x, 3 = AY + my,

4,

2 = Bs + -W,Y,

4,

(6)

where -/l = diag[A, B] and the eigenvaluesof A are just -Xi ,..., --h, . Let K > 1, then ([I], Chap. IX, Ex. 5.l(ii), Corollary 5.2) there exists a C’Kchange of variables x = x, y = y, w = .z -&,Y),

&,y)=o(lx,rl)

as I~,YI+O,

284 defined

VENT1

in a neighborhood

of (x, y, z) = 0 and transforming

(6) into

ji = -x 9 = AY + Y(x, y, w + g(x, Y)),

zi = Bw + W(x, y, w), where

W(x, y, 0) = 0. Thus

it suffices to prove Theorem

2 for

*‘=-xx,

(7)

j = AY + y(%Y?g(%Y))7

since, if y = C(x) defines an invariant curve of (7), then y = 4(x), z = g(x, C(x)) define an invariant curve of (6). Hence, we may assume at the outset that the real parts of the eigenvalues of -A are all less than or equal to -1. Let us assume this and that (4) is again rewritten as (6) where now the real parts of the eigenvalues of A, B are >, <-K, respectively. (We can do this because of (2); note that if K = 1 then -A = B and A is absent.) Because of Condition (3) we may assume (cf. [I], Chap. IX, Section 13) that Y(x,y, x), Z(x,y, x) = o(] x,y, z 1”) as 1 x,y, z 1 -+ 0. If -A # B then, by [3], Theorem 2, there exists a CK change of variables x = x, ?J = Y - g(x, y, z), x = z, defined

in a neighborhood

of (x, y, z) = 0 and transforming

(6) into

$ = -x , it = Av,

t = Bx + .2(x, v, x), where 2~ CK in a neighborhood o(] x, v, z IK) as I x, v, .z 1 + 0. Thus

R = -x,

of (x, v, z) = 0 and g, g((x, v, z) = it suffices to prove Theorem 2 for

s = Bz + .i?((x,0, z),

i.e., the case -A = B; this follows immediately from [3], Lemma 1. (Two misprints in [3] should be corrected as follows: in the second line of Lemma 1, 2 replace replace “0 < Y” by “0 < Y”, and in the first line of Theorem “Lemma 1” by “Lemma 2.“)

Remurk. If (3) holds for all sets of nonnegative integers m. ,..., m, satisfying 2 < CF=,-, mi , then a slight modification of the above argument yields Cm or analytic solutions of (1) according as Y is of class Cm or analytic, respectively.

CR

SOLUTIONS

TO BRIOT-BOUQUET

285

SYSTEMS

REFERENCES

I. HARTMAN, P., “Ordinary Differential Equations.” Wiley, 2. SANSONE, G. AND CONTI, R., “Non-Linear Differential New York, 1964. 3. Vmr~r, R., Linear normal forms of differential equations. 182-194.

New York, Equations.”

1964. Macmillan,

1. By.

Eqs. 2 (1966),