P E R T U R B A T I O N M E T H O D I N T H E T H E O R Y OF

N O N L I N E A R O S C I L L A T I O N S

A H M E D ALY K A M E L

Dept. of Aeronautics and Astronautics, Stanford University, Stanford, Calif., U.S.A.

(Received 12 February, 1970)

Abstract. Asymptotic recurrence formulas for treating nonlinear oscillation problems are presented. These formulas are based on a Lie transform similar to that described by Deprit for Hamiltonian systems. It is shown that the basic formulas have essentially the same forms as those obtained by Deprit and by the present author in the Hamiltonian case.

1. Introduction

M a n y prob lems in the theory of nonl inear osci l lat ions can be fo rmula ted as a mul t i -

d imens iona l system of o rd ina ry differential equat ions

o o

~9 = g (y; e) = ~ g(') (y) , (1)

n=O

where y represents the state vector o f the system, e is a smal l parameter , the do t

denotes the to ta l derivative o f y with respect to the independen t var iable t, and g(")(y) are analy t ica l vectors o f the unknown state vector y.

Because any n o n a u t o n o m o u s system of differential equat ions

2 = h (z, t; e )= n! h(') (z, t) (2)

n=0

can be reduced to the au tonomous form of Equa t ion (1) by defining

,[;] [hll , = , and g f ' ) = ; n ~ > l , (3)

f rom now on, only Equa t ion (1) will be referred to for bo th a u t o n o m o u s as well as

n o n a u t o n o m o u s cases.

A c o m m o n feature o f the p rob lems presented by (1) is tha t they canno t be solved

exactly for a rb i t r a ry e, bu t their solut ion is known when e = 0. This suggests the search

for a near - ident i ty t r ans fo rma t ion y ~ x in the fo rm of power series

8n

y = x + x . @ ,

n = l

(4)

Celestial Mechanics 3 (1970) 90-106. All Rights Reserved Copyright �9 1970 by D. Reidel Publishing Company, Dordrecht-Holland

P E R T U R B A T I O N M E T H O D IN T H E T H E O R Y OF N O N L I N E A R OSCILLATIONS 9 1

so that the resulting system of differential equations oo

:c = f (x ; 5) = ~ f11(x), (5)

n = 0

will contain some specific terms (long-period terms, periodic terms of assigned type, and so on) and, in general, will be more tractable than (1).

This paper proposes a new method based on a Lie transform generated by a vector W that plays the same role as the generating function in canonical transformations depending on a small parameter proposed by Deprit (1969). Because of this similarity, it will be shown that the basic formulas take essentially the same forms as those obtained by Deprit and by Kamel (1969).

The principal advantage of the present method over other existing methods of averaging (Krylov-Bogoliubov, 1961; Morrison, 1966; Musen, 1965) is that, for any vector F in the form

oo

F (y; 5) = n.T F~") (y)' (6a)

11=0

the present method offers a recursive scheme that expresses this vector in the form oo

F [y (x; 5); 5] = ~.. F, (x), (6b)

11=0

beginning with F o (x)= F ~~ (x)= IF w) (y)]r=x. Equation (4)is now a special case of (6) in which one takes F~~ F~11)(y)=0 (n~> 1), Fo(x)=x , and F11(x)=x11(x) (n~ 1). It will also be shown that this same recursive algorithm can be used backward to compute (6a) from (6b) and, thus, one can construct the inverse transformation

oo

x = y + ~ y(11)(y), (7)

1 t = 1

by taking Fo(x)=x , F, (x)=0 (n~> 1), F(~ and r(11)(y)=y(11)(y) (n>~ 1). This inverse transformation is useful in constructing the approximate integrals of

motion when some of the elements x resulting from Equation (5) are constants. It is also useful in obtaining the initial conditions of (5) from those of (1). In addition, a by-product of the present method is the construction of Equation (7) if (4) is given, or vice versa (also, one can construct (6b) if (4) and (6a) are given).

Once the fundamental algorithm for Equation (6) is available, note that the relation between 9 ~") and f , of (1) and (5) can be obtained. First, differentiate Equation (4) with respect to t; thus,

oo

= + \ " (8)

11=1

9 2 AHMED ALY KAMEL

Substituting )~ and 2 from Equations (1), (5) and expressing g (y; e) in terms of x by using (6) and equating the coefficients of likewise powers of e obtain

gn (x) = f , (x) + C~ T x "In-m (X), (9) m = l

where n~

CT,- m!(n - m)!"

In further expositions, the fundamental algorithms are derived and examples will be presented.

2. The Recursive Algorithms

Consider any indefinitely differentiable vector F(x; e) that can be expressed in terms of x and e as a Taylor series

oo

F(x; e) = ~. F,(x), (10a)

n = O

where

F.(x) = ~ F(x; e) ~=o

In terms of y and e, this takes the form

ao

F Ix (y; e); ~] = ~ F<")(y), n = O

where

(10b)

where oo

dx ~ e n _ _ = y ( n + 1)

de nT. (Y) t t =O

is obtained from Equation (7). In view of Equations (6), this can be written as a Lie

d n

5) . F(")(Y)=I~-"F(x; ]~=o Xmy

Note that (O/Oe) F(x; e) denotes the derivative of F(x; e) with respect to e, keeping x fixed, and that (d/de)F(x; e) denotes its derivative with respect to e, keeping only y fixed. As a result, the relation between (a/ae) F(x; e) and (d/de) F(x; e) is

F - - F ( x ; e) = F(x ; e) + F(x ; e) , (11) de ~ L~ A de

P E R T U R B A T I O N M E T H O D IN T H E T H E O R Y OF N O N L I N E A R O S C I L L A T I O N S 93

transform

where dx/de = W (x ; s),

8 n 13 n

W(x; e) = ~ y(n+l)(y) = n~T. W,,+a (x).

11=0 n = 0

Combining Equations (11) and (12), obtains

d F (x; e) + LwF (x; e), F ( x ; e ) = •

(12)

(13)

(14)

where L w is the Lie derivative generated by W and defined by

[ ~ F ( x ; e ) l . W ( x ; e ). (15) Lwe(x; ~) = G

Substituting for F(x; ~) and W(x; e) by their expansion forms of Equations (lOa) and (13), obtains

d F (x; e) = F(,1)(x) (16) d~ ~. '

n = 0

where

F~.l)(x) = F.+l(x ) + ~ C~,L,,+IF,-,,(x), m = 0

and C,~ is as defined in Equation (9). By denoting F,(x) by F(,~ and by induction, one obtains

where

d~ ~ F ( x ; ~ ) = ~., (x); k/>1, r i m 0

F(k)(x) = v(k-1)r~ ~, r " r ~(k-1) � 9 + 1 ~ J + ~ , . ~ m + 1" . - , . ( X ) ; m = 0

(17)

k > ~ l , n~>0. (18)

Except for the redefinition of the operator Li, this recursive relation is essentially the same as the Deprit equation for Hamiltonian systems (see Kamel, 1969). In this recursive relation, and in reference to Equations (10) and (17)

F(. ~ (x) = F n (x), F(o k) (x) = F (k) (x) = [F (k) (y)]r =x- (19)

Thus, this recursive relation can be used to construct the desired relation between F (") and F, of Equations (6a) and (6b). This can be best visualized in the forward

94 A~MED ALY KAMEL

triangle of Figure 1, for example,

F (~) = F~ + L f o

F~ 1) = F 2 Jr L1F t + L2Fo

F(2) = F(li) + L1F(1) (20)

F(2 i) = F 3 + LiF2 + 2L2Ft + LaFo

F~ z) = F2 (1) -I- L1F~ i) + L2 F(~)

F (a) = F~ 2) + L t F ( 2 ) .

Note that the Deprit Equation (19) is useful in constructing F (") from F,. To construct F, f rom F ("), this same equation is written in the backward form as

n--1 F ( k ) _ lZ,(k + 1 ) _ ~ n - l l" ~ , ( k ) . - . . _ ~ ~] n > / 1 k t > 0 . (21) " ~ m ~ m + l ~ n - m - 1 ,

m=O

and can be visualized from the backward triangle of Figure 1.

Fig. 1.

= F (~ F.. F (~ eo o i = F ~ --

II)___~ F(2 ) F2--*E F2-_

V a ----~V~ - - - - -~V, - - - .F F 3 :

F (I)

F! ' ) !,(2)~. F (a'

b

Recurs ive T rans fo rma t ion o f an Analy t ic Vector under a Lie t ransform. - (a) forward tr iangle; (b) backward triangle.

It was shown by the present author that the backward triangle can be reduced to a simpler form in which all operations are carried out in the vertical direction (as in Figure 2) so that, if F ( ~ for some i, then the new intermediate functions Fj, i = 0 for all j . In addition, this new triangle has only one direction for the construction of F (") from F, as well as the construction of F. f rom F ("). This can be achieved if one observes that successive elimination of the vectors on the right-hand side of Equation (21) eventually leads to F, (k) in terms of F (k+"), F (k+"-l), ..., and F (k). As a result, the form of F(, k) can be assumed as

F(, k) = ~ C~Nf(k+"-J); n >1 1, k>l O, (22) j = 0

where N o = 1, and N i (j~> 1) is a linear operator that is a function of L i, Lj_j , ..., and

PERTURBATION METHOD IN T I E THEORY OF NONLINEAR OSCILLATIONS 9 5

Fig. 2.

E -F 1~ 0,0- . . . .

F m . . . . F,, 0 . . . . FO, I: F (I}

F 2 . . . . F2, o . . . . Fi, , . . . . FO,2 :F

E E -F (s) Fs . . . . F3,o . . . . F2,t . . . . 1,2"'" O,3-

Triangle for the visualization of the algorithm of Equations (26).

L 1. For Nj, substitution of Equation (22) into (21) yields the recursion relation

J Nj Z j -1 =- - Cm_IL,,Ni_m; j>~ l .

m = l

For example,

(23)

N o = l

N 1 = - - L1 N 2 = - - L 2 + L1L1 N 3 = -- L 3 + L 1 (L2 -- L1L1) + 2L2L1 �9

(24)

For k = 0, Equation (22) yields

F. = L C~Ny("-J); n/> 1. (25) j = O

Also, if NiF(~ is defined as Fj, ~, then Equation (25) becomes

where

and

F.= L C~Fj,._j, (26a) j = O

J Fj~ Z j -1 (26b) , = - C,._ 1L,,,Fs_,,,. ~,

m = l

Fo,. = F (") . (26c)

The principal formula, Equation (26b), now contains all the operations, and these can be seen in Figure 2. In this triangle, there are no cross computations; i.e., each element is computed only in terms of the elements above it in the same column. This admits that, if F (i) =0 , then Fj, i = 0 for allj~> 1 and, as a result, this column will drop out from the computations. Similarly, Equation (26a) can be visualized as a certain combination of elements in the horizontal direction. This combination allows

96 At-nVIED ALY I~A~nm

the construct ion of F, f rom F ("), or vice versa. For example, in the vertical direction,

F1, 0 = - La F(~

F2, 0 = - L1F1, 0 - L2 F(~ F3, 0 = - LtF2, 0 - 2L2F1, 0 - L3 F(~ Fi, i = -- L i F(i) F2, ~ = - L1Fi , i - L2 F(1)

F1, 2 = - L1F(a) ;

(27)

in the horizontal direction of Figure 2,

F~ = F~, o + F(1)

F2 = F2, o + 2F1, 1 + F(2) F3 =/73, o "k 3F2, i -k 3F1, 2 ''{- F(3).

(28)

3. General Expansions

By employing the recursive algori thms of the last section, the mappings of Equat ions (4) and (7) and the relation between 9 a n d f o f Equat ions (1) and (5) can be constructed. Using Equat ions (26a) and (26b) with F ( ~ F(" )=0 , and F o ( x ) = x obtains

X n ( X ) = F , ( x ) in the fo rm of

n--1

=- - - - - C m _ l L m X n _ m ( X ) , ( 2 9 ) x.(x) W.(x) Z .-1 m = l

Compar i son of (13) with (10) and using (26) lead to

n - 1

y(") (y) = W . ( y ) - ~. C ~ - l y j , n _ i ( y ) , (30) . /=1

where j

= - Cm- 1Lmyi - m, i (Y) Y~,i(Y) E j - 1 r n = l

Y O , i = y ( 1 )

and

When bo th forward and inverse mappings are desired, a useful representat ion in which one needs only to construct the triangle for Yi, ~ o f Equat ion (30) can be obta ined by eliminating x - y between Equat ions (4) and (7). This leads to

oo co

x . ( x ) = - ~ y (" ) (y ) . (31)

n=l n=i

where

P E R T U R B A T I O N M E T H O D I N T H E T H E O R Y OF N O N L I N E A R O S C I L L A T I O N S 97

In view of Equations (10), (26), (30), and (31), obtain

n - - 1 n - - 1 x.(x) = - W . ( x ) - ~. Cj-ayj,._j(x), (32)

j=l

where yy, i(x) = [Ys, i(Y)]y=x, and yy, iCY) is as defined in (30). For the relation between g a n d f o f (1) and (5), substitution for g. of (9), in terms

of g("), by using (26) with F = g leads to

n ~[ -(~xJ~'f ._~(x)]. (33) f . (x) = g~"~ (x) + c j o~ , . - j (~) \ ~ )

1=1

This relation contains W. as an arbitrary vector to be chosen to achieve a desired form forf . (x) . To separate this W. from the rest of the terms in (33), we write it as

n - J -

<'~foW.---- fn--g(n)- t - tg) ' fo--On, O-'P C V L t U J ' f n - j - g j , . - d , j = l

(34) where

and

~. = [X.]w.:o #.,o = [g.,o]w.=o

1 #j,i Z j-S = - - Cm- 1Lmgj- m, i

m = l

go, i ~ g(i).

Equation (34) is directly applicable to nonlinear resonant problems in which

y = [0] , (35a)

+ (35b) P = g = ~v (~, 0; ~) '

where u(e, 0; ~) and v(e, 0; e) are 2~-periodic in each component of 0. It is desirable to transform to a new vector

so that the resulting f . (n ~> 1) will contain only certain "slowly varying" combinations of the 0 elements. Equation (34) can be used to define the W.'s successively so as to remove all 'short period' terms from thef . ' s ; such a W. is unique up to an arbitrary additive long-period vector. It should be mentioned that, in the present case, the W.'s

98 AHMED kEY KAMEL

are easily obtainable through solutions of simple linear partial differential equations of first order.

Once these W.'s are obtained, one can construct the mappings (4) and (7) by using Equations (30) and (32); one can also construct (6a) from (6b), or vice versa, by using (26).

Note that a by-product of the present method is the construction of (4) from (7), or vice versa. In this case, Equation (30) will lead to W. which, then, is employed to compute x, from (32). To construct (6b) from (4) and (6b), one may use (29) to define IV. and then use (26) to obtain F..

An alternative form of (34) can be obtained if one eliminates x. between (34) and (29). This can be shown to lead to

n--1

"~foWn=fn-g (n)'}- Z [C7 - 1 ' n - 1 , 1Eifn_i- Cj gj,,_j]; n/> 1, (37) j=l where j

! g j , i Z ~ j - 1 T, , = -- I'~ra-- 1Lrag j -m, i

m = l

C CjF= ~ .Wj \ a x /

and t go, i = g ( 0 .

For an intrinsic proof of this formula, see Henrard (1970). The above algorithm can be visualized from the triangle of Figure 3, with Fo, ~ = 9("),

F,= f,, and 11--1

F,(l_) 1 = F, + I '"- i t ' ~" = "- 1f- - (38) ""J-- l~jXn--J Z C j j ,n - - j j = l j = 0

analogous to (18) and (22). Note that, if 9(i) =0, then 9), i=0 for all j , and the corre-

sponding column drops out from the computations. [ ! ] I X ] It is interesting to note that, in Equation (37), if y is replaced by , x by ,

Fo

F,

F2

(0) _ _(0) F , 0 " FO,O = I"

i

F; eL)--- F; m 0,1 =F

'~ FII,I . . . . 0 ,2 =

-,,, F =r- 2 . . . . F2, , . . . . F,, 2 . . . . o,3=

Fig. 3. Triangle for the visualization o f the algorithm of Equat ion (37).

P E R T U R B A T I O N M E T H O D I N T H E T H E O R Y O F N O N L I N E A R O S C I L L A T I O N S 99

and W by - and if 9 can be generated f rom a Hamiltonian

- - K <n) (y, Y, t) ; K ( y , Y, t; e) = n!

n = 0

that is, if

then f can also be generated from a Hamiltonian

H(x, x , t; ~) = ~ / - r . (x , x , t) ; n = 0

that is,

f ~ - - ~

so that (37) reduces to the scalar form (Kamel, 1969):

(39a)

(39b)

where

n - 1

DW. Fin _ K(.) + ~ i_r._lr, ,~r . _ j -- L~j._l--s...._j -- Cj K j . . _ j ] , (40) Dt j=l

_

z:~v = ~ \ ax / \ o x / \ o~ / OW. OWn - - - 12'.I-lo Dt Ot

J KJ l Z I " J - I T" K"- , ~ - - ' ~ J m - - l Z ' a m J X j - m , i

m = l

Ko, i = K{O,

which again has the same form as Equation (37). Also, it can be stated that the Hamiltonian case is an important special case of the present theory in which the analysis of the multi-element vector is reduced to the analysis of a single scalar function, the Hamiltonian.

4. Examples

Consider the nonlinear differential equation

oo

q + q = ~ h (")(q, q).

n = l

(41)

100 AHMED ALY KAMEL

Let 0 = P, (42)

then, Equation (41) can be reduced to the form

(t = p oo

-ae" (43 t /~ = _ q § ~ . h(/1)(q, P ) -

/1=1

In what follows, it will be shown how the obtained formulas can be applied, in two ways, to obtain approximate solutions for (43).

For the first approach, a transformation from the rectangular coordinates (q, p) to the polar coordinates (A, 4) is considered, i.e.,

q = A sin q5 (44a) p = A cos q~. (44b)

Under such transformation, Equation (43) takes the new form

oo

A = ~ h (/11 (AS1, AC1) C 1

/1=1

1 ~ e/1 49 = 1 -- ~4 ~. h(") (ASt, ACt) S t , (45)

n = i

where S/1denotes sinn~) and C/1denotes cosnrk. Define y= [~], then Equations (1) and (45) yield

g(~ = [01]

9(/11(Y) = h(")(ASt, ACt) S a [ ' n >~ 1. (46) J

Now, it is desired to transform y into a new vector x = [~1 so that the resulting

system, Equation (5), contains only the 'secular' terms. For this purpose, we may use Equation (37). For O (~ (=f0) of the form given in Equation (46), this algorithm can be written as

~ /1--1 n--1 t c?~ -- f /1(x)-9( /1)(x)+j:a • [ C ~ - I L S f " - j - C j Oj,/1-j]; n>~l (47)

and W, is defined successively to eliminate the short-period terms fromfn. Once these W/1's are computed, one can construct the mapping y ~ x as well as its inverse x ~ y , by using Equations (4), (7), (30), and (32). Also, one can construct any function of y in terms of x, e.g., q and p of (44).

PERTURBATION METHOD IN TH~ THEORY OF NONLINEAR OSCILLATIONS 101

As an application of this procedure, consider Van der Pol's equation

q + q = e(1 - q2) Cl. (48)

In this case,

h~ p) = (1 - qZ)p (49a) h (n) (q, p) = 0, n >~ 2 (49b)

and, thus,

g(~ = [01] (50a)

WA /" A2\ A A 3 -1

L s:- -s4 _J g(")=O; n~>2. (50c)

Up to second order, (37) and (47) yield

fo = g(~ = [01] (51a)

OW1/(~ = f l (x) -- g (1) (x) (51b)

OWz/O~J = ;/'2 (x) + E 1 [ f t (x) + gO)(x)]. (51c)

Choosingfl(x ) as the secular terms of 9(1)(x) obtains

f l (x) = 1 - -~- , (52a)

0

then, Equations (50b) and (51b) yield

4 2 32 4 Wl = 1 / ~{2\ .~2 , (52b)

1-T)e2- g where Sn denotes sinn~ and C, denotes cosn(~. Computing L'~(fl +g(1)), choosing f2 to eliminate its secular part, and using (51c) to obtain Wz, result in

f2 = 2 4 0 ] (53a) _ 1 ~ 8 + 3 X ~ _ � 8 8 - X 5 X 3 X 3 --1

384 e6 - ~ C 4 - ~ ( 7 - 3~2) C2

W2 = .g~4 _ z ~ 2 / / x~2,~ ~ 2 / / z~2,~ �9 (53b)

---- 3~ S6 -"}- 6-4 t l - - T)~'-~4--B2tl "It- 2 - ) '2

102 AnMED ALY KAMEL

Equations (5), (51a), (52a), and (53a) now yield

:? = 2- \ - 4 ] (54)

1 + 52( - k + _ A 1 6 A ,

To express q of Equation (44a) in terms of A and/~, one uses (26) with F (~ =AS1, F(")=0 (n~> 1), denoting F, by q,, to obtain

82

q = qo + eql + ~ q2, (55a)

where qo = AS1 (55b)

q ~ = - L ~ q o = ~ - 1-- C t - ~ - C 3 (55c)

q2 = - L2qo - L lq l A 3

=i6X (1 - / [ 2 + 64-- , ' s 3"4~ ~ + 16 (1 -- ~ z ~ 2) ~3 -- ~-~36 -~5S5 �9 (55d)

In view of (44a) and (54), a periodic solution is obtained when

.~=

where r is a condition on q.

q =

2 (56a)

1 - i 6 t + O ( e ) , (56b)

free function of the small parameter e to be chosen to satisfy some As a result, Equations (55) and (56) lead to

/~ _ ,~2

2SI - ~ C3 +3-2 (391 + 3~qa _ ~r (57)

To choose ~ (e) to satisfy 0(0)= 0, we use the inverse transformation. In view of (44),

~b (0)= re/2. To find ~(0) in terms of qS(0), we use Equation (5)with y = [~] , x = [ ~

and use (30) for the computations of y(")(y). This leads to

rc 3 4, . . . . (58)

2 8

The corresponding q then takes the form

q = ( 2 - 3~s 2) cos0 + �88 sin0 + ~6 e2 c o s 3 0 - -

where 0 = 1 - t.

8 sin30 - ~6e 2 cos50,

4 (59)

PERTURBATION METItOD IN THE THEORY OF NONLINEAR OSCILLATIONS 103

By comparing the above equation with Deprit et al. (1967), one finds that the coefficient of cos0 is [2-(~2/8)-1 rather than [2-(3/32)e2]. The difference is the result of some feedback from the third-order analysis because the periodicity condition will yield [A = 2 - (~z/32)3 rather than 2.

In the second approach, for obtaining approximate solutions of (43), we will show how the operations involved in the perturbation treatment can be carried out directly in terms of the cartesian variables q and p. This can be achieved if we let

and

__ [Asinr162 Y = [A cos

r~]l = [ A sin X = l p j L~cos~J"

Comparing Equations (43) and (60a) with (1) yields

and

Now, let

(60a)

(60b)

[W,1] (62) W.= W.~"

Using this definition and realizing that q = - ap/O~ and t5 = dq/3~, Equation (37) leads to

= (63a) OWn2

L ~

where [F.1] . - 1

= - [ C j - I E J . - j + c j g~,._j.]. (63b) F.2 L g<.)+ y, . - 1 , ~.-1 , j = l

Equation (63a) leads to

O~ 2 dr Wnl = ~ ~- F.2 (64a)

OW.1 W,2 - F,~. (64b)

aft

Referring to Equations (5) and (60b), f . must be constructed as

[a.(X) sin ~ + b.(A-) cos/~] f" = [a. (X) cos/~ - b. (A) sin ~ J '

(65)

[ 0)] = ; n/> 1. (61b)

h (") (q, P

104 AHM~D ALY KAMEL

and, because fo (x) = g (0) (x), oo

= ~.y a. (A) (66a)

n = l oo

1 ~ e " = 1 + ~ ~ b, (A-). (66b)

n = l

In view of Equations (63b), (64), and (65), a,(A) and b.(A) must be chosen to eliminate the coefficients of sin~; and cos~ from 8F,1/O~+F,2 of Equation (64a); otherwise, W,1 will be unbounded as ~ (or t ) ~ 0% and the perturbation expansion will not be valid for all ~ (or t). Note that the operations involved in obtaining F. 1 and F,z of (63b) are carried out in terms of q and p. For the computations of W,t and W,2, q and/5 are substituted in terms of A and q5. After these computations have been performed, return from .,t and ~ to 4 and b to obtain W,x (q, p) and Wn2 (q, Iff) which are to be used in the operations of the next step in obtaining (F.+~)~ and (F,+~)z. The cycle is repeated until the desired order of perturbation is reached; then,

Equations (66) are employed to obtain ~ and ~. The relation between the vector y of Equation (60a) and the vector x of Equation (60b) can be obtained through Equations (4) and (29).

The transformation of the ordinary differential Equation (41) to the partial differ- ential Equations (61) is similar, in spirit, to the methods of Linstedt-Poincar6 (1957), Kevorkian (1966), and Nayfeh (1965). Here, however, all the time scales are contained in the angle variable q5. Indeed, if

d ~ e " 8

d-t = 7 ! & (67)

n = 0

is substituted in Equations (66), then

- - = 0, - - = 1 (68a) ~to 0to

a.(A), at. x b . ( d ) ; n ~ > l , (68b) St.

which should lead, up to nth order, to

Cl = ~]o(to, tx, ..., t ,) ----- A(tl , t 2 .... , t , ) s in~ (t0, t 1 ..... t,) (69a) oo

q = q (~; e) = ~ q.( to , tl . . . . , t . ) . (69b)

n = 0

Equations (67) and (69b) are the basic formulas in the method of multiple time scales (Nayfeh, 1965; Sandri, 1963).

PERTURBATION METHOD IN ~ { E THEORY OF NONLINEAR OSCILLATIONS 105

As an applicat ion of Equat ions (60) to (66), consider the linear differential equat ion

+ q = - 2e{}, (70)

whose exact solution is

q = .go e-`t sin [(X - e2) */2 t + / / ;o ] , (71)

where X o and ~;o are arbi t rary constants. By using Equat ions (41), (60) to (66), and (70), it can be verified that the sixth-order analysis leads to

g(2' = [~], f2= [-~_], Wz=~_] (72c)

g(')= [a]' f3= [a]' W,= [~] (72d)

g(4)= [a], f4= [-~_], W4= [:p] (72e)

g(5~ = [:] , j.s = [:] , Ws = [~4 ] (72f)

g(6)= [~], f6 = [--:~1, 1476= [1200/~]. (72g)

Note that the intermediate functions

[02~ ] f f l , 1 =

[Oo] g 4 , 1 ----

g), i = 0, i~> 2, j >~ 1. Also, it was found that

[~] [ 06~] 92,1 = , 9 3 , 1 ~ ,

, [ o ] gs, t = - 90 "

Thus, in view of Equat ions (65) and (66)

~2

~=1 2

g4 /~6

8 16

The solution o f (74) can be writ ten as

/1 ~--_ ,~0 e-et

~ = 1 t+~o . 2 8 -16

(73)

(74a)

(74b)

(75a)

(75b)

106 AHMED ALY KAMrL

Using Equat ions (4), (29), and (72), with y = [p q] and x = [pq_ -] obtains

q = q (76a)

p = - e~ + 1 2 8 1-6 p" (76b)

Referr ing to Equat ions (60b) and (75), Equat ion (76a) leads to

q = . ~ o e-~*sin 1 2 8 ]-6 t + ~ 0 , (77)

which, in the frequency of oscillation, agrees with the exact solution to within 0(e8).

Acknowledgments

I would like to express my grat i tude to Dr. Andr6 Depri t , Prof. John V. Breakwell, and Dr. Sayed Hassan for their contr ibut ions to this paper .

This research was suppor ted by the Nat iona l Aeronaut ics and Space Adminis t ra t ion, under Cont rac t No. N s G 133-61.

References

Bogoliubov, N. N. and Mitropolsky, Y. A.: 1961, Asymptotic Methods in the Theory o f Nonlinear Oscillations, Gordon and Breach, New York.

Deprit, A.: 1969, Celestial Mech. 1, 12-30. Depfit, A. and Rom, A. R. M.: 1967, Z. angew. Math. Physik 18, 736-747. Henrard, J. : 1970, 'On a Perturbation Theory Using Lie Transforms', Celestial Mech. 3, 107-120. Kamel, A. A.: 1969, Celestial Mech. 1, 190-199. Kevorkian, J. : 1966, 'Space Mathematics', Am. Math. Soc., 206-275. Morrison, J. A. : 1966, Progress in Aeronautics, Vol. 17, Academic Press, New York, pp. 117-138. Musen, P.: 1965, J. Astronaut. Sci. 12, 129-134. Nayfeh, A. H.: 1965, Math. Phys. 44, 364-374. Poincar6, H. : 1957, Les m~thodes nouvelles de la mdcanique c~leste, Vol. III, Dover Publications,

New York. Sandri, G.: 1963, Ann. Phys. 24, 332-379.