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On periodic orbits of nonlinear dynamical systems with many degrees of freedom
Physica A 181 (1992) 385-395
PHYSICA
North-Holland
On periodic orbits of nonlinear dynamical systems with many degrees of freedom* G a m a l M. M a h m o u d Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt Received 17 December 1990 Revised manuscript received 19 August 1991
The purpose of this article is to extend our investigation to periodic orbits of nonlinear dynamical systems with many degrees of freedom. The method of generalized averaging and the indicatrix method are developed. An example is studied to illustrate our techniques. The analytical expressions are in good agreement with numerical results computed by the indicatrix method for this example.
1. Introduction
In recent years, there has systems of multiple degrees have been known to play a [4-7]. The aim here is to study systems of the form
been an increased interest in nonlinear dynamical of freedom [1-3, 6-9]. Periodic solutions (orbits) very important role in the study of these systems periodic orbits of a class of nonlinear dynamical
2
JCi + OOiXi + Agi(x, )C) Pi(~Qit) : 0 ,
(1)
i = 1, 2 . . . . , n, to/> 0, where x --- (xl, x 2. . . . , x , ) , A ~- (xl, x2, • - • , x , ) , gi is a nonlinear function of x and A, Pi is a periodic function of t, h is a small parameter, and dots represent as usual differentiation with respect to t. Eq. (1) arises in numerous physical applications (e.g. the displacements of colliding particle beams in high energy accelerators, vibrations of shells, parametically forced oscillations, and pendulum with vibrating length) [6-11]. Certain limiting cases of eq. (1) are well known to mathematicians, physicists and engineers and have been treated at length in the literature (see for example refs. [3, 6-8, 11]). • The manuscript was presented at the Workshop on Nonlinear Problems in Future Particle Accelerators, Capri, Italy, April 19-25, 1990. 0378-4371/92/$05.00 © 1992- Elsevier Science Publishers B.V. All rights reserved
386
G.M. Mahmoud / Periodic orbits ~f nonlinear systems
Section 2 presents the development of the generalized averaging method [6]. Using this technique we derive approximate expressions for the amplitude and phase of the periodic components x~(t), X2(t ) ..... X,,(t) of the solutions of eq.
(a). Section 3 contains the extension of the indicatrix method [6]. This numerical approach is used to study the existence of the periodic orbits of eq. (1). Finally, in section 4, the case n - 3 is studied to illustrate our techniques. G o o d agreement is found between theoretical predictions and numerical results computed by the method of section 3. Currently, we are in the process of studying the stability properties of the periodic solutions of eq. ( 1 ), n ~> 2. This is considerably more complicated than the case n = 1 [6], and will appear in future publications.
2. Theoretical development
In this section we develop a generalized averaging method [6] to study periodic solutions of systems governed by the equation of motion 5~i+w]xi+Agi(x,.i)Pi(~2it)=(I,
i-1,2
.....
n.
2)
The method starts from the so-called generating solutions x, - a i cos(wit + (hi) ,
-fi = - a i w i sin(wit + qS,) ,
i = 1. . . . .
n .
3)
which satisfy eq. (2) to zero order ( a - 0 ) . In order to solve cq. (2), it is assumed that the constants of integration a, and (hi depend on time, so that in (3) a , - ~ a i ( t ), chi--~&i(t). Then inserting (3) into (2) we get a system of equations which we solve for 6i(t), &i(t) to obtain (for more details see ref. [7])
d a(t) dS( )= A(R(v) (~(t) S(v) )
(4a)
where v = ( . ( t ) . ,~(t). cS(t), c~(1), t ) ,
.(t) = [a,(t). a~(t) ..... R-
a,,(t)l'.
4~(t) - [,Sl(t), ,f_~(t)..... ,b,,(t)l',
gl(u) l't(.(2~t) sin(w~t + d~ ) . . . . .
I g,,(v) P,,(~2,,t)sin(w,t+ &,,) w,,
G.M. Mahmoud / Periodic orbits of nonlinear systems =
38;
g,(v) P , ( a , t ) cos(w,t + ~b,)... ,
1 ]' anw----~,g , ( v ) P,(J~nt) cos(w.t + ~bn)
(4bj
(with [ . . . ] ' denoting transpose). Eq. (4a) with (4b), can now be solved by an extension of the technique ot generalized averaging as follows: Casting the original equation (4) in the form dy = dt A f ( y , t),
y = [a(t),
(5a)
~b(t)] t ,
we set
f ( y , t) = ?(u) + f(u, t) , u--= [a1(t,) , a2(t2) . . . . .
an(t,),
= [A,, A 2 . . . . , A . , q), . . . .
4a,(t,)
....
,
~b,,(t.)l'
(Sb)
, ~.]t,
where the vector functions f(u) and f(u, t) contain the terms of smallest frequency (plus constants) and the terms of higher frequency respectively. We associate with eq. (4) the reduced system du dt
AF(u)
(6a)
y = u + AG(u, t).
(6b)
where
where a bar or a tilde over a function denotes its averaged or oscillatory part in t, respectively. Expand F(u) and G(u, t) in powers of A as follows:
&u) = P,(u) + ~F2(~) + ' " ,
(7a)
6(u, t) = ~,(u, t) + ~62(u, t) + . - -
(7b)
and substitute from eq. (7) into eq. (6), using eq. (5) to obtain, upon equating like powers of A,
388
G.M. Mahmoud / Periodic orbits o f nonlinear systems
/~l(~) : f(u' t) '
~'2([g) : OH ('))~ f j~(/~, t) tit, . • • ,
(~a)
G,(u, t)= f f(u, t)dt .....
(Sb)
F r o m eq. (6b), the a p p r o x i m a t e formulas for the amplitude and phase of the periodic c o m p o n e n t s xt . . . . . x,,, of the solutions of eq. (1) are written as a i ( t ) = A i + A G , , , , ( H , l) + ~ ( A 2 )
,
i = 1,2 . . . . . 4(0
= q'~ + a G ~ , ( u ,
n,
(9)
l) + ~ ( a ~ ) ,
with d ~ ( , , , t) - - [ d , , , .
d~,,,, . . - ,
d~,, ,, , d~,,,, , . . . .
d
a+,,]
~,
k=
1 ........ 9
n
For a m o r e detailed account of generalized averaging, the reader is referred to refs. [6, 7].
3. Numerical development In this section, we present the extension of the indicatrix method [6] in order to be applied to nonlinear systems having the form (l/l) Taking in eq. (10) A¢:0, the sets of points [(xE(t~.t~+ T~), 2 1 ( t l , t ~ + T~)) . . . . . (x,,(t,,, t,, + T,,), 5:,,(t,, t,, + T,,))] with periods "El 2 ~ / % . . . . . T,, =2rr/w,,, form closed curves parameterized by ti¢[0,27r/.(li] in the spaces (xt, ~i'~) . . . . . (x,,, 2,,) respectively, which may or may not pass by the specified initial conditions. These closed curves are called the indicatrices. By varying slightly o~, ~o2~, . . . , w,,,-" and A, we obtain the periodic components x 1, x . . . . . . x,, when the indicatrices in the space (x~, 2~) . . . . . (x,,, ~,,) are observed to pass by the prescribed initial conditions at some values of t I , Q .....
t,~.
In this way we obtained the numerical values of w~. w ~ w~, A at which periodic c o m p o n e n t s x~, x 2 . . . . . x,, of the solutions of eq. (10) with periods T l, T_,. . . . . T,,, respectively pass by the specified initial conditions. For more details the reader is again referred to refs. [6, 7]. However, to
G.M. Mahmoud / Periodic orbits of nonlinear systems
389
m a k e the presentation here more self-contained we have added an appendix where the indicatrix m e t h o d is applied to a single nonlinear O D E order of the form (10).
. An example
To illustrate the techniques of sections 2, 3, we apply them here to the following system: 2 Xl + ~ l X l + A(XlX2 + X ~ ) C O S t = O,
(11)
A(x~ + x2x3)cos2t= 0
(12)
2 + £2 + ~2X2
2 + A(X~ + £3 + W3X3
x ~ ) c o s 3 t = O.
(13)
The main methods of this paper are not affected by the precise form of gi(x, £c) and Pi(~2d) (cf. eq. (1)). Performing the calculations in (4), one obtains the following for eqs. (11)-(13): d (a(t)~=A(R(v)]
dt
~(t)/
S(v)/'
(14a)
where
v = (a(t), a(t), 4~(t), 4~(t), t), &(t) = [~b,(t), ~b2(t), (53(t)1',
a(t) = [ai(t), a2(t), a3(t)]' , R = [R,, R2, Rs]t ,
S = [Si, 82, S3] t ,
R~, S~ (i = 1, 2, 3) are sums of trigonometric functions with arguments
thj.k ~- (% + k)t + ~ j ,
j, k = 1, 2, 3 ,
(14b)
1 l [ala 2 sin(2$,.,)+- ~tt2.0) q- a,2 sin ~tl, 1 q- a 21 sin(2O~ .o + O,,,)] , R 1= ~w
R2= ~
1
2
-
{a,[2 sin 02.2 + sin($2.2 +- 2G..)]
+ a2a3[sin ~.2 + sin(2$2., +- q6.0)]} , _ [(2a~ + a~) sin 03,3 + a3~ sin 3~3., + a 2 sin(q6.1 + 262.~) ] , R3 = _8wsl
(14c)
390
G.M. Mahmoud / Periodic orbits of nonlinear systems
S1= ~
1
[2a 2 cos 62., + a2 c ° s ( 2 G . , -+ 4-'>t ) + 3a, cos t/,,.,
+ a, eos(2&,.o + 4h.i)] . S~
1 {2a2~cos 4'2.2 + ael c ° s ( 2 G . . -+ qh +) 80).a-, "- + + a2a3[ 2 c ° s 4'3.2 + cos(6s..-+ 26>, )]} . 1
- [(3a~ + 2a~) cos ~/q.~ + a~ cos 3qq., + a~ cos(tO3. , + 2 G . , ) I $ 3 - 8w3a 3 From eq. (14) small frequency components occur at 0)t ~ - t g , l ,
w. ~ 1 , 2 .
w 3.~ 1 , 2 , 3.,
~2
¢03~3
0)3+20)-, . . ~ 9 3.
2wt-+0).~1,2. (15)
Choosing i
0)1~3
,
0) 2
and
(16)
(other values of o)i can be treated similarly) the reduced system (6a) becomes du
dt-
AFI(/~)
-~- O ' ( a 2 )
17a)
"
where u
[ A , , A 2, A , . @,, @2. q0~]t,
(17b)
F , ( u ) = [t~,,, /~,2. F,~, Ft4. FI~, />,~,]t .
PE, - A21 sin(2qtt.,, + qll t) ~0)1
]~1 ~ __
'
"
El ~ -
with
= --A~ sin qt+,, 4o)~
- -1 (2A~ + A2~) sin qq.3. 80) 3
A, / ~ 4 - 8w-, cos(2q-',., + q-", t) •
A~ .
F,s-
4w.A 2
COS1/t~,,
8 A 3 w ~ (3A~ + 2A~) cos q q , .
q~.k
(0)] - k ) t + qO] .
(17c)
G.M. Mahmoud / Periodic orbits of nonlinear systems
391
T o satisfy the conditions dAi/dt = 0, i = 1, 2, 3, we m a k e the corresponding contributions of ,~, (u) vanish by setting 2 % , o + %,, = (360, - 1)t + 3qOl = ll"ff , % , 2 = (602 -- 2)t + 42 = 12"rr,
l 1 , 12, l 3 = 0, 1,2 . . . . .
(18)
~ , 3 = ( % - 3)t + q~3 = / 3 "rr , F r o m eq. (18) one obtains
601
-
-
1 3
d~, dt
d~ 2 '
% = 2
dt
d~O3 '
% = 3
dt
'
(19)
F r o m eqs. (17), (19), we get the following expressions for a)~ to first o r d e r in a: w 21 = -19- ¼ A A , ( - 1 ) " + G(A2),
(20a)
2 w 2 =4-
1 A~ ~ A ~ ( - 1 ) ' 2 + •(A2),
(20b)
2 603 = 9 -
~
1
A(-1)z3(3A~ + 2A22) + G(A2).
(20c)
For different values of l~, 12, l 3 = 0, 1, we pick different branches in (o92~, A), (~o~, a), (o)~, A) planes, along which periodic solutions of the system ( 1 1 ) - ( 1 3 ) can be found, with initial conditions Xl(tl)= A,, X2(t2) = A 2 , x3(/3) = A 3 , and "~l (tl) = X2(t2) = "1C3(t3) = 0. These solutions have periodic x,(t) c o m p o n e n t with period Tj = 6rr, periodic X2(t ) with period T 2 = ~ and periodic c o m p o n e n t X3(t ) with period T 3 = 3ft. Their overall period, therefore, is T - - T~ = 6 T , = 9 T 3 = 6-rr. T o verify the validity of the analytical results we apply the numerical technique of section 3 (see also the appendix). By varying slightly o92,, w~, o9~ and A, we obtain the periodic components x I, x 2, x 3 when the indicatrices are observed to pass by our initial conditions (e.g. A I = A , = A.~ = 0.5) for different values of tj, t2, t3, see fig. 1. As expected from the analytical approach we obtained 6rr-periodic solutions with Xl(t ), X2(/), x3(t ) c o m p o n e n t s oscillating with periods 6~r, av, 3~ (w~~ ~ ,j, ~ w~_ ~ 4, w~ ~ 9) respectively. Finally, expressions (20) are c o m p a r e d with the results of this technique and very good a g r e e m e n t is found for A ~< 0.1, see table I.
392
G.M.
Mahmoud
/ Periodic orbits of nonlinear systems
{; {hOf i
00nl
k-
0 0003 j,
~o +
Q 0007
Z
0 0006
I
o ooo3
--
.x
EJ
(
/
/
\
>
i
\
\
#
\ "\\
000n!
/
+ 0 000,
/-
/ \
-x
-\
/ /
0 0002 00b91 //
-o
___i__
0091
L
~,98
~
i
0499
,
_
0 501
Xl(ll~tl*
~"
__L
0 S
,__J 0 502
~
0 49986
0.50002
049094
X2(12,12
0 I:}(~06
r~
/
/
0
6 1[}
0.5001
+ "iT)
0ooo2t /
ca
\,
)
£ ' o ooo ?
/
@onoI
\
x ,,
0 000 l, -00006]
A •x
00002
,
6 49992
0 /.9998
0.50004
X 3 (t 3,t3~
o 4%
213 T I t
o
~99
o.6
o%,
0%
X I { t I , t I ~6 /11
0 -O.OOO~ -0.0002
i
+ -
0001} 3
,-:, -o ooo,
-00004 .p<
- 00006 -O000(,_L
0 L9986
-O 0002
/ I__~__L~
-0.0006
,
0,L9994 Xz{t2~t2
050002 +]Tl
0 9001
049992
0499-38 X 3 (t3~t3
0 L0~34 +2/3TF}
Fig. 1. Indicatrix plots of eq. (10) for A = 0.001, showing x , ( t , ) = x ~ ( Q ) - x~(t~) - 0.5, t~ C [IL 2w I, t~ C [0, wl t~ • [0, ~w], with ki(ti) = 0, and (a) w ~ = 0 . 1 1 0 9 8 6 : ( 0 . 5 , 0 ) at t , - 0 , (b) w ~ - 3 . 9 9 9 6 5 : ( 0 . 5 , 0 ) at t , = 0 , (c) w ~ - 8 . 9 9 9 4 4 7 : ( 0 . 5 , 0 ) at t , = l ) , (d) w ~ = 0 . 1 1 1 2 3 6 : ( 0 . 5 , 0 ) at t , - v . (el w z~- 4.00036: (0.5, 0) at t ~ - ~w, and (f) ~o~- 9.000693:(0.5, 0) al t ~ - 1.09956.
393
G.M. Mahmoud / Periodic orbits of nonlinear systems
Table I Results for l I = 12 = 13 = 1 (t~ = 7, t 2 = ½~, t3 = 1.09956) and l I = 12= l 3 = 0 (t 1 = t 2 = t 3 = 0). A
Analytical results (eq. (20)) 2
0.001 0.005 0.010 0.050 0.100
0.111236 0,110986 0.111736 0.110486 (I.112361 0.109861 0.117361 0.104861 0.123610 0.098610
Numerical results
2
4.00025 3.99975 4.00125 3.99875 4.00250 3.99750 4.01250 3.98750 4.02500 3.97500
2
9
9.000625 8.999375 9.003125 8.996875 9.006250 8.993750 9.031250 8.968750 9.062500 8.937500
0.111236 0.110986 0.111736 0.110486 0.112360 0.109851 0.11740(I 0.104662 0.123600 0.097700
9
4.00036 3.99965 4.00183 3.99825 4.00370 3.99655 4.01790 3.983395 4.03520 3.96850
2
9.000693 8.999447 9.003250 8.996895 9.00635 8.99377 9.03050 8.968573 9.06247 8.93702
Appendix In this a p p e n d i x we p r e s e n t an e x a m p l e for t h e case n = 1 (cf. eq. (1)) to i l l u s t r a t e t h e i n d i c a t r i x m e t h o d (for m o r e e x a m p l e s for n = 1 , 2 , see refs. [6, 7]). W e will a p p l y h e r e t h e indicatrix m e t h o d for finding p e r i o d i c s o l u t i o n s o f a d i f f e r e n t i a l e q u a t i o n of t h e f o r m 5? + co2x + A cos [t(x + x2)] = 0 ,
A small,
(A.1)
w h e r e t h e u n p e r t u r b e d f r e q u e n c y w t a k e s values close to 1 / k , k = 1, 2 , . . . . By using the i n d i c a t r i x m e t h o d [6], we will d e t e r m i n e t h e v a l u e s o f t h e p a r a m e t e r s w 2, A at which p e r i o d i c o r b i t s of ( A . 1 ) with p e r i o d 2"rrk pass by a s p e c i f i e d initial c o n d i t i o n , e.g. ( x ( t o ) , x(t0)) = (0.5, 0). C l e a r l y , for A = 0, all p o i n t s ( x ( t o ) , £c(to)) can s e r v e as initial c o n d i t i o n s for 2"rrk p e r i o d i c solutions. H o w e v e r , for A ~ 0 , the set of p o i n t s ( x ( t o, t o + T ) , k(t~, t 0 + T ) ) with T = 2ark lie on a c l o s e d curve p a r a m e t r i z e d by G E [0, 2-rr] w h i c h m a y o r m a y n o t pass by o u r specified initial c o n d i t i o n (0.5, 0). T h i s c l o s e d curve is c a l l e d the i n d i c a t r i x a n d is p l o t t e d h e r e in fig. 2a for A = 0.1 a n d w ~ 1 / k = ~1 for which it clearly d o e s n o t pass t h r o u g h (0.5, 0). By v a r y i n g slightly w 2 for a given A, we o b t a i n the d e s i r e d p e r i o d i c o r b i t s w h e n t h e i n d i c a t r i x is o b s e r v e d to pass by o u r initial c o n d i t i o n at s o m e v a l u e s of t o. T h i s can h a p p e n , of c o u r s e , for m o r e t h a n o n e v a l u e of t o ~ [0, 2-rr], see figs. 2b, 2c w h e r e t o = 0 a n d 7r r e s p e c t i v e l y for the case k = ~ ( o t h e r v a l u e s of k can be similarly studied). W e can s t u d y in this w a y t h e e x i s t e n c e of p e r i o d i c o r b i t s with T = 2k,u of eq. (A.1).
394
G.M. Mahmoud / Periodic orbits of nonlinear system,~
The values of w e with a up to a ~<0.1) at which eq. (A.I) has 6w-periodic orbits passing through the initial condition (0.5,0), are obtained by the indicatrix method as follows shown in table A.I.
.1
A
÷ 4#°
"o
I\.
4J
(a)
j/
J
-.1
.0
x(to,to+6n )
.23 /./
1.0
J
f
\
A
M
J
40
(b)
-.23 .0
X(to,to+6n)
1.0
.11 A ÷
,> • Q " - - ~
f.-J'//)
(c)
-.11 •4
X(to,to+6.)
.7
Fig. 2. l n d i c a t r i x plots of eq. ( A . I ) for a = 0 . 1 and t,,~[(),2-rr~ for (a) 0~:=0.11097, ~o- =0.09047, (where (0.5,0) at t~,=0), (c) ~o-~ 0.11867 (where (0.5.0) at t,,=w).
(b)
G.M. Mahmoud / Periodic orbits o f nonlinear systems
395
Table A.1 A
~o2
to
0.001
0.1113 0.1108
~r 0
0.01
0.11235 0.10973
~r 0
0.05
0.11573 0.10275
7r 0
0.1
0.11867 0.09047
-rr 0
References [1] J. Kevorkian, Perturbation techniques for oscillatory systems with slowly varying coefficients, SlAM Review vol. 29 (1987). [2] C.-C. Chi and R.M. Rosenberg, On damped nonlinear dynamics systems with many degrees of freedom, Int. J. Nonlin. Mech. 20 (1985) 371-384. [3] D.E. Gilsinn, A High Order Generalized Method of Averaging, SIAM J. Appl. Math. 42 No. 1 (1982). [4] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer Berlin, 1983). [5] M. Lieberman and A. Lichtenberg, Regular and Stochastic Motion (Springer Berlin, 1983). [6] G.M. Mahmoud and T. Bountis, Synchronized periodic solutions of a class of periodically driven nonlinear oscillators, J. Appl. Mech. 55 (1988) 721-727. [7] T. Bountis and G.M. Mahmoud, Synchronized periodic orbits in beam-beam interaction models of one and two spatial dimensions, Particle Accelerators 22 (1987) 129-147. [8] H. Sprysl, Internal resonance of non-linear autonomous vibrating systems with two degrees of freedom, J. Sound Vib. 112 (1) (1987) 63-67. [9] P.R. Sethna, Coupling in certain classes of weakly nonlinear vibrating systems, in: Int. Symp. on Nonlinear Differential Equations and Nonlinear Mechanics, J. Lasalle and S. Lefschetz, eds. (Academic Press, New York, 1963). [10] R. Cohen and I. Porat, Coupled torsional and transverse vibration of unbalanced rotor, J. Appl. Mech. 52 (1985) 701-705. [111 A.N. Nayfeh and D.T. Mook, Nonlinear Oscillations (Wiley, New York, 1979).
PHYSICA
North-Holland
On periodic orbits of nonlinear dynamical systems with many degrees of freedom* G a m a l M. M a h m o u d Department of Mathematics, Faculty of Science, Assiut University, Assiut, Egypt Received 17 December 1990 Revised manuscript received 19 August 1991
The purpose of this article is to extend our investigation to periodic orbits of nonlinear dynamical systems with many degrees of freedom. The method of generalized averaging and the indicatrix method are developed. An example is studied to illustrate our techniques. The analytical expressions are in good agreement with numerical results computed by the indicatrix method for this example.
1. Introduction
In recent years, there has systems of multiple degrees have been known to play a [4-7]. The aim here is to study systems of the form
been an increased interest in nonlinear dynamical of freedom [1-3, 6-9]. Periodic solutions (orbits) very important role in the study of these systems periodic orbits of a class of nonlinear dynamical
2
JCi + OOiXi + Agi(x, )C) Pi(~Qit) : 0 ,
(1)
i = 1, 2 . . . . , n, to/> 0, where x --- (xl, x 2. . . . , x , ) , A ~- (xl, x2, • - • , x , ) , gi is a nonlinear function of x and A, Pi is a periodic function of t, h is a small parameter, and dots represent as usual differentiation with respect to t. Eq. (1) arises in numerous physical applications (e.g. the displacements of colliding particle beams in high energy accelerators, vibrations of shells, parametically forced oscillations, and pendulum with vibrating length) [6-11]. Certain limiting cases of eq. (1) are well known to mathematicians, physicists and engineers and have been treated at length in the literature (see for example refs. [3, 6-8, 11]). • The manuscript was presented at the Workshop on Nonlinear Problems in Future Particle Accelerators, Capri, Italy, April 19-25, 1990. 0378-4371/92/$05.00 © 1992- Elsevier Science Publishers B.V. All rights reserved
386
G.M. Mahmoud / Periodic orbits ~f nonlinear systems
Section 2 presents the development of the generalized averaging method [6]. Using this technique we derive approximate expressions for the amplitude and phase of the periodic components x~(t), X2(t ) ..... X,,(t) of the solutions of eq.
(a). Section 3 contains the extension of the indicatrix method [6]. This numerical approach is used to study the existence of the periodic orbits of eq. (1). Finally, in section 4, the case n - 3 is studied to illustrate our techniques. G o o d agreement is found between theoretical predictions and numerical results computed by the method of section 3. Currently, we are in the process of studying the stability properties of the periodic solutions of eq. ( 1 ), n ~> 2. This is considerably more complicated than the case n = 1 [6], and will appear in future publications.
2. Theoretical development
In this section we develop a generalized averaging method [6] to study periodic solutions of systems governed by the equation of motion 5~i+w]xi+Agi(x,.i)Pi(~2it)=(I,
i-1,2
.....
n.
2)
The method starts from the so-called generating solutions x, - a i cos(wit + (hi) ,
-fi = - a i w i sin(wit + qS,) ,
i = 1. . . . .
n .
3)
which satisfy eq. (2) to zero order ( a - 0 ) . In order to solve cq. (2), it is assumed that the constants of integration a, and (hi depend on time, so that in (3) a , - ~ a i ( t ), chi--~&i(t). Then inserting (3) into (2) we get a system of equations which we solve for 6i(t), &i(t) to obtain (for more details see ref. [7])
d a(t) dS( )= A(R(v) (~(t) S(v) )
(4a)
where v = ( . ( t ) . ,~(t). cS(t), c~(1), t ) ,
.(t) = [a,(t). a~(t) ..... R-
a,,(t)l'.
4~(t) - [,Sl(t), ,f_~(t)..... ,b,,(t)l',
gl(u) l't(.(2~t) sin(w~t + d~ ) . . . . .
I g,,(v) P,,(~2,,t)sin(w,t+ &,,) w,,
G.M. Mahmoud / Periodic orbits of nonlinear systems =
38;
g,(v) P , ( a , t ) cos(w,t + ~b,)... ,
1 ]' anw----~,g , ( v ) P,(J~nt) cos(w.t + ~bn)
(4bj
(with [ . . . ] ' denoting transpose). Eq. (4a) with (4b), can now be solved by an extension of the technique ot generalized averaging as follows: Casting the original equation (4) in the form dy = dt A f ( y , t),
y = [a(t),
(5a)
~b(t)] t ,
we set
f ( y , t) = ?(u) + f(u, t) , u--= [a1(t,) , a2(t2) . . . . .
an(t,),
= [A,, A 2 . . . . , A . , q), . . . .
4a,(t,)
....
,
~b,,(t.)l'
(Sb)
, ~.]t,
where the vector functions f(u) and f(u, t) contain the terms of smallest frequency (plus constants) and the terms of higher frequency respectively. We associate with eq. (4) the reduced system du dt
AF(u)
(6a)
y = u + AG(u, t).
(6b)
where
where a bar or a tilde over a function denotes its averaged or oscillatory part in t, respectively. Expand F(u) and G(u, t) in powers of A as follows:
&u) = P,(u) + ~F2(~) + ' " ,
(7a)
6(u, t) = ~,(u, t) + ~62(u, t) + . - -
(7b)
and substitute from eq. (7) into eq. (6), using eq. (5) to obtain, upon equating like powers of A,
388
G.M. Mahmoud / Periodic orbits o f nonlinear systems
/~l(~) : f(u' t) '
~'2([g) : OH ('))~ f j~(/~, t) tit, . • • ,
(~a)
G,(u, t)= f f(u, t)dt .....
(Sb)
F r o m eq. (6b), the a p p r o x i m a t e formulas for the amplitude and phase of the periodic c o m p o n e n t s xt . . . . . x,,, of the solutions of eq. (1) are written as a i ( t ) = A i + A G , , , , ( H , l) + ~ ( A 2 )
,
i = 1,2 . . . . . 4(0
= q'~ + a G ~ , ( u ,
n,
(9)
l) + ~ ( a ~ ) ,
with d ~ ( , , , t) - - [ d , , , .
d~,,,, . . - ,
d~,, ,, , d~,,,, , . . . .
d
a+,,]
~,
k=
1 ........ 9
n
For a m o r e detailed account of generalized averaging, the reader is referred to refs. [6, 7].
3. Numerical development In this section, we present the extension of the indicatrix method [6] in order to be applied to nonlinear systems having the form (l/l) Taking in eq. (10) A¢:0, the sets of points [(xE(t~.t~+ T~), 2 1 ( t l , t ~ + T~)) . . . . . (x,,(t,,, t,, + T,,), 5:,,(t,, t,, + T,,))] with periods "El 2 ~ / % . . . . . T,, =2rr/w,,, form closed curves parameterized by ti¢[0,27r/.(li] in the spaces (xt, ~i'~) . . . . . (x,,, 2,,) respectively, which may or may not pass by the specified initial conditions. These closed curves are called the indicatrices. By varying slightly o~, ~o2~, . . . , w,,,-" and A, we obtain the periodic components x 1, x . . . . . . x,, when the indicatrices in the space (x~, 2~) . . . . . (x,,, ~,,) are observed to pass by the prescribed initial conditions at some values of t I , Q .....
t,~.
In this way we obtained the numerical values of w~. w ~ w~, A at which periodic c o m p o n e n t s x~, x 2 . . . . . x,, of the solutions of eq. (10) with periods T l, T_,. . . . . T,,, respectively pass by the specified initial conditions. For more details the reader is again referred to refs. [6, 7]. However, to
G.M. Mahmoud / Periodic orbits of nonlinear systems
389
m a k e the presentation here more self-contained we have added an appendix where the indicatrix m e t h o d is applied to a single nonlinear O D E order of the form (10).
. An example
To illustrate the techniques of sections 2, 3, we apply them here to the following system: 2 Xl + ~ l X l + A(XlX2 + X ~ ) C O S t = O,
(11)
A(x~ + x2x3)cos2t= 0
(12)
2 + £2 + ~2X2
2 + A(X~ + £3 + W3X3
x ~ ) c o s 3 t = O.
(13)
The main methods of this paper are not affected by the precise form of gi(x, £c) and Pi(~2d) (cf. eq. (1)). Performing the calculations in (4), one obtains the following for eqs. (11)-(13): d (a(t)~=A(R(v)]
dt
~(t)/
S(v)/'
(14a)
where
v = (a(t), a(t), 4~(t), 4~(t), t), &(t) = [~b,(t), ~b2(t), (53(t)1',
a(t) = [ai(t), a2(t), a3(t)]' , R = [R,, R2, Rs]t ,
S = [Si, 82, S3] t ,
R~, S~ (i = 1, 2, 3) are sums of trigonometric functions with arguments
thj.k ~- (% + k)t + ~ j ,
j, k = 1, 2, 3 ,
(14b)
1 l [ala 2 sin(2$,.,)+- ~tt2.0) q- a,2 sin ~tl, 1 q- a 21 sin(2O~ .o + O,,,)] , R 1= ~w
R2= ~
1
2
-
{a,[2 sin 02.2 + sin($2.2 +- 2G..)]
+ a2a3[sin ~.2 + sin(2$2., +- q6.0)]} , _ [(2a~ + a~) sin 03,3 + a3~ sin 3~3., + a 2 sin(q6.1 + 262.~) ] , R3 = _8wsl
(14c)
390
G.M. Mahmoud / Periodic orbits of nonlinear systems
S1= ~
1
[2a 2 cos 62., + a2 c ° s ( 2 G . , -+ 4-'>t ) + 3a, cos t/,,.,
+ a, eos(2&,.o + 4h.i)] . S~
1 {2a2~cos 4'2.2 + ael c ° s ( 2 G . . -+ qh +) 80).a-, "- + + a2a3[ 2 c ° s 4'3.2 + cos(6s..-+ 26>, )]} . 1
- [(3a~ + 2a~) cos ~/q.~ + a~ cos 3qq., + a~ cos(tO3. , + 2 G . , ) I $ 3 - 8w3a 3 From eq. (14) small frequency components occur at 0)t ~ - t g , l ,
w. ~ 1 , 2 .
w 3.~ 1 , 2 , 3.,
~2
¢03~3
0)3+20)-, . . ~ 9 3.
2wt-+0).~1,2. (15)
Choosing i
0)1~3
,
0) 2
and
(16)
(other values of o)i can be treated similarly) the reduced system (6a) becomes du
dt-
AFI(/~)
-~- O ' ( a 2 )
17a)
"
where u
[ A , , A 2, A , . @,, @2. q0~]t,
(17b)
F , ( u ) = [t~,,, /~,2. F,~, Ft4. FI~, />,~,]t .
PE, - A21 sin(2qtt.,, + qll t) ~0)1
]~1 ~ __
'
"
El ~ -
with
= --A~ sin qt+,, 4o)~
- -1 (2A~ + A2~) sin qq.3. 80) 3
A, / ~ 4 - 8w-, cos(2q-',., + q-", t) •
A~ .
F,s-
4w.A 2
COS1/t~,,
8 A 3 w ~ (3A~ + 2A~) cos q q , .
q~.k
(0)] - k ) t + qO] .
(17c)
G.M. Mahmoud / Periodic orbits of nonlinear systems
391
T o satisfy the conditions dAi/dt = 0, i = 1, 2, 3, we m a k e the corresponding contributions of ,~, (u) vanish by setting 2 % , o + %,, = (360, - 1)t + 3qOl = ll"ff , % , 2 = (602 -- 2)t + 42 = 12"rr,
l 1 , 12, l 3 = 0, 1,2 . . . . .
(18)
~ , 3 = ( % - 3)t + q~3 = / 3 "rr , F r o m eq. (18) one obtains
601
-
-
1 3
d~, dt
d~ 2 '
% = 2
dt
d~O3 '
% = 3
dt
'
(19)
F r o m eqs. (17), (19), we get the following expressions for a)~ to first o r d e r in a: w 21 = -19- ¼ A A , ( - 1 ) " + G(A2),
(20a)
2 w 2 =4-
1 A~ ~ A ~ ( - 1 ) ' 2 + •(A2),
(20b)
2 603 = 9 -
~
1
A(-1)z3(3A~ + 2A22) + G(A2).
(20c)
For different values of l~, 12, l 3 = 0, 1, we pick different branches in (o92~, A), (~o~, a), (o)~, A) planes, along which periodic solutions of the system ( 1 1 ) - ( 1 3 ) can be found, with initial conditions Xl(tl)= A,, X2(t2) = A 2 , x3(/3) = A 3 , and "~l (tl) = X2(t2) = "1C3(t3) = 0. These solutions have periodic x,(t) c o m p o n e n t with period Tj = 6rr, periodic X2(t ) with period T 2 = ~ and periodic c o m p o n e n t X3(t ) with period T 3 = 3ft. Their overall period, therefore, is T - - T~ = 6 T , = 9 T 3 = 6-rr. T o verify the validity of the analytical results we apply the numerical technique of section 3 (see also the appendix). By varying slightly o92,, w~, o9~ and A, we obtain the periodic components x I, x 2, x 3 when the indicatrices are observed to pass by our initial conditions (e.g. A I = A , = A.~ = 0.5) for different values of tj, t2, t3, see fig. 1. As expected from the analytical approach we obtained 6rr-periodic solutions with Xl(t ), X2(/), x3(t ) c o m p o n e n t s oscillating with periods 6~r, av, 3~ (w~~ ~ ,j, ~ w~_ ~ 4, w~ ~ 9) respectively. Finally, expressions (20) are c o m p a r e d with the results of this technique and very good a g r e e m e n t is found for A ~< 0.1, see table I.
392
G.M.
Mahmoud
/ Periodic orbits of nonlinear systems
{; {hOf i
00nl
k-
0 0003 j,
~o +
Q 0007
Z
0 0006
I
o ooo3
--
.x
EJ
(
/
/
\
>
i
\
\
#
\ "\\
000n!
/
+ 0 000,
/-
/ \
-x
-\
/ /
0 0002 00b91 //
-o
___i__
0091
L
~,98
~
i
0499
,
_
0 501
Xl(ll~tl*
~"
__L
0 S
,__J 0 502
~
0 49986
0.50002
049094
X2(12,12
0 I:}(~06
r~
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6 1[}
0.5001
+ "iT)
0ooo2t /
ca
\,
)
£ ' o ooo ?
/
@onoI
\
x ,,
0 000 l, -00006]
A •x
00002
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6 49992
0 /.9998
0.50004
X 3 (t 3,t3~
o 4%
213 T I t
o
~99
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o%,
0%
X I { t I , t I ~6 /11
0 -O.OOO~ -0.0002
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+ -
0001} 3
,-:, -o ooo,
-00004 .p<
- 00006 -O000(,_L
0 L9986
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0,L9994 Xz{t2~t2
050002 +]Tl
0 9001
049992
0499-38 X 3 (t3~t3
0 L0~34 +2/3TF}
Fig. 1. Indicatrix plots of eq. (10) for A = 0.001, showing x , ( t , ) = x ~ ( Q ) - x~(t~) - 0.5, t~ C [IL 2w I, t~ C [0, wl t~ • [0, ~w], with ki(ti) = 0, and (a) w ~ = 0 . 1 1 0 9 8 6 : ( 0 . 5 , 0 ) at t , - 0 , (b) w ~ - 3 . 9 9 9 6 5 : ( 0 . 5 , 0 ) at t , = 0 , (c) w ~ - 8 . 9 9 9 4 4 7 : ( 0 . 5 , 0 ) at t , = l ) , (d) w ~ = 0 . 1 1 1 2 3 6 : ( 0 . 5 , 0 ) at t , - v . (el w z~- 4.00036: (0.5, 0) at t ~ - ~w, and (f) ~o~- 9.000693:(0.5, 0) al t ~ - 1.09956.
393
G.M. Mahmoud / Periodic orbits of nonlinear systems
Table I Results for l I = 12 = 13 = 1 (t~ = 7, t 2 = ½~, t3 = 1.09956) and l I = 12= l 3 = 0 (t 1 = t 2 = t 3 = 0). A
Analytical results (eq. (20)) 2
0.001 0.005 0.010 0.050 0.100
0.111236 0,110986 0.111736 0.110486 (I.112361 0.109861 0.117361 0.104861 0.123610 0.098610
Numerical results
2
4.00025 3.99975 4.00125 3.99875 4.00250 3.99750 4.01250 3.98750 4.02500 3.97500
2
9
9.000625 8.999375 9.003125 8.996875 9.006250 8.993750 9.031250 8.968750 9.062500 8.937500
0.111236 0.110986 0.111736 0.110486 0.112360 0.109851 0.11740(I 0.104662 0.123600 0.097700
9
4.00036 3.99965 4.00183 3.99825 4.00370 3.99655 4.01790 3.983395 4.03520 3.96850
2
9.000693 8.999447 9.003250 8.996895 9.00635 8.99377 9.03050 8.968573 9.06247 8.93702
Appendix In this a p p e n d i x we p r e s e n t an e x a m p l e for t h e case n = 1 (cf. eq. (1)) to i l l u s t r a t e t h e i n d i c a t r i x m e t h o d (for m o r e e x a m p l e s for n = 1 , 2 , see refs. [6, 7]). W e will a p p l y h e r e t h e indicatrix m e t h o d for finding p e r i o d i c s o l u t i o n s o f a d i f f e r e n t i a l e q u a t i o n of t h e f o r m 5? + co2x + A cos [t(x + x2)] = 0 ,
A small,
(A.1)
w h e r e t h e u n p e r t u r b e d f r e q u e n c y w t a k e s values close to 1 / k , k = 1, 2 , . . . . By using the i n d i c a t r i x m e t h o d [6], we will d e t e r m i n e t h e v a l u e s o f t h e p a r a m e t e r s w 2, A at which p e r i o d i c o r b i t s of ( A . 1 ) with p e r i o d 2"rrk pass by a s p e c i f i e d initial c o n d i t i o n , e.g. ( x ( t o ) , x(t0)) = (0.5, 0). C l e a r l y , for A = 0, all p o i n t s ( x ( t o ) , £c(to)) can s e r v e as initial c o n d i t i o n s for 2"rrk p e r i o d i c solutions. H o w e v e r , for A ~ 0 , the set of p o i n t s ( x ( t o, t o + T ) , k(t~, t 0 + T ) ) with T = 2ark lie on a c l o s e d curve p a r a m e t r i z e d by G E [0, 2-rr] w h i c h m a y o r m a y n o t pass by o u r specified initial c o n d i t i o n (0.5, 0). T h i s c l o s e d curve is c a l l e d the i n d i c a t r i x a n d is p l o t t e d h e r e in fig. 2a for A = 0.1 a n d w ~ 1 / k = ~1 for which it clearly d o e s n o t pass t h r o u g h (0.5, 0). By v a r y i n g slightly w 2 for a given A, we o b t a i n the d e s i r e d p e r i o d i c o r b i t s w h e n t h e i n d i c a t r i x is o b s e r v e d to pass by o u r initial c o n d i t i o n at s o m e v a l u e s of t o. T h i s can h a p p e n , of c o u r s e , for m o r e t h a n o n e v a l u e of t o ~ [0, 2-rr], see figs. 2b, 2c w h e r e t o = 0 a n d 7r r e s p e c t i v e l y for the case k = ~ ( o t h e r v a l u e s of k can be similarly studied). W e can s t u d y in this w a y t h e e x i s t e n c e of p e r i o d i c o r b i t s with T = 2k,u of eq. (A.1).
394
G.M. Mahmoud / Periodic orbits of nonlinear system,~
The values of w e with a up to a ~<0.1) at which eq. (A.I) has 6w-periodic orbits passing through the initial condition (0.5,0), are obtained by the indicatrix method as follows shown in table A.I.
.1
A
÷ 4#°
"o
I\.
4J
(a)
j/
J
-.1
.0
x(to,to+6n )
.23 /./
1.0
J
f
\
A
M
J
40
(b)
-.23 .0
X(to,to+6n)
1.0
.11 A ÷
,> • Q " - - ~
f.-J'//)
(c)
-.11 •4
X(to,to+6.)
.7
Fig. 2. l n d i c a t r i x plots of eq. ( A . I ) for a = 0 . 1 and t,,~[(),2-rr~ for (a) 0~:=0.11097, ~o- =0.09047, (where (0.5,0) at t~,=0), (c) ~o-~ 0.11867 (where (0.5.0) at t,,=w).
(b)
G.M. Mahmoud / Periodic orbits o f nonlinear systems
395
Table A.1 A
~o2
to
0.001
0.1113 0.1108
~r 0
0.01
0.11235 0.10973
~r 0
0.05
0.11573 0.10275
7r 0
0.1
0.11867 0.09047
-rr 0
References [1] J. Kevorkian, Perturbation techniques for oscillatory systems with slowly varying coefficients, SlAM Review vol. 29 (1987). [2] C.-C. Chi and R.M. Rosenberg, On damped nonlinear dynamics systems with many degrees of freedom, Int. J. Nonlin. Mech. 20 (1985) 371-384. [3] D.E. Gilsinn, A High Order Generalized Method of Averaging, SIAM J. Appl. Math. 42 No. 1 (1982). [4] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields (Springer Berlin, 1983). [5] M. Lieberman and A. Lichtenberg, Regular and Stochastic Motion (Springer Berlin, 1983). [6] G.M. Mahmoud and T. Bountis, Synchronized periodic solutions of a class of periodically driven nonlinear oscillators, J. Appl. Mech. 55 (1988) 721-727. [7] T. Bountis and G.M. Mahmoud, Synchronized periodic orbits in beam-beam interaction models of one and two spatial dimensions, Particle Accelerators 22 (1987) 129-147. [8] H. Sprysl, Internal resonance of non-linear autonomous vibrating systems with two degrees of freedom, J. Sound Vib. 112 (1) (1987) 63-67. [9] P.R. Sethna, Coupling in certain classes of weakly nonlinear vibrating systems, in: Int. Symp. on Nonlinear Differential Equations and Nonlinear Mechanics, J. Lasalle and S. Lefschetz, eds. (Academic Press, New York, 1963). [10] R. Cohen and I. Porat, Coupled torsional and transverse vibration of unbalanced rotor, J. Appl. Mech. 52 (1985) 701-705. [111 A.N. Nayfeh and D.T. Mook, Nonlinear Oscillations (Wiley, New York, 1979).