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Bifurcation of 4:1 subharmonics in the nonlinear mathieu equation
MECHANICS RESEARCH COMMUNICATIONS 0093-6413/82/040233-08503.00/0
Vol. 9(4),233-240,1982. Printed in the USA. Copyright (c) Pergamon Press Ltd
BIFURCATION OF 4:1 SUBHARMONICS IN THE NONLINEAR MATHIEU EQUATION Leslie Anne Month and Richard H. Rand Dept. Mechanical Engineering, Univ. California, Berkeley CA 94720 and Dept. Theoretical and Applied Mechanics, Cornell Univ., Ithaca NY 14853 (Received 4 Januarg 1982; accepted for print 15 March 1982)
Introduction I t is well known [ l ] that the linear Mathieu equation, + (8 + ~ cost) u = 0
(l.l)
exhibits 2:1 subharmonics for 6 near I/4 when c is small. Specifically these periodic motions with frequency I/2 occur on the transition curves in the 6-E plane separating regions of s t a b i l i t y from regions of i n s t a b i l i t y . Here ins t a b i l i t y refers to unbounded behavior of motions starting a r b i t r a r i l y close to the zero solution u(t) z 0 (Although eq. ( l . l ) exhibits other i n s t a b i l i ties of the zero solution for ~ near nZ/4 (n = 2 , 3 , 4 , . . . ) , none of these are subharmonics; for n > 2 they are superharmonics.) Several recent works [2], [3], [4] have considered the effects of a nonlinear term on this 2:1 subharmonic by investigating the nonlinear Mathieu equation, U + (8 + ~ cost)u + E~u3 = 0 .
(I.2)
I t was shown that the nonlinearity stabilizes the unstable motion when 6 ~ I/4, c << l , in the sense that for those values of 6,E for which ( l . l ) exhibits unbounded motions, all motions of (I.2) are bounded. For such 6,~ the zero solution of (I.2) is locally unstable but globally stable; the 2:1 resonance is balanced by a s h i f t in frequency caused by the nonlinearity, resulting in a f i n i t e amplitude 2:1 subharmonic. See Fig. I . The purpose of this paper is to investigate the nature of 4:1 subharmonics in eq. ( I . 2 ) , and to compare their behavior with the 2:1 subharmonics studied previously. Lie Transforms The following exposition of Lie transforms, a perturbation method for Hamiltonian systems, is based on the work of Deprit [5] and Kamel [6].
233
234
LESLIE
ANNE M O N T H
and R I C H A R D
I
H. RAND
I
/I/
I/4 FIG. l
Bifurcation of 2:1 subharmonics in the nonlinear ~ t h i e u equation (I.2~, after [4]. Transition curves in the 6-~ plane are 6 = I/4 + ~/2 + O(E~). In each of 3 d i s t i n c t regions sketches are shown of invariant curves in the Poincar~ map corresponding to surface of section t = 0 (rood 2~). Quasistatic motion along line LL produces the double pitchfork bifurcation diagram shown at the top of the Figure. Points on the bifurcation diagram correspond to singular points in the Poincar~ map. The prongs of the pitchfork correspond to 2:1 subharmonic periodic motions of (1.2), while the handle corresponds to the zero solution. S -- stable, U = unstable. Given a Hamiltonian which is dependent upon a small parameter ~, H(x,X,t,c) it
,
(2.1)
i s desired to o b t a i n approximate expressions f o r x ( t , e ) ,
= Hx , X = -H x and where s u b s c r i p t s represent p a r t i a l
X ( t , e ) , where (2.2)
differentiation.
The method involves developing a near i d e n t i t y canonical transformation to new variables y,Y based on the generating equations x' = Wx , X' = -Wx where primes represent d i f f e r e n t i a t i o n with respect to ~, and where W(x,X,t,~)
= W1 + E W2 + O(E 2)
i s a generating f u n c t i o n to be chosen a t our d i s c r e t i o n . following initial
(2.3) (2.4)
Eqs. (2.3) have the
c o n d i t i o n s ( a t c = 0):
x(t,O) = y(t)
,
X(t,O) : Y ( t )
(2.5)
SUBHARMONICS IN N O N L I N E A R MATHIEU EQUATION
235
S
I ,
@u l/
1/16 FIG. 2
Bifurcation of 4:1 subharmonics in the nonlinear Mathieu eq. (1.2) for ~,~ > 0 . Transition curve is given by eq. (3.21). Quasistatic motion along line LL produces the pitchfork bifurcation diagram shown at the top of the Figure. The prongs of the pitchfork correspond to 4:1 subharmonic periodic motions of (1.2), while the handle corresponds to the zero solution. S = stable, U = unstable. Cf. Fig. I . The new variables y, Y are called the Lie transforms of x, X, respectively. I t has been shown that this transformation is canonical [5], [6], and therefore that y, Y satisfy Hamiltonian equations: = Ky ,
Y = -Ky
(2.6)
where the transformed Hamiltonian K i s given by the generating equation H' = Wt
w i t h the i n i t i a l
(2.7)
c o n d i t i o n ( a t E : 0)
H(x,X,t,0)
= K(y,Y,t)
(2.8)
Note t h a t in (2.8) K i s a f u n c t i o n o f the new v a r i a b l e s since x : y and X = Y at E = 0 .
Note also t h a t i f
the f u n c t i o n s y, Y and K o f eqs. ( 2 . 5 ) ,
were to depend upon an independent parameter, Cl say, the canonical o f the t r a n s f o r m a t i o n would be preserved.
(2.8)
property
In p a r t i c u l a r we may choose ~I = ~
and wri te y(t,E)
,
Y(t,~)
and
K(y,Y,t,~)
(2.9)
The formal method is derived by constructing approximate solutions to (2.3), (2.7) via Taylor series.
E.g. l e t us expand x(t,E) in a Taylor series about
236
L E S L I E A N N E M O N T H and R I C H A R D H. R A N D
x(t,~)
: x(t,0)
+ ~x'(t,0)
+ E2x '' ( t , 0 ) / 2 + 0(e 3)
(2.10)
From (2.3) - (2.5), x~(t,0) = Wx(X,X,t,0) = WIy(y,Y,t) D i f f e r e n t i a t i n g the f i r s t eq. of (2.3) with respect to E,
(2.11)
x" ( t , ~ ) = Wxx x ' + WXX Using (2.3) (2.5) this gives
(2.12)
X'
+
WX
(2 13) x" (t,0) = WlyY Wiy WiyY Wly + W2y To condense the notation we represent the Poisson bracket (or Lie derivative) of two functions f and g by (f;g) = fy gy - fy gy (2.14) Then (2.10) becomes x = y + E Wiy + (~2/2)[(Wiy;Wl) + W2y] + 0(e 3) Similarly we find
(2.15)
X = Y - E Wly - (E2/2)[(WIy;Wl) + W2y] + 0(e3)
(2.16)
H = K + ~ W l t + (e2/2)[(Wlt;W I ) + W2t] + 0(E 3)
(2.17)
On the other hand H may also be expanded in a power series in c by writing H(x,X,t,F~) = H0(x,X,t) + E Hl(X,X,t) + 2 H2(x,X,t)/2 + 0(E3)
(2.18)
Here the given functions H0, HI , H2 may themselves be expanded in Taylor series for small ~ (since x,X depend on ~), yielding H0(x,X,t) =H0(Y,V,t) +E (H0;WI)+(E2/2)[(H0;WI);~I) + (H0;W2)] + O(e 3) (2.19) Hl(X,X,t) = Hl(Y,Y,t) + e(HI;W I) + 0(c ) (2.20) H2(x,X,t) = H2(Y,Y,t) + 0(E) (2.21) Finally we can combine (2.17) - (2.21) to obtain the following expression for K(y,Y,t,E) : where
K = K0 + ~ K1 + 2 K0 = H0 K1 = HI + (H0;WI)
K2/2 + 0(c3)
(2.22) (2.23)
Wlt
K2 = H2 + (Ho;W2) + 2(HI;WI) + ((Ho;WI);WI) (WIt;W1) Eq. (2.25) can be simplified using (2.24) to give
(2.24) W2t
(2.25)
K2 = H2 + (Hl + Kl;W l) + (Ho;W2) - W2t (2.26) Note that in (2.23) - (2.26) HO, Hl , H2, Wl , W2 are functions of y,Y,t. Application To Nonlinear Mathieu Equation The Hamiltonian for eq. (I .2) is H = u2/2 + (6 + E cost) u2/2 + E ~ u4/4 In order to investigate 4:1 subharn~nics we set
(3.1)
SUBHARMONICS IN N O N L I N E A R MATHIEU EQUATION
6 = I l l 6 + ~ 61 + 2
237
62 + 0 ( 3 )
(3.2)
We begin by transforming to action-angle variables x,X for the E = 0 problem, = ~/2
cos x ,
u = ~
sin x
(3.3)
Substitution of (3.3) into (3.1) puts H in the form of (2.18) with H0 = XI4
(3.4)
Hl = 4X(61 + cos t) sin 2 x + 16~ X2 sin 4 x 2 H2 = 8X 62 sin x
(3.5) (3.6)
Next we perform the Lie transform described in section 2. y,Y as in (2.15),(2.16).
We replace x,X by
The new Hamiltonian K(y,Y,t,~) is given by (2.22)
with K0 = H0 = Y/4
(3.7)
Kl is obtained from eq. (2.24), which becomes Kl = Hl - L Wl where
L =~ +
and where Hl i s
(3.8)
@~
given by (3.5) with x,X replaced by y,Y.
to eliminate all angle terms y from KI.
We choose Wl so as
This requires that
Wl = 2 Y [ s i n t - 2 ( 6 1 +4~Y) s i n 2 y + ~ Y s i n 4 y + s i n ( 2 y - t ) - ( I / 3 )
sin(2y+t)]
whereupon Kl = 261Y + 6~Y2
(3.9) (3.10)
Similarly, K2 is found from (2.26), K2 = H2 + (Hl + Kl;W l ) - L W2 Again we try to choose W2 so as to eliminate all
(3.11) the angle terms y from K2.
This time, however, i t turns out that the r i g h t hand side of (3.11) contains a term of the form cos (4y - t) which l i e s in the null space of the operator L and hence cannot be removed from K2.
We find that
K2 = 4Y[62 + ~- - 461 2 - 4861~Y - 136~2y2 + 4~Y cos (4y - t ) ]
(3.12)
In order to obtain an approximate f i r s t integral we make a final canonical transformation to ~,J defined by = y - t/4 ,
J = Y
(3.13)
This transformation is based on the generating function F(y,J,t) = J(y - t/n)
(3.14)
and hence the new Hamiltonian K is given by = K + Ft = K - J/4
(3.15)
= ~[2J(61 + 3~J)] + ~212J(62 + § - 461 2 - 4861~J (3.16) - 136~2j 2 + 4~J cos 4~)] + 0(~ 3)
238
L E S L I E A N N E M O N T H and R I C H A R D H. R A N D
Note that F, does not depend e x p l i c i t l y on t and hence is a f i r s t integral of the system
= Kj ,
J = -F,~
(3.17)
The s i n g u l a r p o i n t s o f (3.17) correspond via ( 3 . 1 3 ) , to 4:1 subharmonic p e r i o d i c motions of ( I . 2 ) .
(2.15),
(2.16),
(3.3)
To f i n d them we r e q u i r e J and
to vanish. J =O:>Either
J = 0 or sin 4~ = 0 , ~ = n~/4
(3.18)
We ignore J=O which corresponds to the zero s o l u t i o n of (1.2) and s u b s t i t u t e = n~/4 i n t o 2 = 0 : ~ 6 1 + 6~J + ~[62 + ~ + 8m(-l)nj] + 0(3) = 0 Eq. (3.19) i s a q u a d r a t i c on J.
0(I).
2
- 9661~J - 408~2j2
(3.19)
For given ~ = n~/4, one o f the roots of
(3.19) turns out to be of O ( I / e ) . p o i n t s coming from i n f i n i t y
461
We omit c o n s i d e r a t i o n of such s i n g u l a r
since our p e r t u r b a t i o n method r e q u i r e s J to be
The o t h e r r o o t of (3.19) may be w r i t t e n
in the form
- ~ ( - l ) n 6 1 ]} + O(c 2) Now f o r real
solutions u(t),
t h i s r e q u i r e s to 0 ( I )
J must be non-negative ( c f .
t h a t 61 5_ 0 .
In the t r a n s i t i o n
required to O(E) t h a t 62 + 2/3 < 0 .
(3.20)
(3.3)).
For m,e > 0
case 61 = 0 , i t
is
This shows t h a t there is a t r a n s i t i o n
curve in the 6-c plane given by 6 = 1/16 - (2/3)E 2 + 0(~ 3) For m,z > 0 p o i n t s l y i n g to the l e f t i c s , corresponding r e s p e c t i v e l y
(3.21) of t h i s curve e x h i b i t
to n odd and even.
about the s i n g u l a r p o i n t s ( 3 . 1 8 ) , ( 3 . 2 0 ) instability
two 4:1 subharmon-
L i n e a r i z a t i o n of (3.17)
determines t h e i r
stability:
f o r n even (~ = 0 , ~/2 , ~ , 3~/2) and s t a b i l i t y
we f i n d
f o r n odd
(~ = ~/4 , 37/4 , 57/4, 7~/4). These r e s u l t s may be p i c t u r e d by transforming
back to u , u , t
algebra and sketch the r e s u l t s of s e t t i n g t = 0 in the f i r s t sponding to ( 3 . 1 6 ) .
This gives the i n v a r i a n t
.
We omit the integral
corre-
curves in an approximation to
the Poincar~ map corresponding to the surface of section t = 0 (n~d 2~). Fig. 2. Discussion In this section we compare the behavior of the 4:1 subharmonics derived in section 3 with the 2:1 subharmonics studied previously.
See
SUBHARMONICS
IN NONLINEAR MATHIEU EQUATION
Let us compare Figs. 1 and 2.
239
While the 2:1 subharn~nic in Fig. 1 has a
region of local i n s t a b i l i t y of the zero solution associated with i t , no such region occurs in Fig. 2.
The origin remains locally stable as one crosses
the transition curve in Fig. 2.
Moreover the bifurcations of each of the two
2:1 subharmonics present in the left-most region of Fig. l occur at d i s t i n c t points (as one crosses each of the two transition curves).
In Fig. 2, how.-
ever, both 4:1 subharmonics bifurcate simultaneously. The relationship between these two bifurcations can be better understood by considering the linear Mathieu eq. ( l . l ) .
As noted in the Introduction, ( l . l )
has 2:1 subharmonics occurring on each of the two transition curves in Fig. I. We now show, however, that ( l . l )
has two l i n e a r l y independent 4:1 subharmonics
occurring on the single transition curve of Fig. 2.
This follows from (3.16),
(3.17) with ~ = O, which give J = Jo ' ~ = ~ t + ~0 where whereupon,
(4.1)
~ = Kj = 2~[61 + ~(62 + 2 _ 4 6 1 2 ) ] neglecting
terms o f 0(~)
,
(4.2)
,
u(t) = 8JV~JO0 sin [(m + ~ ) t + ~0]
(4.3)
In order that (4.3) represent a 4:1 subharmonic we require that = 0=~61 = 0 , 62 = -2/3 , i . e . we are on the transition curve (3.21).
(4.4)
The coexistence of two l i n e a r l y independent 4:1 subharmonics on the transition curve of Fig. 2 suggests that we consider the bifurcation of Fig. 2 to be a l i m i t i n g case of the bifurcation of Fig. l in which the region of i n s t a b i l i t y of Fig. l becomes smaller and f i n a l l y vanishes as the two transition curves are brought closer together and f i n a l l y coalesce. We also note that the analysis leading to Fig. l neglected terms of O(e2) while the analysis in this paper leading to Fig. 2 necessarily had to keep O(e2) terms.
This suggests that the 2:1 subharmonics are more prominent for
small e than the 4:1 subharmonics.
To test this we numerically integrated
eq. (1.2) and generated Poincar~ maps as in Figs. 1,2.
We found that when
6 ~ 1/16 (to the l e f t of the transition curve (3.21)) both the 4:1 subharmonics of Fig. 2 and the 2:1 subharmonics of Fig. l were present.
However the
2:1 occurred for larger amplitudes than the 4:1, and the region enclosed by the separatrices was much larger for the 2:1 than for the 4:1.
We conjecture
that the additional singular point of eq. (3.19) which was of O(I/e) represents the 2:1 subharmonic. Since the l a t t e r occurs at a f i n i t e amplitude for
240
L E S L I E A N N E M O N T H and R I C H A R D H. R A N D
6 ~ 1/16 we are not surprised that the perturbation method sees i t as coming from i n f i n i t y . Finally we note that all the results of this paper are only valid for small ~, not only because the perturbation method requires i t , but also because KAM theory t e l l s us that chaos will replace the invariant tori of Figs. 1,2 as is increased [7]. References I. 2. 3. 4. 5. 6. 7.
J.J. Stoker, Nonlinear Vibrations, Interscience Publishers (1950). A.H. Nayfeh and D.T. ~ok, Nonlinear Oscillations, Wiley (1979). B.V. Chirikov, A Universal Instability of ~ny-Dimensional Oscillator Systems, Physics Reports 52(5), 265-379 (1979). C.A. Holmes and R.H. Rand, Coupled Oscillators as a ~del for Nonlinear Parametric Excitation, Mechanics Research Comm. 8(5), 263-268 (1981). A. Deprit, Canonical Transformations Depending on a Small Parameter, Celestial Mechanics l , 12-30 (1969). A.A. Kamel, Perturbation Theory Based on Lie Transforms, NASA Contractor Report CR-1622 (1970). V.I. Arnold, Mathematical Models of Classical Mechanics, Springer (1978).
Vol. 9(4),233-240,1982. Printed in the USA. Copyright (c) Pergamon Press Ltd
BIFURCATION OF 4:1 SUBHARMONICS IN THE NONLINEAR MATHIEU EQUATION Leslie Anne Month and Richard H. Rand Dept. Mechanical Engineering, Univ. California, Berkeley CA 94720 and Dept. Theoretical and Applied Mechanics, Cornell Univ., Ithaca NY 14853 (Received 4 Januarg 1982; accepted for print 15 March 1982)
Introduction I t is well known [ l ] that the linear Mathieu equation, + (8 + ~ cost) u = 0
(l.l)
exhibits 2:1 subharmonics for 6 near I/4 when c is small. Specifically these periodic motions with frequency I/2 occur on the transition curves in the 6-E plane separating regions of s t a b i l i t y from regions of i n s t a b i l i t y . Here ins t a b i l i t y refers to unbounded behavior of motions starting a r b i t r a r i l y close to the zero solution u(t) z 0 (Although eq. ( l . l ) exhibits other i n s t a b i l i ties of the zero solution for ~ near nZ/4 (n = 2 , 3 , 4 , . . . ) , none of these are subharmonics; for n > 2 they are superharmonics.) Several recent works [2], [3], [4] have considered the effects of a nonlinear term on this 2:1 subharmonic by investigating the nonlinear Mathieu equation, U + (8 + ~ cost)u + E~u3 = 0 .
(I.2)
I t was shown that the nonlinearity stabilizes the unstable motion when 6 ~ I/4, c << l , in the sense that for those values of 6,E for which ( l . l ) exhibits unbounded motions, all motions of (I.2) are bounded. For such 6,~ the zero solution of (I.2) is locally unstable but globally stable; the 2:1 resonance is balanced by a s h i f t in frequency caused by the nonlinearity, resulting in a f i n i t e amplitude 2:1 subharmonic. See Fig. I . The purpose of this paper is to investigate the nature of 4:1 subharmonics in eq. ( I . 2 ) , and to compare their behavior with the 2:1 subharmonics studied previously. Lie Transforms The following exposition of Lie transforms, a perturbation method for Hamiltonian systems, is based on the work of Deprit [5] and Kamel [6].
233
234
LESLIE
ANNE M O N T H
and R I C H A R D
I
H. RAND
I
/I/
I/4 FIG. l
Bifurcation of 2:1 subharmonics in the nonlinear ~ t h i e u equation (I.2~, after [4]. Transition curves in the 6-~ plane are 6 = I/4 + ~/2 + O(E~). In each of 3 d i s t i n c t regions sketches are shown of invariant curves in the Poincar~ map corresponding to surface of section t = 0 (rood 2~). Quasistatic motion along line LL produces the double pitchfork bifurcation diagram shown at the top of the Figure. Points on the bifurcation diagram correspond to singular points in the Poincar~ map. The prongs of the pitchfork correspond to 2:1 subharmonic periodic motions of (1.2), while the handle corresponds to the zero solution. S -- stable, U = unstable. Given a Hamiltonian which is dependent upon a small parameter ~, H(x,X,t,c) it
,
(2.1)
i s desired to o b t a i n approximate expressions f o r x ( t , e ) ,
= Hx , X = -H x and where s u b s c r i p t s represent p a r t i a l
X ( t , e ) , where (2.2)
differentiation.
The method involves developing a near i d e n t i t y canonical transformation to new variables y,Y based on the generating equations x' = Wx , X' = -Wx where primes represent d i f f e r e n t i a t i o n with respect to ~, and where W(x,X,t,~)
= W1 + E W2 + O(E 2)
i s a generating f u n c t i o n to be chosen a t our d i s c r e t i o n . following initial
(2.3) (2.4)
Eqs. (2.3) have the
c o n d i t i o n s ( a t c = 0):
x(t,O) = y(t)
,
X(t,O) : Y ( t )
(2.5)
SUBHARMONICS IN N O N L I N E A R MATHIEU EQUATION
235
S
I ,
@u l/
1/16 FIG. 2
Bifurcation of 4:1 subharmonics in the nonlinear Mathieu eq. (1.2) for ~,~ > 0 . Transition curve is given by eq. (3.21). Quasistatic motion along line LL produces the pitchfork bifurcation diagram shown at the top of the Figure. The prongs of the pitchfork correspond to 4:1 subharmonic periodic motions of (1.2), while the handle corresponds to the zero solution. S = stable, U = unstable. Cf. Fig. I . The new variables y, Y are called the Lie transforms of x, X, respectively. I t has been shown that this transformation is canonical [5], [6], and therefore that y, Y satisfy Hamiltonian equations: = Ky ,
Y = -Ky
(2.6)
where the transformed Hamiltonian K i s given by the generating equation H' = Wt
w i t h the i n i t i a l
(2.7)
c o n d i t i o n ( a t E : 0)
H(x,X,t,0)
= K(y,Y,t)
(2.8)
Note t h a t in (2.8) K i s a f u n c t i o n o f the new v a r i a b l e s since x : y and X = Y at E = 0 .
Note also t h a t i f
the f u n c t i o n s y, Y and K o f eqs. ( 2 . 5 ) ,
were to depend upon an independent parameter, Cl say, the canonical o f the t r a n s f o r m a t i o n would be preserved.
(2.8)
property
In p a r t i c u l a r we may choose ~I = ~
and wri te y(t,E)
,
Y(t,~)
and
K(y,Y,t,~)
(2.9)
The formal method is derived by constructing approximate solutions to (2.3), (2.7) via Taylor series.
E.g. l e t us expand x(t,E) in a Taylor series about
236
L E S L I E A N N E M O N T H and R I C H A R D H. R A N D
x(t,~)
: x(t,0)
+ ~x'(t,0)
+ E2x '' ( t , 0 ) / 2 + 0(e 3)
(2.10)
From (2.3) - (2.5), x~(t,0) = Wx(X,X,t,0) = WIy(y,Y,t) D i f f e r e n t i a t i n g the f i r s t eq. of (2.3) with respect to E,
(2.11)
x" ( t , ~ ) = Wxx x ' + WXX Using (2.3) (2.5) this gives
(2.12)
X'
+
WX
(2 13) x" (t,0) = WlyY Wiy WiyY Wly + W2y To condense the notation we represent the Poisson bracket (or Lie derivative) of two functions f and g by (f;g) = fy gy - fy gy (2.14) Then (2.10) becomes x = y + E Wiy + (~2/2)[(Wiy;Wl) + W2y] + 0(e 3) Similarly we find
(2.15)
X = Y - E Wly - (E2/2)[(WIy;Wl) + W2y] + 0(e3)
(2.16)
H = K + ~ W l t + (e2/2)[(Wlt;W I ) + W2t] + 0(E 3)
(2.17)
On the other hand H may also be expanded in a power series in c by writing H(x,X,t,F~) = H0(x,X,t) + E Hl(X,X,t) + 2 H2(x,X,t)/2 + 0(E3)
(2.18)
Here the given functions H0, HI , H2 may themselves be expanded in Taylor series for small ~ (since x,X depend on ~), yielding H0(x,X,t) =H0(Y,V,t) +E (H0;WI)+(E2/2)[(H0;WI);~I) + (H0;W2)] + O(e 3) (2.19) Hl(X,X,t) = Hl(Y,Y,t) + e(HI;W I) + 0(c ) (2.20) H2(x,X,t) = H2(Y,Y,t) + 0(E) (2.21) Finally we can combine (2.17) - (2.21) to obtain the following expression for K(y,Y,t,E) : where
K = K0 + ~ K1 + 2 K0 = H0 K1 = HI + (H0;WI)
K2/2 + 0(c3)
(2.22) (2.23)
Wlt
K2 = H2 + (Ho;W2) + 2(HI;WI) + ((Ho;WI);WI) (WIt;W1) Eq. (2.25) can be simplified using (2.24) to give
(2.24) W2t
(2.25)
K2 = H2 + (Hl + Kl;W l) + (Ho;W2) - W2t (2.26) Note that in (2.23) - (2.26) HO, Hl , H2, Wl , W2 are functions of y,Y,t. Application To Nonlinear Mathieu Equation The Hamiltonian for eq. (I .2) is H = u2/2 + (6 + E cost) u2/2 + E ~ u4/4 In order to investigate 4:1 subharn~nics we set
(3.1)
SUBHARMONICS IN N O N L I N E A R MATHIEU EQUATION
6 = I l l 6 + ~ 61 + 2
237
62 + 0 ( 3 )
(3.2)
We begin by transforming to action-angle variables x,X for the E = 0 problem, = ~/2
cos x ,
u = ~
sin x
(3.3)
Substitution of (3.3) into (3.1) puts H in the form of (2.18) with H0 = XI4
(3.4)
Hl = 4X(61 + cos t) sin 2 x + 16~ X2 sin 4 x 2 H2 = 8X 62 sin x
(3.5) (3.6)
Next we perform the Lie transform described in section 2. y,Y as in (2.15),(2.16).
We replace x,X by
The new Hamiltonian K(y,Y,t,~) is given by (2.22)
with K0 = H0 = Y/4
(3.7)
Kl is obtained from eq. (2.24), which becomes Kl = Hl - L Wl where
L =~ +
and where Hl i s
(3.8)
@~
given by (3.5) with x,X replaced by y,Y.
to eliminate all angle terms y from KI.
We choose Wl so as
This requires that
Wl = 2 Y [ s i n t - 2 ( 6 1 +4~Y) s i n 2 y + ~ Y s i n 4 y + s i n ( 2 y - t ) - ( I / 3 )
sin(2y+t)]
whereupon Kl = 261Y + 6~Y2
(3.9) (3.10)
Similarly, K2 is found from (2.26), K2 = H2 + (Hl + Kl;W l ) - L W2 Again we try to choose W2 so as to eliminate all
(3.11) the angle terms y from K2.
This time, however, i t turns out that the r i g h t hand side of (3.11) contains a term of the form cos (4y - t) which l i e s in the null space of the operator L and hence cannot be removed from K2.
We find that
K2 = 4Y[62 + ~- - 461 2 - 4861~Y - 136~2y2 + 4~Y cos (4y - t ) ]
(3.12)
In order to obtain an approximate f i r s t integral we make a final canonical transformation to ~,J defined by = y - t/4 ,
J = Y
(3.13)
This transformation is based on the generating function F(y,J,t) = J(y - t/n)
(3.14)
and hence the new Hamiltonian K is given by = K + Ft = K - J/4
(3.15)
= ~[2J(61 + 3~J)] + ~212J(62 + § - 461 2 - 4861~J (3.16) - 136~2j 2 + 4~J cos 4~)] + 0(~ 3)
238
L E S L I E A N N E M O N T H and R I C H A R D H. R A N D
Note that F, does not depend e x p l i c i t l y on t and hence is a f i r s t integral of the system
= Kj ,
J = -F,~
(3.17)
The s i n g u l a r p o i n t s o f (3.17) correspond via ( 3 . 1 3 ) , to 4:1 subharmonic p e r i o d i c motions of ( I . 2 ) .
(2.15),
(2.16),
(3.3)
To f i n d them we r e q u i r e J and
to vanish. J =O:>Either
J = 0 or sin 4~ = 0 , ~ = n~/4
(3.18)
We ignore J=O which corresponds to the zero s o l u t i o n of (1.2) and s u b s t i t u t e = n~/4 i n t o 2 = 0 : ~ 6 1 + 6~J + ~[62 + ~ + 8m(-l)nj] + 0(3) = 0 Eq. (3.19) i s a q u a d r a t i c on J.
0(I).
2
- 9661~J - 408~2j2
(3.19)
For given ~ = n~/4, one o f the roots of
(3.19) turns out to be of O ( I / e ) . p o i n t s coming from i n f i n i t y
461
We omit c o n s i d e r a t i o n of such s i n g u l a r
since our p e r t u r b a t i o n method r e q u i r e s J to be
The o t h e r r o o t of (3.19) may be w r i t t e n
in the form
- ~ ( - l ) n 6 1 ]} + O(c 2) Now f o r real
solutions u(t),
t h i s r e q u i r e s to 0 ( I )
J must be non-negative ( c f .
t h a t 61 5_ 0 .
In the t r a n s i t i o n
required to O(E) t h a t 62 + 2/3 < 0 .
(3.20)
(3.3)).
For m,e > 0
case 61 = 0 , i t
is
This shows t h a t there is a t r a n s i t i o n
curve in the 6-c plane given by 6 = 1/16 - (2/3)E 2 + 0(~ 3) For m,z > 0 p o i n t s l y i n g to the l e f t i c s , corresponding r e s p e c t i v e l y
(3.21) of t h i s curve e x h i b i t
to n odd and even.
about the s i n g u l a r p o i n t s ( 3 . 1 8 ) , ( 3 . 2 0 ) instability
two 4:1 subharmon-
L i n e a r i z a t i o n of (3.17)
determines t h e i r
stability:
f o r n even (~ = 0 , ~/2 , ~ , 3~/2) and s t a b i l i t y
we f i n d
f o r n odd
(~ = ~/4 , 37/4 , 57/4, 7~/4). These r e s u l t s may be p i c t u r e d by transforming
back to u , u , t
algebra and sketch the r e s u l t s of s e t t i n g t = 0 in the f i r s t sponding to ( 3 . 1 6 ) .
This gives the i n v a r i a n t
.
We omit the integral
corre-
curves in an approximation to
the Poincar~ map corresponding to the surface of section t = 0 (n~d 2~). Fig. 2. Discussion In this section we compare the behavior of the 4:1 subharmonics derived in section 3 with the 2:1 subharmonics studied previously.
See
SUBHARMONICS
IN NONLINEAR MATHIEU EQUATION
Let us compare Figs. 1 and 2.
239
While the 2:1 subharn~nic in Fig. 1 has a
region of local i n s t a b i l i t y of the zero solution associated with i t , no such region occurs in Fig. 2.
The origin remains locally stable as one crosses
the transition curve in Fig. 2.
Moreover the bifurcations of each of the two
2:1 subharmonics present in the left-most region of Fig. l occur at d i s t i n c t points (as one crosses each of the two transition curves).
In Fig. 2, how.-
ever, both 4:1 subharmonics bifurcate simultaneously. The relationship between these two bifurcations can be better understood by considering the linear Mathieu eq. ( l . l ) .
As noted in the Introduction, ( l . l )
has 2:1 subharmonics occurring on each of the two transition curves in Fig. I. We now show, however, that ( l . l )
has two l i n e a r l y independent 4:1 subharmonics
occurring on the single transition curve of Fig. 2.
This follows from (3.16),
(3.17) with ~ = O, which give J = Jo ' ~ = ~ t + ~0 where whereupon,
(4.1)
~ = Kj = 2~[61 + ~(62 + 2 _ 4 6 1 2 ) ] neglecting
terms o f 0(~)
,
(4.2)
,
u(t) = 8JV~JO0 sin [(m + ~ ) t + ~0]
(4.3)
In order that (4.3) represent a 4:1 subharmonic we require that = 0=~61 = 0 , 62 = -2/3 , i . e . we are on the transition curve (3.21).
(4.4)
The coexistence of two l i n e a r l y independent 4:1 subharmonics on the transition curve of Fig. 2 suggests that we consider the bifurcation of Fig. 2 to be a l i m i t i n g case of the bifurcation of Fig. l in which the region of i n s t a b i l i t y of Fig. l becomes smaller and f i n a l l y vanishes as the two transition curves are brought closer together and f i n a l l y coalesce. We also note that the analysis leading to Fig. l neglected terms of O(e2) while the analysis in this paper leading to Fig. 2 necessarily had to keep O(e2) terms.
This suggests that the 2:1 subharmonics are more prominent for
small e than the 4:1 subharmonics.
To test this we numerically integrated
eq. (1.2) and generated Poincar~ maps as in Figs. 1,2.
We found that when
6 ~ 1/16 (to the l e f t of the transition curve (3.21)) both the 4:1 subharmonics of Fig. 2 and the 2:1 subharmonics of Fig. l were present.
However the
2:1 occurred for larger amplitudes than the 4:1, and the region enclosed by the separatrices was much larger for the 2:1 than for the 4:1.
We conjecture
that the additional singular point of eq. (3.19) which was of O(I/e) represents the 2:1 subharmonic. Since the l a t t e r occurs at a f i n i t e amplitude for
240
L E S L I E A N N E M O N T H and R I C H A R D H. R A N D
6 ~ 1/16 we are not surprised that the perturbation method sees i t as coming from i n f i n i t y . Finally we note that all the results of this paper are only valid for small ~, not only because the perturbation method requires i t , but also because KAM theory t e l l s us that chaos will replace the invariant tori of Figs. 1,2 as is increased [7]. References I. 2. 3. 4. 5. 6. 7.
J.J. Stoker, Nonlinear Vibrations, Interscience Publishers (1950). A.H. Nayfeh and D.T. ~ok, Nonlinear Oscillations, Wiley (1979). B.V. Chirikov, A Universal Instability of ~ny-Dimensional Oscillator Systems, Physics Reports 52(5), 265-379 (1979). C.A. Holmes and R.H. Rand, Coupled Oscillators as a ~del for Nonlinear Parametric Excitation, Mechanics Research Comm. 8(5), 263-268 (1981). A. Deprit, Canonical Transformations Depending on a Small Parameter, Celestial Mechanics l , 12-30 (1969). A.A. Kamel, Perturbation Theory Based on Lie Transforms, NASA Contractor Report CR-1622 (1970). V.I. Arnold, Mathematical Models of Classical Mechanics, Springer (1978).