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Normal form investigations of dissipative systems
Mechanics Research Communications, Vol. 21, No. 3, pp. 231-240, 1994 Copyright • 1994 Elsevier Science Ltd Printed in the USA. All fights reserved 0093-6413/94 $6.00 + .00
Pergamon
NORMAL FORM INVESTIGATIONS OF DISSIPATIVE SYSTEMS
Diana M urray Dept. of Physics, SUNY Stony Brook Stony Brook, NY 11794 emaih [email protected]
(Received 17 September 1993; acceptedfor print 26 October 1993)
Introduction
We are interested in studying dissipative systems that are perturbed harmonic oscillators:
dx --=y+e
G(x,y;e)
dt
dy = - x + e H ( x , y ; 8) , dt
(I)
[el
using the method of normal forms [1,2,3,4,5] to develop an approximate solution as a perturbation expansion in a small parameter. With the method of normal forms one seeks a polynomial near-identity coordinate transformation in which the dynamical system takes the simplest form. The normal form is the equation of motion for the zeroth-order approximation and contains only resonant terms (i.e. terms with the same phase as the zeroth-order term). The method of normal forms has a well-known "lack of uniqueness" [6,7] that led Kahn and Zarmi [8] to develop the method of minimal normal forms (MNF's) which we have applied to conservative systems [9]. Kahn and Zarmi have shown that a judicious choice of the zeroth-order approximation can produce a normal form which is far simpler than the one that results from traditional normal form analysis ("usual" normal forms). Here we show that this freedom of choice in tbe normal form expansion can be utilized for dissipative systems, producing compact equations of motion which capture the full characteristics of the flow early on and with great ease of calculation. Dissipative systems are characterized dynamically by the contraction of the phase space area with increasing time onto an attractor of lower dimension. For flows in two dimensions the attractor can be either a fixed point or a limit cycle. The introduction of computer algebra [10,11] enables one to develop algorithms so as to obtain perturbation expansions to high order where the merit of MNF's will be shown to be most striking.
We illustrate the essential elements of the method of normal forms by analyzing a system with cubic damping, i.e. G = 0, H = _y3 in eq (1). We then show how it is applied to the pendulum with cubic damping and to two examples taken from the text by Rand and Armbruster [10]. This treatment is easily generalized to systems 231
232
D. MURRAY
with other nonlinearities. Consider the tbllowmg:
dr
--=y
dt dy_ dt
(2)
x_e>,~ '
x(O) = a, y ( 0 ) = b,
cl
Introducing the diagonalizing transformation, z = x + i y , eq (2) becomes
d~
1 z + - ~" ( z - z * ) ~ 8
--:-i
dt
(3)
where the asterisk denotes complex conjugation. The quantity z is expanded in terms of the zeroth-order approximation, u, in a near-identity transformation:
z=u+e r,(u,u*)+E ~ ~(u,u*)+e'
..
(4)
3 U3(II,I4*)+ .
(5)
r~(u,u*)+
and u satisfies the normal tbrm equation:
du act
-
J u+g
UI(U,U*)+E
2 U2(tt,u*)+£
In the method of normal forms, the ~ , which are polynomials in U and u *, are chosen to annihilate all the non-resonant terms in cq (3). The normal form o f e q (5) will generally contain resonant monomials of the form u"+lu *", with n _> 1 , in each order of the small parameter, g. To determine the J~ and the U i , one
substitutes eqs (4) and (5) into eq (3) and collects terms in each order of oe:
U1 = - i
7~ + 67~ i u -
(l 2 = - J
~ +
U3 = - i
7; +
3u
or~ 6u
o~ 6u
OT, u , - O ~ u 6u
-
&~ ,
I lt-
,
1
,
d'll i U + - ~ ( U - U ) cTu *
dT2i u*+
(u-u*) 2 (7-~*)-~-uU~
dr~ l , 3 i u - Ou "* ' ,, + ~ ( u -
2
,
du*
'
du *
u*) 2 ( l; - T, *) + ~ ( u - u * ) ( ~ - ~ *):
o~i (I~- 6T--~eu * 6u
(6)
NORMAL FORM OF DISSIPATIVE SYSTEMS
233
Note that in eqs (6) the ~ are determined in each order up to a free additive term of the (resonant) form
F . = f . (uu*) u , for which the quantity :
OF.
- i F. +C~F" i u iu*. c3u Ou*
known as the Lie bracket, vanishes. There is no contribution fiom F . to the order n calculation, but, as we will show, it strategically appears in order n+l. In the usual method of normal forms, one chooses F n = 0 for all n. The method of minimal normal forms attempts to make use of the free term from order n to annihilate the resonant contribution in order n+l. Hence one has the possibility of U i = 0 for i > 1 and, in general, the minimal normal form has a more compact structure than the usual normal form. Let us see how this is manifested in our example. The T polynomials that eliminate all the non-resonant terms in equations (6) are, through third order in~? :
2 u , + - - ui T~ = t - •- u 3 - - - 3i u 16 16 32
9
T2 -
U5 +
1024
,3+f~(uu,)u
9 u 4 u * + - - r3i ,u 256 16"'
27 / / / / , 4 3i ,//,3 1024 + -j-~ f t
3i 27 u2u ,3 - ]"g ( f l + 2 f l *)uu ,2 512
3-
3
~/ .s
+ f : (uu*)u
+ 512
3i 279i 6 45 5 (f~9 2048 u7 + 32768 u u * - - -1024 f , u ~' + 16384333iuSu.2+ 256
T3 -
+
8 li 16384
U3U , 4 _
27 256
(f~ +3 f ,)u:u,, - ~3i ~(f2
117i 27 ( l f ~ + f . ) u u , 4 + 32768 u2u*5 - 2 5 6
9/ 8129
u*' +A(uu*)u
+4fl ,)U4U, +
(f12 +f2)u3
+ 2 f 2 * +2f~f~ * + f l *'-)uu*:
27i UU , 6 + 15 + 3i ( _ . ~ )'2 * + f l "2) u*3 + 4096 512
fl
*U
5
234
D. MURRAY
This results in the following normal form:
du_ dt
iu-g- 3 u:u*+g 2 (
(fl-fl*)
u2u*+ 2 7 i u ~ u , 2 )
8
256 (7a)
+~.3 (
.
3(2f~ :+f~fl*-3L+L*) .
.
u2u*+ 27i(fl+f,) 128 .
3u,2
567 11411,3 ) 8192
The quantities F 1 = f l u and F 2 = f2 u , fiee functions from ]; and 7~ respectively, can tx' chosen to 9i annihilate the second and third order terms (f~ = - - - u u 64
,
, and ,~ =
675
u2u ,2 ). Similarly, higher
8192
order resonant contributions to the MNF are eliminated to yield the tbllowing expression correct to all orders:
MNF:
du
dt
-
3
iu--E 8
u2u*
(Vo)
Ill the "usual" case the normal tbrm includes resonant contributions at each order. Through sixth order in ,5." . tbr example, one then obtains
Usual NF:
--=du - i u dt
3 .
~ 27
g -u"tt*+g 8
iu3u,2 --g
256
~ 567 ,3 __ lt4ll 8192
+g4
7965
jtl~ll,4
262144 (7c)
- c 5- - l62721 ~6U 4194304
,5 q-oe t'
214083
iltTU ,0
268435456
Writing u = p e ' ¢, eq (7b) gives
dp dt
-
3 3 gp and 8
dO dt
(8)
--=c0=l
which are both easily solved. Similarly. eq (7c) gives
dp dt
-
3 ~. p3 - - - g 567 8192
8
3 pT_ - 62721 - g ~ p 4194304
Ii
+O(~') (9)
and
- - = c o = l - - - ~ ' d ~27b dt 256
2p4
- - c 7 9 6 5 4p8 262144
214083 ~.~,p~2+0(c7) 268435456
Indeed it is found that all of the resonant contributions to eq (Tb) can be killed by free terms, whereas eq (7c) is continually updated.
NORMAL FORM OF DISSIPATIVE SYSTEMS
235
In dissipative systems there is frequency (co) as well as radius (19) updating. As can be seen in this example, the benefit of MNF's is twofold- to all orders the fundamental frequency of oscillation has no update,
4 =t+40
or
co=l,
(10)
and the radius equation has a simple closed-form solution,
P=~l+~6po2t Po
(ll)
On the other hand, the radius and phase equations for the usual case quickly become calculationally unwieldy. The constants/9 0 and 4 0 in eqs (10 &ll) are, for both methods, updated in each order and found by implementing the initial condition (x(0)=a, y(0)=b). Once the T~ are kaaownthrough the desired order of calculation, z (or equivalently, x) is reconstructed. By inverting the near-identity transformation (eq(4)), we find u(0)and u * (0) in terms of x ( 0 ) and y ( 0 ) . Then/9 0 and 4 0 are determined from the relation
u(0) = P0 exp(-i 40 ).
The
1A/1 + 3/4 Cpo2I dependence of,O(/)dominates the character of the normal
form approximate solutions. Both MNF and usual results are in good agreement with the numerical solution obtained from a fourth-order Taylor series procedure for positive and negative damping. Fig. la shows a comparison of the numerical solution with that obtained from a second-order M NF calculation for 6 = 3/4 and x(0)=l, y(0)=0. Fig. lb illustrates that there is a finite but small mismatch of the nomal form solution from the exact solution for short times, but for longer times the two become indistinguishable. In fig. 2, the numerical and MNF results are compared for negative damping ( oe = -1/20). From eq (11), the approximate solution is expected to blow up at
~25.
1
0.8
0.6
0.4
,4.0 0.2
AAAAAAAAAAAAAAAAAAAAAAAAA, A/VVVVVVVVVVVVVVVVVVVVVV'me 0
41.2
-0.4
41.6
FIG. la. Comparison of numerical and 2nd-order MNF solutions to eq (2) with positive damping displaying the long-time validity of the MNF approximation. ( oe = 3/4, x(0)=l, y(0)=0).
236
D. MURRAY
time 8
lo
:~ I° ~F--',~_.~. FIG. lb. Mismatch, M, of the MNF approximate solution depicted ira Fig. la obtained when the approximation is substitued back into the differential equation.
FIG. 2. Comparison of mlmerical and 2nd-order MNF solutions to the s)stem ofeq (2) with negatiw" 1
damping. ( ~ = - - 2 ~ , x ( O ) = l,y(O) = 0).
N O R M A L FORM OF DISSIPATIVE SYSTEMS
237
To be fair, MNF's cannot always give such a simple equation for the radius update, but in almost all of the cases we have studied it contained at most two terms, a result that holds to all orders. The usual normal form will generally contain an infinite number of terms. This distinction is important since, for instance, the normal forms of equations are very useful in bifurcation analyses [12,13].
One exception to this is when the damping
term is linear, since such a resonance can never be annihilated by a free term.
As an example of when the MNF terminates a~er two terms, we analyze the following system:
dx
m=y
dt
02)
dy = - x - c (x 3 + y3 ) dt By utilizing the free terms in the MNF expansion we can eliminate the imaginary part of the second order resonance and all higher-order (n > 2) resonant terms. The MNF and usual radius and phase equations are determined by heeding the procedure outlined above: MNF's:
UsualNF's:
dp 3--_ p3
3 c2 p5 3
3-5
e2p5 . . 3 . oo3p . 7 171 +~'4 p 9 2061
d - ~ - P = - t p 3 3. at 8
32
4096
32768
_ ~ 5 p l l __153339 [_~6 p13 1216185__ + O ( z 7) 1048576
MNF's:
Usual NF's:
--= d#
1+8p2_
3
'
dt
8
d#= l+cp2
3_c2p 8
dt
8388608
4 39 + 6 3 p 6 . 375 . 128 4096
+ e 6 p 12 3237459 - - + O ( e 16777216
6.4 p 8 . 4623 65536
6s p 10 17187 262144
7)
We comment that by implementing the method of M NF's for the pendulum with cubic damping d2x
dx 3
dt 2 + ( ~ - )
+sinx=0
,
238
D. MURRAY
we obtain a similar result. Here, the resonant contributions of all the nonlinearities greater than cubic can be annihilated yielding radius and phase equations that are qualitatively the same as those for the system ofeq (12). Considering only small amplitudes, we Taylor series expand the sine function arid introduce the scaling, X
) -~
X, to obtain the equation of motion:
dT+X+e{(
)3
}+
6
....
2
e . (_1). ~ _0 (2n+l)!
The M N F radius and phase equations are
dp
epS3+s:
dt
8
p
_
_
64
2Z dt
16
As can be seen, the M N F results, exact to all orders, are a great simplification and improvement over the usual results.
In the spirit of generality we show that similar results hold for a system with general quadratic, cubic, quartic and quintic nonlinearities. The fbllowing is based on an example taken from Rand and Armbruster [10, p55fl]:
dx
~-=y+e(A~.
x 2 + A , xy + A.w y 2 ) + ~'2 (A:o:~ x"s + A,:v, x'-y+ A.~, xy 2 +Am' yS)
+83(Axa~
x 4 +A~:~, x3y+...)+84(Ax.oa~ x 5 +Aax ~, x a y + . . . ) (13)
d y = - x + e ( B , ~ , x 2 +B dt
xy+B., y2)+o~2 ( B "
+e3(Bx~
x ~+B
y x3y+...)+~
x 3 +B~., x 2 y + B , ~ xy 2 +B m, yS) "
•
'"
4( B . . . . x 5 +B~,:~ x x y + . . . )
where A and B are complex constants. Note that the systems o f e q s (2) and (12) are special cases of this one. We find that the method of MNF's yields an equation of motion of the same form as the normal form ofeq(12):
du -
-
itl+oo2ii2tt*C+84tlstl*2D
dt where the constants C and D are fimctions of the constants, A ,=~,..., B ~ ..... from eq (13). This M N F result is exact to all orders, while the usual normal form is updated in each even order.
NORMAL FORM OF DISSIPATIVE SYSTEMS
239
This analysis is also extendable with some restrictions to higher dimensions. Another example from Rand and Armbruster [10, p 28ff] depicts a feedback control system modeled by three differential equations:
dr ~-=y ay --
dt dw dt
---- - - X - -
--
£aX'W
W + £t~X
2
One might seek to reduce this to a two-dimensional problem via the method of center manifolds [10, chp 2] wherein only information about the steady state is obtained. Applying the method of MNF's to the full set of equations, we can derive simple transformed equations of motion and track the transient motion as well.
MNFresult:
du
__
:
_ i u _ e 2 ~ 0 ( 1 - - ~l -l )i ,u 2u . - e 4 - -73 a" 2000
tl311,2
dt dq =_q+ c2 a-~-uu*q dt 10
where u and q are the zeroth-order approximations to the variables z = x + iy and w. This result is exact to all orders.
The hope in performing an analytical calculation is to derive a qualitative understanding of the dynamics of the original system from the transformed, simplified equation of motion (the normal form). We have shown examples of dissipative perturbed harmonic oscillator systems where the method of M NF's yields simpler, more compact equations of motion than the traditional NF method. Conceptually and computationally, it is desirable to have the radius and phase equations fixed early in the calculation. In many cases one can exploit the freedom associated with the method of MNF's to tailor the normal form to match a specific goal- for example, to extend dynamical intuition, to track the transient motion or to simply obtain closed-form expressions which characterize the motion. We are in the progress of extending this analysis to parametric systems, and plan to report on these investigations in the near future.
Acknowledgements: I would like to express my appreciation to Professor Peter B. Kahn and Professor Yair Zarmi for their help and encouragement, to Professor Richard H. Rand for kindly reading the manuscript and to the Institute for Pattern Recognition for financial support.
240
D. M U R R A Y
References 1. V.I. Arnold, Geometrical methods in the theory of ordinaU diflbrential equations, Springer-Verlag, NY (1988) 2. J. Guckenheimer and p.j. Holmes, Nonlincar oscillations, dynamical systems and bithrcations ofveetor fields, Sprinwr-Verlag, NY (1983) 3. S. Wiggins, Introduction to applied nonlinear d}namical systems and chaos, Springer-Verlag, NY (1990) 4. A.H. Nayfeh, Method of normal forms, Wiley-Interscience, NY (1993) 5. L. Jezequel and C.H, Lawrence, J. Sound Vib. 149,429-459 (1991) 6. A. Baider and B.C. Churchill, Proc. R. Soc. Edinburgh See. A 108, 27-33 (1988) 7. A.D. Bruno, Local methods in nonlinear differential equations, Spriuger-Verlag, Berlin (1989) 8. P.B. Kahn and Y. Zarmi, Physica D 54, 65-74 (1991) 9. P.B. Kahn, D. Murray and Y. Zarmi, Roy. So(, Proc. A (19932 10. R.H. Rand and D. Armbruster, Perturbation methods, bifurcation theory and computer algebra, Springer Verlag, NY (1987) 11. H.H. Band and W.L. Keith, Normal tbrm and center manifold calculations on MACSYMA in Applications of computer algebra, B. Pavelle, ed., 309-328, Klumer Academic Publishers, Boston (1985) 12. P,J. Holmes, Physica D 2, 449-481 (1981) 13. J.D. Crawford, Bey. Mod. Phy. 63, no.4,991-1037 (1991)
Pergamon
NORMAL FORM INVESTIGATIONS OF DISSIPATIVE SYSTEMS
Diana M urray Dept. of Physics, SUNY Stony Brook Stony Brook, NY 11794 emaih [email protected]
(Received 17 September 1993; acceptedfor print 26 October 1993)
Introduction
We are interested in studying dissipative systems that are perturbed harmonic oscillators:
dx --=y+e
G(x,y;e)
dt
dy = - x + e H ( x , y ; 8) , dt
(I)
[el
using the method of normal forms [1,2,3,4,5] to develop an approximate solution as a perturbation expansion in a small parameter. With the method of normal forms one seeks a polynomial near-identity coordinate transformation in which the dynamical system takes the simplest form. The normal form is the equation of motion for the zeroth-order approximation and contains only resonant terms (i.e. terms with the same phase as the zeroth-order term). The method of normal forms has a well-known "lack of uniqueness" [6,7] that led Kahn and Zarmi [8] to develop the method of minimal normal forms (MNF's) which we have applied to conservative systems [9]. Kahn and Zarmi have shown that a judicious choice of the zeroth-order approximation can produce a normal form which is far simpler than the one that results from traditional normal form analysis ("usual" normal forms). Here we show that this freedom of choice in tbe normal form expansion can be utilized for dissipative systems, producing compact equations of motion which capture the full characteristics of the flow early on and with great ease of calculation. Dissipative systems are characterized dynamically by the contraction of the phase space area with increasing time onto an attractor of lower dimension. For flows in two dimensions the attractor can be either a fixed point or a limit cycle. The introduction of computer algebra [10,11] enables one to develop algorithms so as to obtain perturbation expansions to high order where the merit of MNF's will be shown to be most striking.
We illustrate the essential elements of the method of normal forms by analyzing a system with cubic damping, i.e. G = 0, H = _y3 in eq (1). We then show how it is applied to the pendulum with cubic damping and to two examples taken from the text by Rand and Armbruster [10]. This treatment is easily generalized to systems 231
232
D. MURRAY
with other nonlinearities. Consider the tbllowmg:
dr
--=y
dt dy_ dt
(2)
x_e>,~ '
x(O) = a, y ( 0 ) = b,
cl
Introducing the diagonalizing transformation, z = x + i y , eq (2) becomes
d~
1 z + - ~" ( z - z * ) ~ 8
--:-i
dt
(3)
where the asterisk denotes complex conjugation. The quantity z is expanded in terms of the zeroth-order approximation, u, in a near-identity transformation:
z=u+e r,(u,u*)+E ~ ~(u,u*)+e'
..
(4)
3 U3(II,I4*)+ .
(5)
r~(u,u*)+
and u satisfies the normal tbrm equation:
du act
-
J u+g
UI(U,U*)+E
2 U2(tt,u*)+£
In the method of normal forms, the ~ , which are polynomials in U and u *, are chosen to annihilate all the non-resonant terms in cq (3). The normal form o f e q (5) will generally contain resonant monomials of the form u"+lu *", with n _> 1 , in each order of the small parameter, g. To determine the J~ and the U i , one
substitutes eqs (4) and (5) into eq (3) and collects terms in each order of oe:
U1 = - i
7~ + 67~ i u -
(l 2 = - J
~ +
U3 = - i
7; +
3u
or~ 6u
o~ 6u
OT, u , - O ~ u 6u
-
&~ ,
I lt-
,
1
,
d'll i U + - ~ ( U - U ) cTu *
dT2i u*+
(u-u*) 2 (7-~*)-~-uU~
dr~ l , 3 i u - Ou "* ' ,, + ~ ( u -
2
,
du*
'
du *
u*) 2 ( l; - T, *) + ~ ( u - u * ) ( ~ - ~ *):
o~i (I~- 6T--~eu * 6u
(6)
NORMAL FORM OF DISSIPATIVE SYSTEMS
233
Note that in eqs (6) the ~ are determined in each order up to a free additive term of the (resonant) form
F . = f . (uu*) u , for which the quantity :
OF.
- i F. +C~F" i u iu*. c3u Ou*
known as the Lie bracket, vanishes. There is no contribution fiom F . to the order n calculation, but, as we will show, it strategically appears in order n+l. In the usual method of normal forms, one chooses F n = 0 for all n. The method of minimal normal forms attempts to make use of the free term from order n to annihilate the resonant contribution in order n+l. Hence one has the possibility of U i = 0 for i > 1 and, in general, the minimal normal form has a more compact structure than the usual normal form. Let us see how this is manifested in our example. The T polynomials that eliminate all the non-resonant terms in equations (6) are, through third order in~? :
2 u , + - - ui T~ = t - •- u 3 - - - 3i u 16 16 32
9
T2 -
U5 +
1024
,3+f~(uu,)u
9 u 4 u * + - - r3i ,u 256 16"'
27 / / / / , 4 3i ,//,3 1024 + -j-~ f t
3i 27 u2u ,3 - ]"g ( f l + 2 f l *)uu ,2 512
3-
3
~/ .s
+ f : (uu*)u
+ 512
3i 279i 6 45 5 (f~9 2048 u7 + 32768 u u * - - -1024 f , u ~' + 16384333iuSu.2+ 256
T3 -
+
8 li 16384
U3U , 4 _
27 256
(f~ +3 f ,)u:u,, - ~3i ~(f2
117i 27 ( l f ~ + f . ) u u , 4 + 32768 u2u*5 - 2 5 6
9/ 8129
u*' +A(uu*)u
+4fl ,)U4U, +
(f12 +f2)u3
+ 2 f 2 * +2f~f~ * + f l *'-)uu*:
27i UU , 6 + 15 + 3i ( _ . ~ )'2 * + f l "2) u*3 + 4096 512
fl
*U
5
234
D. MURRAY
This results in the following normal form:
du_ dt
iu-g- 3 u:u*+g 2 (
(fl-fl*)
u2u*+ 2 7 i u ~ u , 2 )
8
256 (7a)
+~.3 (
.
3(2f~ :+f~fl*-3L+L*) .
.
u2u*+ 27i(fl+f,) 128 .
3u,2
567 11411,3 ) 8192
The quantities F 1 = f l u and F 2 = f2 u , fiee functions from ]; and 7~ respectively, can tx' chosen to 9i annihilate the second and third order terms (f~ = - - - u u 64
,
, and ,~ =
675
u2u ,2 ). Similarly, higher
8192
order resonant contributions to the MNF are eliminated to yield the tbllowing expression correct to all orders:
MNF:
du
dt
-
3
iu--E 8
u2u*
(Vo)
Ill the "usual" case the normal tbrm includes resonant contributions at each order. Through sixth order in ,5." . tbr example, one then obtains
Usual NF:
--=du - i u dt
3 .
~ 27
g -u"tt*+g 8
iu3u,2 --g
256
~ 567 ,3 __ lt4ll 8192
+g4
7965
jtl~ll,4
262144 (7c)
- c 5- - l62721 ~6U 4194304
,5 q-oe t'
214083
iltTU ,0
268435456
Writing u = p e ' ¢, eq (7b) gives
dp dt
-
3 3 gp and 8
dO dt
(8)
--=c0=l
which are both easily solved. Similarly. eq (7c) gives
dp dt
-
3 ~. p3 - - - g 567 8192
8
3 pT_ - 62721 - g ~ p 4194304
Ii
+O(~') (9)
and
- - = c o = l - - - ~ ' d ~27b dt 256
2p4
- - c 7 9 6 5 4p8 262144
214083 ~.~,p~2+0(c7) 268435456
Indeed it is found that all of the resonant contributions to eq (Tb) can be killed by free terms, whereas eq (7c) is continually updated.
NORMAL FORM OF DISSIPATIVE SYSTEMS
235
In dissipative systems there is frequency (co) as well as radius (19) updating. As can be seen in this example, the benefit of MNF's is twofold- to all orders the fundamental frequency of oscillation has no update,
4 =t+40
or
co=l,
(10)
and the radius equation has a simple closed-form solution,
P=~l+~6po2t Po
(ll)
On the other hand, the radius and phase equations for the usual case quickly become calculationally unwieldy. The constants/9 0 and 4 0 in eqs (10 &ll) are, for both methods, updated in each order and found by implementing the initial condition (x(0)=a, y(0)=b). Once the T~ are kaaownthrough the desired order of calculation, z (or equivalently, x) is reconstructed. By inverting the near-identity transformation (eq(4)), we find u(0)and u * (0) in terms of x ( 0 ) and y ( 0 ) . Then/9 0 and 4 0 are determined from the relation
u(0) = P0 exp(-i 40 ).
The
1A/1 + 3/4 Cpo2I dependence of,O(/)dominates the character of the normal
form approximate solutions. Both MNF and usual results are in good agreement with the numerical solution obtained from a fourth-order Taylor series procedure for positive and negative damping. Fig. la shows a comparison of the numerical solution with that obtained from a second-order M NF calculation for 6 = 3/4 and x(0)=l, y(0)=0. Fig. lb illustrates that there is a finite but small mismatch of the nomal form solution from the exact solution for short times, but for longer times the two become indistinguishable. In fig. 2, the numerical and MNF results are compared for negative damping ( oe = -1/20). From eq (11), the approximate solution is expected to blow up at
~25.
1
0.8
0.6
0.4
,4.0 0.2
AAAAAAAAAAAAAAAAAAAAAAAAA, A/VVVVVVVVVVVVVVVVVVVVVV'me 0
41.2
-0.4
41.6
FIG. la. Comparison of numerical and 2nd-order MNF solutions to eq (2) with positive damping displaying the long-time validity of the MNF approximation. ( oe = 3/4, x(0)=l, y(0)=0).
236
D. MURRAY
time 8
lo
:~ I° ~F--',~_.~. FIG. lb. Mismatch, M, of the MNF approximate solution depicted ira Fig. la obtained when the approximation is substitued back into the differential equation.
FIG. 2. Comparison of mlmerical and 2nd-order MNF solutions to the s)stem ofeq (2) with negatiw" 1
damping. ( ~ = - - 2 ~ , x ( O ) = l,y(O) = 0).
N O R M A L FORM OF DISSIPATIVE SYSTEMS
237
To be fair, MNF's cannot always give such a simple equation for the radius update, but in almost all of the cases we have studied it contained at most two terms, a result that holds to all orders. The usual normal form will generally contain an infinite number of terms. This distinction is important since, for instance, the normal forms of equations are very useful in bifurcation analyses [12,13].
One exception to this is when the damping
term is linear, since such a resonance can never be annihilated by a free term.
As an example of when the MNF terminates a~er two terms, we analyze the following system:
dx
m=y
dt
02)
dy = - x - c (x 3 + y3 ) dt By utilizing the free terms in the MNF expansion we can eliminate the imaginary part of the second order resonance and all higher-order (n > 2) resonant terms. The MNF and usual radius and phase equations are determined by heeding the procedure outlined above: MNF's:
UsualNF's:
dp 3--_ p3
3 c2 p5 3
3-5
e2p5 . . 3 . oo3p . 7 171 +~'4 p 9 2061
d - ~ - P = - t p 3 3. at 8
32
4096
32768
_ ~ 5 p l l __153339 [_~6 p13 1216185__ + O ( z 7) 1048576
MNF's:
Usual NF's:
--= d#
1+8p2_
3
'
dt
8
d#= l+cp2
3_c2p 8
dt
8388608
4 39 + 6 3 p 6 . 375 . 128 4096
+ e 6 p 12 3237459 - - + O ( e 16777216
6.4 p 8 . 4623 65536
6s p 10 17187 262144
7)
We comment that by implementing the method of M NF's for the pendulum with cubic damping d2x
dx 3
dt 2 + ( ~ - )
+sinx=0
,
238
D. MURRAY
we obtain a similar result. Here, the resonant contributions of all the nonlinearities greater than cubic can be annihilated yielding radius and phase equations that are qualitatively the same as those for the system ofeq (12). Considering only small amplitudes, we Taylor series expand the sine function arid introduce the scaling, X
) -~
X, to obtain the equation of motion:
dT+X+e{(
)3
}+
6
....
2
e . (_1). ~ _0 (2n+l)!
The M N F radius and phase equations are
dp
epS3+s:
dt
8
p
_
_
64
2Z dt
16
As can be seen, the M N F results, exact to all orders, are a great simplification and improvement over the usual results.
In the spirit of generality we show that similar results hold for a system with general quadratic, cubic, quartic and quintic nonlinearities. The fbllowing is based on an example taken from Rand and Armbruster [10, p55fl]:
dx
~-=y+e(A~.
x 2 + A , xy + A.w y 2 ) + ~'2 (A:o:~ x"s + A,:v, x'-y+ A.~, xy 2 +Am' yS)
+83(Axa~
x 4 +A~:~, x3y+...)+84(Ax.oa~ x 5 +Aax ~, x a y + . . . ) (13)
d y = - x + e ( B , ~ , x 2 +B dt
xy+B., y2)+o~2 ( B "
+e3(Bx~
x ~+B
y x3y+...)+~
x 3 +B~., x 2 y + B , ~ xy 2 +B m, yS) "
•
'"
4( B . . . . x 5 +B~,:~ x x y + . . . )
where A and B are complex constants. Note that the systems o f e q s (2) and (12) are special cases of this one. We find that the method of MNF's yields an equation of motion of the same form as the normal form ofeq(12):
du -
-
itl+oo2ii2tt*C+84tlstl*2D
dt where the constants C and D are fimctions of the constants, A ,=~,..., B ~ ..... from eq (13). This M N F result is exact to all orders, while the usual normal form is updated in each even order.
NORMAL FORM OF DISSIPATIVE SYSTEMS
239
This analysis is also extendable with some restrictions to higher dimensions. Another example from Rand and Armbruster [10, p 28ff] depicts a feedback control system modeled by three differential equations:
dr ~-=y ay --
dt dw dt
---- - - X - -
--
£aX'W
W + £t~X
2
One might seek to reduce this to a two-dimensional problem via the method of center manifolds [10, chp 2] wherein only information about the steady state is obtained. Applying the method of MNF's to the full set of equations, we can derive simple transformed equations of motion and track the transient motion as well.
MNFresult:
du
__
:
_ i u _ e 2 ~ 0 ( 1 - - ~l -l )i ,u 2u . - e 4 - -73 a" 2000
tl311,2
dt dq =_q+ c2 a-~-uu*q dt 10
where u and q are the zeroth-order approximations to the variables z = x + iy and w. This result is exact to all orders.
The hope in performing an analytical calculation is to derive a qualitative understanding of the dynamics of the original system from the transformed, simplified equation of motion (the normal form). We have shown examples of dissipative perturbed harmonic oscillator systems where the method of M NF's yields simpler, more compact equations of motion than the traditional NF method. Conceptually and computationally, it is desirable to have the radius and phase equations fixed early in the calculation. In many cases one can exploit the freedom associated with the method of MNF's to tailor the normal form to match a specific goal- for example, to extend dynamical intuition, to track the transient motion or to simply obtain closed-form expressions which characterize the motion. We are in the progress of extending this analysis to parametric systems, and plan to report on these investigations in the near future.
Acknowledgements: I would like to express my appreciation to Professor Peter B. Kahn and Professor Yair Zarmi for their help and encouragement, to Professor Richard H. Rand for kindly reading the manuscript and to the Institute for Pattern Recognition for financial support.
240
D. M U R R A Y
References 1. V.I. Arnold, Geometrical methods in the theory of ordinaU diflbrential equations, Springer-Verlag, NY (1988) 2. J. Guckenheimer and p.j. Holmes, Nonlincar oscillations, dynamical systems and bithrcations ofveetor fields, Sprinwr-Verlag, NY (1983) 3. S. Wiggins, Introduction to applied nonlinear d}namical systems and chaos, Springer-Verlag, NY (1990) 4. A.H. Nayfeh, Method of normal forms, Wiley-Interscience, NY (1993) 5. L. Jezequel and C.H, Lawrence, J. Sound Vib. 149,429-459 (1991) 6. A. Baider and B.C. Churchill, Proc. R. Soc. Edinburgh See. A 108, 27-33 (1988) 7. A.D. Bruno, Local methods in nonlinear differential equations, Spriuger-Verlag, Berlin (1989) 8. P.B. Kahn and Y. Zarmi, Physica D 54, 65-74 (1991) 9. P.B. Kahn, D. Murray and Y. Zarmi, Roy. So(, Proc. A (19932 10. R.H. Rand and D. Armbruster, Perturbation methods, bifurcation theory and computer algebra, Springer Verlag, NY (1987) 11. H.H. Band and W.L. Keith, Normal tbrm and center manifold calculations on MACSYMA in Applications of computer algebra, B. Pavelle, ed., 309-328, Klumer Academic Publishers, Boston (1985) 12. P,J. Holmes, Physica D 2, 449-481 (1981) 13. J.D. Crawford, Bey. Mod. Phy. 63, no.4,991-1037 (1991)