P e r g a m o n

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

NORMAL FORM INVESTIGATIONS OF DISSIPATIVE SYSTEMS

Diana M urray Dept. of Physics, SUNY Stony Brook Stony Brook, NY 11794 emaih murray@mathlab.sunysb.edu

(Received 17 September 1993; accepted for 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

[ e l < l

(I)

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

d y _ x _ e > , ~ ' dt

x(O) = a, y ( 0 ) = b, c l < l .

(2)

Introducing the diagonalizing transformation, z = x + i y , eq (2) becomes

d~ 1 - - : - i z + - ~" ( z - z * ) ~ dt 8

(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 ' r~(u ,u*)+ .. (4)

and u satisfies the normal tbrm equation:

du

act - J u + g U I ( U , U * ) + E 2 U 2 ( t t , u * ) + £ 3 U3(II , I4*)+ . (5)

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 feq (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:

67~ i u - U1 = - i 7~ + 3u

, 1 , d'll i U + - ~ ( U - U ) cTu *

o r ~ (l 2 = - J ~ + I l t -

6u dT2i u*+ (u -u* ) 2 ( 7 - ~ * ) - ~ - u U ~ du* ' d u *

o~ U3 = - i 7; + i u -

6u

dr~ l , 3 " ' ,, + ~ ( u - u*) 2 Ou *

( l; - T, *) + ~ ( u - u * ) ( ~ - ~ *):

OT, u , - O ~ u , o~i 6u - &~ , 2 6u

(I~- 6T--~e u * (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 - i u * .

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~? :

• 3i 2 , i , 3 + f ~ ( u u , ) u T~ = t - - u 3 - - - u u + - - u

16 16 32

9 9 4 3i 3 27 U 5 + u u * + - - r , u - T2 - 1024 256 1 6 " ' 512

3i u2u ,3 - ]"g ( f l + 2 f l *)uu ,2

27 3i 3 .s + f : (uu*)u / / / / , 4 , / / , 3 ~/

1024 + -j-~ f t + 512

3i 279i 6 45 5 T 3 - 2048 u7 + u u * - - - f , u + 32768 1024 ~'

16384333i uSu.2+ 256 ( f ~ 9 +4fl ,)U4U, + (f12 +f2)u3

+ 8 li 27 U3U ,4 _

16384 256 (f~ +3 f ,)u:u,, 3i - ~ ~ ( f 2 + 2 f 2 * +2f~f~ * + f l * ' -)uu*:

117i 27 ( l f ~ + f . ) u u , 4 + 3i ( _ . + 32768 u2u*5 - 2 5 6 ~ )'2 *+ f l "2) u*3 +

27i 15 UU , 6 + fl * U 5

4096 512

9/ u * ' + A ( u u * ) u

8129

2 3 4 D . M U R R A Y

This results in the following normal form:

3 d u _ i u - g - u : u * + g 2 ( ( f l - f l * ) u2u*+ 2 7 i u ~ u , 2 ) dt 8 256

+~.3 ( 3 ( 2 f ~ : + f ~ f l * - 3 L + L * ) u 2 u * + 2 7 i ( f l + f , ) 3u,2 567 . . . . 11411,3 ) 128 8192

(7a)

The quantities F 1 = f l u and F 2 = f2 u , fiee functions from ]; and 7~ respectively, can tx' chosen to

9i , 675 annihilate the second and third order terms (f~ = - - - u u , and ,~ = u2u ,2 ). Similarly, higher

64 8192

order resonant contributions to the MNF are eliminated to yield the tbllowing expression correct to all orders:

du 3 MNF: - i u - - E u2u* (Vo)

dt 8

Ill the "usual" case the normal tbrm includes resonant contributions at each order. Through sixth order in ,5." .

tbr example, one then obtains

3 . ~ 27 ~ 567 ,3 7965 Usual NF: --=du - i u - g - u " t t * + g - iu3u,2 - -g _ _ lt4ll +g4 j t l ~ l l , 4

dt 8 256 8192 262144

- c 5 62721 ,5 214083 ,0 - - l ~ 6 U q-oe t' i l tTU 4194304 268435456

(7c)

Writing u = p e ' ¢, eq (7b) gives

dp 3 3 dO - g p and - - = c 0 = l

dt 8 dt

which are both easily solved. Similarly. eq (7c) gives

(8)

dp 3 p3 567 3 pT_ 62721 - Ii - ~. - - - g - - g ~ p

dt 8 8192 4194304 + O ( ~ ' )

and - - = c o = l - - - ~ ' d ~ b 27 2p4 - - c 7 9 6 5 4p8 214083 ~.~,p~2+0(c7) dt 256 262144 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.

(9)

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,

Po (ll) P=~l+~6po2t 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 kaaown through 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 ( o e = -1/20). From eq (11), the approximate solution is

expected to blow up at 1

0.8

0.6

0.4

,4.0 0.2

0

41.2

-0.4

41.6

~25.

A AAAAAAAAAAAAAAAAAAAAAAAAA, /VV VVVVVVV VVVVVVV VVVVVV 'me

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. ( o e = 3/4, x(0)=l, y(0)=0).

236 D. MURRAY

:~ I° ~F--',~_.~. - 8

t ime lo

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).

NORMAL 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

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:

02)

MNF's : dp 3 c2 p5 3 3--_ p3 3-5

U s u a l N F ' s : d -~ -P= - t p 3 3 . . . . . e2p5 3 oo3p 7 171 +~'4 p 9 2061 at 8 32 4096 32768

_ ~ 5 p l l __153339 [_~6 p13 1216185__ + O ( z 7)

1048576 8388608

- - = 3 MNF's : d# 1 + 8 p 2 _ ' dt 8

3 _ c 2 p 4 39 6 3 p 6 375 6 4 p 8 4623 6s 10 17187 Usual NF's: d # = l + c p 2 + . . . . p dt 8 128 4096 65536 262144

+ e 6 p 12 3237459 - - + O ( e 7) 16777216

We comment that by implementing the method of M NF's for the pendulum with cubic damping

d2x dx 3 dt 2 + ( ~ - ) + s i n x = 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:

d T + X + e { ( )3 }+ e . (_1). ~ _ 0 6 . . . . 2 (2n+l)!

The MN F radius and phase equations are

dp e p S 3 + s : p _ _

dt 8 64

2Z dt 16

As can be seen, the MNF 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)

+ 8 3 ( A x a ~ x 4 +A~:~, x3y+...)+84(Ax.oa~ x 5 +Aax ~, x a y + . . . )

d y = - x + e ( B , ~ , x 2 +B xy+B. , y2)+o~2 ( B x 3 +B~., x 2 y + B , ~ xy 2 +B m, yS) d t " " • ' "

4 B x 5 + e 3 ( B x ~ x ~ + B y x 3 y + . . . ) + ~ ( . . . . +B~,:~ x x y + . . . )

(13)

where A and B are complex constants. Note that the systems o feqs (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 - - i t l+oo2i i2 t t*C+84t ls t l*2D

dt where the constants C and D are fimctions of the constants, A ,=~,..., B ~ . . . . . from eq (13). This MNF 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:

d r ~-=y ay - - ---- - - X - - £ a X ' W

dt dw

- - W + £ t ~ X 2

dt

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.

~0 l l i , 2 . 4 73 _ _ : t l 3 1 1 , 2 MNFresu l t : du _ i u _ e 2 ( 1 - - ~ - ) u u - e - - a " dt 2000

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

o r d e r s .

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. MURRAY

References

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