Nonlinear dynamics: A tutorial on the method of normal forms Peter B. Kahn and Yair Zarmi Citation: Am. J. Phys. 68, 907 (2000); doi: 10.1119/1.1285895 View online: http://dx.doi.org/10.1119/1.1285895 View Table of Contents: http://ajp.aapt.org/resource/1/AJPIAS/v68/i10 Published by the American Association of Physics Teachers Additional information on Am. J. Phys. Journal Homepage: http://ajp.aapt.org/ Journal Information: http://ajp.aapt.org/about/about_the_journal Top downloads: http://ajp.aapt.org/most_downloaded Information for Authors: http://ajp.dickinson.edu/Contributors/contGenInfo.html Downloaded 05 May 2013 to 198.91.37.2. Redistribution subject to AAPT license or copyright; see http://ajp.aapt.org/authors/copyright_permission
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Nonlinear dynamics: A tutorial on the method of normal formsPeter B. Kahn and Yair Zarmi Citation: Am. J. Phys. 68, 907 (2000); doi: 10.1119/1.1285895 View online: http://dx.doi.org/10.1119/1.1285895 View Table of Contents: http://ajp.aapt.org/resource/1/AJPIAS/v68/i10 Published by the American Association of Physics Teachers Additional information on Am. J. Phys.Journal Homepage: http://ajp.aapt.org/ Journal Information: http://ajp.aapt.org/about/about_the_journal Top downloads: http://ajp.aapt.org/most_downloaded Information for Authors: http://ajp.dickinson.edu/Contributors/contGenInfo.html
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Nonlinear dynamics: A tutorial on the method of normal formsPeter B. KahnDepartment of Physics and Astronomy, State University of New York at Stony Brook, Stony Brook,New York 11794
Yair Zarmia)
Department of Energy and Environmental Physics, The Jacob Blaustein Institute for Desert Research,Ben-Gurion University of the Negev, Sede Boqer Campus, 84990, Israel
~Received 11 August 1999; accepted 8 February 2000!
The undergraduate physics curriculum is usually restricto the study of linear systems. However, it is clear that,pecially with the introduction of computer algebra softwasuch asMAPLE and MATHEMATICA ,1,2 many nonlinear prob-lems can be discussed with tools available to studentshave completed their junior year of study. There exist vgood numerical techniques to analyze aspects of nonlinmotion. Numerical and analytical methods complement eother. It is with this in mind that we have written a tutoriarticle, designed to show how one analyzes simple nonlinproblems by an analytical perturbation technique duePoincare´,3 known as ‘‘the method of normal forms’’~NF!.4–9
This approach opens up the possibility of a richer investition that combines numerical and analytical methods. Nmal forms have been successfully applied to a broad strum of problems, including oscillation and vibration9
astronomy,10,11 accelerator design,12–15 nuclear magneticresonance,8 fluid flow,16 and optics.17 We illustrate the effec-tiveness of the normal form expansion by using it to obtperturbative solutions for a broad class of nonlinear differtial equations. Before the development of symbolic algecomputer programs, calculations were generally carriedonly through first or second order, and methods such asav-eraging and multiple scaleswere used extensively. Higherorder calculations involve significant algebraic complexand really cannot be done by hand. However, with the advof readily available computer algebra software the calcutions are straightforward to organize, and the fundamesimplicity and beauty of the normal form expansion leadsits appeal.
Our basic goal is to show how the method of normal forallows one to develop a consistent perturbation expanthat, for a given class of problems, captures the esseelements of the solution. In carrying out this program,will have to re-think many of our standard proceduresmost of our experience is based on linear problems andtend to use techniques that worked there in treating nonlinproblems. Often this leads to disappointment. For exam
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in contrast to linear systems, in a series solution for a nlinear problem, the coefficients are not ‘‘determined finallyat a given order of the expansion. They are usually updaas one computes higher-order terms. Also, the principlesuperposition is no longer valid. With this noted and reconizing that nonlinear systems encompass an enormous rof phenomena, it is perhaps best to begin by study‘‘weakly nonlinear systems.’’ These are systems in whithe basic or unperturbed problem is linear and solvablewhich a small nonlinear term proportional toe has beenadded.
Our procedures then lead to perturbation expansionsfollow the true motion with a given error for a reasonabperiod of time. The time validity can be a delicate issue aleads to an extensive error analysis. We avoid dwelling oby restricting our consideration to generating approximsolutions that have an error no bigger thanO(e2t). Thismeans that if the timet5O(1), the error is O(e2), and ift5O(1/e), the error is O(e). The basic aspects of thmethod of normal forms are fully revealed at this levelapproximation and the algebra is not too oppressive. Wthis understood, we will terminate all our expansions aftermost,O(e2) and neglectO(e3) effects. Those interested inmore extensive analysis as well as a detailed error anaare referred to the literature.4–6,8,18
We begin, in Sec. II, with the basic ideas of the methodnormal forms that originated in Poincare´’s Ph.D. thesis andwere further developed in the early years, primarilyBirkhoff and subsequently by Arnold, Siegel, and otheThe fundamental idea of normal forms is that, with propattention given to the structure of the spectrum of the linterms in a nonlinear differential equation, one introduce‘‘formal’’ near-identity transformation, in the neighborhooof a fixed point, to reduce the nonlinear flow to its most baelements, yielding equations that are more amenableanalysis. The resulting equation is said to be in its ‘‘normform.’’ A critical component involves separating the nonlinear differential equations under consideration into classespending on the location of the eigenvalues, in the compplane, of the unperturbed linear problem. The normal fo
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analysis for the simpler case, when the eigenvalues liePoincare´ domain, is treated after the introduction of the idof ‘‘resonance.’’ Then, we continue the developmentturning our attention to the much richer case when the eigvalues lie in a Siegel domain. The simplest paradigm for ttype of problem is discussed in Sec. IV, when we presentbasic equations, for the cubic oscillator known as ‘‘the Dfing oscillator.’’ We give explicit expressions for the displacement including numerical comparisons. In orderbuild confidence we illustrate the method in Sec. V throuthe analysis of six different types of problems, each illustring a different aspect of the method of normal forms. Tvarious systems can be separated into five classes.
~1! Systems in which all the eigenvalues of the unpturbed linear problem have negative real part. The eigenues are said to lie in aPoincaredomain, and the normal formand the perturbation expansion have a particularly simcharacter leading to a convergent expansion~examples 1–3!.
~2! Systems in which the unperturbed problem is a sinsimple harmonic oscillator which is perturbed by a changethe potential, so that the system remains conservative.phase curves are changed, but remain closed. Alternativthe harmonic oscillator can be perturbed by dissipative teleading to damped motion or limit cycles~examples 4–8!.
~3! A problem in which there is one zero eigenvalue athe rest have negative real part. This problem is amenaba normal form analysis, and this technique may, for thevestigator’s purposes, prove to be far superior to themethodof the center manifold.19 The latter method is only effectiveafter the transients have decayed and is limited to initial aplitudes that are ‘‘very close’’ to the fixed points. By way ocontrast, the method of normal forms with only slightly moalgebraic complexity is applicable to problems that havenite initial amplitudes, and it can follow the transient behaior until the system enters a region of finite-time blowuphas decayed onto the center manifold~example 9!.
~4! We briefly treat coupled nonlinear oscillators. Thproblems are inherently much more complex, and a thoroanalysis is beyond the scope of this text.8 However, it isstraightforward to obtain the first- and second-order appromations and, for particular systems, to obtain some genresults~example 10!.
~5! Nonautonomous systems have the capacity to exhvery complex behavior. We restrict ourselves to a discussof these systems in a region where one still has regulartion that is describable by a perturbation treatment. Thiscludes, of course, chaotic motion. However, it does covebroad range of behavior~examples 11 and 12!.
We conclude, in Sec. VI, with a brief introduction tonew integral formulation of the method of normal forms, fperturbed one-dimensional oscillatory systems, tsmoothly connects with the method of Poincare´–Lindstedt.20
II. NORMAL FORM EXPANSION: GENERAL IDEAS
We begin by considering an equation of the form
d
dtx5Ax1eF~e,x!, ~1!
wherex is an n-dimensional vector andA is a diagonaln3n constant matrix, with eigenvaluesl1 ,l2 ,...,ln . Theperturbation is writteneF(e,x), wheree is a small param-eter, ueu!1, andF is an n-dimensional vector field whosterms are polynomials or functions with Taylor series exp
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sions inx of degree at least one. The vectorx may containcoordinates, velocities, and possibly the time. If the forexplicitly contains the time, we say that the system is noautonomous. If it is free of the time, it is said to be autonmous. We discuss, in the examples, both classes of syst
As a first step, refer to the unperturbed linear componof our equations and plot the location of the eigenvaluesthe matrixA in the complex-l plane. If they all lie such thatone can construct a straight line through the origin so tthey fall on one side, we say that the eigenvalues lie i‘‘Poincaredomain.’’ If such a construction is impossible, wsay that the eigenvalues lie in a ‘‘Siegel domain.’’ When teigenvalues lie in a Poincare´ domain, there are only a finitenumber of possible resonance relations and consequethere can never be an ‘‘eigenvalue update.’’ The eigenvalassociated with the original variables are the same as fornormal form variables and it then follows that the associaequations can be solved sequentially. If the eigenvalues lia Siegel domain, one encounters much greater calculaticomplexity and there is the possibility that there will beinfinite number of resonant relations and eigenvalue updcan occur.
Let us begin by considering an example based on Eq.~1!.Consider a three-dimensional nonlinear system of the folloing form:
dxi
dt5l ixi1 (
n51
N
ena i , jkmn x1
j x2kx3
m ,
i 51,2,3, j 1k1m>2, ueu!1. ~2!
The three eigenvalues are taken to be distinct and the sparameter is used to order the terms in the expansion.
In the method of normal forms, we introduce a smootransformation, a ‘‘near identity transformation~NIT!,’’ ofvariables to a new set that yields equations that are easisolve. The transformation is written as
xi5ui1(n
enTni ~u1 ,u2 ,u3!. ~3!
The zero-order approximations are denoted byui . The Tfunctions, the generators of the transformation, dependthe ‘‘u variables.’’ Whene vanishes, the transformation becomes the identity transformation.
The T functions will be constructed to eliminate as materms in Eq.~2! as possible. There is freedom in the choiof these functions, but as this is an introductory article,will not discuss this aspect of the expansion and instead rthe reader to the literature.8,21–25 The u’s satisfy their owndynamical equation,
d
dtui5U0
i 1(n
enUni , U0
i 5l iui . ~4!
Observe that, except for its lowest order component,U0i
5l iui , the right-hand side of Eq.~4! is not known. We willsoon show how to construct an expansion in which all coponents are known, and hence it will be possible to solvetheu’s. It is critical to solve theu equation~the normal form!as it constitutes a ‘‘skeleton’’ equation, and the entire dy-namics is revealed by ‘‘dressing it up.’’
If we know theu’s, we can construct the perturbation bsuccessively constructing the generators. To gain someperience, we begin with Eq.~2!, considering the equation fothe first variable through second order:
908P. B. Kahn and Y. Zarmi
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dx1
dt5l1x11 (
n51j 1k1m>2
N
ena1,jkmn x1
j x2kx3
m , ~5a!
x15u11eT111e2T2
1, ~5b!
d
dtu15U0
11eU111e2U2
1. ~5c!
Notation: The components of the vectors are given as sscripts. In theT’s andU’s, superscripts refer to componenand subscripts refer to the order of the perturbation.
Substitute the NIT given by Eq.~5b!, utilizing the dynami-cal equation given by Eq.~5c!, into both sides of Eq.~5a!and retain terms through second order.
The left-hand side:
d
dtx15U0
11eU111e2U2
11eS U01
]T11
]u11U0
2]T1
1
]u21U0
3]T1
1
]u3D
1e2S U01
]T21
]u11U0
2]T2
1
]u21U0
3]T2
1
]u3D
1e2S U11
]T11
]u11U1
2]T1
1
]u21U1
3]T1
1
]u3D . ~6a!
The right-hand side:
l1$u11eT111e2T2
1%1e (j 1k1m>2
a1,jkm1 u1
j u2ku3
m
1e2 (j 1k1m>2
a1,jkm1 $ jT1
1u1j 21u2
ku3m1ku1
j T12u2
k21u3m
1mu1j u2
ku3m21T1
3%1e2 (j 1k1m>2
a1,jkm2 u1
j u2ku3
m . ~6b!
Subtract Eq.~6b! from Eq. ~6a! and identify the terms thaare proportional to different powers ofe. In O(e0), we ob-tain
U015l1u1 . ~7!
Collecting all the first-order terms, we obtain
U115l1T1
12S U01
]T11
]u11U0
2]T1
1
]u21U0
3]T1
1
]u3D
1 (j 1k1m>2
a1,jkm1 u1
j u2ku3
m . ~8!
Define the ‘‘Lie bracket’’~LB! as
@U0 ,T1#1[(i 51
3 S T1i
]U01
]ui2U0
i]T1
1
]uiD . ~9a!
In our example, the matrixU is diagonal. Therefore, usinEq. ~7!, Eq. ~9a! has the simpler form
@U0 ,T1#15l1T112S U0
1]T1
1
]u11U0
2]T1
1
]u21U0
3]T1
1
]u3D . ~9b!
Equation~8! can now be written as
U115@U0 ,T1#11 (
j 1k1m>2a1,jkm
1 u1j u2
ku3m . ~10!
The termU01 is the zero-order term that is associated w
the unperturbed linear problem of the first component.~This
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follows because the associated eigenvalue is different fzero.! We will now see under what conditions we can compute theU functions, beginning withU1
1. ~The superscriptassociated with the square bracket indicates the compoassociated with each term. Here it is ‘‘1,’’ indicating that ware finding the first component. The Lie bracket playssame role in this approach as does the Poisson brackclassical mechanics.!
In O(e2), we find
U215@U0 ,T2#12S U1
1]T1
1
]u11U1
2]T1
1
]u21U1
3]T1
1
]u3D
1 (j 1k1m>2
a1,jkm1 $ jT1
1u1j 21u2
ku3m1ku1
j T12u2
k21u3m
1mu1j u2
ku3m21T1
3%1 (j 1k1m>2
a1,jkm2 u1
j u2ku3
m . ~11!
Observe that the second-order LB,@U0 ,T2#1, has thesame structure as the first-order LB. This pattern persistall orders. We postpone a consideration of the second-oterms until Sec. IV, where we discuss motion in a consertive field of force.
In the problems we consider, the interaction is a polynmial in its arguments and with this information, we wichoose theT functions to have a similar structure.Choosingthe T functions to mimic the interaction leads to a singcondition on the structure of the Lie bracket, and consquently determines the entire structure of the perturbatexpansion. If you decide to stop at first order, you needsolve for the explicit form of the T functions. But, youneed them when you do a higher order calculation. It is thisform that plays a critical role in determining which termsthe expansion can be eliminated. Referring to our interacgiven by Eq.~2!, we write
T115 (
j 1k1m>2b1,jkmujvkwm. ~12!
First order: Let us now look at the LB for the first component of the vector:
@U0 ,T1#15 (j 1k1m>2
b jkm1 u1
j u2ku3
m$l12~l1 j 1l2k1l3m!%.
~13!
Equation~10! now becomes
U115 (
j 1k1m>2b jkm
1 u1j u2
ku3m$l12~l1 j 1l2k1l3m!%
1 (j 1k1m>2
a1,jkm1 u1
j u2ku3
m . ~14!
To simplify the normal form, Eq.~4!, one wants to con-structT1
1 such thatU11 is made to vanish. However, from Eq
~14!, we see that this is not possible for terms for which
l15l1 j 1l2k1l3m. ~15!
Equation~15! is called a ‘‘resonance relation’’~note that theintegers,j , k, m must each be greater than or equal to zer!.
We eliminate in Eq.~14! monomialsu1j u2
ku3m for which
Eq. ~15! is not satisfied, by choosing
909P. B. Kahn and Y. Zarmi
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def
nr
oainas
r oio
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s:
th
n-
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lue
thon,
cted
sily
xi-n re-
istheri-ngng
ferm
b jkm1 52
a1,jkm1
$l12~l1 j 1l2k1l3m!%. ~16!
The remaining terms in Eq.~14! are attributed toU11 and
constitute theO(e) update of the normal form, Eq.~4!.
III. RESONANCE: POINCARE´ AND SIEGELDOMAINS
We now give examples of resonance, followed by thexamples where the eigenvalues lie in a Poincare´ domain andnine where the eigenvalues fall in a Siegel domain. Adtional examples and further discussion are found in R4–9 and 18.
Definition of resonance: A resonance occurs whenever aeigenvalue,ls , is a linear combination with positive integecoefficients, involving the other eigenvalues,
ls5m1l11m2l21¯msls1¯mnln , ( mi.1.
~17!
A. The Poincaredomain
In general, the eigenvalues of the unperturbed linear prlem are complex numbers. They are said to lie in a Poinc´domain, if in the complex plane one can draw a straight lthat separates all the eigenvalues from the origin. The bideas are embodied in the Poincare´–Dulac theorem,4 whichstates that, under such conditions, only a finite numberesonant relations are possible. The normal form equatcan then be solved sequentially and hence the problemreduced to the construction of generators with known arments.
Illustration:~a! l153; l255; l357. Using Eq.~15! we see that there
is no resonant combination.~b! l151; l252. Only one resonant combination occu
in the l2 equation whenm152, m250.~c! l152; l256; l3514. The least eigenvalue,l152,
cannot yield a resonance. The next larger eigenvalue,l2
56, can be formed from the combinationl253l1 . Finally,the greatest eigenvalue has three resonant combinationl3
57l1 ; l354l11l2 ; l35l112l2 .We show how one uses the resonance condition to find
dynamical equation for theu variable.Example 1: Given the equation with two negative eige
values,
x52x1ex2y,
y522y1e$x21xy%, ~18!
l1521, l2522.
As we can draw a straight line through the origin wiboth eigenvalues on one side, the eigenvalues lie in a Pcaredomain. The unperturbed equation is in diagonal forIntroduce the near identity transformation
x5u1 (n>1
enTn~u,v !, y5v1 (n>1
enSn~u,v ! ~19a!
and a normal form
du
dt5U01 (
n>1enUn ,
dvdt
5V01 (n>1
enVn . ~19b!
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i-s.
b-reeic
fnsis-
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Construct theT and S functions to mimic the following in-teraction:
T15 (k,m>0
k1m.1
bk,m1 ukvm, S15 (
k,m>0k1m.1
gk,m1 ukvm. ~20!
Referring to Eq.~10!, the Lie bracket takes the form
@U0 ,T1#u5H l1•T12U0
]T1
]u2V0
]T1
]v J5H 21•T111•u
]T1
]u12•v
]T1
]v J5 (
k,m>0k1m.1
bk,mukvm$211k12m%. ~21!
It is clear that there are no values ofk or m for which theLB vanishes. ~The casem50, k51 is not acceptable.!Hence, one can chooseT functions such that allUn , n.0,will vanish. One then has
du
dt5U052u. ~22!
This equation is solved byu5u(0)exp(2t).Now, construct the LB associated with the eigenva
equal to 2. We have
@U0 ,S1#v5H l2•S12U0
]S1
]u2V0
]S1
]v J5H 22•T111•u
]S1
]u12•v
]S1
]v J5 (
k,m>0k1m.1
gk,mukvm$221k12m%. ~23!
The only term that can vanish in this LB is the one wik52, m50. Hence, if such a term appears in the interactiits contribution to the analog of Eq.~10! in the present casecannot be eliminated. Our example has been construwith ak52, m50 term~i.e.,x2 in they equation!, so that theresonant term is present. Our skeleton equation is easolved.
As the generators,T andS, have as their argumentsu andv, we can now use the NF expansion to obtain an appromation to the desired accuracy. The procedure has beeduced to algebra for which a symbolic computer programideally suited. The details regarding how one recoverssolution to a given problem, in terms of the original vaables, are illustrated in detail in the analysis of the Duffioscillator given in Sec. IV and in many of the succeediexamples.
Example 2: Consider
x52&x1e~nlt!,
y522&y1e~nlt!, ~24!
z523&z1e~nlt!.
We use~nlt! to indicate nonlinear polynomial terms odegree two or higher. There is no possible resonance tassociated with the eigenvalue52&; there is one possibleterm associated with the eigenvalue522&; and there are
910P. B. Kahn and Y. Zarmi
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three possible terms associated with the eigenvalue23&.Thus, after introducing the NIT from variables~x,y,z! to~u,v,w! and considering the associated generating functito be polynomials that mimic the form of the nonlinear iteraction, our skeleton equations will have the following geeral form:
u52&u,
v522&v1eAu2, ~25!
w523&w1eBuv1e2Cu3.
The coefficientsA, B, Cdepend on the detailed structurethe nlt, in Eq. ~24!. There are no more possible resonaterms. We solve these equations sequentially to obtain
u~ t !5u~0!exp~2&t !,
v~ t !5$v~0!1eAu~0!2t%exp~22&t !, ~26!
w~ t !5H w~0!1~eBu~0!v~0!1e2Cu~0!3!t
1e2B
2u~0!3At2J exp~23&t !.
Example 3: Consider
x5x2e$x212xy%, y52y2e$2y21xy%. ~27!
Introduce the NIT from~x,y! to ~u,v!. Observe that, asthere is nox2 term in they equation, the only possible resonance in thev equation (u2) is absent. Therefore, our skeeton equations are the following unperturbed linear eqtions:
u5u, v52v. ~28!
B. The Siegel domain
If the eigenvalues do not lie in a Poincare´ domain, they aresaid to lie in a Siegel domain.
Illustration: l153; l252; l3521. It is straightforwardto see that there are an infinite number of resonant combtions for eachof the eigenvalues. For example,
Additional resonant terms are easily constructed.In the study of Hamiltonian systems we encounter pr
lems in which the eigenvalues are purely imaginary. Forample, the simple harmonic oscillator has eigenvalues,l1
51 i andl252 i . There are an infinite number of possibresonant terms. For any integern, one hasl151 i 5(n11)i 2ni andl25 i 5ni2(n11)i .
These situations are much richer. They lead to ‘‘eigenvue update’’ and also have the capacity to generate chaorbits. All the examples considered hereafter are associwith problems whose eigenvalues lie in a Siegel domain
IV. THE DUFFING OSCILLATOR
In conservative systems with a single degree of freedthe period,T, and, hence, the fundamental frequency,v, are
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affected by the nonlinearity. Specifically,v becomes differ-ent from the unperturbed frequency,v0 : It depends on thesmall parameter,e, and on the amplitude of oscillations,a.The NF expansion amounts to the parallel evaluation, orby order, of two expansions ine. The first is the expansion othe displacement,x(t), and the second—of the updated frquency,v(e,a). We demonstrate the procedure through tDuffing oscillator, which is a linear harmonic oscillator, sujected to a cubic perturbation:
d2x
dt21x1ex350, x~0!5A, x~0!50. ~30a!
We first convert Eq.~30a!, a second-order differentiaequation, into a two-dimensional vector equation by intducingy[ x:
d
dt S xyD5S 0 1
21 0D S xyD2eS 0
x3D , y[ x. ~30b!
It is traditional to diagonalize the matrix of the lineaproblem by introducingz5x1 iy , and its complex conjugatez* 5x2 iy . Our equation becomes
z52 iz2 i e~z1z* !3
8. ~31!
The equation forz* is just the complex conjugate of Eq~31!.
We introduce a near-identity transformation from the vetor (z,z* ) to a vector (u,u* ) and generators that are functions of u andu* :
z→u1 (n51
enTn~u,u* !, z* →u* 1 (n51
enTn* ~u,u* !.
~32!
A. Essential points
The near identity transformation has a variety of aspec~1! The quantity ‘‘u’’ is the ‘‘zero-order approximation.’’
Its dynamical equation is called the ‘‘normal form:’’
du
dt5U01 (
n51enUn~u,u* !, U052 iu. ~33!
~We need only consider the dynamical equation for the vable, u, as u* , its complex conjugate, obeys the complconjugated equation.! In generating the time dependencethe zero-order term, the normal form accounts for the upding of the fundamental frequency,v(e,a). That there is afrequency update is manifested by the fact that not allUi ,i.0 can vanish.
~2! TheT functions carry information regarding the highharmonics as well as the updating of the coefficients offundamental component. They depend on (u,u* ). Once wehave solved the dynamical equation foru to the desired or-der, we can insert it into the expressions of theT functionsand obtain an approximation toz(t). One tries to choose theT functions so as to make the normal form as simplepossible. There is freedom in the choice of theT functionsbecause of the inherent freedom in the choice of the zpoint about which to perform an expansion. For examplewe choose theT functions all equal to zero, we have thidentity transformation. In this article, to keep things
911P. B. Kahn and Y. Zarmi
cense or copyright; see http://ajp.aapt.org/authors/copyright_permission
e
ethdiths
nte
ha
nauil
atrdms
y
es
osethepe-
e.
ice
and
simple as possible, we do not discuss the ‘‘freedom issuwhich is discussed at length in the literature.8,21–25
~3! The ‘‘resonance condition’’ is a key component of thnormal form expansion. This relationship, satisfied byeigenvalues of the unperturbed linear problem, wascussed in Sec. III, when we introduced the concept ofPoincare´ and Siegel domains. In the case of the Duffing ocillator, for all integersn.0, the two eigenvalues,l152 iand l251 i , obey the resonance relationship,i 5(n11)i1n(2 i ). To each integer,n, there corresponds a resonaterm of the form@(uu* )nu#. Resonant terms may contribuin every order of the expansion of the normal form.
Construction of the latter amounts to identifying whicresonant terms actually appear in every order of the expsion.
~4! The NIT is generally not convergent, but is aasymptotic series. This needn’t worry us as we usually cculate a few terms in the expansion and a satisfactory bofor the error. The convergence issue is discussed in detathe literature.4,8,18
Important: In this article we terminate the expansionsecond order and do not indicate the error term. Higher oterms follow a similar pattern and with the advent of coputer algebra programs one can develop the expansion aas desired.
B. Normal form expansion for the Duffing oscillator
The procedure is implemented in a series of steps.Step 1: Refer to Eq.~31!, and substitute the near-identit
transformation~NIT! given by Eq. ~32!. Expand the left-hand-side throughO(e2):
dz
dt5
du
dt1eS ]T1
]u
du
dt1
]T1
]u*du*
dt D1e2S ]T2
]u
du
dt1
]T2
]u*du*
dt D5$U01eU11e2U2%1eS ]T1
]uU01
]T1
]u*U0* D
1e2S ]T2
]uU01
]T2
]u*U0* D1e2S ]T1
]uU11
]T1
]u*U1* D .
~34a!
Step 2: Now substitute the expansion forz given by Eq.~32! into the right-hand side of Eq.~31!:
dz
dt52 i $u1eT11e2T2%2 i e
~u1u* !3
8
2 i e23~u1u* !2
8~T11T1* !. ~34b!
Step 3: Equate Eqs.~34a! and ~34b!. Collecting thee0
terms, we findU052 iu, which embodies the eigenvaluequation for the unperturbed problem, given here in termthe u variable. But,du/dtÞ2 iu unless allUi50, i .0. Weset up and solve, order by order in the small parameter,e, aseries of equations for theUi .
Step 4: Organizing Eqs.~34a! and ~34b! we have
U15H 2 iT12U0
]T1
]u2U0*
]T1
]u* J 2 i~u1u* !3
8. ~35!
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,’’
es-e-
t
n-
l-ndin
er-far
of
The structure of the term in braces in Eq.~35! is independentof the interaction. It is the Lie bracket~LB! for this problem.In every order, the LB term is of the following form:
H 2 iTn2U0
]Tn
]u2U0*
]Tn
]u* J . ~36!
We postulate for theT functions a form that mimics thepolynomial structure of the perturbation. In thenth order,Tn
are polynomials of power (2n11) in u andu* :
Tn5 (k,m>0
k1m52n11
akmuku* m. ~37!
Substituting Eq.~37! in Eq. ~36!, we observe that the LBtakes the form
@U0 ,T1#u52 i (k,m>0
k1m53
akmuku* m@12k1m#. ~38!
We can kill all the terms in Eq.~35!, except those of theform u2u* . So, in the implementation of the normal formexpansion, we have reduced the problem to solving for thT functions that kill all nonresonant terms associated withinteraction and then solving a dynamical equation of a scial form.
Step 5: Substitute Eq.~37! into Eq. ~35! and solve for thea coefficients so as to ‘‘kill’’ as many terms as possiblSolving we find
a3051
16, a1252 316, a0352 1
32,~39!T15 1
16 u32 316 uu* 22 1
32 u* 31a21u2u* .
Important: The coefficienta21 has a ‘‘zero multiplier’’and hence is undetermined. We follow the traditional choof setting a2150, which Bruno18 calls ‘‘the distinguishedchoice.’’ Other choices can be used, and lead to compactcomputational effective approximations.24,25
Step 6: So, by choosing ourT functions with the coeffi-cients given by Eq.~39! and substituting into Eq.~35!, ourfirst-order correction todu/dt is given by
U152 i 38u
2u* ,
~40a!du
dt52 iu2 i e
3
8u2u* .
If we write u5r exp(2iw), we have
dr
dt50,
dw
dt52 iv52 i S 11e
3
8r2D ,
~40b!u5r exp~2 i @11e 3
8r2#t !.
The radius,r, is constant, and the frequency,v, has been‘‘updated.’’
Step 7: At this level of approximation, we have
z5u1e~ 116 u32 3
16uu* 22 132 u* 3!,
~41a!z* 5u* 1e~ 1
16 u* 32 316 u2u* 2 1
32 u3!,
x~ t !5~z1z* !
25r cosS F11e
3
8r2G t D
1er3~2 316 cos~ t !1 1
32 cos~3t !!. ~41b!
912P. B. Kahn and Y. Zarmi
cense or copyright; see http://ajp.aapt.org/authors/copyright_permission
rm
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ow
eecto
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We have not updated the arguments of the first-order teas there is already a factore as a multiplier.
A central aspect of the method is the ability to updatefrequency, accounting for the effect of the perturbation. Ththe full oscillatory nature of the solution is expressed in eaorder of the expansion.
Step 8: We calculate the second-order correction to shthe general structure of the expansion. From Eqs.~34a! and~34b!, we obtain
U25H 2 iT22]T2
]uU02
]T2
]u*U0* J
1S 2]T1
]uU12
]T1
]u*U1* 2 i
3~u1u* !2
8~T11T1* ! D .
~42!
As we knowT1 , given by Eq.~39!, andU1 , given by Eq.~40a!, we can computeT2 andU2 :
T25 31024u52 15
256 u4u* 1 69512 u2u* 31 21
1024uu* 42 1512 u* 5,
U25 i 51256 u3u* 2, ~43!
v511e 38r
22e2 51256 r4.
With the computation ofT2 andU2 , our expression forx(t)becomes
x~ t !5r cos~@11e 38 r22e2 51
256 r4#t !
1er3~2 316 cos~@11e 3
8 r2#t !1 132 cos~3@11e 3
8r2#t !!
1e2r5~ 69512 cos~ t !2 39
1024cos~3t !1 11024cos~5t !!. ~44!
Comment on the solution: The phase is updated, but thradius,r, remains constant throughout the expansion, refling energy conservation. The results can be generalizedwider spectrum of interactions, some of which we will stuin the succeeding examples.
C. Satisfaction of the initial conditions
The relation between the radiusr and the turning pointx(0)5A is updated as we include more terms in the expsion. Begin with Eq.~41b!. This yields
x~0!5A5r~12e 532 r2!. ~45a!
Proceeding to next order, referring to Eq.~44!, we have
x~0!5A5r~12e 532 r21e2 25
256 r4!. ~45b!
Inverting Eqs.~45a! and~45b! for r, we see that, as we gto higher orders, the expression forr incorporates higherorder terms in the expansion.
D. Numerical comparison
We chooseA5p/3 ande521/6, the values considered ithe literature. Note that the expansion parameter in the Dfing problem is reallyeA252p2/54. The exact solution tothe Duffing equation is given as a Fourier expansion intext by Davis,26 pages 291–297. It is
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s
esh
t-a
-
f-
e
x~ t !51.054 09 cos~vt !20.006 94 cos~3vt !
10.000 046 cos~5vt !1hot,~46!
v50.928 447.
~Higher order terms in the series expansion are denoted‘‘hot.’’ ! Referring to the relationship betweenr and A, andour parameter values, Eqs.~41b! and ~44! are written as
O~e!: x~ t !51.050 19 cos~vt !20.005 48 cos~3vt !,
v~approximate!511e 38 A250.9315, ~47a!
r5A~11e 532 A2!51.0173.
O~e2!: x~ t !51.053 32 cos~vt !20.006 62 cos~3vt !
10.000 029 cos~5vt !,
v~approximate!511e 38 A22e2 21
256 A450.9287, ~47b!
r5A~11e 532 A22e2 25
1024A4!51.016 4.
Notice that the Fourier coefficients are updated as we cpute higher order terms in the expansion. This is characistic of nonlinear problems.
We comment in passing that a particular choice of the fa21 term inT1 and corresponding choices of the free termshigher orders yield what Kahn and Zarmi24 have called a‘‘minimal normal form.’’ The latter displays closer agreement between the approximate and true solution thanobtains with other choices. We do not discuss the issue offree terms in the normal form expansion, as it takes us tooafield. However, it is important when shifting focus froanalytical to numerical methods, particularly in the analyof higher-dimensional systems. When numerical and anacal techniques are combined, minimal normal forms provebe a powerful approach.27
E. Error accumulation
A simple test of the quality of a perturbation expansionthe degree to which the approximation deviates from thesolution. A first-order normal form approximation eliminatespurious secular terms of the formet which, if present, gen-erate a linearly growing error over time intervals ofO(1/e).~A term that combines a power of ‘‘t’’ multiplied by a trigo-nometric function is called a ‘‘secular term.’’ We introducthe word ‘‘spurious’’ to indicate that it is an artifact of thexpansion and is not present in this form in the full solutioThe word ‘‘secular’’ comes from the French word for cetury and indicates that the effect of the term in astronomiproblems is important at these long times.! However, a first-order approximation cannot, in general, guarantee thesence of higher order secular terms, e.g., of the forme2t. Toeliminate the latter, one must extend the normal form anasis to second order, increasing the time validity of the expsion. To demonstrate this point, we show in Fig. 1 the mmatch between the full solution of Eq.~30a!, with A5p/3ande521/6, and the first- and second-order approximatiogiven by Eqs.~47a! and ~47b!, respectively.
913P. B. Kahn and Y. Zarmi
cense or copyright; see http://ajp.aapt.org/authors/copyright_permission
einngtioousean
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V. EXAMPLES ILLUSTRATING ASPECTS OF THEMETHOD OF NORMAL FORMS
In this section we study problems in which the eigenvaluof the linear unperturbed problem lie in a Siegel domaNow that we have completed the analysis of the Duffioscillator, we observe that the first-order skeleton equafor many systems can be found by inspection, i.e., withcalculating theT functions. This is a very nice result becauit enables you to just look at the structure of an equationobtain some information about the system’s behavior.~In thefollowing, we give the expressions for a fewT functions toillustrate aspects of the expansion and to facilitate the calation of higher-order terms.!
A. Other conservative oscillators
Example 4: The quintic oscillator. When the perturbatiois an odd power of the displacement, the first-order equatcan be found by inspection. Consider
d2x
dt21x1ex550, x~0!5A, x~0!50. ~48a!
Referring to our experience with the Duffing oscillator, wknow that, in first order, we can construct aT function thatkills all the terms except the one of the form@u3u* 2#, whosecoefficient is the binomial coefficient in the expansion of tquintic perturbation. The amplitude is constant and the phis given by
w5@11e 516 A4#t. ~48b!
Example 5: The quadratic oscillator. In this case, the peturbation is an even power and hence the frequency updapostponed until second order,
d2x
dt21x1ex250, x~0!5A, x~0!50,
~49!z52 iz2 i e
~z1z* !2
2.
There are no resonant terms in first order, so we can cstruct aT function that kills all the terms:
U15H 2 iT11 iu]T1
]u2 iu*
]T1
]u* J 2 i~u1u* !2
4,
~50a!T15 14 u22 1
2 uu* 2 112 u* 2⇒U150.
Proceeding to second order, we find
Fig. 1. Mismatch,m, between the full solution of Eq.~30a!, A5p/3,e521/6, and the NF approximations.~a! Mismatch inO(e) approximation;~b! mismatch inO(e2) approximation.
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s.
nt
d
u-
ns
se
-is
n-
T25 124 u31 5
24 uu* 22 148 u* 3,
~50b!U25 i 5
12 u2u* ⇒v5$12e2 512 A2%.
Theuu* term in the first-orderT function indicates a shiftin the zero point of oscillation, as a consequence ofasymmetry of the potential. The oscillator does not speequal amounts of time in each quadrant of the motion.
B. Examples with dissipation
Example 6: We discuss a simple harmonic oscillator wia small cubic damping term. In first order, there is only aplitude damping. Frequency updating occurs in second orThe equation is
d2x
dt21x1e x350, 0,e!1. ~51!
We are interested in the change in the behavior ofsystem for relatively short times, to lowest order in the smparametere. Don’t perform any calculation! Rather, refer tthe following argument. We transform to thez variables andthen introduce the NIT. As the perturbation is cubic, we ptulate a cubic polynomial for theT1 function. Inspection ofthe LB shows that theT1 function can kill all the termsexcept for the one of the form@u2u* #. Thus,U1 takes theform
U152 38 u2u* . ~52!
Our skeleton equation is
u52 iu2e 38 u2u* ,
~53a!u5r exp~2 iw!, r52e 3
8r3, w50,
which is solved by
u5r0
A11e 34r0
2texp~2 i t !. ~53b!
The radial component is damped and, in this order, therno change in the phase. Observe how easy it is to getleading term in the expansion. TheT functions are the sameas for the Duffing oscillator, Eq.~39!, except for multiplica-tive factors, as one is killing terms of the same characHowever, as the factors contain minus signs and ‘‘i’s’’ youcannot take them over without calculation. We find that27
T15 116 iu32 3
16 iuu* 21 132 iu* 3. ~54!
C. Limit cycles
Example 7: Van der Pol28 developed some of the seminarguments that led to a systematic attack on limit-cycle prlems, which play a significant role in nonlinear dynamics.@A‘‘limit cycle’’ is a closed curve in the ‘‘(velocity, position)plane’’ or ‘‘phase plane’’ that acts as an attractor for outward and inward going spirals.# Van der Pol studied multi-vibrator circuits in which the resistive element alteredcharacteristics as a function of the current. The circuit exhited sustained oscillations that were stable against smallturbations, leading to the concept of negative-feedbackplifiers. Sometimes oscillators of this type are called seexcited oscillators because, if the system initially has a v
914P. B. Kahn and Y. Zarmi
cense or copyright; see http://ajp.aapt.org/authors/copyright_permission
tfo
ve
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lueudeof
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ionuf-ur-orni-otices,ndo-
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ro-
small amplitude~i.e., energy!, it draws upon its environmenso as to increase its amplitude. The differential equationthe Van der Pol oscillator is
d2x
dt21x5e@12x2# x, 0,e!1,
~55!
z52 iz1e~z2z* !
2 S 12~z1z* !2
4 D .
If we begin with small amplitudes, we have negatidamping and the oscillations grow. Then, asx crosses 1, thecoefficient of the velocity changes sign, and the amplitustarts to decrease. But, as the amplitude crosses 1, agadamping factor changes sign. So, we see a dynamic interbetween positive and negative damping. Eventually,leads to a stable ‘‘limit cycle’’ of radius approximately equto 2. Trajectories associated with amplitudes greater thaare attracted to the limit cycle from without and trajectorassociated with amplitudes less than 2 are attracted to it fwithin.
Refer to Sec. III and review aspects of the argumentssociated with Poincare´ and Siegel domains. In the Van dePol problem, the eigenvalues of the linear system lie inright half-plane, at (e6 i ). As a straight line that separatethem from the origin can be drawn, one might conclude tthe eigenvalues lie in a Poincare´ domain and apply thePoincare´–Dulac theorem. However, in doing this, the lineterm now includes a part of the perturbation which is coparable in size to the nonlinear perturbation. This approleads to a nonuniform NF expansion. Thus one cannotclude the lineare term in the unperturbed equation. Consquently, this is a Siegel domain problem, as the unpertureigenvalues are located at6 i .
A first-order NF analysis gives the basic results withoudetailed calculation. We introduce thez variables, followedby the NIT. The underlying unperturbed system is the simharmonic oscillator, leading to a LB resonance conditionwhich terms of the form@uku* m# for k5m11 cannot bekilled by theT function. The algebra is straightforward anin the original perturbation, we have two such terms:k51,m50; k52, m51. Our skeleton equation becomes
u52 iu1eS u
22
u2u*
8 D . ~56a!
We introduceu5r exp(2iw) and have
w50, r5e1
2rF12
r2
4 G ,~56b!
T15 i 14 u* 2 i 1
16 u32 i 116 uu* 22 i 1
32 u* 3.
The radial equation has two stationary solutions:r50 andr52. A stability analysis of the radial equation reveals ththe origin is an unstable stationary point: The solutionrepelled away from it. On the other hand,r52 is an attrac-tor. An ‘‘exchange of stability’’ occurs between the repusion from the origin as the amplitude grows and the attrtion to the finite limit cycle radius.
Example 8: The Rayleigh oscillatorAnother limit cycle oscillator has been studied by Lo
Rayleigh to explain forced vibrations. Rayleigh29 developedan approximation procedure for handling maintained vibtions that resulted from the interplay of negative and posit
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etheayis
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m
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-h-
-d
a
en
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-
-e
damping terms. He mentions that this situation occurs ivariety of systems, including organ pipes, violin stringelectromagnetic tuning forks, etc. It is closely related toVan der Pol oscillator. The basic equation is
d2x
dt21x5eF12
1
3x2G x, 0,e!1,
~57a!z52 iz1e
~z2z* !
2 S 11~z2z* !2
12 D .
Everything proceeds in parallel to the Van der Pol case. Tfirst-order equation for the radius of the limit cycle is idetical to that of the Van der Pol oscillator, but theT functionhas a different structure:
u52 iu1eS u
22
u2u*
8 D ,
u5r exp~2 iw!, w50, r5e1
2rF12
r2
4 G , ~57b!
T15 i 14 u* 1 i 1
48 u32 i 116 uu* 21 i 1
96 u* 3.
Again, we have limit cycle oscillations with an amplitudequal to 2 in the zero-order approximation.
The analysis and results obtained in examples 4–8 cataken over almost unchanged into more complicated prlems. However, it takes some practice to recognize thetern of the perturbation expansion.
D. A zero eigenvalue problem
Example 9: We discuss a system with one zero eigenvaand one negative eigenvalue. One finds that the amplitassociated with the zero eigenvalue decays as a powert,and the amplitude associated with the negative eigenvdecays as a modified exponential function oft. If one waits‘‘long enough’’ so that the transient aspects of the solutbecome unimportant, the exponential will have decayed sficiently and the long-time motion then takes place on a sface~i.e., manifold! associated with the zero eigenvalue. Fproblems of this class, the ‘‘method of the center mafold’’ 19 has been developed. It only captures the asymptbehavior. However, the method of normal forms appliwithout modification, to problems with zero eigenvalues ayields the transient behavior, as well as the asymptotic mtion over the center manifold. Thus, if you are willing twork through the algebra, it seems sensible to use NFcapture the time evolution of the entire solution. As an eample, consider the equations found in the textManneville:16
dx
dt52
1
2exy, 0,e!1,
~58!dy
dy529y1
1
2ex2.
We proceed as before. The near-identity transformation tathe form
where we have distinguished the two generators,S and T.We limit ourselves to a second-order calculation. The ze
915P. B. Kahn and Y. Zarmi
cense or copyright; see http://ajp.aapt.org/authors/copyright_permission
av
,er
e
outingval-
l-
er
by
dice
cal-s
p-
order approximations satisfy dynamical equations,
u5U01eU11e2U2 , U050,~60!
v5V01eV11e2V2 , V0529v.
Substituting the NIT in thex equation yields
u5U01eU11e2U2
521
2e$uv1euS11euT1%2eU0
]T1
]u2eV0
]T1
]v
2e2U0
]T2
]u2e2V0
]T2
]v2e2U1
]T1
]u2e2V1
]T1
]v. ~61a!
We have a companion equation from they equation,
v5V01eV11e2V2
529v29eS129e2S21 12e$u212euT1%
2eU0
]S1
]u2eV0
]S1
]v2e2U0
]S2
]u
2e2V0
]S2
]v2e2U1
]S1
]u2e2V1
]S1
]v. ~61b!
Collecting theO(e) terms in theu equation yields
U1521
2uv2U0
]T1
]u2V0
]T1
]v52
1
2uv19v
]T1
]v. ~62!
We see that the choice
T15uv18
~63!
leads toU150. Similarly, we have, in thev equation,
V1529S111
2u219
]S1
]v. ~64!
We see that the choice
S15u2
18~65!
leads toV150.Returning to Eq. ~61a!, inserting the values for
T1 , S1 , U1 , V1 , we solve forU2 :
U2521
2 H 1
18u31
u2v18 J 19v
]T2
]v52
1
36u3. ~66a!
Solving for V2 , we have
V2529S21u2v18
19v]S2
]v5
u2v18
. ~66b!
It is straightforward to solve forT2 andS2 :
T251
2•182 uv2, S250. ~66c!
Constructing the dynamical equations, and solving, we h
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e
u52 136 e2u3, v529v1 1
18 e2u2v,
u~ t !5u~0!
A11 118e
2u~0!2t, ~66d!
v~ t !5v~0!exp~29t !ln$11 118 e2u~0!2t%.
Finally, we recover the solutions,x(t) and y(t,), with anerror O(e3):
x~ t !5u1 118 euv1 1
648 e2uv21O~e3!,~67!y~ t !5v1 1
18 eu21O~e3!.
Observe thatu(t) decays as an inverse power,not expo-nentially, while v(t) decays exponentially. Asymptoticallywhen the exponential,v(t), has died out, one is on the centmanifold associated with theu variable. ‘‘The method of thecenter manifold’’ concentrates on uncovering this long-timbehavior.19
One sees that the normal form analysis can be carriedfor a problem with one zero eigenvalue and the remainones negative, just as for problems with nonzero eigenues.
E. Phase locking
Example 10: We introduce a model of two-coupled oscilators, introduced by Mitropolskii and Samoilenko,30 that ex-hibit ‘‘phase locking,’’ a phenomenon encountered in lasphysics:
x11x15ez1x11e~12x1x2!x1 ,~68a!x21x25ez2x21e~12x1
2!x2 , e.0.
The frequency mismatch is incorporated by thez terms. Theequations are analyzed in Ref. 8. We proceed as beforeintroducing thez variables, obtaining
z152 iz11 12 i ez1~z11z1* !1 1
2 e~z12z1* !
2 18 e~z1
22z1*2!~z21z2* !,
~68b!z252 iz21 12 i ez2~z21z2* !1 1
2 e~z22z2* !
2 18 e~z1
212z1z21z1*2!~z22z2* !.
We introduce the NIT,
z15u11eT11~u1 ,u2!, z25u21eT1
2~u1 ,u2!. ~69!
The superscript on theT functions indicates the associatecomponent. All terms can be killed by an appropriate choof the T functions, except those of the form (u1
n11u1*n)
and (u2n11u2*
n) and mixed monomials of the form(u1
pu1*qu2
r u2*s), with (p2q1r 2s)51. However, limiting
ourselves to a first-order calculation, there is no need toculate theT functions. The first-order normal form equationare
u152 iu1~12ez1!1 12 eu12 1
8 eu12u2* ,
~70!u252 iu2~12ez2!1 12 eu21 1
8 eu12u1* 2 1
4 eu1u1* u2 .
To exhibit the phase locking, we introduce the polar reresentation:
u15r exp~2 iw!, u25r exp~2 iq!, V5w2q.~71!
916P. B. Kahn and Y. Zarmi
cense or copyright; see http://ajp.aapt.org/authors/copyright_permission
be
er-
q
r
ip-e
tio
ck
te
nis
.
der
-
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-
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ce,onicdic
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ith
Our dynamical equations become
r5 12 er2 1
8 er2r cos~V!, w512 12 ez11 1
8 err sin~V!,
r 5 12 er 2 1
8 er2r @112 sin2~V!#, ~72a!
q512 12 ez22 1
8 er2 sin~2V!.
As a direct consequence of the equality of the unperturfrequencies, we have a single independent equation forV:
V5~ w2q !52 12 e~z12z2!1 1
8 er@r sin~V!1r sin~2V!#.~72b!
To find when ‘‘phase locking’’ occurs, i.e., when thphasesw andq vary at the same rate, with a constant diffeence between them, we need
V50. ~72c!
As a first step, we seek the fixed points. Referring to our E~72a! and ~72b!, we have
r50⇒cos~V!54
rr, r 50⇒112 sin2~V!5
4
r2 ,
~73!V50⇒~z12z2!5 1
4 @r2 sin~2V!1rr sin~V!#.
The values ofr, r, andV change as the frequency shiftD[z12z2 is varied. The fact that cos(V) is bounded yields
0<cos~V!<1⇒rr>4, 2p
2<V<1
p
2. ~74a!
The bound forr is found to be
0<sin2~V!<1⇒ 2
)<r<2. ~74b!
If the frequency shift is small, we obtain, to the leading oders inD,
V5 13 D, r52~12 1
9 D2!, r 52~11 16 D2!. ~75!
To arrive at the last result requires some algebraic manlation. Refer to Eq.~73! and expand the trigonometric functions throughO(V2). The result then follows. Thus we sethat asD moves away from 0,r decreases below 2 andrincreases above 2.
Observe that we have gained considerable informawithout solving for theT functions.
F. Nonautonomous systems
Example 11: Modified Rayleigh oscillatorConsider the Rayleigh oscillator in which the feedba
term is modified:
x1x5e x~12 x2 cos2 t !, 0,e!1. ~76a!
We transform the two-component nonautonomous sysinto a four-component autonomous one by introducingz andv variables:
z5x1 iy , z* 5x2 iy , v5exp~2 i t !, v* 5exp~1 i t !.~76b!
Thev variable is introduced to more easily identify resonaterms in the interaction.~Its equation is known and thereno near-identity transformation associated with it.! Watchhow the introduction of thev variable simplifies the analysisOur dynamical equations become
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d
s.
-
u-
n
m
t
z52 iz1e@ 12~z2z* !1 1
32~v1v* !2~z2z* !3#. ~76c!
Now introduce the NIT forz and for z* . Noting that v5exp(2it), the resonant terms that appear in the first-ornormal form are the following monomials:u, uu* 2v2,u2u* vv* , andu3v* 2. The skeleton equation is found without computing theT functions:
u52 iu1e@ 12 u1 1
32#$3uu* 2v226u2u* vv* 1u3v* 2%.~77!
Introducing polar coordinates and extracting explicitly tfast time dependence, we have
u5r exp~2 i t !exp~2 iw!,
r5e 12 r@11 1
8 r2$2 cos~2w!23%#, ~78!
w52 116er2 sin~2w!.
Observe thatw is the slowly varying part of the phase ofu.The fixed points are
~1!r50, ~2!w50, r258, ~3!w5p
2, r25
8
5.
~79!
The pointr50 is unstable. To examine the stability of thsecond fixed point, write
r2581j, w501h ~80!
and retain linear terms inj andh. We find
j52ej, h52eA8h, ~81!
indicating that this fixed point is locally stable. Now, examine the third fixed point:
r258
51j, w5
p
21h,
~82!j52ej, h51eA 8
125 h
and conclude that the third fixed point is locally unstable.r andw tend to the stable fixed point,u approaches the limitcycle radiusr5A8.
Example 12: The Mathieu equationThe Mathieu equation occurs in many areas of scien
e.g., solid state and plasma physics. It describes a harmoscillator whose frequency is modulated by a small perioperturbation:
x1x@v21e cos~2t !#50, ueu!1. ~83!
Often the equation is solved by numerical techniques,it is amenable to a normal form analysis. Let us assumethe unperturbed frequency,v, is close to an integer,n. In thisexample, we choosen51. The analysis for other values ofnfollows the same pattern. We introduce a small parametemas a measure of the closeness ofv to 1:
v2511m. ~84!
Equation~83! becomes
x1x@11m1e cos~2t !#50, 0,e!1, umu!1. ~85a!
Our first step is to diagonalize the unperturbed system wthe introduction of the variablesz and z* . Our dynamicalequation becomes
917P. B. Kahn and Y. Zarmi
cense or copyright; see http://ajp.aapt.org/authors/copyright_permission
mn
an
ecy
--
rmh
thede
tht-
-a
ofa-in-
l
ng
sing
r
.
z52 iz2 im
2~z1z* !2 i
e
2~z1z* !cos~2t !. ~85b!
Introducingn5exp(2it), we convert Eq.~85b! into an au-tonomous system with two degrees of freedom:
z52 iz2 im
2~z1z* !2 i
e
2~z1z* !~n21n* 2!, n52 iv.
~85c!
The NIT is constructed with two independent small paraeters,m and e. We construct only the first-order expansioand, hence, do not solve for theT functions,
The NIT has two indices, indicating the appropriate expsion parameter.
Keep in mind that the solution to the equation for thenvariables is known; hence there is no associated NIT. Idtifying all the terms that resonate with the natural frequenwe have as the skeleton equation
u52 iu2 i 12 mu2 i e 1
4 n2u* . ~87!
Writing u5 f exp(2it), we have
f 52 i 12 m f 2 i 1
4 e f * , f * 51 i 12 m f * 1 i 1
4 e f . ~88!
Equation ~88! is solved by exponentials of the formexp(lt) with l satisfying the equation
l21$ 12m%22$ 1
4e%250. ~89!
The solution is unstable forl2.0. For neutrally stable oscillations we needl2,0. The first zone of stability is therefore
umu,e
2. ~90!
VI. AN INTEGRAL EQUATION APPROACH TONORMAL FORMS FOR CONSERVATIVE SYSTEMSWITH ONE DEGREE OF FREEDOM
In this section we show how to construct the normal foexpansion by means of an integral equation. The methodyet to be fully developed, and we raise it here to givereader an opportunity to explore a path not yet well travelWe begin with the Duffing equation, written in terms of thz variables:
z52 iz2 i e~z1z* !3
8, z~0!5A. ~91!
We can convert this to an integral equation by moving‘‘ iz’’ term to the left-hand side, and introducing the integraing factor, exp(1it), that completes the integration and mimics the unperturbed angular frequency,v51. This leads to
z~ t !5A exp~2 i t !2 ie
8 E0
t
exp@2 i ~ t2t8!#~z1z* !3dt8.
~92!
It seems natural to substituteA exp(2it8) for z(t8) andA* exp(it8) for z* (t8) within the integral in order to construct a first-order approximation. However, as plausible
918 Am. J. Phys., Vol. 68, No. 10, October 2000
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-
-
n-,
ase.
e-
s
this is, it generates a spurious secular term of the form
2 ie
8A2A* @ t exp~2 i t !#. ~93!
In developing our alternative approach to the methodnormal forms, we exploit the information that the perturbtion updates the frequency. To illustrate the basic ideas,troduce the NIT through first order, allowing for anO(e2)error:
z5u1eT1~u,u* !, z* 5u* 1eT1* ~u,u* !. ~94!
The frequency is updated fromv51 to v511ea1 . Withthis in mind, Eq.~91! takes the form
dz
dt52 ivz1 i ea1z2 i e
1
8@z1z* #3. ~95!
We formally solve Eq.~95! by converting it to an integraequation of the form
z~ t !5z~0!exp~2 ivt !1 i E0
t
exp~2 iv@ t2t8# !
3$ea1z~ t8!2e 18@z~ t8!1z* ~ t8!#3%dt8. ~96!
Next introduce the near-identity transformation, obtainithroughO(e)
u~ t !1eT1@u~ t !,u* ~ t !#
5~u~0!1eT1@u~0!,u* ~0!# !exp~2 ivt !
1 i eE0
t
exp~2 iv@ t2t8# !H a1u21
8@u1u* #3J dt8.
~97!
Collecting theO(e0) terms, we find
u~ t !5u~0!exp~2 ivt !. ~98!
With Eq. ~98! for u(t), within the integral, the terms
a1u2 38u
2u* ~99!
are proportional to exp(2ivt8), and hence generate spuriousecular terms after integration. To kill the secular generatterm we must choosea15(3/8)u(0)u* (0). With thischoice, we evaluate the integral, obtaining
T1~u,u* !5T1@u~0!,u* ~0!#exp~2 ivt !
15
32vu~0!3 exp~2 ivt !
11
vu~0!3S 1
16exp~23ivt !
23
16exp~ ivt !2
1
32exp~3ivt ! D . ~100!
As this is anO(e) approximation, one should writev51 inall components ofT1 . However, to construct the next ordeterm,T2 , one has to take into account thee dependence ofv.For this reason, we have included thev dependence in Eq~100!. Note that the sum of the first two terms in Eq.~100!constitutes a free resonant term, of the formbu2u* . Withthis done, and re-writing our expression in terms ofu andu* , we have
918P. B. Kahn and Y. Zarmi
cense or copyright; see http://ajp.aapt.org/authors/copyright_permission
somtha
therceesn-
mbe
ee
doa
luaTdioti
Byavodallehpad
hen
re
e
y
al
nd
-
al
r,’’
a-
-
a
al
s
ar
m
nd
or-ct.
-
ter. A
s
sD
ra-
s
T1~u,u* !5 116 u32 3
16 uu* 22 132 u* 31bu2u* , ~101!
which is identical to theT function given in Eq.~39! thatwas obtained using the traditional method of normal formFor a conservative system of a single degree of freedresults obtained by the integral method and the usual meagree in all aspects. But, the approaches are differentsometimes it is of value to see the same thing from anoperspective. In the integral method, we do not introducdynamical equation for theu variable. Potential seculagenerating terms are identified and killed by frequencorrections. The generators play a passive role and sto ‘‘just fall out.’’ We note that the integral method ireally a straightforward variation of the method of Poicare–Lindstedt.20
VII. CONCLUSIONS
We have illustrated aspects of the method of normal foras applied to a variety of examples. When the unperturlinear system has its eigenvalues in a Poincare´ domain, thereare only a finite number of possible resonant terms, andgenvalue update is excluded. On the other hand, if the eigvalues of the unperturbed linear system lie in a Siegelmain, then an infinite number of terms in the expansion myield resonant combinations in the Lie bracket. Eigenvaupdate is possible, and the method of normal forms separthe resonant and nonresonant terms in the expansion.former becomes the dynamical equation for the zero-orapproximation, i.e., the skeleton equation. With this equatsolved, its elements become components of the generafunctions that flesh out the structure of the solution.studying a variety of simple perturbed systems, we hgained insight into the simplicity and beauty of the methof normal forms. The lowest order calculations are genereasy to do by hand and the higher order ones are amenabanalysis by a computer algebra program. The methodbroad applicability and the interested reader, with this preration, should be able to initiate original investigationswell as to pursue some of the more technical aspectscussed in the references.
ACKNOWLEDGMENTS
We thank Diana Murray for her critical comments on tmanuscript. In addition, we have benefited greatly from ahave incorporated the incisive comments of one of ourerees.
a!Also at: Physics Department, Ben-Gurion University of the Negev, BeSheva, Israel.
1R. H. Rand and D. Armbruster,Perturbation Methods, Bifurcation Theorand Computer Algebra~Springer-Verlag, New York, 1987!.
2R. H. Rand,Topics in Nonlinear Dynamics with Computer Algebra~Gor-don and Breach, Amsterdam, 1994!.
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.,
odndera
ym
sd
i-n--yetesheernng
e
lyto
asa-sis-
df-
r-
3H. Poincare´, New Methods of Celestial Mechanics, originally published asles Methodes Nouvelles de la Mechanique Celeste~Gauthier-Villars, Paris,1892, AIP Press, New York, 1993!.
4V. I. Arnold, Geometrical Methods in the Theory of Ordinary DifferentiEquations~Springer-Verlag, New York, 1988!, 2nd ed.
5J. Guckenheimer and P. J. Holmes,Nonlinear Oscillations, DynamicalSystems, and Bifurcations of Vector Fields~Springer-Verlag, New York,1988!.
6S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems aChaos~Springer-Verlag, New York, 1990!.
7A. Nayfeh,Method of Normal Forms~Wiley, New York, 1993!.8P. B. Kahn and Y. Zarmi,Nonlinear Dynamics~Wiley, New York, 1998!.9V. M. Starshinskii,Applied Methods in the Theory of Nonlinear Oscillations ~Mir, Moscow, 1977!.
10G. I. Hori, ‘‘Theory of General Perturbations with Unspecified CanonicVariables,’’ Publ. Astron. Soc. Jpn.18, 287–296~1966!.
11A. Deprit, ‘‘Canonical Transformations Depending on a ParameteCelest. Mech.1, 12–13~1969!.
12A. J. Dragt and J. M. Finn, ‘‘Lie Series and Invariant Functions for Anlytic Symplectic Maps,’’ J. Math. Phys.17, 2215–2227~1976!.
13A. J. Dragt and E´ . Forest, ‘‘Computation of Nonlinear Behavior of Hamiltonian Systems using Lie Algebraic Methods,’’ J. Math. Phys.24, 2734–2744 ~1983!.
14E. Forest and D. Murray, ‘‘Freedom in Minimal Normal Forms,’’ PhysicD 74, 181–196~1994!.
15E. Forest,Beam Dynamics: A New Attitude and Framework~Harwood,Amsterdam, 1998!.
16P. Manneville,Dissipative Structures and Weak Turbulence~Academic,New York, 1990!.
17A. J. Dragt, ‘‘Lie-Algebraic Theory of Geometrical Optics and OpticAberrations,’’ J. Opt. Soc. Am.72, 372–379~1982!.
18A. D. Bruno, Local Methods in Nonlinear Differential Equation~Springer-Verlag, New York, 1989!.
19J. Carr,Applications of Centre Manifold Theory~Springer-Verlag, NewYork, 1981!.
20P. B. Kahn,Mathematical Methods for Scientists and Engineers: Lineand Nonlinear Systems~Wiley, New York, 1990!.
21M. Kummer, ‘‘How to Avoid Secular Terms in Classical and QuantuMechanics,’’ Nuovo Cimento Soc. Ital. Fis., B1, 123–148~1971!.
22J. C. Liu, ‘‘The Uniqueness of Normal Forms via Lie Transformations aits Application to Hamiltonian Systems,’’ Celest. Mech.36, 89–104~1985!.
23A. Baider and R. C. Churchill, ‘‘Uniqueness and Non-Uniqueness of Nmal Form Expansions for Vector Fields,’’ Proc. R. Soc. Edinburgh, SeA: Math. 108, 27–33~1988!.
24P. B. Kahn and Y. Zarmi, ‘‘Minimal Normal Forms in Harmonic Oscillations with Small Nonlinear Perturbations,’’ Physica D54, 65–74~1991!.
25P. B. Kahn, D. Murray, and Y. Zarmi, ‘‘Freedom in Small ParameExpansion for Nonlinear Perturbations,’’ Proc. R. Soc. London, Ser443, 83–94~1993!.
26H. T. Davis,Introduction to Nonlinear Differential and Integral Equation~Dover, New York, 1962!.
27P. B. Kahn and Y. Zarmi, ‘‘Reduction of Secular Error in Approximationto Harmonic Oscillator Systems with Nonlinear Perturbations,’’ Physica118, 221–249~1998!.
28B. Van der Pol, ‘‘On Oscillation Hysteresis in a Simple Triode Genetor,’’ Philos. Mag.43, 700–719~1922!.
