ComputationofthenormalformsforgeneralM-DOF ... · ComputationofthenormalformsforgeneralM-DOF systemsusingmultipletimescales.PartI:autonomoussystems PeiYu*,SonghuiZhu1 DepartmentofAppliedMathematics,FacultyofScience

www.elsevier.com/locate/cnsns

Communications in Nonlinear Science

and Numerical Simulation 10 (2005) 869–905

Computation of the normal forms for general M-DOFsystems using multiple time scales. Part I: autonomous systems

Pei Yu *, Songhui Zhu 1

Department of Applied Mathematics, Faculty of Science, The University of Western Ontario, WSC 123,

London, Ontario, Canada N6A 5B7

Received 24 April 2004; received in revised form 17 June 2004; accepted 17 June 2004

Available online 6 August 2004

Abstract

This paper is concerned with the symbolic computation of the normal forms of general multiple-degree-

of-freedom oscillating systems. A perturbation technique based on the method of multiple time scales, with-

out the application of center manifold theory, is generalized to develop efficient algorithms for systemati-

cally computing normal forms up to any high order. The equivalence between the perturbation technique

and Poincare normal form theory is proved, and general solution forms are established for solving orderedperturbation equations. A number of cases are considered, including the non-resonance, general resonance,

resonant case containing 1:1 primary resonance, and combination of resonance with non-resonance.

‘‘Automatic’’ Maple programs have been developed which can be executed by a user without knowing com-

puter algebra and Maple. Examples are presented to show the efficiency of the perturbation technique and

the convenience of symbolic computation. This paper is focused on autonomous systems, and non-auton-

omous systems are considered in a companion paper.

� 2004 Elsevier B.V. All rights reserved.

Keywords: Non-linear vibration; Normal form; Center manifold; Non-resonance; Resonance; Computer algebra;

Maple

1007-5704/$ - see front matter � 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.cnsns.2004.06.004

* Corresponding author. Tel.: +1 519 661 2111x88783; fax: +1 519 661 3523.

E-mail address: [email protected] (P. Yu).1 Present address: Department of Mathematics and Computer Science, Benedict College, Columbia, SC 29204, USA

mailto:[email protected]

870 P. Yu, S. Zhu / Communications in Nonlinear Science and Numerical Simulation 10 (2005) 869–905

1. Introduction

Nonlinear dynamical system theory is playing important role in almost all the areas of scienceand engineering because real world systems are indeed nonlinear. The theory has been applied tomechanics, aeronautics, electrical circuits, control systems, population problems, economics,financial systems, stock market, ecological systems, etc. In particular, during the past decadeapplications of the theory in solid and structural mechanics as well as fluid mechanics have ap-peared, and there is now widespread interest in the engineering and applied science communitiesin complex behavior such as bifurcations and chaos (for example, see [1–5]). While a relativelycomplete theory was developed for linear systems, nonlinear systems remained largely inaccessi-ble. Although various analytical and numerical methods have been developed for nonlinear sys-tems, better and more efficient computational methods are still needed.Normal form theory often proves useful to transform a problem to its simplest form before try-

ing to solve it. A lot of the early applications were in models from celestial mechanics. More re-cently, normal form theory has been widely used in the analysis of vibration and bifurcation fornonlinear vibrating systems [3–12]. The basic idea of the method of normal forms is applying aseries of near identity nonlinear transformations to systematically construct a simple form ofthe original system. The simplified system keeps the dynamic characteristics of the original system,and thus the analysis of the dynamical behavior becomes simpler. Normal forms are generally notuniquely defined and computing explicit normal forms in terms of the coefficients of the originalsystem is not easy. In the past few years, symbolic computation of normal forms using computeralgebra has received considerable attention (e.g., see [2,13–20]). The method of normal form isusually employed together with center manifold theory [21] which uses the same idea of successivenonlinear transformations. In general, given a nonlinear system, center manifold theory is appliedbefore using normal form theory. However, there exist methods which combine the two steps intoone unified procedure (e.g., see [15,19,20]).This paper presents a perturbation technique which combines the method of multiple time

scales (MTS) [4,8] and harmonic balancing [1] to study non-linear vibration and bifurcation prob-lems. Huseyin and Lin [11] used this approach to obtain the explicit formulae of simplified differ-ential equations (which are actually normal forms) up to first order approximation. Later, thismethod was extended to compute the normal forms of Hopf and generalized Hopf bifurcationsup to an arbitrary order [15]. This method does not need the application of center manifold theoryand can been directly applied to general n-dimensional systems. Moreover, user-friendly symbolicprograms written in Maple was developed [15], which can be executed ‘‘automatically’’ on a com-puter system. The crucial part in the computation of normal forms using a computer algebra sys-tem is memory problem. A computer may quickly run out of its memory if an inefficientcomputation approach is used. For example, it is difficulty to obtain a fifth order normal formfor a 3-dimensional system using matrix approach even with a fast computer. Therefore, develop-ing efficient methodologies for computing normal forms is necessary.Another difficulty in the application of normal forms is that many end users may not be famil-

iar with normal form theory and may be not good in coding symbolic programs. However they dowant to apply a method or a program to study their own specific problems which usually havelarge dimensions. Therefore, not only the computational efficiency of a method, but also the eas-iness to use the method needs to be considered. Yu [15] has developed a perturbation method to

P. Yu, S. Zhu / Communications in Nonlinear Science and Numerical Simulation 10 (2005) 869–905 871

provide ‘‘automatic’’ symbolic programs for computing the normal forms of Hopf and general-ized Hopf bifurcations. Later this method was extended to consider double Hopf bifurcations[19,20]. The Maple programs developed in these papers only require a user to prepare a verystraightforward input file. The source code and sample examples are available on the website:http://pyu1.apmaths.uwo.ca/~pyu/pub/index/sofware.html.In this paper, the perturbation technique combined with the MTS method is generalized to

compute the normal forms and the associated nonlinear transformations for general multiple-de-gree-of-freedom (M-DOF) autonomous systems. The Jacobian of such a system contains at leastone pair of purely imaginary eigenvalues in order to apply the MTS approach [15]. Although theMTS method has been widely used in vibration analysis for several decades, it has only been re-cently applied for systematically computing normal forms [15,19,20]. The MTS method is believedto yield correct normal forms by comparisons with existing results (e.g., see [12]), however no rig-orous proof, to our best knowledge, has been given in literature to show that the MTS is indeed togenerate normal forms. This paper presents a proof to confirm that the normal form obtainedusing the MTS is indeed equivalent to that derived by Poincaree normal form theory—bothare based on the concept of resonant terms. Then general solution forms are established for solv-ing ordered perturbation equations, which provides guidelines for developing symbolic algo-rithms. A number of cases are considered, including the non-resonance, general resonance,resonant case containing 1:1 primary resonance, and combination of resonance with non-reso-nance. Further, efficient algorithms and user-friendly Maple programs are developed.The perturbation method and formulations for general autonomous systems are described in

the next section. Section 3 considers the non-resonant case and proves the equivalence betweenthe MTS approach and Poincare normal form theory. Section 4 is devoted to derive solutionforms for various resonant cases. The combination cases of resonance with non-resonance areconsidered in Section 5. Section 6 outlines the algorithms for computing normal forms. Examplesare presented in Section 7 to show the applicability of the technique. Finally, concluding remarksare given in Section 8.

2. Perturbation technique based on the MTS

In order to show how to use the perturbation technique based on the MTS method to find thenormal form of differential equations, we first consider a simple, well-known example—van derPol�s equation—before dealing with general n-dimensional systems. The van der Pol�s equationis described by

€xþ xþ �ðx2 � 1Þ _x ¼ 0; ð1Þ
where the dot ‘‘.’’ indicates differentiation with respect to time, � is a small non-negative real num-ber (i.e., 0 6 � � 1). This kind of systems are called weakly nonlinear systems and perturbationmethods can be applied to find approximate periodic solutions.This system has been studied by many researchers. Recently, this equation is re-investigated
using, in addition to the regular (direct) perturbation method, four frequently used perturbationapproaches: Lindstedt–Poincare procedure, time averaging, multiple time scales and intrinsic har-monic balancing [22]. It has been shown that the regular perturbation method yields unbounded

http://pyu1.apmaths.uwo.ca/~pyu/pub/index/sofware.html


solution which contains secular terms. The Lindstedt–Poincare procedure cannot be used for sta-bility analysis though it produces the same accurate approximation as that obtained using theMTS and intrinsic harmonic balancing approaches. The first order time averaging method hasthe simplest solution procedure and can be used for stability analysis, but its solution is less accu-rate. The intrinsic harmonic balancing technique, unlike the other three perturbation methods,does not require solving differential equations. However, this approach needs to construct the‘‘normal form’’ (governing equations) for stability analysis, which is usually not straightforward,in particular, for high codimensional systems. Moreover, the ‘‘normal form’’ obtained using theintrinsic harmonic balancing is only valid up to the leading order term. Therefore, this approach isnot suitable for finding higher order normal forms.The MTS method can be used to find not only the approximate solutions but also the normal

forms. More importantly, its procedure in finding higher order normal forms is systematic, and itsformulas can be easily implemented using computer algebra systems. It is shown [22] that theMTS is the best approach among the above four mentioned perturbation methods for the studyof nonlinear oscillating systems, in particular, for computing the normal forms. Thus, the MTSperturbation technique is adopted and generalized in this paper for computing the normal formsof general M-DOF higher dimensional nonlinear vibrating systems.To apply the MTS method, one begins with introducing the new independent variables

T k ¼ �kt for k ¼ 0; 1; 2; . . . ð2Þ

It follows that the derivatives with respect to t become expansions in terms of the partial deriva-tives with respect to Tn according to

d

dt¼ dT 0

dto

oT 0þ dT 1

dto

oT 1

þ dT 2dt

o

oT 2þ � � � ¼ D0 þ �D1 þ �2D2 þ � � � ;

d2

dt2¼ D2

0 þ 2�D0D1 þ �2ðD21 þ 2D0D2Þ þ � � � ; etc:

ð3Þ

where Di (i = 1, 2, . . .) denotes o/oTi.Next, assume that the solution of the van der Pol� Eq. (1) is represented by an expansion in the

form of

xðt; �Þ ¼ x0ðT 0; T 1; T 2; . . .Þ þ �x1ðT 0; T 1; T 2; . . .Þ þ �2x2ðT 0; T 1; T 2; . . .Þ þ � � � ð4Þ

Note that the number of independent time scales used in the solution depends upon the order towhich the expansion is carried out. For example, if the expansion is expanded to O(�2), then T0, T1and T2 are needed. In general, if we want to find the approximate solution up to order O(�

n), thenthe scaled times T0, T1, . . ., Tn should be used.Applying the above formulas (3) and (4) to system (1) and balancing the like powers of � results

in the following ordered perturbation equations:

�0 : D20x0 þ x0 ¼ 0; ð5Þ

�1 : D20x1 þ x1 ¼ �2D1D0x0 � ðx20 � 1ÞD0x0; etc: ð6Þ


The solution for the �0 order Eq. (5) can be expressed as

x0 ¼ aðT 1; T 2; . . .Þ cos½T 0 þ /ðT 1; T 2; . . .Þ a cosðhÞ: ð7Þ
Then substitute this solution into the �1 order Eq. (6) to obtain
D20x1 þ x1 ¼ 2D1a� a 1� 1

4a2

� �� sinðT 0 þ /Þ þ 2aD1/ cosðT 0 þ /Þ þ 1

4a3 sin 3ðT 0 þ /Þ:

ð8Þ
Eliminating the secular terms, which may appear in solution x1, requires that
D1a ¼ 1

2a 1� 1

4a2

� �;

D1/ ¼ 0;

ð9Þ

and thus Eq. (8) becomes

D20x1 þ x1 ¼

1

4a3 sin 3ðT 0 þ /Þ ð10Þ

which, in turn, yields the solution

x1 ¼ � 1

32a3 sin 3ðT 0 þ /Þ: ð11Þ

Hence, the approximate solution up to first order is obtained as

xðt; �Þ ¼ a cosðt þ /Þ � �

32a3 sin 3ðt þ /Þ: ð12Þ

It should be noted that we only find the particular solution from Eq. (10) since we can leave thehomogeneous solution part to be included in a and /. In fact, if we add the homogeneous solution,given by a1 cosðt þ /1Þ where a1 and /1 are determined from the initial conditions, to the partic-ular solution (11), we can see that the homogeneous solution can be indeed combined with the firstterm of solution (12).Finally, the governing equations for the amplitude a and the phase / of the above periodic solu-

tion can be obtained, up to O(�) order, as follows:

dadt

¼ oaoT 1

oT 1ot

þOð�2Þ ¼ �D1aþOð�2Þ � �

2a 1� 1

4a2

� �ð13Þ

and

dhdt

¼ 1þ d/dt

¼ 1þ o/oT 1

oT 1ot

þOð�2Þ ¼ 1þ �D1/ þOð�2Þ � 1: ð14Þ

These two equations are in fact the normal form of the van der Pol�s equation up to � order.Now we turn to consider an n-dimensional autonomous system, described by

_x ¼ Jxþ f ðxÞ; x 2 Rn; f : Rn ! Rn; ð15Þ
where J is an n · n Jacobian matrix, and Jx is the linear part of the system. The function f rep-resents the nonlinear terms, and is assumed to be analytic. The f and its first derivative vanish atthe origin 0, indicating that 0 is an equilibrium (fixed point) of the system. J is given by


J ¼A0 0 0

0 A1 0

0 0 A2

264

375; ð16Þ

where A0 is a 2n0 · 2n0 matrix with purely imaginary eigenvalues, given by

A0 ¼

0 �x1 0 0 . . . 0 0

x1 0 0 0 . . . 0 0

0 0 0 �x2 . . . 0 0

0 0 x2 0 . . . 0 0

. . . . . . . . . . . . . . . . . . . . .

0 0 0 0 . . . 0 �xn0

0 0 0 0 . . . xn0 0

26666666666664

37777777777775; ð17Þ

and the eigenvalues of A1 and A2 have negative real parts, which implies that the center manifoldof the system has 2n0 dimension. The eigenvalues of J are in Siegel domain [23] and one encoun-ters much greater computation complexity.Matrix A1 is an n1 · n1 matrix having negative real eigenvalues:

A1 ¼�a2n0þ1 . . . 0

. . . . . . . . .

0 . . . �a2n0þn1

264

375; ð18Þ

and A2 is an n2 · n2 matrix whose eigenvalues are complex conjugate with negative real parts:

A2 ¼

�a2n0þn1þ1 �x2n0þn1þ1 0 0 . . . 0 0

x2n0þn1þ1 �a2n0þn1þ1 0 0 . . . 0 0

0 0 �a2n0þn1þ3 �x2n0þn1þ3 . . . 0 0

0 0 x2n0þn1þ3 �a2n0þn1þ3 . . . 0 0

. . . . . . . . . . . . . . . . . . . . .

0 0 0 0 . . . �an�1 �xn�1

0 0 0 0 . . . xn�1 �an�1

26666666666664

37777777777775; ð19Þ

where xk (k = 1, 2, . . ., n0), ap (p = 2n0 + 1, 2n0 + 2,. . ., 2n0 + n1) and aq (q = 2n0 + n1 + 1, 2n0 +n1 + 3, . . ., n � 1) are positive, and 2n0 + n1 + 2n2 = n.For the convenience of the analysis using the MTS, one may write Eq. (15) in the component

form:

_x2i�1 ¼ �xix2i þ f2i�1ðxÞ;_x2i ¼ xix2i�1 þ f2iðxÞ ði ¼ 1; 2; . . . ; n0Þ;

ð20Þ

_xp ¼ �apxp þ fpðxÞ ðp ¼ 2n0 þ 1; . . . ; 2n0 þ n1Þ; ð21Þ


_xq ¼ �aqxq � xqxqþ1 þ fqðxÞ;_xqþ1 ¼ xqxq � aqxqþ1 þ fqþ1ðxÞ ðq ¼ 2n0 þ n1 þ 1; 2n0 þ n1 þ 3; . . . ; n� 1Þ:

ð22Þ

Based on the above equations, the MTS can be employed to find the normal forms. First, assumethat the solution of system (15) is given in the form

xjðt; �Þ ¼ �xj;1ðT 0; T 1; . . .Þ þ �2xj;2ðT 0; T 1; . . .Þ þ � � � ðj ¼ 1; 2; . . . ; nÞ; ð23Þ
and then substituting Eq. (23) into Eqs. (20)–(22) with the aid of Eqs. (2) and (3) and balance thelike powers of � to obtain the following ordered perturbation equations:
�1 : D0x2i�1;1 ¼ �xix2i;1;

D0x2i;1 ¼ xix2i�1;1 ði ¼ 1; 2; . . . ; n0Þ; ð24ÞD0xp;1 ¼ �apxp;1 ðp ¼ 2n0 þ 1; . . . ; 2n0 þ n1Þ; ð25ÞD0xq;1 ¼ �aqxq;1 � xqxðqþ1Þ;1;

D0xðqþ1Þ;1 ¼ xqxq;1 � aqxðqþ1Þ;1 ðq ¼ 2n0 þ n1 þ 1; 2n0 þ n1 þ 3; . . . ; n� 1Þ; ð26Þ

�2 : D0x2i�1;2 ¼ xix2i;2 � D1x2i�1;1 þ f2i�1;2ðx1Þ;D0x2i;2 ¼ �xix2i�1;2 � D1x2i;1 þ f2i;2ðx1Þ ði ¼ 1; 2; . . . ; n0Þ; ð27ÞD0xp2 ¼ �apxp2 þ fp2ðx1Þ; ðp ¼ 2n0 þ 1; . . . ; 2n0 þ n1Þ; ð28ÞD0xq;2 ¼ �aqxq;2 � xqxðqþ1Þ2 þ fq;2ðx1Þ;D0xðqþ1Þ;2 ¼ xqxq;2 � aqxðqþ1Þ;2 þ fðqþ1Þ;2ðx1Þðq ¼ 2n0 þ n1 þ 1; 2n0 þ n1 þ 3; . . . ; n� 1Þ; etc: ð29Þ

where x1 represents the first order approximations of x, xi,j represents the jth order approximationof xi, and

fi;2 ¼ d2½fiðx1; x2; . . .Þ=�=d�2j�¼0 ð30Þ
which are functions of xi,1 only. In general, function fi,k only involves the ordered approximationsxi,1, xi,2, . . ., xi,k � 1 which have been solved from the previous (k � 1) perturbation equations. Itshould be noted that unlike Eq. (4) which starts with zero order term, the solution form (23) startswith first order term. This is because the van der Pol�s equation has � in the nonlinear term, whilefor the general nonlinear system (15), usually the first step is to use scaling x!�x to separate dif-ferent order terms, and then use the solution form starting with zero order term. Here the first stephas been included in solution form (23).Before giving a detailed analysis on how to find the solutions of the ordered perturbation equa-
tions and the normal forms, we need to discuss resonance conditions which determine the patternof the solutions and normal forms. Depending upon the ratios of the frequencies involved in theoscillating system, the system can be classified as non-resonant case or resonant case. In general,for a given system with frequencies x1x2,. . .,xn0

, if the condition
Xn0i¼1
mixi 6¼ 0 ð31Þ


is held for any n integers m1, m2, . . ., mn0(n0 P 2) that are not all zero, then the system is said to

be non-resonant. For example, if the system has two pairs of purely imaginary eigenvalues and theratio of frequencies x1 and x2 is an irrational number, then this system is non-resonant.In contrast, if there exist some n0 integers m1, m2, . . ., mn0

which are not all zero such that thefollowing relation:
Xn0
i¼1mixi ¼ 0 ð32Þ

is satisfied, then the system is called resonant. This definition of resonance implies that for a res-onant system which has n0 (n0 P 2) pairs of purely imaginary eigenvalues, there exist at least twofrequencies xi and xj (1 6 i, j 6 n0, i 6¼ j) such that their ratio is a fraction number, i.e.,

xi

xj¼ m1

m2

; ð33Þ

where mi and mj are positive integers.Note that the definitions for non-resonance and resonance given above are defined in pure

mathematics. In solving physical and engineering problems, the right-hand-side of Eq. (32) is as-sumed to be an O(�), term. In other words, if the condition (32) is roughly satisfied, one may callthe system resonant; otherwise, it is non-resonant.From the computation point of view, in particular for the computation of the normal form and

its associated nonlinear transformation, resonant and non-resonant cases have to be treated dif-ferently. In general, non-resonance is simpler than resonant cases. Thus we shall first discuss thenon-resonant case in the next section.

3. Non-resonance and the proof for the MTS technique

For the non-resonant case, condition (31) is satisfied. To find the normal form of the system, westart with Eq. (24). Differentiating the first equation of (24) and then substituting the second equa-tion into the resulting equation yields

D20x1;1 þ x2

1x1;1 ¼ 0 ð34Þ
which is a free vibrating system with the solution
x1;1 ¼ R1ðT 1; T 2; . . .Þ cosðx1T 0 þ /1ðT 1; T 2; . . .ÞÞ R1 cos h1; ð35Þ
where R1 and h1 are the amplitude and phase of motion, respectively. Having found x1,1, one caneasily find the solution of x2,1 as
x2;1 ¼ R1 sin h1 ð36Þ
from the first equation of (24).Similarly, one can find the solution of x2i�1,1 (i = 2, . . ., n0) from Eq. (24) as
x2i�1;1 ¼ Ri cos hi

x2i;1 ¼ Ri sin hi

�ði ¼ 1; 2; . . . ; n0Þ; ð37Þ


where Ri ” Ri(T1, T2, . . .) and hi ” xiT0 + /i(T1, T2, . . .). Thus the first order solutions forx1,1, x2,1, . . ., x2n0,1 are found.Since we are interested in the steady-state (asymptotic) solutions of the system, the �1 order

solutions for xj,1, j = 2n0 + 1,. . .,n, associated with the eigenvalues having real negative parts,are equal to 0, i.e.,

xj;1 ¼ 0; j ¼ 2n0 þ 1; . . . ; n: ð38Þ
It follows from solution (35) that D0R1 = D0/1 = 0. In general, we have
D0Ri ¼ 0

D0/i ¼ 0

�ði ¼ 1; 2; . . . ; n0Þ: ð39Þ

Next, to solve the �2 order perturbation Eq. (27), one may apply the above procedure and sub-stitute the �1 order solutions into (27) to obtain the following second order non-homogeneousODE:

D20x1;2 þ x2

1x1;2 ¼ �D1D0x1;1 � D1x2;1 þ D0f1;2 þ f2;2: ð40Þ
Note that the right-hand-side of Eq. (40) is a polynomial of the first order solutions xi,1(i = 1,2,. . .,n0), so the solution of x1,2 can be expressed by a finite Fourier series
x1;2 ¼X2s¼0

C1;p1p2...pn0cos

Xn0i¼1

pihi

!þ E1;p1p2...pn0 sin

Xn0i¼1

pihi

!" #; ð41Þ

where pi (i = 1, 2, . . .,n0) are integers and s ¼Pn0

i¼1jpij. Similarly, in general, one can obtain thefollowing equation from Eq. (27):

D20x2j�1;2 þ x2

j x2j�1;2 ¼ �D1D0x2j�1;1 � D1x2j;1 þ D0f2j�1;2 þ f2j;2; ð42Þ

where j = 1, 2, . . ., n0. The solution of the equation can be written as a Fourier series

x2j�1;2 ¼X2s¼0

Cj;p1p2...pn0cos

Xn0i¼1

pihi

!þ Ej;p1p2...pn0

sinXn0i¼1

pihi

!" #ðj ¼ 1; 2; . . . ; n0Þ; ð43Þ

where Cj,p1p2. . .pn0and Ej,p1p2. . .pn0

are coefficients to be determined.In order to determine the coefficients C and E and thus solve for x2j�1,2, one may substitute Eq.

(43) into Eq. (42) and then balance the harmonics. However, as usual, the resulting equation mayinvolve terms which will generate secular terms in the solution of x2j�1,2. A careful considerationshows that these terms in the jth equation is associated with index pj. Therefore, to eliminate thesecular terms in the solution of x2j�1,2, we must set the coefficient of the terms, which have theindexes pj = 1 but pk = 0, for j, k = 1, 2, 3, . . ., n0, k 6¼ j, zero. This yields a set of algebraic equa-tions to determine D1Rj and D1/j which are called resonant terms and will be retained in the nor-mal form.The solutions for xj,2 (j = 2n0 + 1, . . ., n) can also be found in the same form of Fourier series as

that of x2j�1,2, except that no secular terms would appear. Hence, they can be uniquely determinedby a straightforward harmonic balancing approach.The general solution of the nrth order perturbation equations of the system is given in the fol-

lowing Theorem. A similar study for this case on Hill�s equation has been given in [24].


Theorem 1. The solutions of the nrth order perturbation equations of an autonomous system can beexpressed by a finite Fourier series:

Xnr Xn0 ! Xn0 !" #
xi;nr ¼
s¼0Ci;p1p2...pn0

cosi¼1

pihi þ Ei;p1p2...pn0sin

i¼1pihi ði ¼ 1; 2; . . . ; nÞ; ð44Þ

where pi (i = 1, 2, . . ., n0) are integers and s ¼Pn0

i¼1jpij.Proof. First, we may rewrite the first order solution of Eqs. (24)–(26) as

xi;1 ¼X1s¼0

Ci;p1;p2;...;pn0cos

Xn0j¼1

pjhj

!þ Ei;p1;p2;...;pn0

sinXn0j¼1

pjhj

!; ð45Þ

where s ¼Pn0

j¼1jpjj, Ci,p1,p2,. . ., pn0and Ei,p1,p2,. . .,pn0

are constants.Next, for the second order perturbation equations (27)–(29), it is easy to observe that the

nonlinear terms are second degree polynomials of the first order solutions. So the highest orderterms are xi,1xj,1 (i, j = 1, 2, . . ., n). Therefore, the highest order terms involved in the second ordersolutions are sin hisin hj, cos hicos hj and sin hicos hj, which can be rewritten as� 1

2½cosðhi þ hjÞ � cosðhi � hjÞ, 1

2½cosðhi þ hjÞ þ cosðhi � hjÞ, and 1

2½sinðhi þ hjÞ � sinðhi � hjÞ,

respectively. These terms can be put into a general form sinðP2

s¼0pjhjÞ or cosP2

s¼0pjhj

� �.

Similarly, the highest order terms in the third order perturbation equations come from themultiplication of first and second order solutions, i.e., xi,1xj,2 (i, j = 1, 2, . . ., n). So the highestorder terms in third order solutions are

sinX2s1¼0

pjhj

!cos

X1s2¼0

qkhk

!; sin

X2s1¼0

pjhj

!sin

X1s2¼0

qkhk

!;

cosX2s1¼0

pjhj

!cos

X1s2¼0

qkhk

!; cos

X2s1¼0

pjhj

!sin

X1s2¼0

qkhk

!;

where s1 ¼Pn0

j¼1jpjj and s2 ¼Pn0

k¼1jqkj. Therefore, the highest order terms in the third order solu-tions can be written as sinð

P3

s¼0pjhjÞ or cosðP3

s¼0pjhjÞ.The above procedure can be easily extended to discuss higher order perturbation equations.

More rigorously, one may apply the method of mathematical induction to show that the highestorder terms in the nrth order solutions can be written in the form of sinð

Pnrs¼0pjhjÞ or

cosðPnr

s¼0pjhjÞ. Consequently, the general solution for the nrth order perturbation equations isgiven by the finite Fourier series (44), and only the terms satisfying

s6 nr ð46Þ
are presented in solution (44). This completes the proof.As discussed above, the normal form terms D1Rj and D1/j are obtained from the second order
perturbation equations by removing the secular terms from the solutions. In general, for the nrthorder equations, the normal form terms Dnr�1Rj and Dnr�1/j (j = 1, 2, . . ., n0) are obtained byeliminating the secular terms.


Finally, the normal form of system (15) for the non-resonant case, up to any order, can bewritten in polar coordinates as follows:

dRi

dt¼ oRi

oT 1

oT 1

otþ oRi

oT 2

oT 2

otþ oRi

oT 3

oT 3

otþ � � � ¼ �D1Ri þ �2D2Ri þ �3D3Ri þ � � � ; ð47Þ

Ridhi

dt¼ Ri xi þ

o/i

oT 1

oT 1ot

þ o/oT 2

oT 2

otþ o/oT 3

oT 3ot

þ � � ��

¼ Riðxi þ �D1/i þ �2D2/i þ �3D3/i þ � � �Þ; ð48Þ

for i = 1, 2, . . ., n0. It should be noted that for the non-resonant case, only odd order terms areretained in the normal form implying that

D2k�1Ri ¼ D2k�1/i ¼ 0 ðk is a positive integerÞ: ð49Þ

Note that the subscript 2k � 1 is not the order of the term. h

More specifically we have the following theorem.

Theorem 2. The ‘‘form’’ of the normal form for the non-resonant case of an autonomous system withn0 pairs of purely imaginary eigenvalues, is given by (in polar coordinates)

_Ri ¼ RiP R21;R22; . . . ;R

2n0

� �; ð50Þ

Ri_/i ¼ RiQ R21;R

22; . . . ;R

2n0

� �; ð51Þ

where P and Q are multi-variable polynomials in R21;R22; . . . ;R

2n0

.

Remark. The equations given in (50) and (51) are actually the Poincare normal form for non-res-onant M-DOF oscillating systems.

Proof. It is easy to prove the theorem using complex formulas. Thus introduce the followingtransformations:

x2i�1 ¼1

2ðz2i�1 þ z2iÞ;

x2i ¼1

2Iðz2i�1 � z2iÞ ði ¼ 1; 2; . . . ; n0Þ; ð52Þ

xp ¼ zp ðp ¼ 2n0 þ 1; 2n0 þ 2; . . . ; 2n0 þ n1Þ; ð53Þ

xq ¼1

2ðzq þ zqþ1Þ;

xqþ1 ¼1

2Iðzq � zqþ1Þ ðq ¼ 2n0 þ n1 þ 1; 2n0 þ n1 þ 3; . . . ; n� 1Þ; ð54Þ


where I is the imaginary unit satisfying I2 = �1. It should be noted in the above expressions thatz2i and zq + 1 are the complex conjugates of z2i�1 and zq, respectively. Then Eqs. (20)–(22) can betransformed into complex form as follows:

_z2i�1 ¼ Ixiz2i�1 þ f2i�1 þ If 2i;

_z2i ¼ �Ixiz2i þ f2i�1 � If 2i ði ¼ 1; 2; . . . ; n0Þ; ð55Þ_zp ¼ �apzp þ fp ðp ¼ 2n0 þ 1; 2n0 þ 2; . . . ; 2n0 þ n1Þ; ð56Þ_zq ¼ ð�aq þ IxqÞzq þ fq þ If qþ1;

_zqþ1 ¼ �ðaq þ IxqÞzqþ1 þ fq � If qþ1 ðq ¼ 2n0 þ n1 þ 1; 2n0 þ n1 þ 3; . . . ; n� 1Þ; ð57Þ
where
fi ¼ fiðx1ðzÞ; x2ðzÞ; . . . ; xnðzÞÞ ð58Þ
and xi(z) are given by transformations (52)–(54). Similarly, applying the MTS to the above equa-tions results in complex ordered perturbation equations, which are similar to the real forms (24)–(26).The solutions to the first order complex equations (which can be readily obtained from Eqs.
(55)–(57) by removing f�s terms) are

z2i�1;1 ¼ RieIhi ;

z2i;1 ¼ �z2i�1 ði ¼ 1; 2; . . . ; n0Þ; ð59Þzp;1 ¼ 0 ðp ¼ 2n0 þ 1; . . . ; nÞ; ð60Þ

where �z represents the complex conjugate of z. Ri = Ri (T1, T2, . . .) (i = 1, 2, . . ., n0) are real posi-tive, and hi = xiT0 + /i (T1,T2,. . .).Similar to the real analysis, it can been shown that the nonlinear terms on the right-hand-side

of the complex perturbation equations can be written as polynomials of the first order solutions inthe form of

F ¼X

C Pn0j¼1z

ajj �z

bjj

� �¼X

C Pn0j¼1R

ðajþbjÞj eIðaj�bjÞhj

� �; ð61Þ

where C are complex constants and aj, bj (j = 1, 2, . . ., n0) are integers.For the ith equation (i = 1, 2, . . ., 2n0), the secular term should be in the form of eIhi, and thus it

follows from Eq. (61) that the terms producing the secular terms for the ith equation satisfies

ai � bi ¼ 1;

aj � bj ¼ 0 for j ¼ 1; 2; . . . ; n0; but j 6¼ ið62Þ

Therefore, the powers of the normal form terms for the ith equation are the solution of Eq. (62).Now substituting Eq. (62) into Eq. (61) results in the secular term

S ¼X

CR2biþ1i eIhiPj6¼iR2bjj ð63Þ

which is balanced by the term Dm�1(RieIhi), where m ¼

Pn0j¼1ðaj þ bjÞ ¼ 1þ 2

Pn0j¼1bj. Thus the

normal form terms Dm�1Ri and Dm�1/i can be found from the following equation:

Dm�1Ri þ IRiDm�1hi ¼ S ¼X

CR2biþ1i eIhiPj6¼iR2bjj ð64Þ


which, in turn, yields

Dm�1Ri ¼ Ri

XC1P

n0j¼1R

2bjj ¼ RiPm R21;R

22; . . . ;R

2n0

� �;

RiDm�1hi ¼ Ri

XC2P

n0j¼1R

2bjj ¼ RiQm R21;R

22; . . . ;R

2n0

� �;

ð65Þ

where Pm and Qm are the mth degree polynomials of R21;R22; . . . ;R

2n0. Note that m is an odd number,

implying that only odd order terms are retained in the normal form.This finishes the proof of Theorem 2. h

Similar theorems and proofs for the equivalence of other cases (discussed in the next two sec-tions) can be established, and are not repeated in the paper.

4. Resonant cases

When condition (32) is satisfied, the system exhibits a resonant vibration. In this case, there areat least two frequencies whose ratio is a rational number. In particular, when n0 = 2, the resonancecondition becomes

x1

x2

¼ m1

m2

; ð66Þ

where m1 and m2 are positive integers. A special case is when x1 = x2, which is called 1:1 primaryresonance.Since the above ratio representation is commonly used in practice, we extend it to the case

which involves more than two frequencies. In other words, we want to transfer the general reso-nance condition (32) to a set of ratio representations. To do this, suppose the smallest frequencyof x1, x2, . . ., xn0

is x1, then define

xi : x1 ¼ ri for i ¼ 2; 3; . . . ; n0; ð67Þ
where ri�s are rational numbers and are assumed, without loss of generality, to be in ascendingorder. Further, one may assume, for convenience, that all the frequencies xi are integers. Thisis because one can find the smallest divisor x* such that xi ¼ n�i x
� where n�i �s (i = 1, 2, . . ., n0,r)are positive integers. Then applying a time scaling s = x*t under which the first n0,r frequenciesbecome integers xi ¼ n�i .In the following, we first consider the general case in which all ri�s are distinct, and then consider

the special case �1:1 primary resonance.

4.1. General resonant case

For the general case, the general nrth (nr > 1) order solution is given in the following theorem.

Theorem 3. If system (15) satisfies the resonance condition (67), and assume that x1, x2, . . ., xn0are

all integers and that x1 < x2 < � � � < xn0, then the solution of the autonomous system can be given in

the Fourier series


xi;nr ¼Xs

p¼�s

Ci;p;nr cosðpT 0Þ þ Ei;p;nr sinðpT 0Þ; i ¼ 1; 2; . . . ; n; ð68Þ

where s = nrxn0, and the coefficients Ci,p,nr

and Ei,p,nrare functions of T1, T2, . . .

Proof. The proof is similar as that for Theorem 1. The first order solution to Eqs. (24)–(26) can bewritten as

xi;1 ¼Xxi

p¼�xi

Ci;p cosðpT 0Þ þ Ei;p sinðpT 0Þ; ð69Þ

where Ci,p and Di,p are functions of T1,T2,. . .Next, for the second order perturbation Eqs. (27)–(29), one can see that the nonlinear terms are

second degree polynomials of the first order solutions. So the highest order terms are xi,1xj,1

(i, j = 1, 2, . . ., n), involving trigonometric functions sin xiT0sin xjT0, cos xiT0cos xiT0and sin xiT0cos xiT0, which can be rewritten as � 1

2½cosðxi þ xjÞT 0 � cosðxi � xjÞT 0,

12½cosðxi þ xjÞT 0 þ cosðxi � xjÞT 0 and 1

2½sinðxi þ xjÞT 0 � sinðxi � xjÞT 0, respectively. These

terms can be put into general form sin(pT0) or cos(pT0), where jpj 6 2xn0.

Similarly, the highest order terms in the third order perturbation equations come from themultiplication of the first and second order solutions and order-2 solutions, given in the form ofxi,1xj,2 (i, j = 1, 2, . . ., n), which only involve the functions sin(pT0) and cos(pT0) for jpj 6 3xn0

.Repeating the above procedure to higher order equations shows that the highest order terms inthe nrth order equations are sin(pT0) and cos(pT0) for jpj 6 nrxn0

. Therefore, the general solutionsof the nrth order equations can be written by the Fourier series (68), where

06 jpj6 nrxn0 ð70Þ

implying that the Fourier series is finite, and thus Theorem 3 is proved. h

Similar to the case of non-resonance, one can find the particular solution for each perturbationequation by harmonic balancing and the normal form terms by eliminating the secular terms. Tobe more precise, one can find D1Ri and D1/i by setting the coefficients of the second order equa-tions, associated with p = xi, zero. Then the solution for x1,2 is found by harmonic balance fromthe first equation of the second order perturbation equations, and x2,2 is then determined from thesecond equation. This procedure can be repeatedly applied to find higher order normal form termsDjRi and Dj/i as well as solutions xi,j.

4.2. Resonant case including 1:1 primary resonance

In the previous sub-section we considered the general resonance in which all ri�s are distinct.Now suppose two of the n0 frequencies are equal, i.e., x1 = x2, or r2 = 1, but other ri�s (i >2)are still different. The part associated with x1 = x2 is called 1:1 primary resonance. The 1:1 pri-mary resonance is different from the general resonance and must be treated separately. However,the other part associated with distinct ri�s can be still treated using the procedure presented in theprevious sub-section. The system can be now written as


_x1 ¼ �x1x2 þ x3 þ f1ðxÞ;_x2 ¼ x1x1 þ x4 þ f2ðxÞ; ð71Þ_x2i�1 ¼ �xix2i þ f2i�1ðxÞ;_x2i ¼ xix2i�1 þ f2iðxÞ ði ¼ 2; 3; . . . ; n0Þ; ð72Þ_xp ¼ �apxp þ fpðxÞ ðp ¼ 2n0 þ 1; 2n0 þ 2; . . . ; 2n0 þ n1Þ; ð73Þ_xq ¼ �aqxq � xqxqþ1 þ fqðxÞ;_xqþ1 ¼ xqxq � aqxqþ1 þ fqþ1ðxÞ ðq ¼ 2n0 þ n1 þ 1; 2n0 þ n1 þ 3; . . . ; n� 1Þ; ð74Þ

where fi (i = 1, 2, . . ., n) are polynomials of x without constants and the linear part. It should benoted that the Jacobian of the above system is still given in the form of Eq. (16) but with a dif-ferent A0: 2 3

A0 ¼

0 �x1 1 0 . . . 0 0

x1 0 0 1 . . . 0 0

0 0 0 �x1 . . . 0 0

0 0 x1 0 . . . 0 0

. . . . . . . . . . . . . . . . . . . . .

0 0 0 0 . . . 0 �xn0

0 0 0 0 . . . xn0 0

666666666664

777777777775: ð75Þ

The part associated with the eigenvalue x1 is called non-semi-simple case since it has two one�sin the first two rows. The case without the two one�s is called semi-simple and can be directly trea-ted using the procedure given in the previous sub-section.The separate treatment for the general resonance and the resonant case including 1:1 primary

resonance is due to different scalings used which are necessary for the application of the pertur-bation technique. Uniform scaling is used for the general resonance while non-uniform scaling hasto be applied for the case including 1:1 resonance to match the perturbation orders. For this case,it is straightforward to show that a uniform scaling leads to solving nonlinear differential equa-tions at the first perturbation order. This difficulty is due to the non-semi-simple resonance, thatis, linear terms x3 and x4 are present in the first and second equations (see the Jacobian given byEq. (75)). In order for the perturbation technique to work for the 1:1 non-semi-simple resonantcase, we must use different scalings between x1, x2 and other variables, xi, i = 3, 4, 5, . . ., n. A care-ful consideration shows that the ratio of the perturbation order for x1, x2 and the remaining var-iables should be 2:3 [19]. Thus, let

x1 ! �2x1;

x2 ! �2x2;

xi ! �3xi; i ¼ 3; 4; . . . ; n:

ð76Þ

Then, unlike the general resonance, where the highest degree of harmonics at an order n is con-sistently equal to n + 1, now the kth order terms in the original differential equations may spreadinto different order (>n) perturbation equations due to the different scalings. However, it can beshown that the highest order of the perturbation equation which should be included for the original


nth order terms is 3n � 2. Therefore, the highest order of the harmonics in the nth order equationsmust be 3n � 2. In other words, if an nth order normal form is needed, then the perturbation equa-tions up to 3n � 2 order must be considered. Once this scaling is applied and the solution form isappropriately set, a solution procedure similar to that presented in the previous sub-section can beobtained and the formulas similar to that given in the previous sub-section can be found.It should be noted, unlike the general resonance, the nth order normal form and the associated

nonlinear transformations (solutions) obtained using the above procedure for the 1:1 resonantcase may involve higher order terms which actually do not belong to the nth order expression.This is due to the difference in scaling. This suggests that the difference in scaling has also causedthe lower order terms to spread into higher order perturbation equations, and has caused higherorder terms to appear in lower order (<3n � 2) perturbation equations. This is the fundamentaldifference between the 1:1 primary resonance and the general resonance. In order to removethe terms which do not belong to the nth-oder normal form, one may just simply truncate the nor-mal form and solutions up to nth order. (Note: here the order means the order of the original dif-ferential equations.) This redundant calculation increases the complexity and the time required forthe computation.Now under the scaling (76) the original Eqs. (71)–(74) become

_x1 ¼ �x1x2 þ �x3 þ f 01ðxÞ;

_x2 ¼ x1x1 þ �x4 þ f 02ðxÞ; ð77Þ

_x2i�1 ¼ �xix2i þ f 02i�1ðxÞ;

_x2i ¼ xix2i�1 þ f 02iðxÞ ði ¼ 2; 3; . . . ; n0Þ; ð78Þ

_xp ¼ �apxp þ f 0pðxÞ; ðp ¼ 2n0 þ 1; 2n0 þ 2; . . . ; 2n0 þ n1Þ; ð79Þ

_xq ¼ �aqxq � xqxqþ1 þ f 0qðxÞ;

_xqþ1 ¼ xqxq � aqxqþ1 þ f 0qþ1ðxÞ ðq ¼ 2n0 þ n1 þ 1; 2n0 þ n1 þ 3; . . . ; n� 1Þ; ð80Þ

where the primes indicate that f 0i are at least in �1 order.

To apply the MTS technique to Eqs. (77)–(80), assume that xi (i = 1, 2, . . ., n) are expressed inTaylor series (23), and then substitute them into the above equations and balance the like terms of�, taking into account of formula (2), to obtain

�1 : D0x1;1 ¼ �x1x2;1;

D0x2;1 ¼ x1x1;1; ð81ÞD0x2i�1;1 ¼ �xix2i;1;

D0x2i;1 ¼ xix2i�1;1 ði ¼ 2; 3; . . . ; n0Þ; ð82ÞD0xp;1 ¼ �apxp;1 ðp ¼ 2n0 þ 1; 2n0 þ 2; . . . ; 2n0 þ n1Þ; ð83ÞD0xq;1 ¼ �aqxq;1 � xqxðqþ1Þ;1;

D0xðqþ1Þ;1 ¼ xqxq;1 � aqxðqþ1Þ;1 ðq ¼ 2n0 þ n1 þ 1; 2n0 þ n1 þ 3; . . . ; n� 1Þ; ð84Þ


�2 : D0x1;2 ¼ �x1x2;1 � D1x1;1 þ x3;1 þ f 01;2ðxÞ;

D0x2;1 ¼ x1x1;1 � D1x2;1 þ x4;1 þ f 02;2ðxÞ; ð85Þ

D0x2i�1;1 ¼ �xix2i;1 � D1x2i�1;1 þ f 02i�1;2ðxÞ;

D0x2i;1 ¼ xix2i�1;1 � D1x2i;1 þ f 02i;2ðxÞ ði ¼ 2; 3; . . . ; n0Þ; ð86Þ

D0xp2 ¼ �apxp2 þ fp2ðxÞ ðp ¼ 2n0 þ 1; 2n0 þ 2; . . . ; 2n0 þ n1Þ; ð87ÞD0xq;2 ¼ �aqxq;2 þ xqxðqþ1Þ2 þ f 0

q;2ðxÞ;

D0xðqþ1Þ;2 ¼ xqxq;2 � aqxðqþ1Þ;2 þ f 0ðqþ1Þ;2ðxÞ

ðq ¼ 2n0 þ n1 þ 1; 2n0 þ n1 þ 3; . . . ; n� 1Þ ð88Þetc:

The first order solutions to Eqs. (81) and (82) can be found as

x1;1 ¼ R1 cosðx1T 0 þ /1Þ;x2;1 ¼ R1 sinðx1T 0 þ /1Þ; ð89Þx2i�1;1 ¼ Ri cosðxiT 0 þ /iÞ;x2i;1 ¼ Ri sinðxiT 0 þ /iÞ; ði ¼ 2; 3; . . . ; n0Þ: ð90Þxj;1 ¼ 0 ðj ¼ 2n0 þ 1; 2n0 þ 2; . . . ; nÞ: ð91Þ

The nrth (nr > 1) order solutions are given in the following theorem.

Theorem 4. Assume that x1 = x2 < x3 < � � � < xn0. The solutions to the nrth order equations of the

system governed by Eqs. (71)–(74) are given in the Fourier series:

xi;nr ¼XMp¼�M

Cinrp cosðpT 0Þ þ Dinrp sinðpT 0Þ; ð92Þ

where M = xn0nr.

Proof. The proof is similar to that for Theorem 3 and is thus omitted.In order to eliminate the secular terms that would appear in the solutions of x1,nr

and x2,nr, one

sets the terms, associated with p = x1, zero, which yields the normal form terms Dnr�1R1 andDnr�1/1. Then by harmonic balance, one can find x1,nr

, and x2,nrcan be directly determined from

the nrth order perturbation equation. Similarly, to eliminate the secular terms which mayotherwise appear in the solutions of x2i�1,nr

and x2i,nr(i = 2,3,. . .,n0), one must set the terms,

associated with p = xi, zero to obtain the normal form terms Dnr�1Ri and Dnr�1/i. At the sametime the solutions of x2i�1,nr

and x2i,nrcan be found using harmonic balance.

The procedure described above for finding the solution and normal form of the 1:1 resonancecan be directly extended to other primary resonances like 1:1:1, 1:1:1:1, etc. h


5. Combination of resonance with non-resonance

Having considered the non-resonance and resonances in the previous two sections, we now turnto the case that includes non-resonant modes and resonant modes. Such combinations may havemany choices. Here we shall consider the combination of non-resonance and resonance, in whichsome of frequencies are rationally linked and others are not. Assume that the first n0,r frequenciesare rationally linked and the remaining n0,n = n0 � n0,r frequencies are not rationally linked to anyother frequencies. As usual, one may assume that the first n0,r resonant frequencies are sorted inascending order, i.e., x1 < x2 < � � � < x0,r. Again, without loss of generality, we may assume thatall these frequencies are integers.The MTS procedure used in Section 3 for solving the non-resonant case can still be used here. It

is easy to obtain the first order solutions, given by

x2i�1;1 ¼ Ri cosðxiT 0 þ /iÞx2i;1 ¼ Ri sinðxiT 0 þ /iÞ

�ði ¼ 1; 2; . . . ; n0;rÞ; ð93Þ

x2j�1;1 ¼ Rj cos hj

x2j;1 ¼ Rj sin hj

)ðj ¼ n0;r þ 1; n0;r þ 2; . . . ; n0Þ; ð94Þ

xk;1 ¼ 0 ðk ¼ 2n0 þ 1; 2n0 þ 2; . . . ; nÞ; ð95Þ

where hi = xiT0 + /i (i = 1, 2, . . ., n0,r), Ri = Ri(T1,T2,. . .), and /k = /k(T1,T2,. . .) fori = 1, 2, . . ., n.The solutions for the nrth (nr P 2) order perturbation equations are given in the following

theorem.

Theorem 5. Assume that the first n0,r frequencies of the autonomous system governed by Eqs. (20)–(22) are integers, arranged in the ascending order x1<x2<� � �<n0,r while all other frequencies xi

(i = n0,r + 1, n0,r + 2,. . .,n0) are not rationally linked with any other frequencies. Then the solutions tothe nrth (nr P 2) order perturbation equations are given by the Fourier series:

xi;nr ¼XMs¼0

Ci;nr ;p1þn0;r;p2þn0;r

;...;pn0cos pT 0 þ

Xn0j¼1þn0;r

pjhj

0@

1A

24

þEi;nr ;p1þn0;r;p2þn0;r

;...;pn0sin pT 0 þ

Xn0j¼1þn0;r

pjhj

0@

1A35; ð96Þ

where p and pj (j = n0,r + 1, n0,r + 2,. . .,n0) are all integers. s ¼ jpj þ xn0;r

Pn0j¼1þn0;r

jpjj andM = nrxn0,r

.

Proof. Similar to the proof of Theorem 1, it is easy to know that the form of the Fourier series forthe nrth solutions is given by Eq. (15). What needs to be proved here is the bounds of the index forthe non-zero terms in the Fourier series.


From the proof of Theorem 1, one can find that the non-zero terms in the solutions of the nrthorder equations must satisfy condition (46), i.e.,
Xn0
i¼1jpij6 nr: ð97Þ

For the combination of non-resonance and general resonance, the condition needs to be mod-ified since the first nn0,r

pi�s cannot be separated. ThusPn0;r

i¼1pixi ¼ p from which we obtain

jpj6xn0;r

Xn0;ri¼1

jpij: ð98Þ

But from Eq. (97) one has

xn0;r

Xn0i¼1

jpij6 nrxn0;r : ð99Þ

Finally, substituting Eq. (98) into Eq. (99) yields the bounds for the non-zero terms as

s jpj þ xn0;r

Xn0j¼1þn0;r

jpjj6M nrxn0;r ; ð100Þ

and hence the proof is completed. h

6. Outline of algorithms

The computation of normal forms is very time consuming because the number of operationsincreases very rapidly as the order of normal forms increases. Hand calculation is usually limitedto very low (second or third) order normal forms and yet very easy to make mistakes. Thereforeintroducing computer algebra systems in computing normal forms is necessary, in particular, forhigher order normal forms.Based on the explicit formulae derived in the previous sections, we outline the algorithms for

computing normal forms and the associated nonlinear transformations. The algorithms includethe non-resonant case, general resonance and the resonant case including 1:1 primary resonance.Based on the algorithms, Maple programs have been developed and applied to consider mathe-matical and physical examples.Since the procedures involved in the algorithms for the different cases are similar, we shall dis-

cuss the non-resonant case in detail and omit other cases. Our discussions start with input file,followed by the computation of normal forms and nonlinear transformations.

6.1. Create input file

An input, in general, can be either online input, or a pre-prepared input file. In our algorithms,we use the second approach and keep the preparation required from a user to be minimum. In


fact, the procedure to create an input is straightforward, as shown below. Note that some nota-tions given here are adopted from the Maple programs.

(A) Set the following variables:N0—the number of pairs of purely imaginary eigenvalues of the Jacobian.N1—the number of the real eigenvalues of the Jacobian.N2—the number of pairs of complex conjugate eigenvalues of the Jacobian.N—the dimension of the system, N = 2N0 + N1 + 2N2.norder—the order of the normal forms to compute.

(B) Set the differential equations (vector field) of the system, which are given in Taylor expansionabout the equilibrium, x = 0, i.e., in the form of homogeneous polynomials. See Eqs. (20)–(22).

(C) Set the internal frequencies: xi, i = 1, 2, . . ., N0, which have to be a part of input, since theyare needed in the programs. Here, xi is the natural frequency of the ith pair equations asso-ciated with the purely imaginary eigenvalues.

The following is a sample input file. One can easily follow this sample to create one�s own inputfiles.

N0 := 2:

N1 := 1: # No.of non-zero real eigenvalues

N2 := 1: # No.of complex conjugate eigenvalues

N := 2*N0 + N1 + N2*2: # Dimension of the system

norder := 4: # The order of normal form to compute

#### ORIGINAL DIFFERENTIAL EQUATIONS ####

omg[1] := 1:

omg[2] := 2�(1/2):

beta[1] := 1:

omega[1] := 1:

alpha[1] := 1:

Dx[1] := - omg[1]*x[2]+x[1]�2:

Dx[2] := omg[1]*x[1]+x[1]*x[2]+x[2]*x[3]:

Dx[3] := - omg[2]*x[4]+x[3]�2:

Dx[4] := omg[2]*x[3]+x[4]�2+x[3]*x[1]:

Dx[5] := - beta[1]*x[5] + (x[1])�2:

Dx[6] := - alpha[1]*x[6] - omega[1]*x[7] + (x[1]�2):

Dx[7] := omega[1]*x[6] - alpha[1]*x[7] + (x[2]�2):

where Dx[i] denotes _xi, and the nonlinear terms are expressed in homogeneous polynomialsof xi�s.


6.2. Compute the normal forms and nonlinear transforms

Based on the theorems and the formulas presented in the previous sections, one may use a com-puter algebra system to implement the MTS technique on computers to calculate normal formsand associated nonlinear transformations. The detailed steps are described below.

(A) Read an pre-prepared input file.(B) Scale the original differential equations: xi ! �xi, where � is a small positive perturbation

parameter.(C) Use Eqs. (52)–(54) to transform the original system given in real coordinates (x1, x2, . . ., xn)

to a complex system with coordinates (z1, z2, . . ., zn). The complex system is given by (55)–(57).Transforming real system to complex system can save computational time. This can be ex-plained as follows. Suppose we work with the real system, then, as discussed in Sections3–5, the solutions to the first order perturbation equations are given in functionscos(xiT0 + /i) and sin(xiT0 + /i):

x2i�1;0 ¼ Ri cos hi;

x2i;0 ¼ Ri sin hi ði ¼ 1; 2; . . . ;N0Þ;xk;0 ¼ 0 ðk ¼ 2N0þ 1; . . . ;NÞ;

where Ri ” Ri(T1,T2,. . .) and hi = xiT0 + /i(T1,T2,. . .), i = 1, 2, . . ., N0. Then solving higherorder equations involves substantial operations such as expanding functions cosð

PN0i¼1mihiÞ

and sinðPN0

i¼1mihiÞ, where mi�s (i = 1, 2, . . ., N0) are integers, and their combinations. Thisis very time consuming in symbolic manipulations, in particular, for higher order normalforms. For example, with Maple V, expanding the trigonometric functioncos(5h1 + 3h2 � 7h3) does not take much time. However, recovering the function from the ex-panded terms, using Maple command ‘‘combine’’, takes 1.37 s of CPU time and more than1.75 M of memory on a PC with 512 Mb memory and 450 MHz CPU. Unfortunately, the‘‘combine’’ command is frequently used in dealing with trigonometric functions. This indi-cates that it is impractical to symbolically solve higher order equations which are given inreal coordinates when the dimension of the center manifold of the system is greater than five.Moreover, by using complex formulation, the first 2N0 differential equations associated withpurely imaginary eigenvalues can be reduced to N0 complex equations since the other half isjust the complex conjugate of the N0 equations. This is also true for the last 2N2 differentialequations. This certainly reduces the computation time and memory as well.

(D) Use the MTS technique to obtain the ordered perturbation equations. This can be achievedby first assuming that the solution is given in the form of Taylor series:

zi ¼Xj¼0

�jZi;j i ¼ 1; 2; . . . ;N ;

and then, with the aid of Eqs. (2) and (3), substituting them into the complex system and bal-ancing the like powers of �. The procedure is similar to that of real system.


(E) Compute zero order solution. It is easy to find the general formula of zero order solution forthe complex system, given by

Z2i�1;0 ¼ RieIhi ;

Z2i;0 ¼ Rie�Ihi ði ¼ 1; 2; . . . ;N0Þ;

Zj;0 ¼ 0 ðj ¼ 2N0þ 1; . . . ;NÞ:

The above solution form indicates that by transforming the original real system to complexsystem, we do avoid operations on functions sin(

Pimihi) and cos(

Pimihi). Instead the oper-

ations on complex exponent functions eIðP

imihiÞ are needed. However, it has been found that

such operations using Maple are even more time-consuming than the trigonometry manipu-lations. For example, differentiating a complex function with respect to the complex expo-nential function is much slower than differentiating a real function with respect totrigonometry functions. Moreover, it is very easy for such operations to cause the system‘‘dead’’. However, fortunately, this trouble can be avoided since the complex exponentialfunctions which appear in higher order equations can be expressed as polynomials of the zeroorder solutions. Thus, for example, let eIhi = E_thi, then function e

IðPimihiÞ would become

Pi(E_thi)mi and this makes the computation using Maple much faster.

(F) Solve the first order solutions to the first N0 pairs of equations associated with the purelyimaginary eigenvalues.

(i) Solve the first pair of equations and find the normal form terms D1R1 and D1/1. FromTheorem 1, one may similarly find the general solutions of the complex perturbationequations (order j > 0), expressed by the complex Fourier series:� �

Zk;j ¼Xjþ1s¼0

Cm1;m2;...;mN0eIPN0i¼1

ðmihiÞðk ¼ 1; 2; . . . ;NÞ;

where mi�s are integers and jmij 6 j + 1, s ¼PN0

i¼1jmij and hi = xiT0 + /i (i = 1, 2, . . ., N0). Itshould be noted that the upper bound of s is (j + 1) rather than j for the jth order equations,while for the real system (see Theorem 1) the upper bound for s is nr for the nrth order equa-tions. This difference is due to different scaling used here for complex system. Here, the termsstart at zero order since a pre-scaling x! �x has been used, while the terms in the real systemstart at first order.Setting k = 1 and j = 1, then by harmonic balance, one can find the solution of Z1,1. Settingthe coefficients of the terms, associated with m1 = 1 and mi = 0 (i = 2, 3, . . ., N0), zero yieldsthe normal form terms D1R1 and D1/1. Solution for Z2,1 is just a complex conjugate of Z1,1.It should be pointed out that, in symbolic computation software like Maple, finding the com-plex conjugate of a function is not so easy as one can think of at the first sight. The Maplebuilt-in functions such as conjugate and Re only work when a function is simple. We havethus coded a function ConjFun for computing the complex conjugate of much involved com-plex functions. It is computationally efficient. However, this function only works when theinput function is a polynomial.

(ii) If N0 > 1, set k = 2, . . ., N0, and repeat step (i) for N0 � 1 times.


(G) If N1 > 0, solve the solutions for the first order N1 perturbation equations associated with thenegative real eigenvalues.(i) Solve the first equation associated with real eigenvalue. Similarly, the general solutionform for the jth order can also expressed by a complex Fourier series:

nord

dR1

dR2

dt1

dt2

Zk;j ¼Xjþ1s¼0

Cm1;m2;...;mN0eIPi¼N0

i¼1ðmihiÞ

� �ðk ¼ 2N0þ 1; . . . ; 2N0þ N1Þ;

where s ¼PN0

i¼0jmij 6 ðjþ 1Þ. Set k = 2N0 + 1, j = 1, and then use harmonic balance to findthe solution of Z2N0 + 1,1 from the term involved in the �1 perturbation equation which con-tain Z2N0 + 1,1.(ii) If N1 > 1, repeat step (i) N1 � 1 times for k = 2N0 + 2,. . .,2N0 + N1.

(H) If N2 > 0, find the solutions to the first order N2 pairs of perturbation equations associatedwith the complex conjugate eigenvalues.

(i) Solve the first pair of the last N2 pairs equations. As discussed in chapter 3, the generalsolutions of the last N2 pairs of equations associated with the complex conjugate eigen-values can also be written as complex Fourier series. Therefore, similar to solving anequation associated with the real eigenvalue, by harmonic balancing, one can find thesolution Z2N0 + N1 + 1,1. Solution Z2N0 + N1 + 2,1 is the complex conjugate ofZ2N0 + N1 + 1,1 and can be obtained by calling the function ConjFun.

(ii) If N2 > 1, repeat step (i) N2 � 1 times for k = 2N0 + N1 + 3,2N0 + 5,. . .,N � 1.(I) If norder > 1, repeat steps (F), (G), and (H) until the order norder is reached.(J) Transform the solutions back to real coordinates.(K) Verify the solution and normal form if necessary. The normal form and solution can be ver-

ified as follows: Substitute the normal form and solution back into the original differentialequations and check if the terms cosð

PN0i¼1pihiÞ remained in the residue satisfyPN0

i¼1jpij > ðnorder þ 1Þ. If this condition is satisfied for all equations, then the normal formand the solution are correct; otherwise, they must be wrong.It should be pointed out that normal forms are in general, not unique [3,5]. Therefore, thenormal form and solution obtained using the algorithms developed in this paper may be dif-ferent from that obtained using other approaches.

(L) Save the solution and normal form.

A sample output result is given below.

er := 5;

:= -1/8*R[2]�2*R[1]*2

�(1/2)

-1/28224*R[1]*R[2]�2*2

�(1/2)*(11232*R[1]

�2+1127*R[2]

�2);

:= 11/64*R[1]�2*R[2]

�3*2

�(1/2);

:= 1-1/8*R[2]�2+1/112896*R[2]

�2*(10008*R[1]

�2+343*R[2]

�2);

:= 2�(1/2)-1/84*2

�(1/2)*(3*R[1]

�2+14*R[2]

�2)

-1/592704*2�(1/2)*(64484*R[2]

�4+115227*R[2]

�2*R[1]

�2

+3321*R[1]�4);


The algorithms outlined above can been applied to study the non-resonant case for any nonlin-ear systems that have N0 P 1 pairs of equations associated with purely imaginary eigenvalues,N1 P 0 equations associated with negative real eigenvalues, and N2 P 0 pairs of equations asso-ciated with complex conjugate eigenvalues. The Maple programs can be executed to find normalforms and solutions up to, in principle, any high order. The Maple source code is given in Appen-dix, which can be downloaded from the website: http://pyu1.apmaths.uwo.ca/~pyu/pub/index/sofware.html.

7. Examples

In this section, we shall present two examples: one is a system of mathematical equations andthe other is chosen from physical problems. We use the Maple programs to obtain the normalforms and give bifurcation and stability analysis.

7.1. Example 1—mathematical equations

Consider the following 9-dimensional nonlinear system:

_x1 ¼ �x2 þ 2x2x24 þ x21x5 þ 3x51;

_x2 ¼ x1 � x32 � x22x5 þ x22x6;

_x3 ¼ �ffiffiffi2

px4 þ x21x3 þ

6

5

ffiffiffi2

pþ 1

� �x23x4;

_x4 ¼ffiffiffi2

px3 � 2x22x4 þ 3x35;

_x5 ¼ �ffiffiffi3

px6 þ x2x6 þ x3x25;

_x4 ¼ffiffiffi3

px5 þ x1x5 þ x21x3;

_x7 ¼ �x7 þ ðx1 � x5Þ2;_x8 ¼ �x8 � x9 þ x23;

_x9 ¼ x8 � x9 þ x25:

ð101Þ

The Jacobian matrix of the system evaluated at the origin xi = 0 consists of three pairs of purelyimaginary eigenvalues, ±i, �

ffiffiffi2

pi and �

ffiffiffi3

pi, one real eigenvalue, �1, and one complex conjugate,

�1 ± i. This means that the center manifold of the system has dimension 6. Note that the coeffi-cients of the nonlinear terms contain irrational numbers. Executing our Maple program gives thenormal form up to fifth order:

_R1 ¼ R31 � 38þ 15

16R21 �

1

16R22

� �;

_R2 ¼ R2 � 14R21 þ

15ffiffiffi2

p� 14

101600635R21R

22 � ð7938

ffiffiffi2

pþ 17010ÞR43

h i( );

_R3 ¼ffiffiffi3

p

48400R3 1350

ffiffiffi3

pR41 � 12100R21R

22 þ 13673

ffiffiffi6

pR43

h i;

ð102Þ




_h1 ¼ 1� 1

2R22 �

5ffiffiffi2

pþ 22

16665608910� 2025

ffiffiffi2

p� �R41 þ 10416R42 þ

ffiffiffi3

p1600

ffiffiffi2

p� 7040

� �R21R

23

h i;

_h2 ¼ffiffiffi2

p� 6

ffiffiffi2

pþ 5

40R22 �

97ffiffiffi2

pþ 120

28275200194000� 120000

ffiffiffi2

p� �R41 þ 6627R42

hþ

ffiffiffi3

p698400

ffiffiffi2

p� 864000

� �R21R

23

i;

_h3 ¼ffiffiffi3

p�

ffiffiffi3

p

22R21 �

ffiffiffi3

p

319440075125R41 � 13200R21R

22 � 598950

ffiffiffi6

pR21R

23 þ 223608R22R

23

h i:

ð103Þ
Note that the phases h1, h2 and h3, described by Eq. (103), are decoupled from the amplitudes R1,R2 andR3 (see Eq. (102)), as expected, since system (101) is in non-resonance:x1:x2:x3 ¼ 1:
ffiffiffi2

p:ffiffiffi3

p.

Based on Eq. (102), one can easily obtain the steady-state solutions and their stability condi-tions. Setting _R1 ¼ _R2 ¼ _R3 ¼ 0 yields the following two real solutions:

(I) Periodic solution: R1 = R3 = 0, R2 = any real non-negative value, which includes the originalequilibrium R1 = R2 = R3 = 0.

(II) Periodic solution: R2 = R3 = 0, R21 ¼ 2

5.

The stability of the above steady-state solutions may be obtained by calculating the eigenvaluesof the Jacobian of Eq. (102) on the above solutions. It is easy to find that the eigenvalues for theperiodic solutions given in (I) are zero and thus the stabilities for these solutions cannot be deter-mined, indicating that bifurcation parameters are needed for stability analysis. Solution (II) isunstable since two of the three eigenvalues of the Jacobian evaluated on this solution are real pos-itive. To find the routes of bifurcations from one solution to another, bifurcation parameters(unfolding) are needed. Moreover, it can be seen that the above periodic and quasi-periodic solu-tions are not small, which may violate the assumption for local analysis, indicating again thatbifurcation parameters are needed.

7.2. Example 2—double pendulum

Now we consider a real physical problem—double pendulum system, shown in Fig. 1. The sys-tem consists of two rigid weightless links of equal length l which carry two concentrated masses2m and m, respectively. A follower force P is applied to this system.The system energy for the three linear springs k1, k2 and k3 is assumed to be given in the form of [1]

V ¼ 1

2ðk1 þ k2 þ k3l

2Þh21 þ 2ðk3l2 � k2Þh1h2 þ ðk2 þ k3l2Þh22

� �� 1

6k3l

2ðh1 þ h2Þðh31 þ h32Þ;

ð104Þ
where h1 and h2 are generalized co-ordinates which specify the configuration of the system com-pletely. The kinetic energy T of the system is expressed by
T ¼ ml2

2X23h02

1 þ h022 þ 2h0

1h02 cosðh1 � h2Þ

� �; ð105Þ

where X is an arbitrary value rendering the time variable non-dimensional, and the prime denotesdifferentiation with respect to the non-dimensional time variable s with s = Xt.

k

k

k 1

,

3

2

1d

d2 d3

P

m

2 m

l

l

θ

θ

2

1

P

Fig. 1. A double pendulum system.


The generalized force corresponding to the generalized co-ordinates h1 and h2 may be written as

Q ¼ Pl sinðh1 � h2Þ; ð106Þ
and the damping can be expressed by
D ¼ 1

2d1h

021 þ d2ðh0

1 � h02Þ2

h i� 1

4d3ðh0

1 � h02Þ4; ð107Þ

where d1, d2 represents the linear parts and d3 describe the non-linear parts, respectively. In gen-eral, one may assume that d1, d2, d3 P 0, indicating that the system has positive linear damping,but may have a negative nonlinear damping term. This is physically feasible.With the aid of the Lagrangian equations, in addition, choosing the state variables

z1 ¼ h1; z2 ¼ h01; z3 ¼ h2 and z4 ¼ h0

2; ð108Þ
one can derive a set of first order differential equations up to third order terms as follows:
z01 ¼ z2;

z02 ¼ �12f1 � f2 þ

1

2f4

� �z1 �

1

2g1 þ g2

� �z2 þ f2 �

1

2f4

� �z3 þ g2z4

þ 1

4f1 þ

3

4f2 �

1

3f4

� �z31 þ �3

4f2 �

1

2f3 þ

1

3f4

� �z33 � f5z34

þ f5z32 þ3

4g2 þ

1

4g1

� �z21z2 �

1

2f1 þ

9

4f2 �

1

2f3 � f4

� �z21z3

� 3

4g2z

21z4 �

1

2z1z22 þ

1

2z22z3 � 3f 5z

22z4 þ

1

4f1 þ

9

4f2 � f4

� �z1z23

þ 1

4g1 þ

3

4g2

� �z2z23 �

3

4g2z

23z4 �

1

2z1z24 þ 3f 5z2z

24 þ

1

2z3z24

� 1

2g1 þ

3

2g2

� �z1z2z3 þ

3

2g2z1z3z4;


z03 ¼ z4;

z04 ¼1

2f1 þ 2f 2 � f3 �

1

2f4

� �z1 þ

1

2g1 þ 2g2

� �z2 þ �2f 2 � f3 þ

1

2f4

� �z3

� 2g2z4 �1

2f1 þ

5

4f2 �

1

6f3 �

7

12f4

� �z31 þ

5

4f2 þ

7

6f3 �

7

12f4

� �z33

� 2f 5z32 þ 2f 5z

34 �

1

2g1 þ

5

4g2

� �z21z2 þ f1 þ

15

4f2 �

1

2f3 �

7

4f4

� �z21z3

þ 5

4g2z

21z4 þ

3

2z1z22 �

1

2f1 þ

15

4f2 �

1

2f3 �

7

4f4

� �z1z23 �

3

2z22z3

þ 6f 5z22z4 �

1

2g1 þ

5

4g2

� �z2z23 þ

5

4g2z

23z4 þ

1

2z1z24 � 6f 5z2z

24

� 1

2z3z24 þ g1 þ

5

2g2

� �z1z2z3 �

5

2g2z1z3z4;

ð109Þ

where fi�s and gj�s are dimensionless coefficients, defined as

f1 ¼k1X

2

ml2; f 2 ¼

k2X2

ml2; f 3 ¼

k3X2

m; f 4 ¼

PX2

ml; f 5 ¼

d3X4

ml2; g1 ¼

d1X2

ml2; g2 ¼

d2X2

ml2;

ð110Þ
and f1, f2, f3 P 0 due to physical restrictions, and g1, g2 P 0.It can be shown that when
f1 ¼4

7; f 2 ¼

407

56; f 3 ¼

1

56; f 4 ¼

535

28; f 5 ¼ 1; g1 ¼ g2 ¼ 0; ð111Þ

the Jacobian of system (109) evaluated on the equilibrium zi = 0 has two pairs of purely imaginaryeigenvalues: k1,2 = ±i, k3;4 ¼ �

ffiffiffi2

pi, indicating that the system is in non-resonant with two frequen-

cies x1c = 1 and x2c ¼ffiffiffi2

p.

Next, introducing the linear transformation

z1z2z3z4

8>><>>:

9>>=>>; ¼

0 1621

0 27

ffiffiffi2

p

1621

0 47

0

0 1 0 12

ffiffiffi2

p

1 0 1 0

26664

37775

x1x2x3x4

8>><>>:

9>>=>>;; ð112Þ

into Eq. (109) yields

x01 ¼ �x2 �22

7l1 �

75

28l2

� �x1 �

33

14l1 �

135

28l2

� �x3 �

625

4116x31 þ

351167

2765952x32

� 1215

1372x33 þ

286595

1229312

ffiffiffi2

px34 þ

48575

24696x21x2 �

1125

1372x21x3 þ

9715

5488

ffiffiffi2

px21x4

þ 753923

1843968

ffiffiffi2

px22x4 �

2025

1372x1x23 þ

4215

2744x2x23 þ

7587

5488

ffiffiffi2

px4x23 þ

478909

614656x2x24

þ 14165

4116x1x2x3 þ

8499

2744

ffiffiffi2

px1x3x4;

Table

Bifurc

Bifurc

E.S.

H.B. (

H.B. (

2-D T

3-D T


x02 ¼ x1;

x03 ¼ �ffiffiffi2

px4 þ

74

21l1 �

265

84l2

� �x1 þ

37

14l1 �

159

28l2

� �x3 þ

6625

37044x31 �

3568499

24893568x32

þ 1431

1372x33 �

1015277

36879362ð12Þx34 �

56635

24696x21x2 þ

1325

1372x21x3 �

11327

5488

ffiffiffi2

px21x4

� 2628781

5531904

ffiffiffi2

px22x4 þ

2385

1372x1x23 �

14585

8232x2x23 �

8751

5488

ffiffiffi2

px4x23 �

562705

614656x2x24

� 5475

1372x1x2x3 �

9855

2744

ffiffiffi2

px1x3x4;

x04 ¼ffiffiffi2

px3;

ð113Þ

where li = gi (i = 1, 2) have been chosen as perturbation parameters. Note that the Jacobian of thetransformed system (113) is now in Jordan canonical form (13).Similar to Example 1, the Maple program has be used to find the following normal form:

R01 ¼ R1 � 11

7l1 þ

75

56l2 �

625

10976R21 �

2025

5488R22

� �;

R02 ¼ R2

37

28l1 �

159

56l2 þ

1325

5488R21 þ

4293

10976R22

� �;

ð114Þ

h01 ¼ 1� 6493901

22127616R21 �

1423069

2458624R22;

h02 ¼

ffiffiffi2

p1þ 14046397

22127616R21 þ

2975501

9834496R22

� �;

ð115Þ

1

ation solutions and stability conditions for the double pendulum

ation Solution Stability Slope of critical line

R1 = 0 l2 <8875

l1 L1 : 8875 ) H:B:ðIÞR2 = 0 l2 >

74159

l1 L2 : 74159

) H:B:ðIIÞ

I) R21 ¼ 10976625

� 117

l1 þ 7556

l2& '

Stable L3 : 74783975) 2-D Tori

R2 = 0

x1 ¼ 1þ 714329118820000

l1 � 6493901940800

l2

II) R1 = 0 Unstable

R22 ¼ � 109764293

3728

l1 � 15956

& 'l2

x2 ¼ffiffiffi2

p1� 110093537

107702784l1 þ 2975501

1354752l2

& 'ori R21 ¼ 188944

99375l1 þ 196

25l2 Stable L4 : 300317950

) 3-D Tori

R22 ¼ � 146568831975

l1 þ 19681

l2

x1 ¼ 1þ 10485083223750485680000

l1 � 4701091712700800

l2

x2 ¼ffiffiffi2

p1� 34401875231

201942720000l1 þ 580057817

101606400l2

& 'ori Bifurcating from the critical line L5

L 1

4LL3

L2

0

Stable

Stable H.B.(I)

µ2

µ1

H.B.(II)Unstable

2–D Tori Stable

3-D Tori

Region

for E.S.

Fig. 2. Bifurcation diagram for the double pendulum system.


which can be used to find the steady-state solutions and their stabilities, as well as the sequence ofbifurcations. In fact, we may apply the Maple program developed in [20] to Eqs. (114) and (115) toperform the bifurcation analysis. The results are presented in Table 1 and the bifurcation diagramis shown in Fig. 2, where E.S., H.B. and 2-D (3-D) Tori denote equilibrium state, Hopf bifurcationand 2-dimensional (3-dimensional) Tori, respectively. Note that the bifurcation diagram is re-stricted to the first quadrant of l1–l2 plane since l1, l2 P 0. The double pendulum example exhib-its not only stable periodic solutions, but also stable quasi-periodic motion. These results have beenconfirmed by numerical simulations. The results also indicate that the 2-D torus loses stability atthe critical line L4 and bifurcates into a 3-D torus. The solution and stability for the 3-D torus maybe also found by using the perturbation technique, which are not presented in this paper.

8. Conclusions

A main theorem has been established for the equivalence between the MTS method and Poin-care normal form theory. The previous developed perturbation approach has been generalized tocompute the normal forms of high dimensional autonomous systems. Explicit formulas have beenderived for the solutions of ordered perturbation equations for various singularities. Based on theexplicit solutions, algorithms for computing the normal forms of the various cases are developed.Examples are presented to show the applicability of the method and the efficiency of using Mapleprograms. In particular, an physical problem is studied in detail.

Acknowledgement

This work was supported by the Natural Sciences and Engineering Research Council of Can-ada (NSERC).


Appendix A. Maple source code

In this appendix, the Maple source code for the non-resonant case is listed. A user can executethe source code for one�s own input file. The preparation of an input file can follow the sampleexample given in Section 6. A sample input file can also be found on the website: http://pyu1.ap-maths.uwo.ca/~pyu/pub/index/sofware.html.









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ComputationofthenormalformsforgeneralM-DOF ... · ComputationofthenormalformsforgeneralM-DOF systemsusingmultipletimescales.PartI:autonomoussystems PeiYu*,SonghuiZhu1 DepartmentofAppliedMathematics,FacultyofScience