Differential equations with symbolic computation

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Differential Equations
with
Symbolic Computation

Dongming Wang
Zhiming Zheng
Editors

"IRKHËUSER6ERLAG
"ASEL "OSTON "ERLIN
s

s


Editors:

Dongming Wang
School of Science
Beihang University
37 Xueyuan Road
Beijing 100083
China

Zhiming Zheng
School of Science
Beihang University
37 Xueyuan Road
Beijing 100083
China

and

e-mail:

Laboratoire d’Informatique de Paris 6
Université Pierre et Marie Curie - CNRS
8, rue due Capitaine Scott
75015 Paris
France
e-mail:

2000 Mathematical Subject Classification 34-06; 35-06; 68W30
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Preface
This book provides a picture of what can be done in differential equations with
advanced methods and software tools of symbolic computation. It focuses on the
symbolic-computational aspect of three kinds of fundamental problems in differential equations: transforming the equations, solving the equations, and studying
the structure and properties of their solutions. Modern research on these problems using symbolic computation, or more restrictively using computer algebra,
has become increasingly active since the early 1980s when effective algorithms
for symbolic solution of differential equations were proposed, and so were computer algebra systems successfully applied to perturbation, bifurcation, and other
problems. Historically, symbolic integration, the simplest case of solving ordinary
differential equations, was already the target of the first computer algebra package
SAINT in the early 1960s.
With 20 chapters, the book is structured into three parts with both tutorial
surveys and original research contributions: the first part is devoted to the qualitative study of differential systems with symbolic computation, including stability
analysis, establishment of center conditions, and bifurcation of limit cycles, which
are closely related to Hilbert’s sixteenth problem. The second part is concerned
with symbolic solutions of ordinary and partial differential equations, for which
normal form methods, reduction and factorization techniques, and the computation of conservation laws are introduced and used to aid the search. The last part

is concentrated on the transformation of differential equations into such forms that
are better suited for further study and application. It includes symbolic elimination and triangular decomposition for systems of ordinary and partial differential
polynomials. A 1991 paper by Wen-tsă
u
ă n Wu on the construction of Gră
obner
ă
bases
based on RiquierJanets theory, published in China and not widely available to
the western readers, is reprinted as the last chapter. This book should reflect the
current state of the art of research and development in differential equations with
symbolic computation and is worth reading for researchers and students working
on this interdisciplinary subject of mathematics and computational science. It may
also serve as a reference for everyone interested in differential equations, symbolic
computation, and their interaction.
The idea of compiling this volume grew out of the Seminar on Differential
Equations with Symbolic Computation (DESC 2004), which was held in Beijing,
China in April 2004 (see to facilitate
the interaction between the two disciplines. The seminar brought together active
researchers and graduate students from both disciplines to present their work and
to report on their new results and findings. It also provided a forum for over 50
participants to exchange ideas and views and to discuss future development and
cooperation. Four invited talks were given by Michael Singer, Lan Wen, Wen-tsă
un
Wu, and Zhifen Zhang. The enthusiastic support of the seminar speakers and the


vi
high quality of their presentations are some of the primary motivations for our
endeavor to prepare a coherent and comprehensive volume with most recent advances on the subject for publication. In addition to the seminar speakers, several

distinguished researchers who were invited to attend the seminar but could not
make their trip have also contributed to the present book. Their contributions have
helped enrich the contents of the book and make the book beyond a proceedings
volume. All the papers accepted for publication in the book underwent a formal
review-revision process.
DESC 2004 is the second in a series of seminars, organized in China, on
various subjects interacted with symbolic computation. The first seminar, held in
Hefei from April 24–26, 2002, was focused on geometric computation and a book
on the same subject has been published by World Scientific. The third seminar
planned for April 2006 will be on symbolic computation in education.
The editors gratefully acknowledge the support provided by the Schools of
Science and Advanced Engineering at Beihang University and the Key Laboratory
of Mathematics, Informatics and Behavioral Semantics of the Chinese Ministry of
Education for DESC 2004 and the preparation of this book. Our sincere thanks
go to the authors for their contributions and cooperation, to the referees for their
expertise and timely help, and to all colleagues and students who helped for the
organization of DESC 2004.

Beijing
May 2005

Dongming Wang
Zhiming Zheng


Contents
Symbolic Computation of Lyapunov Quantities and the Second Part
of Hilbert’s Sixteenth Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Stephen Lynch
Estimating Limit Cycle Bifurcations from Centers . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Colin Christopher
Conditions of Infinity to be an Isochronous Center for a Class of
Differential Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Wentao Huang and Yirong Liu
Darboux Integrability and Limit Cycles for a Class of Polynomial
Differential Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
Jaume Gin´
´ and Jaume Llibre
Time-Reversibility in Two-Dimensional Polynomial Systems . . . . . . . . . . . . . . . . . 67
Valery G. Romanovski and Douglas S. Shafer
On Symbolic Computation of the LCE of N -Dimensional Dynamical
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Shucheng Ning and Zhiming Zheng
Symbolic Computation for Equilibria of Two Dynamic Models . . . . . . . . . . . . . . 109
Weinian Zhang and Rui Yan
Attractive Regions in Power Systems by Singular Perturbation Analysis . . . . .121
Zhujun Jing, Ruiqi Wang, Luonan Chen and Jin Deng
Algebraic Multiplicity and the Poincar´
´e Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . .143
Jinzhi Lei and Lijun Yang
Formalizing a Reasoning Strategy in Symbolic Approach to Differential
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Shilong Ma
Looking for Periodic Solutions of ODE Systems by the Normal Form
Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Victor F. Edneral


viii
Algorithmic Reduction and Rational General Solutions of First Order

Algebraic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Guoting Chen and Yujie Ma
Factoring Partial Differential Systems in Positive Characteristic . . . . . . . . . . . . . 213
Moulay A. Barkatou, Thomas Cluzeau and Jacques-Arthur Weil
On the Factorization of Differential Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
Min Wu
Continuous and Discrete Homotopy Operators and the Computation
of Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
Willy Hereman, Michael Colagrosso, Ryan Sayers, Adam Ringler,
Bernard Deconinck, Michael Nivala and Mark Hickman
Partial and Complete Linearization of PDEs Based on Conservation
Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
Thomas Wolf
CONSLAW: A Maple Package to Construct the Conservation Laws
for Nonlinear Evolution Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
Ruo-Xia Yao and Zhi-Bin Li
Generalized Differential Resultant Systems of Algebraic ODEs and
Differential Elimination Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
Giuseppa Carr´
´ Ferro
On “Good” Bases of Algebraico-Differential Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . 343
Wen-tsă
u
ăn Wu
On the Construction of Groebner Basis of a Polynomial Ideal Based
on Riquier–Janet Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
Wen-tsă
u
ăn Wu
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .369

Trends in Mathematics: Differential Equations with Symbolic Computation, 1–22
c 2006 Birkhauser
ă
Verlag Basel/Switzerland

Symbolic Computation of Lyapunov Quantities
and the Second Part of Hilbert’s Sixteenth
Problem
Stephen Lynch
Abstract. This tutorial survey presents a method for computing the Lyapunov
quantities for Lienard
´
systems of differential equations using symbolic manipulation packages. The theory is given in detail and simple working MATLAB
and Maple programs are listed in this chapter. In recent years, the author
has been contacted by many researchers requiring more detail on the algorithmic method used to compute focal values and Lyapunov quantities. It is
hoped that this article will address the needs of those and other researchers.
Research results are also given here.
Mathematics Subject Classification (2000). Primary 34C07; Secondary 37M20.
´
equation, limit cycle, Maple, MATLAB, smallKeywords. Bifurcation, Lienard
amplitude.

1. Introduction
Poincare´ began investigating isolated periodic cycles of planar polynomial vector
fields in the 1880s. However, the general problem of determining the maximum
number and relative configurations of limit cycles in the plane has remained unresolved for over a century. In the engineering literature, limit cycles in the plane can
correspond to steady-state behavior for a physical system (see [25], for example),
so it is important to know how many possible steady states there are. There are

applications in aircraft flight dynamics and surge in jet engines, for example.
In 1900, David Hilbert presented a list of 23 problems to the International
Congress of Mathematicians in Paris. Most of the problems have been solved,
either completely or partially. However, the second part of the sixteenth problem
remains unsolved. Ilyashenko [37] presents a centennial history of Hilbert’s 16th
problem and Li [19] has recently written a review article.


2

Lynch

The Second Part of Hilbert’s Sixteenth Problem. Consider planar polynomial
systems of the form
x˙ = P (x, y),

y˙ = Q(x, y),

(1.1)

where P and Q are polynomials in x and y. The question is to estimate the maximal
number and relative positions of the limit cycles of system (1.1). Let Hn denote
the maximum possible number of limit cycles that system (1.1) can have when P
and Q are of degree n. More formally, the Hilbert numbers Hn are given by
Hn = sup {π(P, Q) : ∂P, ∂Q ≤ n},
where ∂ denotes “the degree of” and π(P, Q) is the number of limit cycles of system
(1.1).
Dulac’s Theorem states that a given polynomial system cannot have infinitely
many limit cycles. This theorem has only recently been proved independently by
Ecalle et al. [13] and Ilyashenko [36], respectively. Unfortunately, this does not

imply that the Hilbert numbers are finite.
Of the many attempts to make progress in this question, one of the more
fruitful approaches has been to create vector fields with as many isolated periodic
orbits as possible using both local and global bifurcations [3]. There are relatively
few results in the case of general polynomial systems even when considering local bifurcations. Bautin [1] proved that no more than three small-amplitude limit
cycles could bifurcate from a critical point for a quadratic system. For a homogeneous cubic system (no quadratic terms), Sibirskii [33] proved that no more than
five small-amplitude limit cycles could be bifurcated from one critical point. Recently, Zoladek [39] found an example where 11 limit cycles could be bifurcated
from the origin of a cubic system, but he was unable to prove that this was the
maximum possible number.
Although easily stated, Hilbert’s sixteenth problem remains almost completely unsolved. For quadratic systems, Songling Shi [32] has obtained a lower
bound for the Hilbert number H2 ≥ 4. A possible global phase portrait is given
in Figure 1. The line at infinity is included and the properties on this line are determined using Poincar´
´e compactification, where a polynomial vector field in the
plane is transformed into an analytic vector field on the 2-sphere. More detail on
Poincare´ compactification can be found in [27]. There are three small-amplitude
limit cycles around the origin and at least one other surrounding another critical
point. Some of the parameters used in this example are very small.
Blows and Rousseau [4] consider the bifurcation at infinity for polynomial
vector fields and give examples of cubic systems having the following configurations:
{(4), 1}, {(3), 2}, {(2), 5}, {(4), 2}, {(1), 5} and {(2), 4},
where {(l), L} denotes the configuration of a vector field with l small-amplitude
limit cycles bifurcated from a point in the plane and L large-amplitude limit cycles


Symbolic Computation of Lyapunov Quantities

3

Limit cycle

Small-amplitude
limit cycles

Figure 1. A possible configuration for a quadratic system with
four limit cycles: one of large amplitude and three of small amplitude.
simultaneously bifurcated from infinity. There are many other configurations possible, some involving other critical points in the finite part of the plane as shown
in Figure 2. Recall that a limit cycle must contain at least one critical point.
By considering cubic polynomial vector fields, in 1985, Jibin Li and Chunfu
Li [18] produced an example showing that H3 ≥ 11 by bifurcating limit cycles out
of homoclinic and heteroclinic orbits; see Figure 2.

Figure 2. A possible configuration for a cubic system with 11
limit cycles.
Returning to the general problem, in 1995, Christopher and Lloyd [7] considered the rate of growth of Hn as n increases. They showed that Hn grows at least
as rapidly as n2 log n.
In recent years, the focus of research in this area has been directed at a
small number of classes of systems. Perhaps the most fruitful has been the Li´´enard


4

Lynch

system. A method for computing focal values and Lyapunov quantities for Li´
´enard
systems is given in detail in the next section. Li´´enard systems provide a very
suitable starting point as they do have ubiquity for systems in the plane [14, 16, 28].

2. Small-Amplitude Limit Cycle Bifurcations
The general problem of determining the maximum number and relative configurations of limit cycles in the plane has remained unresolved for over a century.

Both local and global bifurcations have been studied to create vector fields with
as many limit cycles as possible. All of these techniques rely heavily on symbolic
manipulation packages such as Maple, and MATLAB and its Symbolic Math Toolbox. Unfortunately, the results in the global case number relatively few. Only in
recent years have many more results been found by restricting the analysis to
small-amplitude limit cycle bifurcations.
It is well known that a nondegenerate critical point, say x0 , of center or focus
type can be moved to the origin by a linear change of coordinates, to give
x˙ = λx − y + p(x, y),

y˙ = x + λy + q(x, y),

(2.1)

where p and q are at least quadratic in x and y. If λ = 0, then the origin is
structurally stable for all perturbations.
Definition 2.1. A critical point, say x0 , is called a fine focus of system (1.1) if it
is a center for the linearized system at x0 . Equivalently, if λ = 0 in system (2.1),
then the origin is a fine focus.
In the work to follow, assume that the unperturbed system does not have a
center at the origin. The technique used here is entirely local; limit cycles bifurcate
out of a fine focus when its stability is reversed by perturbing λ and the coefficients
arising in p and q. These are said to be local or small-amplitude limit cycles. How
close the origin is to being a center of the nonlinear system determines the number
of limit cycles that may be obtained from bifurcation. The method for bifurcating
limit cycles will be given in detail here.
By a classical result, there exists a Lyapunov function, V (x, y) = V2 (x, y) +
V4 (x, y) + · · · + Vk (x, y) + · · · say, where Vk is a homogeneous polynomial of degree
k, such that
dV
= η2 r2 + η4 r4 + · · · + η2i r2i + · · · ,

(2.2)
dt
where r2 = x2 + y 2 . The η2i are polynomials in the coefficients of p and q and
are called the focal values. The origin is said to be a fine focus of order k if
η2 = η4 = · · · = η2k = 0 but η2k+2 = 0. Take an analytic transversal through
the origin parameterized by some variable, say c. It is well known that the return
map of (2.1), c → h(c), is analytic if the critical point is nondegenerate. Limit
cycles of system (2.1) then correspond to zeros of the displacement function, say
d(c) = h(c) − c. Hence at most k limit cycles can bifurcate from the fine focus.
The stability of the origin is clearly dependent on the sign of the first non-zero


Symbolic Computation of Lyapunov Quantities

5

focal value, and the origin is a nonlinear center if and only if all of the focal values
are zero. Consequently, it is the reduced values, or Lyapunov quantities, say L(j),
that are significant. One needs only to consider the value η2k reduced modulo the
ideal (η2 , η4 , . . . , η2k−2 ) to obtain the Lyapunov quantity L(k − 1). To bifurcate
limit cycles from the origin, select the coefficients in the Lyapunov quantities such
that
|L(m)|
|L(m + 1)| and L(m)L(m + 1) < 0,
for m = 0, 1, . . . , k − 1. At each stage, the origin reverses stability and a limit
cycle bifurcates in a small region of the critical point. If all of these conditions
are satisfied, then there are exactly k small-amplitude limit cycles. Conversely, if
L(k) = 0, then at most k limit cycles can bifurcate. Sometimes it is not possible
to bifurcate the full complement of limit cycles.
The algorithm for bifurcating small-amplitude limit cycles may be split into

the following four steps:
1. computation of the focal values using a mathematical package;
2. reduction of the n-th focal value modulo a Grobner
ă
basis of the ideal generated by the first n − 1 focal values (or the first n − 1 Lyapunov quantities);
3. checking that the origin is a center when all of the relevant Lyapunov quantities are zero;
4. bifurcation of the limit cycles by suitable perturbations.
Dongming Wang [34, 35] has developed software to deal with the reduction part
of the algorithm for several differential systems. For some systems, the following
theorems can be used to prove that the origin is a center.
The Divergence Test. Suppose that the origin of system (1.1) is a critical point of
focus type. If
∂(ψP ) ∂(ψQ)
+
= 0,
div (ψX) =
∂x
∂y
where ψ : 2 → 2 , then the origin is a center.
The Classical Symmetry Argument. Suppose that λ = 0 in system (2.1) and that
either
(i) p(x, y) = −p(x, −y) and q(x, y) = q(x, −y) or
(ii) p(x, y) = p(−x, y) and q(x, y) = −q(−x, y).
Then the origin is a center.
Adapting the classical symmetry argument, it is also possible to prove the
following theorem.
Theorem 2.1. The origin of the system
x˙ = y − F (G(x)),
where F and H are polynomials, G(x) =
0 for x = 0, is a center.

G (x)
H(G(x)),
2
g(s)ds with g(0) = 0 and g(x) sgn(x) >

y˙ = −
x
0


6

Lynch

To demonstrate the method for bifurcating small-amplitude limit cycles, consider Lienard
´
equations of the form
x˙ = y − F (x),

y˙ = −g(x),

(2.3)

where F (x) = a1 x + a2 x2 + · · ·+ au xu and g(x) = x + b2 x2 + b3 x3 + · · · + bv xv . This
system has proved very useful in the investigation of limit cycles when showing
existence, uniqueness, and hyperbolicity of a limit cycle. In recent years, there have
also been many local results; see, for example, [9]. Therefore, it seems sensible to
use this class of system to illustrate the method.
The computation of the first three focal values will be given. Write

Vi,j xi y j

Vk (x, y) =
i+j=k

and denote Vi,j as being odd or even according to whether i is odd or even and that
Vi,j is 2-odd or 2-even according to whether j is odd or even, respectively. Solving
equation (2.2), it is easily seen that V2 = 12 (x2 + y 2 ) and η2 = −a1 . Therefore,
set a1 = 0. The odd and even coefficients of V3 are then given by the two pairs of
equations
3V
V3,0 − 2V
V1,2 = b2 ,
V1,2 = 0
and
−V
V2,1 = a2 ,
V0,3 = 0,
2V
V2,1 − 3V
respectively. Solve the equations to give
1
2
b2 x3 − a2 x2 y − a2 y 3 .
3
3
Both η4 and the odd coefficients of V4 are determined by the equations
V3 =

−η4 − V3,1 = a3 ,

V3,1 − 3V
V1,3 = −2a2 b2 ,
−2η4 + 3V
−η4 + V1,3 = 0.
The even coefficients are determined by the equations
V2,2 = b3 − 2a22 ,
4V
V4,0 − 2V
V0,4 = 0
2V
V2,2 − 4V
and the supplementary condition V2,2 = 0. In fact, when computing subsequent
coefficients for V4m , it is convenient to require that V2m,2m = 0. This ensures that
there will always be a solution. Solving these equations gives
V4 =

1
(b3 − 2a22 )x4 − (η4 + a3 )x3 y + η4 xy 3
4


Symbolic Computation of Lyapunov Quantities

7

and

1
(2a2 b2 − 3a3 ).
8

Suppose that η4 = 0 so that a3 = 23 a2 b2 . It can be checked that the two sets of
equations for the coefficients of V5 give
η4 =

V5 =

b4
2a2 b2
− 2
5
3

x5 + (2a32 − a4 )x4 y +

8a32
4a4
2a2 b3
−
+
3
3
3

x2 y 3

16a32
8a4
4a2 b3
−
−

y5.
15
15
15
The coefficients of V6 may be determined by inserting the extra condition V4,2 +
V2,4 = 0. In fact, when computing subsequent even coefficients for V4m+2 , the extra
condition V2m,2m+2 + V2m+2,2m = 0, is applied, which guarantees a solution. The
polynomial V6 contains 27 terms and will not be listed here. However, η6 leads to
the Lyapunov quantity
+

L(2) = 6a2 b4 − 10a2 b2 b3 + 20a4 b2 − 15a5 .
Lemma 2.1. The first three Lyapunov quantities for system (2.3) are L(0) = −a1 ,
L(1) = 2a2 b2 − 3a3 , and L(2) = 6a2 b4 − 10a2 b2 b3 + 20a4 b2 − 15a5 .
Example. Prove that
(i) there is at most one small-amplitude limit cycle when ∂F = 3, ∂g = 2 and
(ii) there are at most two small-amplitude limit cycles when ∂F = 3, ∂g = 3,
for system (2.3).
Solutions. (i) Now L(0)=0 if a1 = 0 and L(1) = 0 if a3 = 23 a2 b2 . Thus system (2.3)
becomes
2
x˙ = y − a2 x2 − a2 b2 x3 , y˙ = −x − b2 x2 ,
3
and the origin is a center by Theorem 2.1. Therefore, the origin is a fine focus of
order one if and only if a1 = 0 and 2a2 b2 − 3a3 = 0. The conditions are consistent.
Select a3 and a1 such that
|L(0)|

|L(1)| and L(0)L(1) < 0.

The origin reverses stability once and a limit cycle bifurcates. The perturbations
are chosen such that the origin reverses stability once and the limit cycles that
bifurcate persist.
(ii) Now L(0) = 0 if a1 = 0, L(1) = 0 if a3 = 23 a2 b2 , and L(2) = 0 if
a2 b2 b3 = 0. Thus L(2) = 0 if
(a) a2 = 0,
(b) b3 = 0, or
(c) b2 = 0.


8

Lynch

If condition (a) holds, then a3 = 0 and the origin is a center by the divergence
test (divX = 0). If condition (b) holds, then the origin is a center from result (i)
above. If condition (c) holds, then a3 = 0 and system (2.3) becomes
x˙ = y − a2 x2 ,

y˙ = −x − b3 x3 ,

and the origin is a center by the classical symmetry argument. The origin is thus
a fine focus of order two if and only if a1 = 0 and 2a2 b2 − 3a3 = 0 but a2 b2 b3 = 0.
The conditions are consistent. Select b3 , a3 , and a1 such that
|L(1)|

|L(2)|, L(1)L(2) < 0

and |L(0)|

|L(1)|, L(0)L(1) < 0.

The origin has changed stability twice, and there are two small-amplitude limit
cycles. The perturbations are chosen such that the origin reverses stability twice
and the limit cycles that bifurcate persist.

3. Symbolic Computation
Readers can download the following program files from the Web. The MATLAB
M-file lists all of the coefficients of the Lyapunov function up to and including
degree six terms. The output is also included for completeness. The program was
written using MATLAB version 7 and the program files can be downloaded at
/>under the links “Companion Software for Books” and “Mathematics”.
% MATLAB Program - Determining the coefficients of the Lyapunov
% function for a quintic Lienard system.
% V3=[V30;V21;V12;V03], V4=[V40;V31;V22;V13;V04;eta4],
% V5=[V50;V41;V32;V23;V14;V05],
% V6=[V60;V51;V42;V33;V24;V15;V06;eta6]
% Symbolic Math toolbox required.
clear all
syms a1 a2 b2 a3 b3 a4 b4 a5 b5;
A=[3 0 -2 0;0 0 1 0;0 -1 0 0;0 2 0 -3];
B=[b2; 0; a2; 0];
V3=A\B
A=[0 -1 0 0 0 -1;0 3 0 -3 0 -2;0 0 0 1 0 -1;4 0 -2 0 0 0;
0 0 2 0 -4 0; 0 0 1 0 0 0];
B=[a3; -2*a2*b2; 0; b3-2*a2^2;0;0];
V4=A\B
A=[5 0 -2 0 0 0;0 0 3 0 -4 0;0 0 0 0 1 0;0 -1 0 0 0 0;0 4 0 -3 0 0;

Symbolic Computation of Lyapunov Quantities
0 0 0 2 0 -5];
B=[b4-10*a2^2*b2/3;0;0;a4-2*a2^3;-2*a2*b3;0];
V5=A\B
A=[6 0 -2 0 0 0 0 0;0 0 4 0 -4 0 0 0;0 0 0 0 2 0 -6 0;
0 0 1 0 1 0 0 0;0 -1 0 0 0 0 0 -1;0 5 0 -3 0 0 0 -3;
0 0 0 3 0 -5 0 -3;0 0 0 0 0 1 0 -1];
B=[b5-6*a2*a4-4*a2^2*b2^2/3+8*a2^4;16*a2^4/3+4*a2^2*b3/3-8*a2*a4/3;
0;0;a5-8*a2^3*b2/3;-2*a2*b4+8*a2^3*b2+2*a2*b2*b3-4*a4*b2;
16*a2^3*b2/3+4*a2*b2*b3/3-8*a4*b2/3;0];
V6=A\B
L0=-a1
eta4=V4(6,1)
L1=maple(’numer(-3/8*a3+1/4*a2*b2)’)
a3=2*a2*b2;
eta6=V6(8,1)
L2=maple(’numer(-5/16*a5+1/8*a2*b4-5/24*a2*b2*b3+5/12*a4*b2)’)
%End of MATLAB Program
V3 =
[ 1/3*b2]
[
-a2]
[
0]
[ -2/3*a2]
V4 =
[
1/4*b3-1/2*a2^2]
[ -5/8*a3-1/4*a2*b2]
[

0]
[ -3/8*a3+1/4*a2*b2]
[
0]
[ -3/8*a3+1/4*a2*b2]
V5 =
[
1/5*b4-2/3*a2^2*b2]
[
-a4+2*a2^3]
[
0]
[
-4/3*a4+8/3*a2^3+2/3*a2*b3]
[
0]
[ -8/15*a4+16/15*a2^3+4/15*a2*b3]
V6 =

9


10

Lynch

[
14/9*a2^4+1/6*b5-10/9*a2*a4-2/9*a2^2*b2^2+1/18*a2^2*b3]
[ -11/16*a5-1/8*a2*b4+5/24*a2*b2*b3-5/12*a4*b2+8/3*a2^3*b2]
[

2/3*a2^4+1/6*a2^2*b3-1/3*a2*a4]
[
2/9*a4*b2-5/6*a5+1/3*a2*b4-1/9*a2*b2*b3+16/9*a2^3*b2]
[
-2/3*a2^4-1/6*a2^2*b3+1/3*a2*a4]
[
-5/16*a5+1/8*a2*b4-5/24*a2*b2*b3+5/12*a4*b2]
[
-2/9*a2^4-1/18*a2^2*b3+1/9*a2*a4]
[
-5/16*a5+1/8*a2*b4-5/24*a2*b2*b3+5/12*a4*b2]

L0 =-a1
L1 =-3*a3+2*a2*b2
L2 =-15*a5+6*a2*b4-10*a2*b2*b3+20*a4*b2

The Maple 9 program files can be found at
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# MAPLE program to compute the first two Lyapunov quantities for
# a quintic Lienard system.
restart:
kstart:=2:kend:=5:
pp:=array(1..20):qq:=array(1..20):
vv:=array(1..20):vx:=array(0..20):
vy:=array(0..20):xx:=array(0..20,0..20):
yy:=array(0..20,0..20):uu:=array(0..20,0..20):
z:=array(0..20):ETA:=array(1..20):
pp[1]:=y:qq[1]:=-x:vv[2]:=(x^2+y^2)/2:vx[2]:=x:vy[2]:=y:
for j1 from 0 to 20 do
for j2 from 0 to 20 do
xx[j1,j2]:=0:yy[j1,j2]:=0:
od:od:
# Insert the coefficients for a quintic Lienard system.

xx[0,1]:=1:xx[2,0]:=-a2:xx[3,0]:=-a3:xx[4,0]:=-a4:xx[5,0]:=-a5:
yy[1,0]:=-1:yy[2,0]:=-b2:yy[3,0]:=-b3:yy[4,0]:=-b4:yy[5,0]:=-b5:
for kloop from kstart to kend do
kk:=kloop:
dd1:=sum(pp[i]*vx[kk+2-i]+qq[i]*vy[kk+2-i],i=2..kk-1):
pp[kk]:=sum(xx[kk-i,i]*x^(kk-i)*y^i,i=0..kk):
qq[kk]:=sum(yy[kk-i,i]*x^(kk-i)*y^i,i=0..kk):
vv[kk+1]:=sum(uu[kk+1,i]*x^(kk+1-i)*y^i,i=0..kk+1):
d1:=y*diff(vv[kk+1],x)-x*diff(vv[kk+1],y)+pp[kk]*vx[2]+


Symbolic Computation of Lyapunov Quantities

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qq[kk]*vy[2]+dd1:
dd:=expand(d1):
if irem(kk,2)=1 then dd:=dd-ETA[kk+1]*(x^2+y^2)^((kk+1)/2):
fi:
dd:=numer(dd):x:=1:
for i from 0 to kk+1 do z[i]:=coeff(dd,y,i);
od:
if kk=2 then
seqn:=solve({z[0],z[1],z[2],z[3]},
{uu[3,0],uu[3,1],uu[3,2],uu[3,3]}):
elif kk=3 then
seqn:=solve({z[0],z[1],z[2],uu[4,2],z[3],z[4]},
{uu[4,0],uu[4,1],uu[4,2],uu[4,3],uu[4,4],ETA[4]}):
elif kk=4 then
seqn:=solve({z[0],z[1],z[2],z[3],z[4],z[5]},
{uu[5,0],uu[5,1],uu[5,2],uu[5,3],uu[5,4],uu[5,5]}):
elif kk=5 then
seqn:=solve({z[0],z[1],z[2],uu[6,2]+uu[6,4],z[3],z[4],z[5],z[6]},

{uu[6,0],uu[6,1],uu[6,2],uu[6,3],uu[6,4],uu[6,5],
uu[6,6],ETA[6]}):
fi:
assign(seqn):x:=’x’:i:=’i’:
vv[kk+1]:=sum(uu[kk+1,i]*x^(kk+1-i)*y^i,i=0..kk+1):
vx[kk+1]:=diff(vv[kk+1],x):vy[kk+1]:=diff(vv[kk+1],y):
ETA[kk+1]:=ETA[kk+1]:
od:
print(L1=numer(ETA[4])):a3:=2*a2*b2/3:print(L2=numer(ETA[6]));
L1 = -3 a3 + 2 b2 a2
L2 = 20 b2 a4 - 10 b3 b2 a2 + 6 b4 a2 - 15 a5

The programs can be extended to compute further focal values. The algorithm
in the context of Lienard
´
systems will now be described. Consider system (2.3); the
linearization at the origin is already in canonical form. Write Dk for the collection
of terms of degree k in V˙ . Hence
Dk = y

∂V
Vk
∂V
Vk
−x
−
∂x
∂y

k−1

ar
r=2

∂V
Vk−r+1
∂V
Vk−r+1
+ br
∂x
∂y

xr

.

(3.1)

Choose Vk and η2k (k = 2, 3, . . .) such that D2k = η2k rk and D2k−1 = 0. The focal
values are calculated recursively, in a two-stage procedure. Having determined V


12

Lynch

with ≤ 2k, V2k+1 is found by setting D2k+1 = 0, and then V2k+2 and η2k+2 are
k+1
computed from the relation D2k+2 = η2k+2 x2 + y 2
. Setting D2k+1 = 0 gives

2k + 2 linear equations for the coefficients of V2k+1 in terms of those of V with
≤ 2k. These uncouple into two sets of k + 1 equations, one of which determines
the odd coefficients of V2k+1 and the other the even coefficients. For system (2.3)
the two sets are as follows:
2k−1

V2k−1,2 =
(2k + 1)V
V2k+1,0 − 2V

2k−2

iV
Vi,1 a2k+1−i +
i=2
2k−3

(2k − 1)V
V2k−1,2 − 4V
V2k−3,4 =

2k−4

iV
Vi,3 a2k−1−i +
i=1

..
.

2V
Vi,2 b2k−i
i=0

4V
Vi,4 b2k−2−i
i=0

..
.

=

V1,2k = V1,2k−1 a2 + 2kV
V0,2k b2
3V
V3,2k−2 − 2kV
V1,2k = 0,
and
2k

2k−1

−V
V2k,1 =

iV
Vi,0 a2k+2−i +
i=2
2k−2

2kV
V2k,1 − 3V
V2k−2,3 =

2k−3

iV
Vi,2 a2k−i +
i=1

..
.

Vi,1 b2k+1−i
i=1

3V
Vi,3 b2k−1−i
i=0

..
.

=
2

V2,2k−1 =
4V
V4,2k−3 − (2k − 1)V

1

(2k − 1)V
Vi,2k−1 b3−i

a4−i +
i=1

i=0

2V
V2,2k−1 − (2k + 1)V
V0,2k+1 = 0.
k+1

The coefficients of V2k+2 are obtained by setting D2k+2 = η2k+2 x2 + y 2
odd and even coefficients are given by the following two sets of equations:
2k+1

−η2k+2 − V2k+1,1 =

2k

iV
Vi,0 a2k+3−i +
i=2

V3,2k−1 − (2k + 1)V
V1,2k+1

−kη2k+2 + 3V

2k−2

iV
Vi,2 a2k+1−i +
i=1

..
.

Vi,1 b2k+2−i
i=1

2k−1

−kη2k+2 + (2k + 1)V
V2k+1,1 − 3V
V2k−1,3 =

3V
Vi,3 b2k−i
i=0

..
=
.
= V1,2k a2 + (2k + 1)V
V0,2k+1 b2

−η2k+2 + V1,2k+1 = 0,

. The


Symbolic Computation of Lyapunov Quantities

13

and
2k

(2k + 2)V
V2k+2,0 − 2V
V2k,2 =

2k−1

iV
Vi,1 a2k+2−i +
i=1
2k−2

2kV
V2k,2 − 4V
V2k−2,4 =

2k−3

iV

Vi,3 a2k−i +
i=1

..
.

2V
Vi,2 b2k+1−i
i=0

4V
Vi,4 b2k−1−i
i=0

..
.

=
2

4V
V4,2k−2 − 2kV
V2,2k =

(3.2)
1

iV
Vi,2k−1 a4−i +
i=1

2kV
Vi,2k b3−i
i=0

2V
V2,2k − (2k + 2)V
V0,2k+2 = 0.
V2r+2,2r = 0 if k = 4r+2,
To simplify the calculations for system (3.3) set V2r,2r+2 +V
and V2r,2r = 0 if k = 4r.

4. Results
Lienard
´
systems have proved very useful in the investigation of multiple limit cycles
and also when proving existence, uniqueness, and hyperbolicity of a limit cycle.
Let ∂ denote the degree of a polynomial, and let H(m, n) denote the maximum
number of global limit cycles, where m is the degree of f and n is the degree of g
for the Li´
´enard equation
x
ă + f (x)x + g(x) = 0.
(4.1)
The results in the global case are listed below:
1. In 1928, Lienard
´
[17] proved that when ∂g = 1 and F is a continuous odd
function, which has a unique root at x = a and is monotone increasing for
x ≥ a, then (2.3) has a unique limit cycle.

2. In 1975, Rychkov [30] proved that if ∂g = 1 and F is an odd polynomial of
degree five, then (2.3) has at most two limit cycles.
3. In 1976, Cherkas [5] gave conditions in order for a Li´
´enard equation to have
a center.
4. In 1977, Lins, de Melo, and Pugh [20] proved that H(2, 1) = 1. They also
conjectured that H(2i, 1) = H(2i + 1, 1) = i, where i is a natural number.
5. In 1988, Coppel [10] proved that H(1, 2) = 1.
6. In 1992, Zhifen Zhang [38] proved that a certain generalised Li´´enard system
has a unique limit cycle.
7. In 1996, Dumortier and Chengzhi Li [11] proved that H(1, 3) = 1.
8. In 1997, Dumortier and Chengzhi Li [12] proved that H(2, 2) = 1.
More recently, Giacomini and Neukirch [15] introduced a new method to
investigate the limit cycles of Li´
´enard systems when ∂g = 1 and F (x) is an odd
polynomial. They are able to give algebraic approximations to the limit cycles and
obtain information on the number and bifurcation sets of the periodic solutions


14

Lynch

even when the parameters are not small. Sabatini [31] has constructed Li´
´enard
systems with coexisting limit cycles and centers.
Although the Lienard
´
equation (4.1) appears simple enough, the known global
results on the maximum number of limit cycles are scant. By contrast, if the

analysis is restricted to local bifurcations, then many more results may be obtained.
Consider the Li´´enard system
x˙ = y,

y˙ = −g(x) − f (x)y,

(4.2)

where f (x) = a0 + a1 x + a2 x + · · · + ai x and g(x) = x + b2 x + b3 x + · · · + bj xj ;
ˆ j) denote the maximum number of smalli and j are natural numbers. Let H(i,
amplitude limit cycles that can be bifurcated from the origin for system (4.2) when
the unperturbed system does not have a center at the origin, where i is the degree
of f and j is the degree of g. The following results have been proved by induction
using the algorithm presented in Section 2.
ˆ
1) = i.
1. If ∂f = m = 2i or 2i + 1, then H(m,
ˆ
2. If g is odd and ∂f = m = 2i or 2i + 1, then H(m,
n) = i.
ˆ n) = j.
3. If ∂g = n = 2j or 2j + 1, then H(1,
ˆ
4. If f is even, ∂f = 2i, then H(2i,
n) = i.
ˆ
5. If f is odd, ∂f = 2i+1 and ∂g = n = 2j +2 or 2j +3; then H(2i+1,
n) = i+j.
ˆ 2j) = j.
6. If ∂f = 2, g(x) = x + ge (x), where ge is even and ∂g = 2j; then H(2,

Note that the first result seems to support the conjecture of Lins, de Melo, and
Pugh [20] for global limit cycles. Results 1 and 2 were proven by Blows and
Lloyd [2], and the results 3 to 5 were proven by Lloyd and Lynch [21]. An example illustrating the result in case 5 is given below.
2

i

2

3

Example. Use the algorithm in Section 2 to prove that at most four limit cycles
can be bifurcated from the origin for the system
x˙ = y− a2 x2 + a4 x4 + a6 x6 ,

y˙ = − b2 x2 + b3 x3 + b4 x4 + b5 x5 + b6 x6 + b7 x7 .

Solution. Note in this case that ∂f = 2i + 1 = 5 and ∂g = 2j + 3 = 7, therefore
in this case i = 2 and j = 2. That η4 = L(1) = 2a2 b2 , follows directly from the
computation of the focal values in the second section of the chapter. The computer
programs given earlier can be extended to compute further focal values. This is
left as an exercise for the reader. Let us assume that the reader has computed the
focal values correctly.
Suppose that b2 = 0, then L(2) = a2 b4 . Select a2 = 0, then L(3) = 7a4 b4 .
Next, select b4 = 0, then L(4) = 9a4 b6 . Finally, select a4 = 0, then L(5) = 99a6 b6 .
A similar argument is used if a2 is chosen to be zero from the equation L(1) = 0.
The first five Lyapunov quantities are as follows:
L(1) = a2 b2 , L(2) = a2 b4 , L(3) = 7a4 b4 , L(4) = 9a4 b6 , L(5) = 99a6 b6 .
If a2 = a4 = a6 = 0 with b2 , b4 , b6 = 0, then the origin is a center by the divergence
test. If b2 = b4 = b6 = 0 with a2 , a4 , a6 = 0, then the origin is a center by the

classical symmetry argument.


Symbolic Computation of Lyapunov Quantities

15

From the above, the origin is a fine focus of order four if and only if
a2 b2 = 0,

a2 b4 = 0,

a4 b4 = 0,

a4 b6 = 0,

but
a6 b 6 = 0 .
The conditions are consistent: for example, let b2 = a2 = a4 = b4 = 0 and
a6 = b6 = 1. Select a6 , b6 , a4 , b4 , a2 and b2 such that
|a4 b6 |

1 and a4 a6 < 0,

|a4 b4 |

1 and b4 b6 < 0,

|a2 b4 |

1 and a2 a4 < 0,

|a2 b2 |

1 and b2 b4 < 0,

respectively. The origin has reversed stability four times and so four small-amplitude
limit cycles have bifurcated.

5. A New Algorithm
The author [22] considered the generalized Li´´enard equation
x˙ = h(y) − F (x),

y˙ = −g(x),

(5.1)

where h(y) is analytic with h (y) > 0. It is not difficult to show that the above
results 1. – 6. listed above also hold for the generalized system.
In [23], the author gives explicit formulae for the Lyapunov quantities of generalized quadratic Lienard
´
equations. This work along with the results of Christopher and Lloyd [8] has led to a new algorithmic method for computing Lyapunov
quantities for Lienard
´
systems. An outline of the method is given below and is
taken from [9].
Define
(5.2)
u = 2G(x)sgn(x), u(0) = 0, u (0) > 0,
where

x

x2
+ O(x3 ).
2
0
The function u is analytic and invertible. Denote its inverse by x(u) and let
F ∗ (u) = f (x(u)), then (5.1) becomes
G(x) =

g(ξ)dξ =

u˙ = h(y) − F ∗ (u),

y˙ = −u,

(5.3)

after scaling time by u/g(x(u)) = 1 + O(u).
It turns out that the Lyapunov quantities can be expressed very simply in
terms of the coefficients of F ∗ (u), as shown in [23] and stated in the following
theorem: