Ž .
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
202,
90
107 1996
ARTICLE NO
. 0304
A Numerical Approach to Bifurcations fromQuadratic Centers
Douglas S. Shafer,* Xiaonan Wu, and Andre Zegeling
´
Mathematics Department, Uni
ersity of North Carolina at Charlotte, Charlotte, North Carolina 28223Submitted by Jack K. Hale
Received February 6, 1995A combination of analytical and numerical work is done to analyze bifurcation of limit cycles from nonHamiltonian codimensionthree quadratic centers. The winding curve
C C
of cyclicitythree separatrix cycles, qualitatively located in earlier
3
Ž . Ž . Ž .
work, is determined numerically. Evidence is given that the 2,2 , 3,2 , and 3,3configurations of limit cycles do not bifurcate from this class of quadratic centers.
1996 Academic Press, Inc.
INTRODUCTIONAlthough it is now known that but finitely many limit cycles can appearin the phase portrait of a twodimensional system of ordinary differential
equations whose righthand sides are polynomials 5 , it is not known if
Ž .
there exists a uniform bound
H n
, as called for in Hilbert’s Sixteenth
Problem 4 , on the number of limit cycles of all such systems of degree
n
.
Ž . Ž .
Even for quadratics
n
2 one has at best that, if it exists,
H
2
4 7 .One place to look for large numbers of limit cycles is in their creation bybifurcation from a system possessing one or more centers. In applying thisstrategy within the class of quadratics, the class
Q
NH
of nonHamiltonian,
3
codimensionthree centers is a natural starting point, since it containssystems having two centers, each of which has a period annulus whoseclosure has cyclicity three, i.e., for each center there is a quadraticperturbation causing three limit cycles to bifurcate from the closure of that
Ž
center’s period annulus. ‘‘Codimension three’’ means that the set of parameter values corresponding to these systems is a set of codimension
*To whom all correspondence should be addressed.900022247X
96 $18.00
Copyright
1996 by Academic Press, Inc.
All rights of reproduction in any form reserved.
BIFURCATIONS FROM QUADRATIC CENTERS
91
three in the twelvedimensional parameter space of all quadratics. See 9
.
for more on these systems. In 6 , regions in parameter space wereanalytically located for this class of centers corresponding to systems forwhich at least three limit cycles could certainly be bifurcated. This paper isa continuation of that study, focusing primarily on a numerical study of theintegrals which determine the number of limit cycles which persist fromthe period annuli. The goal is twofold. First it is shown how the integralscan be used to determine the bifurcation diagram after perturbation,specifically by locating the bifurcation sets for multiple limit cycles and forseparatrix cycles. In general these bifurcation sets are difficult to determine even numerically, but our aim is to show that in the center perturbation these sets are a natural starting point for the investigation of thenumber of limit cycles. Also, this approach can be used to study higherorder phenomena which are not described by such integrals, but whosecharacteristics can be shown numerically if the parameters in the unperturbed system are varied as well. For example, the numerical study showsthat the bifurcation of limit cycles from the hyperbola bounding the periodannuli in the unperturbed system resembles the well known mechanism forthe creation of three limit cycles in the Andronov
Hopf bifurcation from athirdorder weak focus.The second goal of this paper is to investigate numerically whether it ispossible to create more than four limit cycles within the class of quadraticsystems from the center case under consideration. By comparing thebifurcation sets for semistable limit cycles surrounding the two differentantisaddles after perturbation, numerical computations indicate a negativeanswer.1. PRELIMINARIESSystems in the class
Q
NH
of nonHamiltonian, codimensionthree quad
3
ratic centers can be placed in the canonical form
x
y
ax
2
by
2
,
˙
Ž .
y
x
1
2
y
. From the point of view of maximizing the number of limit
˙
cycles generated from such a system according to the strategy outlined inthe Introduction, the only region in parameter space of interest is thatcorresponding to systems having two centers, each of whose period annuli
is bounded by one branch of an invariant hyperbola 6 . Thus we henceforth restrict our study to the systems:
x
y
ax
2
by
2
˙
2
a
,
b
R R
a
,
b
a
2,0
b
2 .
4
Ž . Ž .
½
y
x
1
2
y
,
Ž .
˙
1.1
Ž .
SHAFER
,
WU
,
AND ZEGELING
92
1 1
Ž .
After a translation
y
y
, the invariant line
y
of system 1.1 is
1
2 2
Ž .
brought onto the
x
axis. The integral of 1.1 reads, in the new variables
Ž .
x
,
y
:
1
b b
1
b
2
a
2 2
h
H x
,
y
y x
y
y
.
Ž .
1 1 1 1
ž / ž / ž /
a
2
a
1 4
a
1.2
Ž .
1
1
Ž . Ž . Ž .
To each
h
in the interval
h
*
H
0,
h
0,
h
H h
is a
2
Ž .
periodic solution in the unperturbed system 1.1 which surrounds the
Ž
1
srcin. For the other period annulus the interval for
h
is
h
**
H
0,
b
1
.
h
0. The two limits
h
h
* and
h
0 correspond to the bound
2
Ž .
aries of the period annulus surrounding the origin in 1.1 , the center itself
Ž .
on the inside and a branch of the hyperbola 1.2 on the outside,
h
0
respectively. On the Poincare sphere the two period annuli lie inside two
´
heteroclinic connections through two saddles at infinity.Since the parameter space of quadratic systems in which limit cycles canappear has dimension five under the action of affine transformations,
Ž .
systems 1.1 can be thought of as corresponding to a twodimensionalmanifold in this parameter space, so that three parameters suffice to
Ž .
describe the whole bifurcation pattern. For fixed
a
,
b
it is sufficient to
Ž .
consider perturbations with three small parameters
,
,
in parameterspace:
x
y
ax
2
by
2
x
xy
Ž . Ž .
˙
1.3
Ž .
y
x
1
2
y
x
2
.
Ž . Ž .
˙
In this paper we consider bifurcations from the period annulus. This
Ž .
means that
a
,
b
will be taken fixed and
,
, and
will be taken to belinear in a small parameter
. Information is thus possibly lost concerningbifurcations in directions tangent to the twodimensional manifold of
Ž
centers in the full fivedimensional parameter space but only in those
directions, which are not important for the present study; see 6 about
.
these exceptional cases . Since the parameter
takes arbitrarily small
Ž . Ž .
values, the parameter
can be eliminated by setting
. The
1
only information lost by doing so is the situation in which
0; but then
Ž .
the system 1.3 has an invariant line, and hence has at most one limit
cycle 3 . Dropping the subscript in
, the systems under study then are
1
x
y
ax
2
by
2
x
xy
˙
2
½
y
x
1
2
y
x
,
Ž .
˙
0
1,
a
,
b
R R
,
,
2
. 1.4
Ž . Ž . Ž .
BIFURCATIONS FROM QUADRATIC CENTERS
93
Ž .
Thus we consider a twodimensional parameter space
,
corresponding to the perturbed system, and in this plane we investigate the bifurcation sets corresponding to the creation and destruction of limit cycles. It is
Ž .
shown in 6 that the bifurcation of limit cycles from periodic orbits in 1.1
Ž Ž . .
is governed by the zeros of the integral reverting to
x
,
y
coordinates :
1
I h
I h
I h
I h
, 1.5
Ž . Ž . Ž . Ž . Ž .
1 2 3
ž /
2where
Ž .
y h
a
1
1
I h
2
G y
,
h
y dy
'
Ž . Ž . Ž .
H
1 1 1 1
Ž .
y h
1
Ž .
y h
a
1
I h
2
G y
,
h
y dy
'
Ž . Ž . Ž .
H
2 1 1 1
Ž .
y h
1
1.6
Ž .
2 1
a
Ž .
3
Ž .
y h
a
2
1
I h
G y
,
h
y dy
'
Ž . Ž . Ž .
H
ž /
3 1 1 1
3
Ž .
y h
1
and
b b
1
b
2
a
2
G y
,
h
h
y
y
y
.
Ž . Ž .
1 1 1 1
ž / ž / ž /
a
2
a
1 4
a
Ž .
These are the integrals corresponding to perturbations 1.4 . For fixed
h
, a
Ž .
zero of
I h
generally signals creation of a limit cycle from the periodic
Ž . Ž .
orbit
h
corresponding to this value of
h
in 1.2 . The limits of integra
Ž .
Ž .
tion
y h
and
y h
are the two intersections of the periodic orbits with
1 1
Ž .
the
y
axis, determined implicitly by
G y
,
h
0.
1
2. STRUCTURE OF BIFURCATION SETS
Ž . Ž .
The bifurcation set of 1.4 is investigated in the
,
plane introducedin the previous section. In this section we concentrate on the limit cycles
Ž . Ž .
created from the period annulus surrounding
O
0,0 in 1.1 .
Ž . Ž .
P
ROPOSITION
2.1 6 .
For fixed a
,
b
,
the bifurcation set in the
,
plane corresponding to the creation or destruction of limit cycles surrounding
Ž . Ž .
O
0,0
in
1.4
consists of three parts
:
Ž .
1
a straight line
,
corresponding to a heteroclinic separatrix cycle join

ing two saddlepoints on the line at infinity
,
which cycle is composed of a
Ž .
branch of the hyperbola
1.2
and a portion of the line at infinity
;
h
0
SHAFER
,
WU
,
AND ZEGELING
94
Ž .
2
a straight line
,
corresponding to a weak focus
,
created from the
Ž . Ž .
center O
0,0
in
1.1 ;
and
Ž . Ž . Ž . Ž Ž . Ž ..
3
a connected cur
e
h
,
g h
,
f h
,
h
*
h
0,
corresponding to a semistable limit cycle
,
created from the periodic orbit at
Ž . Ž .
le
el H
h in
1.1 .
The components of
h are
:
I
I
I I
1 3 1 3
g h
g h
Ž . Ž .
Ž .
a
,
b
I I
I I
1 2 1 2
2.1
Ž .
1
I I
I I
I I
I I
Ž .
2 3 2 3 1 3 1 3
2
f h
f h
,
Ž . Ž .
Ž .
a
,
b
I I
I I
1 2 1 2
Ž . Ž . Ž . Ž .
where I h
,
I h
,
and I h are defined by
1.6 ,
and the prime denotes
1 2 3
differentiation with respect to h
.
Ž .
What is meant here is that, in case 3 for example, the curve picks outchoices of
and
yielding the direction of approach of the manifold of
Ž .
semistable limit cycles to the manifold of centers at
a
,
b
. An explicit
Ž .
expression for the line of Proposition 2.1 1 is obtained by letting
h
0 in
Ž . Ž
.
1.5 . The limit may be brought inside the integral see 6 , yielding:
1
I
0
I
0
I
0
0. 2.2
Ž . Ž . Ž . Ž .
Ž .
1 2 3
2
Ž . Ž . Ž . Ž .
The three integrals
I
0 ,
I
0 , and
I
0 in 2.2 are integrals over a
1 2 3
Ž .
branch of the hyperbola 1.2 . The possibility of explicitly evaluating
h
0
Ž .
2.2 is discussed at the end of this section. The fact that the bifurcation of
Ž .
separatrix cycles in the
,
plane is represented by a straight line is dueto the transversal intersection of the codimensionone manifold of separa
Ž .
trix cycles with the manifold of centers, defined by 1.1 , in the fivedimen
Ž . Ž
. Ž .
sional parameter space of 1.3 Proposition 1.6 of 6 . In fact, the
,
plane displays the directions of approach in the fivedimensional parameter space of the three types of bifurcation sets described in Proposition 2.1.
Ž . Ž .
The integrals 1.5 therefore determine through 2.2 the direction of bifurcation of separatrix cycles. However, the exact number of limit cyclesbifurcated from the hyperbola does not follow necessarily from theseintegrals. Our approach for studying the perturbation from periodic orbitsgives only a lower bound in this situation, but we conjecture that this is theupper bound as well.
Ž .
The bifurcation line in Proposition 2.1 2 is determined by taking the
Ž .
limit
h
h
* in 1.5 . However, since we already know that the weak focusmanifold in parameter space is
0, we can skip this calculation. For