On Poincaré's Fourth and Fifth Examples of Limit Cycles at Infinity

0.31 MB·English

by  Roland K. W. Roeder

On Poincar´e’s Fourth and Fifth Examples of
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Limit Cycles at Infinity
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n Roland K. W. Roeder
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J University of California, San Diego

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1 January 14, 2001

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Abstract
D
. Errors are found in example problems from Henri Poincar´e’s paper
h
“M´emoiresurlescourbesd´efiniesparune´equationdiff´erentielle.” Exam-
t
a ples four and five from chapter seven and examples one, two, and three
m from chapter nine do not have the limit cycles at infinity predicted by
[ Poincar´e. Instead they have fixed points at every point at infinity. In
order to understandtheerrors madeby Poincar´e, examples four and five
1
are studied at length. Replacement equationsfor thefourth and fifth ex-
v
amples are suggested based on the supposition that terms were omitted
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from Poincar´e’s equations.
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1 Introduction
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0 InthelatenineteenthandearlytwentiethcenturyHenriPoincar´ebegantostudy
/ the qualitative aspects of systems of differential equations. This analysis was a
h
t breakthroughinthefieldbecauseonenolongerhadtoobtainaspecificsolution
a
of the equation in question to understand its general behavior. This manner
m
of analysis was introduced by Poincar´e in his paper “M´emoire sur les courbes
:
v d´efinies par une ´equation diff´erentielle” [8, 1881]. Poincar´e worked primarily
i with systems of two variables and playeda key role in identifying the existence
X
oflimitcycleswiththePoincar´e-Bendixsontheorem. Poincar´ealsosearchedfor
r
a a complete global analysis of a system of two variables; to do so he introduced
analysis at infinity by means of the Poincar´e Sphere. These two aspects of his
analysis join in a behavior known as a limit cycle at infinity.
To illustrate the application of these techniques, Poincar´epresents example
problems in chapters seven and nine. In chapter seven, Poincar´e presents five
example problems. He states that examples three, four, and five have limit
cycles at infinity. (Examples three, four and five throughout the paper will be
referredtoasP3,P4,andP5respectively.) However,whenonechecksPoincar´e’s
assertion, one finds that examples P4 and P5, in fact, do not have limit cycles
at infinity. The errors remains in the collected works [7] and the error in P4
1
remains in a modern work that includes translations of examples P3 and P4.
[3, p. 155].
In chapter nine, Poincar´e presents three additional example problems. He
states that each of these examples have limit cycles at infinity. However, he
makes the same error analyzing these equations as he did for P4 and P5.
Throughoutthis paper,inaneffortto understandwhy Poincar´emade these
errors, we focus on studying examples P4 and P5 from chapter seven. In the
conclusion we will attempt to apply what we have learned about the errors in
P4 and P5 to the examples from chapter nine.
Becausetheanalysisatinfinityisalgebraicallysimpleforeachoftheseprob-
lems, one becomes perplexed at this finding. Surely Poincar´e could not have
madesuchanerror. Withthisbelief,onemightsupposethattherewasasimple
omissionofsomesort. Ifoneweretofindequationsthathavethesamebehavior
as Poincar´e predicts for P4 and P5, and that, were one to drop certain terms,
could become his printed equations, this would serve as a plausible explana-
tion. We lookatP4inthe contextofP3,andconsideraplausiblemodification.
Suchananalysisis encouragedby the stronggeometricsimilaritiesbetweenthe
two systems. The modification restores the limit cycle at infinity, and hence
supports the hypothesis of an error of omission, becoming a key candidate for
the intended fourth example. We will call this example R4. Example P5 does
not fit in the same genre of geometric system and hence no such modification
suggests itself.
Whenone investigatestheuniqueness ofthese “solutions”tothe problemof
finding modifications ofPoincar´e’sequationthat matchhisanalysisfor P4,one
canfindaclassificationofallmodificationsofthe originalequation,ofacertain
type, that result in the correct behavior. There exist a great number of such
modifications; however,this remarkablenon-uniqueness in “solutions” does not
cause skepticism that R4 is the intended equation for the fourth example. It is
seenthatthemodificationtoobtainR4isthesimplestofthemodificationsfound
under the classification, and hence, through application of the philosophical
principle known as Occam’s razor, R4 remains the most plausible equation for
the fourth example.
Based on the success of the technique applied to P4, a similar technique is
applied to P5 to try to determine a possible modification of P5 which yields
the correct behavior. This technique provides a number of modifications which
agreewithPoincar´e’sanalysisforP5thusillustratingthepowerofthetechnique.
One of these, which we shall call R5, is chosen as a suggested replacement to
P5. (No modifications are suggested for the examples from chapter nine.)
Toaidthereaderwiththeanalysisofdifferentialequationsatinfinity,asec-
tion is included presenting the standard techniques for analysis of the behavior
at infinity for a polynomialsystem in rectangular coordinates. A more geomet-
ric analysis is also presented that is quite valuable in many cases in which such
a system simplifies in polar coordinates.
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2 Differential Equations at Infinity
Differential equations that are defined on the plane can be analyzed at infinity
byextendingtheplanetoarepresentationoftheplanethatincludesthe“points
atinfinity.” ThisanalysisbeganwiththeworkofPoincar´eandBendixson[8][2].
ThroughoutthispaperweusePoincar´e’smethodofanalysis. Moremodernand
readableaccountsonthe Poincar´e’smethodareavailableinbooksbyPerko[6],
Lefschetz[4],andMinorsky[5]. Thissectionbeginswithabriefsummaryofthe
standardresultsasinPerko[6,pp. 264-268]. Atheoremispresented,consistent
with the standard techniques, based on polar coordinates, that demonstrates
the geometric nature of these systems at infinity. This theorem often makes it
possible for one to determine by inspection whether a system has a limit cycle
at infinity.
Through the use of projective geometry, the complete behavior of a two
dimensional system of differential equations can be seen in its behavior on a
sphere of finite radius known as the Poincar´e sphere. To do this, one places
the phase plane tangent to the sphere and makes correspond points on the
plane with the points on the sphere by central projection (a point on the plane
correspondswithanantipodalpaironthe sphere.) Onemustnotethatthe line
intersects the sphere at two points; to remove this non-uniqueness, antipodal
pointsareidentifiedonthe Poincar´esphere. Thepointsthatwereatinfinityon
the original plane become points on the equator of the sphere.
From the behavior on the Poincar´e sphere, one can construct the “global
phaseportrait.” Toconsiderthe globalphase portraitofa system,oneprojects
the trajectories from the upper hemisphere orthogonally down onto the plane
that goes through the center of the sphere and is parallel to the original plane.
In this way the complete behavior on the original plane becomes the behavior
on the finite disk of this new plane. Points at infinity become the points at
which the sphere intersects the plane – the boundary of the disk.
Thefirstanalysisthatonewoulddoforasysteminthefiniteplaneistofind
the location of the fixed points. With this as motivation, the primary problem
of analyzing the behavior of a system at infinity is to determine the location of
fixed points, if there are any. One considers the system:
dx
= P(x,y),
dt
dy
= Q(x,y),
dt
whereP andQarepolynomialsofdegreeminxandy. Denoteby P andQ
m m
the homogeneous polynomials consisting of terms of degree exactly m.
Theorem 1 The critical points at infinity for the system above occur at the
points (X,Y,0) on the equator of the Poincar´e Sphere where X2+Y2 =1 and
XQ (X,Y) YP (X,Y)=0 (2.1)
m m
−
or equivalently at the polar angles θ and θ+π which are solutions of
G (θ) cosθQ (cosθ,sinθ) sinθP (cosθ,sinθ)=0. (2.2)
m+1 m m
≡ −
3
This equation has at most m+1 pairs of roots θ and θ +π unless G (θ)
m+1
is identically zero. If G (θ) is not identically zero, then the flow on the
m+1
equator of the Poincar´e sphere is counter-clockwise at points corresponding to
polar angles θ where G (θ)>0 and it is clockwise at points corresponding to
m+1
polar angles where G (θ)<0.
m+1
See Perko [6], section 3.10, Theorem 1.
An importantconsequenceofthis Theorem1is thatanypolynomialsystem
in rectangular coordinates can be extended to the Poincar´esphere.
One should be aware that in the proof of Theorem 1, a new time scale is
definedto study the trajectoriesinthe proximityofthe equator. There aretwo
opposingconventionsusedinthedegeneratecasewhere(2.1)isidenticallyzero.
Themostcommonone,usedbyPerko[6]andMinorsky[5]istore-parameterize
time insucha waythatthe equatoris forcedto consistoftrajectoriesandfixed
points. The second convention, followed by Lefschetz [4] and Poincar´e [8] uses
a different time scale, allowing trajectories to cross the equator (implying that
trajectories in one of the hemispheres will flow in a direction inconsistent with
the flow on the plane.) Because the former method is more common, we will
adhere to it, even though Poincar´e used the latter. (A brief comment will be
made in the analysis of Example P4 about why this does not invalidate our
results.)
Iftherearenofixedpointsatinfinitythereisacycleatinfinity. Iftrajectories
intheproximityofthiscycleatinfinityapproach(orrecedefrom)it,itisalimit
cycle at infinity. A corollary immediately follows:
Corollary 1 Let r denote radial distance in polar coordinates. If the system
considered above has no fixed points at infinity and dr/dt = 0 for r R, for
6 ≥
some R, then there exists a limit cycle at infinity.
While this theorem makes it relatively easy to determine the behavior of a
system at infinity, there is a more geometric analysis that for certain systems
will make the behavior at infinity particularly easy to determine.
Theorem 2 Consider a polynomial system
dx dy
=P(x,y), =Q(x,y) (2.3)
dt dt
with expressions for dr/dt and dθ/dt of order I and J in r as r , respec-
→ ∞
tively. Letk =I J. Then, if k 2,theequatorof thePoincar´e sphereconsists
− ≥
entirely of fixed points. If k 1 then: if G (θ)=0 for all θ there is a cycle
m+1
≤ 6
at infinity. Furthermore, if dr/dt satisfies the conditions of the above corollary,
it is a limit cycle. Otherwise, the equator of the Poincar´e sphere has finitely
many fixed points located at θ such that G (θ)=0, as in Theorem 1.
m+1
Proof of Theorem 2:
Suppose thatP andQ arepolynomialsofdegreem inx andy. We canexpress
this system in polar coordinates, for r =0, as follows,
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dr
=(cosθP +sinθQ )+r(cosθP (cosθ,sinθ)+sinθQ (cosθ,sinθ))+
0 0 1 1
dt
4
+rm(cosθP (cosθ,sinθ)+sinθQ (cosθ,sinθ)).
m m
···
dθ
=r−1(cosθQ sinθP )+(cosθQ (cosθ,sinθ) sinθP (cosθ,sinθ))+
0 0 1 1
dt − −
+rm−1(cosθQ (cosθ,sinθ) sinθP (cosθ,sinθ)).
m m
··· −
Throughout the remainder of the proof we require r =0. The following defini-
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tions will greatly simplify notation:
η (θ) = (cosθP +sinθQ ),
0 0 0
.
.
.
η (θ) = (cosθP (cosθ,sinθ)+sinθQ (cosθ,sinθ)),
I I I
and
ξ−1(θ) = (cosθQ0 sinθP0),
−
.
.
.
ξ (θ) = (cosθQ (cosθ,sinθ) sinθP (cosθ,sinθ)),
J J+1 J+1
−
whereI andJ arethe highestdegreeterms,inr, inthe equationsfordr/dt and
dθ/dt respectively. With this notation we obtain,
dr
= η (θ)+ +rIη (θ),
0 I
dt ···
dθ
= r−1ξ−1(θ)+ξ0(θ)+ +rJξJ(θ).
dt ···
By the definition of k we have k = I J. If we put this system in differential
−
form we obtain
(r−1ξ−1(θ)+ +rJξJ(θ))dr (η0(θ)+ +rIηI(θ))dθ =0. (2.4)
··· − ···
To determine the behavior at infinity, one projects a differential equation
onto the Poincar´esphere. To understand the details of the behavior at infinity,
projectthe upper hemisphereofthe Poincar´esphereontothe cylinderofradius
1withaxisorthogonaltothephaseplanethatistangenttothePoincar´esphere
atthe equator. (We canrestrictourattentionto the upper hemispherebecause
wefollowthestandardconvention,whichdoesnotallowtrajectoriestocrossthe
equator.) The “equator”of the cylinder is the part ofthe cylinder that touches
the sphere. By geometric analysis of the projection from the plane directly to
the cylinder, s= 1/r and correspondingly dr = −1ds. This projection leaves θ
s2
unchanged.
Projecting (2.4) from the plane directly onto the cylinder gives:
1 1 1
(sξ−1(θ)+···+ sJξJ(θ))(−s2 ds)−(η0(θ)+···+ sIηI(θ))dθ =0,
which simplifies to
1 1
(sξ−1(θ)+···+ sJξJ(θ))ds+s2(η0(θ)+···+ sIηI(θ))dθ =0. (2.5)
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Case 1: k 2
In this c≥ase we have I J 2, so I 2 J. We multiply (2.5) by sI−2 to
− ≥ − ≥
clear the denominators obtaining:
(sI−1ξ−1(θ)+ +sI−J−2ξJ(θ))ds+(sIη0(θ)+ +ηI(θ))dθ =0 (2.6)
··· ···
However, this equation would indicate trajectories crossing the equator (from
one side or the other) for all points (θ,0) where η (θ) = 0. Because we wish
I
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to maintain the same convention as in Theorem 1, the equator is required to
consist only of fixed points and trajectories moving along the equator. As a
result, the trajectories crossing the equator would lead to a violation of the
uniqueness of solutions to (2.6) at these points (θ,0). To resolve this, we must
multiply (2.6) by an additional value of s, a reparameterization of time that
causes these trajectories to slow down so that they do not cross the equator.
We obtain:
(sIξ−1(θ)+ +sI−1−JξJ(θ))ds+(sI+1η0(θ)+ +sηI(θ))dθ =0(2.7)
··· ···
With regards to this new time scale, τ, (2.7) can be expressed as the system:
ds
= (sI+1η (θ)+ +sη (θ)),
0 I
dτ − ···
dθ
= (sIξ−1(θ)+ +sI−J−1ξJ(θ)). (2.8)
dτ ···
Because k =I J 2, at s=0, we have:
− ≥
ds dθ
=0, =0.
dτ dτ
We conclude that every point on the equator is a fixed point.
Case 2: k 1
≤
We have I J 1 and J I 1. We multiply (2.5) by sJ to clear the
− ≤ ≤ −
denominators. Inthiscase,notrajectoriesapproachtheequatorinafinitetime,
hence there is not need for the re-parameterizationdone in case 1.
We obtain:
(sJ+1ξ−1(θ)+ +sξJ−1(θ)+ξJ(θ))ds+(sJ+2η0(θ)+ +s2−(I−J)ηI(θ))dθ =0.
··· ···
Expressing this in the form of a system, with respect to the new time scale τ,
we obtain:
ds
= s2−(I−J)(sIη (θ)+ +η (θ)),
0 I
dτ − ···
dθ
= (sJ+1ξ−1(θ)+ +ξJ(θ)).
dτ ···
So, on the equator we have s=0 and
ds dθ
=0, =ξ (θ).
J
dτ dτ
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So, the fixed points at infinity are for θ such that ξ (θ) = 0. It is easy to see
J
that, in this case, ξ (θ)=G (θ), as defined in Theorem 1. Hence, the fixed
J m+1
points at infinity are at θ such that
G (θ)=cosθQ (cosθ,sinθ) sinθP (cosθ,sinθ)=0.
m+1 m m
−
We conclude that if G (θ) =0 for all θ then there is a cycle at infinity, and
m+1
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that, if nearby trajectories approach that cycle (or recede from it), as in the
corollary to Theorem 1, then it is a limit cycle.
This concludes a proof of Theorem 2.
Oneshouldnoticethatintheproofofthistheoremonefindsthatforthecase
k 1 one has G (θ)=ξ (θ), the highest order term in r of the dθ/dt equa-
m+1 J
≤
tion. This makes this theorem particularly useful – one can often tell whether
a system has a limit cycle at infinity by merely expressing the system in polar
coordinates.
One should also note that in the proof of case 1, if k > 2, equation (2.8)
givesthatatrajectoryapproaching(orrecedingfrom)anypointontheequator
does so orthogonally to the equator.
3 The Poincar´e Examples
To enable the reader to understand Poincar´e’sfourth and fifth examples, three
of Poincar´e’s examples from chapter seven of his paper [8, pp 274-281] are
presented. Poincar´e’sthirdexample, P3,worksasanintroductionto his fourth
and fifth examples and serves as an example of a system that has a limit cycle
at infinity, the behavior that Poincar´eclaims for P4 and P5. Examples P4 and
P5arethenpresentedwithdemonstrationoftheerrorinhisanalysisatinfinity.
At the end of this section, the errorin Poincar´e’sanalysis of his examples from
chapter nine is briefly discussed.
Example P3
Poincar´e considers the equation:
dx dy
= .
x(x2+y2 1) y(x2+y2+1) y(x2+y2 1)+x(x2+y2+1)
− − −
This differential form of the equation is equivalent to the more familiar form of
a system as noted in [6, p. 266].
dx
= x(x2+y2 1) y(x2+y2+1),
dt − −
dy
= y(x2+y2 1)+x(x2+y2+1).
dt −
The geometric aspects of this system become far more obvious in polar form:
dr
= r(r2 1),
dt −
dθ
= r2+1.
dt
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Poincar´epresents the key qualitative features of this system:
The origin: (x,y) = (0,0) is a fixed point. One has a stable spiral at the
•
origin.
A repelling limit cycle at radius r =1 centered about the origin.
•
An attracting limit cycle at infinity.
•
All of the features in the finite plane are easily established, so only the features
of this system at infinity are analyzed in detail. Poincar´eestablishes that there
is a limit cycle at infinity because he claims that there are no fixed points at
infinity: “Iln’yaaucunpointsinguliersurl’´equateur,quiestunecaract´eristique
et quiest parcons´equentun cycle limite” [8, p. 278]. He omits this calculation,
so we verify it here.
WeapplyTheorem2toshowthattherearenofixedpointsatinfinity. From
theexpressionfortheP3inpolarcoordinates,wefindI =3andJ =2.Further,
ξ (θ) = 1 for all θ, demonstrating the existence of a cycle at infinity. Because
2
dr/dt>0 when r 2, this cycle is a limit cycle.
≥
Example P4
Poincar´e’s fourth example can be considered a more complicated version of
his third example. Poincar´ewrites the equation as:
dx
=
x(x2+y2 1)(x2+y2 9) y(x2+y2 2x 8)
− − − − −
dy
,
y(x2+y2 1)(x2+y2 9)+x(x2+y2 2x 8)
− − − −
which he transforms into the system:
dx
= x(x2+y2 1)(x2+y2 9) y(x2+y2 2x 8),
dt − − − − −
dy
= y(x2+y2 1)(x2+y2 9)+x(x2+y2 2x 8).
dt − − − −
Poincar´estates the key features of the system:
The origin: (x,y) = (0,0) is a fixed point. One has an unstable spiral at
•
the origin.
A: (x,y) = (1/2,√35/2), and B: (x,y)= (1/2, √35/2) are fixed points:
• −
anunstablenodeandasaddlerespectively,attheintersectionofthecircles
defined by x2+y2 9=0 and x2+y2 2x 8=0.
− − −
An attracting limit cycle of radius r =1 centered about the origin.
•
Two heteroclinic orbits connecting fixed points A to fixed point B along
•
the circle x2+y2 9=0.
−
An attracting limit cycle at infinity.
•
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A
C3
C1
C2
B
Figure 3.1: Sketch of the global phase portrait corresponding to Poincar´e’s
analysis of example P4. Circles C ,C , and C , which are referred to later in
1 2 3
this paper, are labeled. (The dashed line for C indicates that it is a nullcline
3
and not a trajectory.)
All of the stated features in the finite plane can be easily established. (A
similaranalysisisappliedtoaslightlymorecomplicatedsysteminthefollowing
sectionnumberfive.) Thegeometricnatureofthesystembecomesclearinpolar
coordinates. The system becomes:
dr
= r(r2 1)(r2 9),
dt − −
dθ
= r2 2rcosθ 8.
dt − −
ThephaseportraitcorrespondingtoPoincar´e’sanalysisisincludedinFigure3.1.
He claims that, as in P3, there are no fixed points at infinity, and hence a limit
cycle. However, when one does the calculations to check Poincar´e’s assertion,
one finds that this is not the case.
To check the assertion, one applies Theorem 2 to the system. By looking
at the the expression for P4 in polar coordinates, one finds that I = 5 and
J =2,hencek =3andeverypointatinfinity isafixedpoint. Thereisnolimit
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Figure 3.2: The correct global phase portrait corresponding to Example P4.
(Figures3.2and4.1werecalculatednumericallyusingaRungeKuttamethod.)
cycleatinfinity because everypoint onthe equatoris afixed point. (If onehad
usedthe secondconvention,whichwasusedby Poincar´e,onewouldhavefound
trajectoriescrossingtheequator. Withthisconvention,thetrajectoriescrossing
the equator eliminate the possibility of a cycle on the equator. All reference to
this second convention will be dropped for all of the following examples.) The
correct global phase portrait for the system is presented in Figure 3.2.
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