Infinite singular points of quadratic systems in the plane

NonlinearAnalysis, Theory,Methods&Applications, Vol. 24, No. 6, pp. 895-927, 1995 Copyright © 1995ElsevierScienceLtd Printed in Great Britain. All rights reserved 0362-546X/95 $9.50+ .00

Pergamon 0362-546X(94)00124-3

INFINITE SINGULAR POINTS OF QUADRATIC IN THE PLANE

SYSTEMS

J. W. REYN and R. E. KOOIJ Delft University of Technology, Faculty of Technical Mathematics and lnformatics, Mekelweg 4, P.O. Box 5031, 2628 CD Delft, the Netherlands

(Received 24 March 1993; received in revised form 3 January 1994; received for publication 31 May 1994) Key words and phrases: Quadratic systems of ordinary differential equations in the plane, singular points, behavior at infinity. INTRODUCTION

Consider the quadratic system o f differential equations = aoo + aloX + a o l Y +

a2o X2 +

allxy +

ao2Y 2 -

P(x,y),

(*)

JP = boo + blo X + b o l Y + bE0 X2 + b l l X Y + b0EY 2 -- Q(x,y),

where " = d / d t and a o, b U ~ ~. Solutions of (,) can be studied in the phase plane x, y. The qualitative theory of differential equations in the plane is concerned with the question of finding the partition of the coefficient space (a o, bo) = R 12, corresponding to the topologically different phase portraits of (.). There exists a vast literature on phase portraits of quadratic systems [1, 2], often with an emphasis on specific aspects, such as limit cycle behaviour, and often for particular classes of systems, usually brought into a simpler analytical form by means of a linear transformation. Conditions directly on the coefficients in (,), such that the system has certain properties, e.g. the existence of a center in the system, were also given. In this context one should refer to Curtz [3-6], and the work done in the school of Sibirskii, where these conditions are formulated in terms of affine invariants [7]. However, there remains the problem of a systematic approach to the ordering of the phase portraits of (.) and the determination of the partition of the coefficient space (a o, bo) e R 12 corresponding to this ordering. Now, a phase portrait is topologically characterized by the number, location and character of its singular points (both in the finite part of the plane and at infinity), its separatrix structure and the location of periodic solutions, if any, in particular of its limit cycles. When trying to define classes of quadratic systems, it seems natural to use the properties of the singular points for a first ordering principle and as a basis for a partition of the coefficient space (a/j, bu). In this context, the question arises as to what singularities are possible at infinity, and what their relation is to the coefficients in the system of differential equations. In this paper an answer to this question is given. Value is attached to create a reliable source for a detailed description of the possible singular points at infinity, and their relation to the coefficients a Uand b o. The only restriction is that b2o is taken to be equal to zero, which enables the study of the infinite singular point as being located at the ends of the x-axis. In particular, the index of the point and its multiplicity in tangential and transversal direction with respect to the Poincar6 circle are also determined. There are only a few papers exclusively concerned with singular points at infinity of quadratic systems. Some are in Russian [8, 9], the majority in Chinese [10-14], not easily accessible and of 895

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J . W . REYN and R. E. KOOIJ

limited circulation [15]. In these papers, possible combinations of singular points at infinity are also investigated, a question that has not been taken up in the present paper. The description of the singular points, however, was not given in enough detail to serve the goal we have in mind. Some aspects of our results could nevertheless be fruitfully compared with previous results. In particular, a careful check is made with the results of Hu Qinxun and Lu Yugi [12], and of Call i Vicens [15], and initial errors have, thereby, been removed. A definition of a singular point at infinity would be such that it is in the direction ).x - lzy = 0, with ).,/z ~ •, )2 +/12 ~ 0 for which the flow at infinity is in that same direction, thus L r - / z ) = 0. These directions are, therefore, given by doE ).3 + ( a l l -- boE)).Efl + (020 - bll))., t/E - bEofl 3 = 0.

(1)

Since (1) is of odd degree and has real coefficients, there is at least one real pair ().,/1) which satisfies it. Then, without loss of generality, we may put bE0 = 0, SO that there is a singular point in the direction of the x-axis, since the transformation £ = Ax +/zy, P = -/zx + 2),, ~E d" //2 ~ 0, where ).,/1 e [P satisfy (1), leads to a system (.) with bE0 = 0. By using the Poincar6 transformation z = 1/x, u = y / x , this singular point may then be defined to be the point z = u = 0 of the transformed system

Zit = U' = b l o Z + ( - a E o + b l l ) U + boo zE + ( - a l o

-- aolZU 2 -- aoozEu - a02 u3

d- b o l ) z u + ( - - a l l q- boE)u E

(**)

The point (0, 0) is called a singular (or critical or equilibrium) point, and simple (or first-order or elementary) if, in (0, 0), P(z, u) = 0 and Q(z, u) = 0 have exactly one common zero. It is called multiple (or higher order) if P(z, u) = 0 and Q(z, u) = 0 have several c o m m o n zeros; then its multiplicity equals the number of common zeros in that point. It is also equal to the number of simple singular points that bifurcate from the multiple singular point upon a perturbation of the coefficients in (**) that leaves only simple (complex or real) singular points in a neighbourhood of (0, 0). The eigenvalues of the coefficient matrix of (**), linearized near (0, 0) are 2 z = -aEo and A~ = -aEo + b11. Since z -- 0 is an orbit of (**) it follows that (0, 0) is an elementary node N if aEo(aEo - bll) > 0 and an elementary saddle S if aEo(aEo - bll) < 0. For the higher order singular points we distinguish the following cases: (i) aEo = b~l ~ 0, thus ).z ;~ 0, ;t u = 0; there is a one dimensional center manifold tangent to the Poincar6 circle (z ----0); (ii) aEo = 0, bll ~ 0, thus ).z = 0, ).u ~ 0; there is a one dimensional center manifold transversal to the Poincar6 circle; (iii) aEo = bl~ = 0, blo ~ 0, thus ).~ = 2u = 0; there is no one dimensional center manifold, but the higher order singular point is represented by (**) containing a linear term; it will be called a nilpotent singularity; and (iv) aEo = bll = 0, b~o = 0 thus ).z = ).~ -- 0; there is no one dimensional center manifold and the higher order singular point is represented by (**) having no linear terms; it will be called a highly degenerate singularity.

Infinite singular points

897

The first three cases will be investigated using the results of the analysis in [16], which are obtained by a blowing up method. For the last case the blow up will be presented in the present paper. If 2z ;~ 0, the singular point will be called transversally hyperbolic with respect to the Poincar6 circle (z - 0), whereas for )-z -- 0 the point will be called transversaily nonhyperbolic. Similarly, if ;~u ~ 0, the singular point will be called tangentially hyperbolic with respect to the Poincar6 circle, and if ).u = 0 tangentially nonhyperbolic. If a singular point is transversally nonhyperbolic, one or more finite singular points can be bifurcated from such a point upon a perturbation of the coefficients. We introduce a notation that indicates this property. A multiple singular point at infinity is indicated by M~,q, where i indicates the index of the point, p the maximum number of simple finite singular points that can be bifurcated upon a perturbation of the coefficients and q the maximum number of simple singular points on the Poincar6 circle that remain after such a perturbation. The multiplicity of the singular point equals p + q. Thus M,0,2 ° indicates a tangentially nonhyperbolic singular point of second order with index 0, M °1, l a transversally nonhyperbolic second order singular point with index O, and SO o n .

Apart from real also complex singular points are possible. They occur in pairs and their number at infinity is at most two. Since at most four finite singular points are possible and, therefore, at most four can be generated out of a pair of complex singular points at infinity there exist the possibilities C~,1,° C1,,° and C °, 1 Although the index for a complex singular point is not defined, the superscript 0 will be used to indicate that a complex singular point does not give a contribution when calculating the sum of the multiplicities of singular points of system (.). One of the properties of a quadratic system that enters in the division in classes of such systems is its finite multiplicity rag, being the sum of the multiplicities of the singular points in the finite part of the plane. Of various singularities presented in this paper, in particular of those of high order, can be stated that they can only occur in a class with a specific value of ms. It will be mentioned if this is the case. Also, for these more complicated and less common singular points, a more detailed description will be given, and expressed in their notation. The character of such a singular point will be given by listing the sectors around the point, starting at the positive u axis and going in anticlockwise direction around the singular point. An elliptic sector will be indicated by E, a hyperbolic sector by H and a parabolic sector by P. The notion of a parabolic sector will be used in a restricted sense, as containing all points in a neighbourhood of the singular point through which there is an orbit approaching the singular point in the same direction. An elementary node would thus be indicated by PPI1, where the subscript indicates that the multiplicity equals 1 and the superscript that the index also is equal to I. If the parabolic sector extends over an interval ~t - ~ -< tPu of the polar angle ~0 around the singular point and all orbits of the sector approach the point along ~0t the notation will be extended to P, and similarly to P if all orbits in the sector approach the point along ¢Pu- If ~Pt -- ¢u, then the opening angle of the sector is zero and the sector will be indicated by/3. With regard to an elliptic sector it should be noted that local investigation near a singular point alone cannot determine the extent of an elliptic sector. Whether or not parabolic sector(s) are present adjacent to it can only be determined using global arguments. As a result the notation E means that no statement is made with regard to the existence of parabolic sector(s) adjacent to the elliptic sector. If, on occasion, a global argument is available to verify that there is no parabolic sector near the elliptic sector, this will be indicated by IE if this is certain for the lower value of the polar angle, and by EI for the higher value, whereas IEI means that on both .

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J . W . REYN and R. E. KOOIJ

sides o f the elliptic sector there is no parabolic sector. If there certainly are parabolic sectors adjacent to E, they will be indicated as P E , E P or P E P . For elementary points, higher order points with a one dimensional central manifold and nilpotent singular points short notations will be used as well, as will be indicated below. In order to indicate the conditions, corresponding to a specific character o f the singular point, use will be made o f the following determinants, which also (in a different notation) appear in [3, 12] aooblo - axoboo,

c~3 = aoobo~ - aolboo,

c14 = aoob2o - a2oboo,

c~5 = aoobH - a~xboo,

c~6 = aoob02 - ao2boo,

C23

c24 = alob20 - a20blo ,

C25 = a l O b l l - aHb~o ,

C26 =

c34 = aolb2o - a 2 o b o l ,

C35 = aOlbll - allb01 ,

c36 = a o l b o 2 - aozb01 ,

C45 = a 2 0 b l l -- a l l b 2 0 ,

c46 = a20b02 - ao2b20 ,

c56 = a l l b 0 2 - a o 2 b H .

C12 =

I. S I N G U L A R

POINTS WITH A CENTER MANIFOLD

TANGENT

a~obo~ - ao~b~o,

=

alob02 - ao2bm,

TO THE POINCARI~ CIRCLE

If a2o = b n : 0, there is Az ~ 0, A,, = 0 and (**) has a center manifold tangent to the Poincar~ circle. This can be seen by applying the transformation Z = z, ~ = u + (blo/a2o)Z to (**) so that the system ---

1 Z' = Z = Z + a l o a 2 0

a20

-

a l l b l o z 2 + a~lZf~ + aooa20 - aola20blo + ao2620Z3

a2o

a2o

a230

+ aol a2o a2o-2ao2 blo Z2~ + __Z/~2a20a02

- z + P2(L t2), 1

,

u

(2)

-a20boo + aEobolblo - b20bo2z2 + aloa20 - a20bol - a n b l o + 2bo2bloz~

40

a2o 2

+ al~ - b02 122 + aooa20 - aola20b~o + ao2btoz2/~ + aola20 - 2ao2bloz~2 + __a02~3 a20 a230 a220 a2o -= Q2(Z, ~),

is obtained, which can be analysed following the classification given in [16], in particular by applying theorem 65 on p. 340. With Z = ~(~) -= 0 as solution o f Z + P2(z, ~) -- 0 follows

~/(~) - Q2(~(~), ~) =

a l l -- b02/~ 2 + a°---~2/~3. a2o

a2o

Infinite singular points

899

It then follows that for al~ - b02 # 0, the singular point M°,2 is a second-order saddle node SN~0,2 with hyperbolic sectors adjacent to the positive u axis if a20(all - b02) < 0 and the negative u axis if a20(all - b02) > 0. If aH - b20 = 0, the singular point M~, 3 is a third order node Nd, 3 if ao2a20 > 0 and a third-order saddle point So~ if ao2a20 < 0. Since such a point is tangentially third order, it is the only singular point at infinity in the system wherein it occurs, and since it cannot bifurcate finite singular points there follows that m / = 4 for a system containing such a point. If, finally, all - - b 0 2 = a 0 2 = 0 , the right-hand sides of (2) have the common factor z, which shows that the Poincar6 circle consists of singular points. The results of this section are summarized in Fig. 1.

/

UJ

al 1"b02 # 0

sNO,2

U,

al 1"bo2 = 0

a2oao2>O

J

J

II,

z

rnf=4

S0,3 ao2=O: all points on the Poincaro circle are singular Fig. 1. Singular points with a center manifold tangent to the Poincar6 circle (a2o = b H # 0).

900

J . W . REYN and R. E. KOOIJ 2. S I N G U L A R

POINTS WITH A CENTER MANIFOLD POINCARI~ CIRCLE

TRANSVERSAL

TO THE

If a2o = 0, b n ~ 0 there is )-z = 0, 2 u # 0 and (**) has a center m a n i f o l d transversal to the Poincar6 circle. This can be seen by applying the t r a n s f o r m a t i o n g -- z, ~ = bxoz + b n u to (**) so that the system _ _1~ t

2

= Z --

bll

C25 ~2 --

--~-Zuall - - + -aoob21 + a o l b l o b H - a o z b m 3 3

b~l

'ql

+ -aolbn

b]l

+ 2ao2bm b~l

32/~ _

ao__£-/~2 b~l Z

-- P2(7,, u),

__1 ~' = u = fi + boob~l + boeb~o - b m b m b i 1 3 2 bl 1

+ -c25 + bolbll - 2blobo2 zu--

b21

+ -au

b21

+ bo2~2 -aoob~l + a o l b m b l l - aoeb~o b21 + b~ 1

(3)

Z2f'l

+ - a o l bl 1 -1- 2a02 blo 3/~2 _ a0_...22/~3 b]l -

b~l

r, + O2(e, r,),

is obtained, which can be analysed following the classification given in [16], in particular by applying t h e o r e m 65 on p. 340. With ct = boob21 + bo2b~o - bol b m b l l we distinguish between a # 0 and ct = O. 2.1. ~ -- boobEl + boEb210 - b o l b l o b l l # 0 I f a # 0, near (0, 0), ~ + QE(Z, u) = 0 m a y be a p p r o x i m a t e d by ~ = q~(z-) = fll 32 + Yl 3 3 + 51 ~4, where o/

fll =

o/

bE1 ,

Yl = -~11(-c25 + bolbll - 2blobo2),

o/

51 = b16--~[ - ( - c 2 5 + b01 b n - 2blob02) 2 + b~l(-aoob21 + a o l b l o b l l - aoEb20) + or(all - b02)]. T h e n it follows that ~u(z-) = P2(z, tp(z-)) = fl2g 2 + y2g 3

/~2-

+

¢~2: -1- eEZ5, where

c25 b21 '

-b21Cls Y2 =

c~2 =

+ blobllC35 +

b20c56

b41 c¢(anc2s + b n c 3 s + 2b10c56) b 611

'

t~[~c56 - all(--b21c15 + blobllC35 + b120c56) + (Czs - b o l b u + 2bmbo2)(allc25 + b n c a s + 2bmc56)] e2 = ball

Infinite singular points

901

According to t h e o r e m 65, p. 340 in [16] then, if c25 # 0, the singular point MI°~ is a secondorder saddle node SN°,I with hyperbolic sectors for z > 0 if c25 > 0 and for z < 0 if c25 < 0. I f cz5 = 0, the singular point M~ ~ is a third-order node N 1 ~ if b 41 72 = -bE~ c15 + b10 bll c~5 + b20c56 > 0 and a third-order saddle point S~,~ if b41~,2 = S-b211c15 + blobllc35 + b20c56 < O. If c25 -b21c15 + blobllC35 + b20c56 ~ 0, the singular point M 3,1 ° is a f o u r t h - o r d e r saddle n o d e S N ° l i f b l l C 3 5 + 2bloc56 ;~ 0 with hyperbolic sectors f o r z < Oifotbll(bllc35 + 2bloc56 ) > 0 and for z > 0 if Otbll(bllC35 + 2b10c56) < 0. If c25 -: -bElc15 + b10b~c35 + b20c56 = b11c35 + 2b10c56 = 0, the singular point M~,~ is a fifth-order node NI,1 if cs6 > 0 and a fifth-order saddle $4~] if c56 < 0. A singular point M~,~ can only occur in a system with m: = O. If c25 = -bElcx5 + blob~lC3s + b~0c56 = bllC3~ + 2b~0c56 = c56 = 0, (*) has a h y p e r b o l a with singular points t h r o u g h z = u = 0.

2.2. o~ = boob21 + bo2b20 - bolblobll = 0 If ~ = 0, we have fi = ~p(z--)= 0 as solution through (0, 0) of fi + Q2(z, rE) -- 0. T h e n it follows as before that C2_._.~5~2

~'(z-) = PE(Z, ~p(z--)) = - b21 Z +

-b21c15 + blobllC35 + b20c56~3. b~l

According to t h e o r e m 65, p. 340 in [16] then if c25 76 0, the singular point M ° , I is a second-order saddle node SN°,~, with hyperbolic sectors for z > 0 if c25 > 0 and for z < 0 if c25 < 0. If c25 = 0, the singular point M~, 1 is a third-order node N~, 1 if b41~2 = -bElcls + b10bllc35 + b210c56 > 0 and a third-order saddle SZ,~ if b4xy2 -- -b21c1~ + blobllCa5 + bEoc56 < 0. I f c25 = -bE1 c15 + blobllC3s + bEoc56 = 0, (,) is an essentially linear system with z = u = 0 on the line o f singularities. The results o f this section are s u m m a r i z e d in Fig. 2.

3. N I L P O T E N T S I N G U L A R P O I N T S

If a2o = bH = 0, ,l.z = 2u = 0. We consider first the case b~o ~ 0 so that (**) has a linear term. In order to follow the analysis given in [16], in particular theorems 66 and 67, we write (**) as

1 z, 1 --U' bid

//alo

al,

aooz2

a01

a02 u2'~ •

boo 2 - a i d q- bol --all at- bo2 U2 _ aol z u 2 _ a o o z 2 u _ a02 g3 = il ~- Z + - + ZU + b~o z b~o blo ~10 ~10 ~10

-- Z + Q2(z, u).

(4)

Near (0, 0), z + Q2(z, u) = 0 m a y be a p p r o x i m a t e d by Z :

(p(U) ~- 0/2 u2 -I- 0£3u3 -['- 0/4//4 ~ 0/5 uS,

902

J . W . REYN and R. E. KOOIJ

U,

c25¢0 .......................................................... sNO, 1

U~

c25=0 ?2>0 1 N2,1 ~'2<0 U,

~2=0

~=/=0 62#0 . ~

J

}'-1 S2,1

] s.°,, U~

62= 0 2

ll m' "°

C56>0

I

2

N4,1

~= boob11+bo2blo-bolblob11

u

b1172= -bllC15+blobllC35+bloC56 2 2 b~1~2= ~(allC25+b11c35+2b10c56) c56<0

~_ mf=O

s~1 ~ c56=0: (*) has a hyperbola with singular points through z=u=O = O: (*) degenerates into an essentially linear system with z=u=O on line of singular points Fig. 2. Singular points with a centre manifold transversal to the Poincar6 circle (az0 = 0, b=l ~ 0).

Infinite singular points

903

where a l l -- bo2 bl 0 ,

0/2 --

ao2 alo - bol 0/2, 0/3 = b l o + blo

0/4

0/5

boo 2 blo 0/2

:

----

alo - bol 0/3 + a°---L bl ° 0/2, blo

+

= - 2 boo b l o 0/2 0/3

+

alo - bol

blo

a01 a00 0/2 0/4 -- ~ 1 0 0/3 -- -b- l o 2,

so that ~u(u) = PE(~(u), u) = a3u 3 + a 4 u ' + a s u 5 + f16/./6, a p p r o x i m a t e l y , w h e r e all(all

a3 = a4 =

alo 0/2 _ a l l -blo

05 ~

-- bo2 )

b~o

2

"~ alo lbO

a02

blO0/3-

0/2,

a02 0/20/3 -

a 6 = _2 aol

b lo 0/2 ct3

lbO

all 0/3 -

0/, -

aol 0/2 2,

-b-l °

_ 2alo aoo 0/3 alo 0/2 a02 all b lo 0/2 0/4 -- bl---o 2 - bl-~ 3 - ~10 0/4 - -bl- ° 0/5,

and tr(u) = O/OzP2(¢(u), u) + O/Ou Q2((p(u), u) : blU + bl --

b2 u2, approximately,

where

+ 2bo2

-3all

blo - - 4 a o 2 b l o + ( a l l - boE)(-3alo + bol) b2 :

b2o

We use theorems 66 and 67 o f [16] in the simplified version (p. 365), and distinguish between a l l -- b02 ~ 0 a n d a l l -- b02 = 0.

For some cases it is necessary to extend the blow up procedure used in [16] to obtain m o r e i n f o r m a t i o n on the singular points in this section. This is due to the fact that it should be questioned whether z - 0, being a solution o f (4), also is a separatrix o f a hyperbolic sector or not. F o r this extended blow up we use polar coordinates z = r cos 0, u = r sin 0, 0 _< 0 < 2n, and

(**),

w i t h flEO = b l l --- O, b e c o m e s

r ' = b l o r c o s O sin 0 + r E [ - a l o c o s 3 0 + ( - a l l

+ bo2 ) sin30 + (boo - b 0 2 ) c o s 20 sin 0

+ (bol - alo) cos 0 sin20] + r 3 [ - a o l cos 0 sin 0 - aoo C O S 2 0

--

ao2 sinEO],

O' = blo cos 2 0 + r[boo cos20 + bol cos 0 sin 0 + bo2 sin20] cos O.

(5)

904

J.W. REYN and R. E. KOOIJ

On r = 0, since bid # 0, there are only singular points in (0, zt/2) and (0, 3rt/2). Near (0, rt/2) we write, with ~ = 0 - n/2, approximately,

r' = -blor~ - (all -- bo2)r 2,

~' = -bo2r~ + bl0~ 2,

(6)

whereas, near (0, 3n/2), with ( = 0 - 3n/2, (5) is approximately written as

r' = -blor( + (all -- boz)r 2,

~' = - b o 2 r ~ q- blo~ 2.

3.1. Tangentially second-order, nilpotent singular points (al i

-

-

(7)

hOE ~ 0)

In order to apply either theorem 66 or 67 of [16] we must distinguish between all ~ 0 and all = 0. 3.1.1. all ;~ 0 In this case a3 ;~ 0 and we may apply theorem 66 of [16] with k = 3, m = 1. Then follows, if a l l ( a l l - bo2)< 0 (a3 > 0), that the singular point M ~ is a saddle S~, 1, with three hyperbolic sectors for z > 0 if bid(all - bo2) > 0 and for z < 0 if bid(all - bo2) < 0. I f a l l ( a l l - b02) > 0 ( a 3 < 0), there is bl ~ 0, sinceb I = 0impliesall(all - b02) = -)-all21 • 0, so n = 1 and 2 = b 2 + 4(m + 1)a3 = (axl - 2boE)E/b2o >- O. According to theorem 66 of [16] the singular point has an elliptic and a hyperbolic sector. The elliptic sector is in the half plane z > 0 and the hyperbolic sector in the half plane z < 0 if bid(all -- b02 ) > 0, and the reversed situation occurs for b l 0 ( a l l - bo2) < 0. We should now investigate the location of the separatrices of the hyperbolic sector, compared with z -= 0 and will use polar coordinates, leading to equations (5)-(7). In Fig. 3 the results of the blow up are shown in the r, 0 plane and the z, u plane. Use can be made of the known location of the elliptic and hyperbolic sectors. Furthermore, it may be seen that there are three directions r = k ( in which paths are approaching the singular points (0, rt/2) and (0, 3n/2) in the r, 0 plane. These are k = 0 and k = ao for both points and k = (-2blo)/(all - 2b02) for (0, rt/2) and k = 2blo/(all -- 2b02 ) for (0, 3n/2). From this the phase portraits in the r, 0 plane m a y be completed and the phase portrait in the z, u plane determined. The singular point Mll,2 is a third-order elliptic saddle EH12 or HE~,2 or a third-order elliptic saddle node EPHp1,2 or PHPE1,2 . For all(all -- 2b02) < 0 there is a parabolic sector in the half plane, which contains the hyperbolic sector and for all(all - 2b02) -> 0 there is no such parabolic sector. 3.1.2. all ---- 0 In this case a3 -- 0 and a4 = (boE/b~o)C26 7~ 0 if c26 ;~ 0. We may apply theorem 67 of [16] with k = 4, m = 2, n -- 1 and the singular point M~°2 is a fourth-order saddle node SN~2,2 having a hyperbolic sector with opening angle r~ for z < 0 if bo2 bid < 0 and for z > 0 if b0Ebl0 > 0. A hyperbolic sector with opening angle 0 is situated at u > 0 for bo2 bl0 c26 < 0 and at u < 0 for bo2bloc26 > 0. In order to show that the separatrix between the hyperbolic and parabolic sector coincides with z -= 0 use will again be made of polar coordinates and (5)-(7). Use can be made of the known location of the hyperbolic sectors. For the three directions r = k~ in which paths are approaching the singular points (0, ~z/2) and (0, 31r/2) m a y be found k = 0, k = do and k -- blo/bo2 for (0, zt/2) and k = - ( b l o / b o 2 ) for (0, 3rt/2). F r o m this the phase portrait in the r, 0 plane may be completed and the phase portrait in the z, u plane determined. Then follows that the singular point M°,2 is a fourth-order saddle node SN°,2 of type H_PH2°,2 for c26 > 0, bo2blo < 0, of type H H P ~ 2 for c26 ~ 0, bo2blo < 0, o f type _P/-/H2°,2 for c26 > 0, bo2blo > 0 and of type HPH°,2 for c26 < 0, bo2blo > 0.

2

blo(al1"b02)<0,al 1(al1"2b02)>-0

r,~/) ~ ~ _ . _ _

bl0(a11-b02)>0, a11(al1-2b02)>-0

2

~,~/{/~

U

z

Fig. 3.

(EH)11,2

(HE)11,2

r 3~ 2

2n

bl 0 (al 1-b02)<0'

2

al 1(al£2b02)<0

2

i i

2~

blo(al1-b02)>0, a11(al1-2b02)<0

_~ 2

U

(E~H-_P)11,2

(~H~E)11.2 ,= o.

906

J . W . REYN and R. E. KOOIJ

I f c26 = 0, then a3 = a4 = 0 and a5 = (b22/b4o)C23.If c23 # 0 we m a y apply theorem 66 with k = 5, m = 2. Then it follows, that if c23 > 0, the singular point M 3,2 -~ is a fifth-order saddle S ~ with three hyperbolic sectors for z > 0 if bl0b02 < 0 and for z < 0 if bl0b02 > 0. I f c23 < 0 (as < 0), since also b 1 # 0, SO ?/ = 1 < m = 2, it follows that the singular point has an elliptic and a hyperbolic sector. The elliptic sector is in the half plane z > 0 and the hyperbolic sector in the half plane z < 0 if bm0b02 < 0 and the reversed situation occurs for blobo2 > 0. In order to show that the separatrices of the hyperbolic sector coincide with z - 0, as before, polar coordinates and (5)-(7) are used. Use is made of the known location of the elliptic and hyperbolic sectors. For the three directions along which paths can approach the singular points (0, rt/2) and (0, Mt/2) the same is found as for c26 # 0. Then follows that the separatrices of the hyperbolic sector coincide with z - 0. The singular point M12 is a fifth-order elliptic saddle (or elliptic saddle node) HEI,2 for boEbl0 < 0, and EH~. 2 for b0Ebl0 > 0. I f c26 = ¢23 = 0, then a3 = a4 = as = 0 and a 6 = (b32/bSlo)C12. If c12 ~ 0 we may apply theorem 67 with k -- 6, m = 3, n = 1. Then follows that the singular point M4°,2 is a sixth-order saddle n o d e SN°,2 . The analysis given for the fourth-order saddle n o d e SN°2,2may be repeated with ¢26 replaced by -c12. Then follows that the singular point is of type HHf~°,2 for c~2 > 0, boEblo < O, of type HP_H°,2 for c12 < O, boEblo < O, of type HPH°,2 for c12 > O, boEblo > 0 and o f type P_HH°2 for c12 < O, bo2blo > O: I f c26 = c23 = c12 = O, (**) has a parabola through (0, O) with singular points. Under the conditions of Section (3.1), (.) reads -~ = aoo + a l o x + flolY ÷ a l l x Y + ao2Y 2 ~

P(x,y),

)' = boo + blox + holY + bo2Y2 - Q(x, y), with blo ~ 0, a l ~ - b02 ;~ 0. I f b02 = 0, then a H # 0 and there are two transversally nonhyperbolic singular points since all isoclines 3.P(x,y)+ pQ(x,y)= 0 have the two asymptotic directions y - 0 and a l l x + ao2Y = 0 in c o m m o n , and no statement can be made with regard to the finite multiplicity the system should have wherein a singularity occurs corresponding to these conditions. If b02 # 0, the only transversally nonhyperbolic singular points are at the end of the x-axis and the finite multiplicity may be easily determined. The results of Section 3.1 are summarized in Figs 4, 5.

3.2.

Tangent&lly third-order, nilpotent singular points (all - bo2 = O)

In this case a3 = 0 and a4 = -(allao2/b2o). Suppose first a02 # 0, then we may apply theorem 67 with k -- 4, m = 2, n = 1 if a~l # 0, since bl = --(all/blo). Then follows that the singular point M ° 3 is a fourth-order saddle node SN°3 having a hyperbolic sector with opening angle 7t and a hyperbolic sector with opening angle 0 both for z < 0 if blobo2 > 0 and both for z > 0 if blobo2 < 0. Furthermore, the hyperbolic sector with opening angle 0 is located in u < 0 for a02bo2 > 0 and in u > 0 for ao2bo2 < 0. In order to determine the location of the separatrices of the hyperbolic sectors with respect to z - 0, we again use a blow up with polar coordinates and (5)-(7). As a result the singular point may be found to be of type PHHP°3 for aoEblo > 0, aoEa11 > 0, of type PHH°3 for aoEblo > 0, a o 2 a l l < 0, of type _PH/-/_PI°3 for ao2blo < 0, ao2all > 0 and of type HHP_I°3 for aozblo < 0, ao2all < O.

Infinite singular points

all(a11-bo2)
Z

907

rnf=3

sil2 al l(all"bo2)>O 1 M1,2 a11(a11-2b02)->O a11(a11-2b02)
I f all = 0 then a3 = a4 = 0 and a5 = -(a2o2/b2o) < 0; then we m a y apply t h e o r e m 66 o f [16] with k = 5, m = 2, n = 2 since bl = 0, b2 = -4(ao2/blo) ;~ O. With ). = b22 + 4(m + 1)as = 4(ag2/b21o) > O, m = n then follows that the singular point M2~,3 is a fifth-order n o d e N 2,3 1 of type Pp1,3 for ao2b~0 > 0 and o f type P__PP~,3for a02b~o < 0. I f a02 = 0, z -- 0 is a line o f singular points, so the Poincar6 circle consists o f singular points. The results o f Section 3.2 are summarized in Fig. 6. The values for my follow directly f r o m the fact that the points are tangentially third order.

4. H I G H L Y

DEGENERATE

SINGULAR

POINTS

These points occur if a2o = bH = b~0 = 0, then Az = 2 , = 0 and (**) has no linear terms. They only occur in systems with my = 0, 1 or 2, so with few finite singular points, and are at least o f order 4. E q u a t i o n (**) m a y n o w be written as

z' = -z(a~oz + a l l U + aooZ 2 + aolzU + ao2U2), u' = booz 2 + (-a~o + boOzU

+

(--all

+

bo2)U 2 -

flolZU 2 -

aoozZu

(8) -

ao2 u3.

F o r the blow up we use polar coordinates z = r cos 0, u = r sin 0, 0 _< 0 < 2zt, and (8) becomes 1

- r' = 1: = r [ - a l o cos 0 + (--all + b02) sin 0 + (boo - b02) COS20sin 0 + bol cos 0 sin20] r

+ r2[--aol COS 19sin 0 -- aoo cos20 -- a02 sin20], 1

-- 0 ' ~--- 0 = (boo c o s 2 0 -~- bol c o s O sin 0 + bo2 sin20) cos O. r

(9)

908

J . W . REYN and R. E. KOOIJ

mr=2

(226=/=0 ...........................

023o................ j/j z

mf=l

-1 S3,2

c23 <0 ...............

~ u

1

mf=l

M3,2 u~

C23=0

zNO2

ml=0

O12=/=0

c12=0:(*)hasaparabolathroughz=u=0 filled with singular points

Fig. 5. Tangentially second-order, nilpotent singular points (a2o = b H = 0, blo # 0, al~ - bo2 # 0), all = 0 .

For r = 0 there are at least two singular points: (0, n / 2 ) and (0, 3n/2). Near (0, rr/2) we write, with ~ = 0 - rt/2, a p p r o x i m a t e l y , /; = - - ( a l l

--

bo2)r - a02 r 2

+

(alo - bol)r~,

~ = - b o 2 ~ + bol~ 2 - boo~ 3,

whereas, near (0, 3rt/2) with ( = 0 - 3zt/2, we a p p r o x i m a t e (9) by /: = ( a l l -- b o 2 ) r -

aozr z + (-alo

+ bol)r~,

~ = b02 ~ -

bol~ 2 +

boo~ 3.

distinguish b e t w e e n t h e c a s e s : (i) all - b02 # 0 , b02 # 0, (ii) all - bo2 # 0, bo2 = 0, (iii) all - bo2 = 0 , bo2 ~ 0 , (iv) a l l -- bo2 = 0, bo2 = 0. We

Infinite singular points

909

mf=3

SN~,3

al 1=0

mr=2

N21,3

ao2=O: Poincar6 circle consists of singular points Fig. 6. Tangentially third-order, nilpotent singular points (a2o =

bll

= 0,

b~o ;~ 0, a ,

-

bo2 = 0 ) .

4.1. Tangentially second-order highly degenerate singular points occurring in systems with one transversally nonhyperbolic singular point; (a~ - bo2 # 0, bo2 # 0) In this case, (.) m a y be written as ~c = ado + alox + ao~y + allxY

+

ao2Y 2,

= boo + bolY + boEY2, with bo2 ~ 0 and we distinguish, with A = b2~ - 4boobo2 , between A < 0, A = 0 and A > 0. A p a r t f r o m the singular points 0 = l r t and 0 -- art on r - 0, equation (9) has on r = 0 singular points in 0 = 0+, being the solutions o f boo c0s20 + bo~ cos 0 sin 0 + bo2 sin20 = 0. The value for my follows easily f r o m the fact that there is only one transversally nonhyperbolic point. (In Section 4.1 we understand this to m e a n also two coinciding transversally nonhyperbolic points.) 4.1.1. A < 0 There are only two real singular points on r = 0 : 0 = rt/2 and 0 = 3rt/2, b o t h elementary saddle points if bo2(a~l - bo2) < 0 and both elementary nodes if bo2(au - bo2) > 0. As a result for b o 2 ( a l l - b 0 2 ) < 0 we have the singular point H H ° and for b o 2 ( a . - b o 2 ) > 0 the singularity E E 2. In order to determine the multiplicity m we must distinguish between a2o + a2~ ~ 0 and a12o + a21 = 0. For a2o + a121 ~ 0 we have m = 2, 2, so the singular point is o f fourth order. For a20 + a12~ = 0 we have to distinguish between c 2 6 - c13c36 ;;~ 0 and c26 - c13c36 -- 0. In the first case since bo2(all - bo2 ) = -b022 < 0, we have the singularity H H ° 2 . In the second case, (8) is degenerate, and for a 2 + a21 + a22 + b20 + b21 + bEE ~ 0,

910

J.W. REYN and R. E. KOOIJ

bo2(al 1bo2)
a2o+a21# 0 ......................

ml=2

Iz MO,2

a20+a21=O

uI

C26-C13C36# 0

ml=O

M40,2 2 c 16-c13c36 =0

(**) is essentially linear U~

mf=2

bo2(a11-b02)>O ........................................

M2,2 Fig. 7. Tangentially second-order highly degenerate singular points occurring in systems with one transversally nonhyperbolic singular point; A < 0.

after dividing out the c o m m o n factor aooz 2 + aolZU + ao2 u2 a n d / o r booz 2 ÷ bolT.u ÷ bo2 u2 (8) becomes linear, and (0, 0) is an o r d i n a r y point or a elementary singular point. See Fig. 7.

4.1.2. A = 0 A p a r t f r o m the singular points on r - 0 at 0 = n / 2 and 0 -- 3n/2 there are singular points at 0 = 0o, 0 _< 0o < n, and 0 = 0o + n with tan 0o = -(bol/2bo2). Near (0, 00) we write, with = 0 - 0o, (9) a p p r o x i m a t e l y as t: = - 2 a l ° b ° 2 + al~bol r c o s 0o + -4bo2c16 + 2bolC36r2 2bo2 bo21 + 4b22

alobol + 2allbo2 r~cosOo, 2bo2

= b2°l + 4b2°2 cos 0o~ 2, 4bo2 and similarly near (0, 0 o + n). As for A < 0, the singular points for (0, n/2) and (0, 3tr/2) are b o t h elementary saddle points if bo2(all - bo2 ) < 0 and b o t h elementary nodes if bo2(all - bo2) > 0.

Infinite singular points

911

4.1.2.1. If -2alobo2 + allbol ~ 0, the singular points, 0 = 0o and 0 = 00 + n are second-order saddle nodes with a center manifold along r -- 0. If bo2(a H - bo2) < 0, for -2alobo2 + allbo~ > 0 the singular point is a saddle node o f type /-/_P/-/P2°2 and for -2alobo2 + aHbol < OoftypePHPH°,2. If bo2(aH - bo2) > 0, f o r - 2 a l o b o 2 + aHboi > 0 t h e singular point is o f type PEPE2,2 and for -2alobo2 + a l l b o l < 0 o f type EPEP222 .

4.1.2.2. If -2alobo2 + allbol = 0 we distinguish between fl - -4bo2c16 + 2bolc36 ~ 0 and fl = 0. I f f l # 0, the singular points in 0 = 0o and 0 = 0o + n are fourth-order singularities with three specific directions r = k~, where k = (all + bo2)(bo21 + 4b22) cos 0o)/4flb22, 0 and oo. The flow in these directions determines the character of these singular points. Apart from the sign o f bo2(all - bo2) also the signs o f boE(all + bo2) and fl then have to be taken into account. If boE(all - bo2) < 0 and bo2(all + bo2) > 0 the singular point is o f type HP_PH°2 for flbo2 < 0 and o f type/3H/-/_P°,2 for flbo2 > 0; if boE(all - bo2) < 0 and bo2(all + bo2) < 0 the singular point is o f type /D/-/_P/°H_P4°,2for flbo2 < 0 and o f type /sH_PPH_P4°2 for fib02 > 0; if boE(all - bo2) > 0 and bo2(all + bo2) > 0 the singular point is o f type P_EEp2,2 for flbo2 < 0 and o f type Eff~PE2,2 for fib02 > 0; whereas bo2(all - bo2) > 0 and boE(all + bo2) < 0 cannot occur since this would imply b022 < allb02 < -b22, with b02 ;~ 0. If all + bo2 = 0 then boE(a11 - b02) = -2b22 < 0 and a singular point appears which can be considered as a limiting case o f the first two cases; the point is o f t y p e H_P/3H°,2 for/~bo2 < 0 and o f type/3H/-/P4°,2 for flbo2 > 0. I f f l = -4bo2c16 + 2bolC36 = 0, ( , ) is an essentially linear system with z = u = 0 on the line o f singularities. The results o f Section 4.1.2 are given in Fig. 8.

4.1.3. A > 0 Apart f r o m the singular points on r - 0 at 0 = rt/2 and 3rr/2 there are singular points at 0 = 0±, 0 < 0± < n, and 0 = 0± + rt with tan 0± = (-b01 + x/b21 - 4booboz)/2bo2. Near (0, 0±) we write, with ~ = 0 - 0±, (9) approximately as f = q~: cos O+_r+ q~ cos20±r 2,

~ = q~ cos 0±~,

and similarly near (0, 0± + rt). Here 1

2

q~: = 2-~o2 [ - alobo2 + allbol :1:all~/b21 - 4boobo2], 1

q~ = 4-~o2 [-4bo2c16 + 2bolc36 7= 2c36~b21 - 4boobo2],

q~ = 4-~2o2[-2bol(b21 - 4boobo2) :1:(2b21 + bEE -- 4boobo2)~/b2o~ - 4boobo2]. As for A <_ 0 the singular points for (0, rt/2) and (0, 3rt/2) are both elementary saddle points for bo2(all - bo2) < 0 and both elementary nodes if bo2(all - bo2) > 0.

912

J . W . REYN and R. E. KOOIJ

-2al obo2+al 1bol =/=0

bo2(a11-bo2)
Mo2

mf=2

bo2(a11-bo2)>O .......................

mr=2

bo2(al 1b02 )<0

mf=O

"2alobo2+al lb01=0 -4b02c 16+2b01036 =~0

bo2(al 1+b02)->0

I' "M°,2 °T mf=O

bo2(al 1+bo2) <0

MO,2

bo2(all-b02)>O

mf=O

bo2(all+b02) >0 2 M4,2

-4bo2c16+2b01c36=O: (*)degenerates into an essentially linear system with z=u=O on the line of singular points Fig. 8. Tangentially second-order highly degenerate singular points occurring in systems with one transversally nonhyperbolic singular point; A = 0.

Infinite singular points

913

The singular points in (0, 0±) and (0, 0+ + n) are also elementary if qgqo ~ 0 and

q~q2 ~ O. Now 4q~q2 -- -A(16b2o + 2bo21 + b2z + 2A) < 0 and y-- bo2qoqo =a20b02 + a211boo aloallbol. ÷

- -

n

We distinguish between y ~ 0 and y = O. 4.1.3.1. y = alZobo2 + al21boo - aloallbol ;~ O. Let bo2 < 0, then since tan0+ - t a n 0 _ = 1/bo2 ~/b21 - 4boobo2 < 0, the possibilities for 0+_ are: (i) 0 < 0 + < 0 _ < n / 2 ; (ii) 0 < 0 _ < n/2 0, then all < 0 and since q~- - qo = -(all/bo2)~/b21 - 4boobo2 < 0, the possibilities for q~ are: (i) q~ < 0, qo < 0, thus y < 0 and because of q~" + qo = 1/bo2(-2alobo2 + allboO < O, follows -2alobo2 + a~lbo~ > 0; (ii) q~- > 0, qo > 0, thus y < 0 and -2alobo2 + altbol < 0; (iii) q~- < 0, qo > 0, thus y > 0. For each combination the character of the singular points r = 0 can now be determined, as well as the singularity in u = z = 0. It appears that any of the choices for 0_+ leads to the same phase portrait and reversal of the signs of bo2 and a H - bo2 leads to flow reversal. It can then be shown that for the case bo2(all - bo2) > 0, if bo2Y > 0 the singular point M2,2 is of the type EP_[~Epp2,2 for -2ao2bo2 + a~lbo~ < 0 and of type P_PEP_pE2,=for -2alobo2 + allbol > 0; if bo2Y < 0 the singular point M°,z is of type PHPH°,2 . Now let al~ - bo2 > 0, thus bo2(all - bo2) < 0, then the possibilities for qo are (i) qg < 0, qo < 0, thus y < 0 and, because of q~ + qo = 1/bo2(-2alobo2 + allb01) < 0, follows -2alobo2 + a l l b o l > 0 , (ii) q~ > 0 , qo > 0 , thus y < 0 and - 2 a l o b o 2 + a l l b 0 1 < 0 , (iii) qg < 0, qo > 0, thus y > 0 and because of qg - qo = -(all/bo2)~/b21 - 4boobo2, there follows allbo2 > 0 and (iv) q~- > 0, qo < 0, thus y > 0 and al~bo2 < 0. For each combination the character of the singular points on r = 0 can again be determined, as well as the singularity in u = z = 0. It appears that any of the permitted choices for 0± leads to the same phase portrait and reversal of the signs of bo2 and all - bo2 again leads to flow reversal. It can then be shown that if bo2(all - b02) < 0, if bo2Y > 0 the singular point M2°,2 is of type HPHP°,2 for -2alobo2 + allbol > 0, and of type PHPH°,2 for --2alob02 + allbo~ < 0, if bo2Y < 0 the singularity M2~2 is of type HHHHHHf22 for all b02 > 0 and of type PEP/SE/32 2 for all bo2 < 0. See Fig. 9. 4.1.3.2. y -= a20b02 + a21boo - aloallb01 = 0. As in the previous section, let again be b02 < 0, then there are the same possibilities for 0+: (i) 0 < 0 + < 0 _ < n / 2 ; (ii) 0 < 0_ < n / 2 < 0÷ < n; (iii) n / 2 < 0+ < 0_ < n. First, consider the case that either q~- or qo ~.0, whereas bozqgqo = Y = 0, then all # 0 and if q~- = 0, then q~- = 6 and if qo = 0 then q~- = 6, where 6 = 1/(all bo2)(aloc36 - allcl6). First, let 6 ;~ 0.

914

J.W. REYN and R. E. KOOIJ U,

bo2(a11-b02)>0

mf=2

b027>0 ................ M2,2

b027<0 ................

mr=2

II Z

0 M2,2 U,

// bo2(a11-b02)
b02~>0 ................

mr=2

II

J U,

b02"(
mr=2

al 1b02>0 -2 M2,2

al 1bo2
f•~

ml=2 2

M2,2 Fig. 9. Tangentially second-order highly degenerate singular points occurring in systems with one transversally nonhyperbolic point; A > 0, y -= a2oboz + a21 boo - a l o a . bo~ ;~ O.

F u r t h e r , let a l l -- b02 < 0, thus bo2(all - b02) > 0 a n d a~l < 0, t h e n since q~ - qo = - 4boob02 < 0, the various possibilities for q~ are: (i) qo = O, q~ < O; (ii) qo > O, q~ = O. F o r each c o m b i n a t i o n the character o f the singular p o i n t s o n r = 0 c a n n o w be d e t e r m i n e d , as well as the singularity in u = z = O. It appears that a n y o f the choices for 0± leads to the same phase p o r t r a i t a n d reversal o f the signs o f b02 a n d a l l - b02 leads to flow reversal. It c a n t h e n -(all/bo2)~/b21

Infinite singular points

915

be shown that if bo2(all - b02 ) > 0, qff = 0 the singular point is o f type PHPPE~,2 for d)* -- bo2t~ > 0, o f type PPEP_H~,2 for t~* < 0, whereas for qo = 0 the singular point is o f type EPPHpI,2 for t~* > 0 and o f type PHPEP~,2 for J* < 0. N o w let a H - bo2 > 0, thus bo2(all - bo2 ) < 0, then since

q~--qo-

all x/b21 - 4boobo2 bo2

the possibilities for q~ are: (i) qo = 0, q~- < 0, then allb02 > 0; (ii) qo = 0, q~ > 0, then all b02 < 0; (iii) qo < 0, q~ = 0, then allb02 < 0; (iv) qo > 0, qg = 0, then allb02 > 0. F o r each c o m b i n a t i o n o f p a r a m e t e r s the characters o f the singular points on r = 0 can n o w be determined, as well as the singularity in u = z = 0. As before the choices for (9o and signs o f b02 and a H - b02, keeping bo2(a~l - b02) < 0, are irrelevant for the phase portrait. It can then be shown, that if bo2(aH - b02) < 0, qo = 0, for a~b02 > 0 then the singular point is o f type HHHPH~ 1 if ~* < 0 and of type PHHHH~.~2if J* > 0, for aHb02 < 0 then the singular point is o f type PEP_HP31,2if t~* > 0 and o f type HPPEP_~,2,if ~* < 0, whereas if bo2(all - b02 ) < 0, q~- = 0, for aHb02 > 0 then the singular point is o f type HPHHH~2 if ~* < 0 and of type H H H H P ~ if d* > 0, for aHb02 < 0 the singular point is o f type PEP_pHI,2 if ~* < 0, and of type PHPEP~,2 if ~* > 0. If ~ - l/(altbo2)(aloc36 - allc16) = 0, (.) is an essentially linear system with z = u = 0 on the line o f singularities. N o w consider the case that qo = qff -- 0, then al~ = 0 and bo2(aH - b02) = -b22 < 0, thus on r - 0, 0 = rt/2 and 3zt/2 are b o t h elementary saddles. In order to investigate the singular points at 0~ there should be considered the cases: (i) q~- < 0, q~ < 0; (ii) q]- < 0, q~- > 0; (iii) q l > 0, q~ < 0; (iv) q l > 0, q~ > 0. P e r f o r m i n g the analysis as before it can then be shown that if bo2ql 2 - ql+ = c26 c13c36--E1 > 0 the singular point is o f type PHHP4°,2 if boE(-2boEC16 + b01c36 ) > 0 and o f type HPPH°2 if boE(-2boEC16 + bol c36) < 0, whereas if E 1 < 0 the point is o f type PEP_HHH°,2 if c36 > 0 and o f type HHHPEP°,2 if c36 < 0. Finally, it can be r e m a r k e d that if E~ -- 0 system (,) degenerates. If c36 ~ 0 the system is essentially linear with z -- u = 0 on the line o f singular points; if c36 = 0 system (.) has a degenerate p a r a b o l a with singular points through u = z = 0. See Fig. 10. 4.2. Tangentially second-order highly degenerate singular points occurring in systems with two transversally nonhyperbolicpoints; (all - b02 ~ 0, b02 = 0) In this case, (.) m a y be written as

Jc = aoo + aloX + aoly + allxy + ao2Y 2, P = boo + boly,

916

J . W . REYN and R. E. KOOIJ

al 1=~0

8*=f 0

mf=l

bo2(al 1b02 )>0

bo2(al 1-bo2)<0

al 1bo2>O

••

mf=l X~

-1 M3,2

mf=l

a11bo2
a11=0 .............................

Ei
8" = !all (a10c36-allC16) 2 E1=c16-c13c36

mf=O

EI>O ................

EI=O

~ \

A<)f

z

mf=O

MO,2

036=#0:(*) is essentially linear with z=u=O on line of singular points C36=0: (*) has a degenerale parabola with singular points

Fig. 10. Tangentially second-order highly degenerate singular points occurring in systems with one transversally nonhyperbolic point; A > 0, ? --- a20b02 + a~l boo - a l o a l l bol = O.

Infinite singular points

917

with all ~ 0. This system has two different transversally nonhyperbolic singular points, being in the directions y -- 0 and allx + ao2Y = 0, respectively. The finite multiplicity my o f the system m a y be easily determined. Rewriting (**) yields Z' = - - z ( a l o Z + a l l U +

U' = booz 2 + ( - a l o +

(20oz2 + aolZU + ao2u2),

boOzu

-

all/./2

-

aolZU 2 -

a o o z 2 u --

a02 u3,

and the blow up (9) gives i: = r[-a~o cos 0 - a~l sin 0 + boo cos20 sin 0 + bo~ cos 0 sin20] + r 2 [ - a o l cos 0 sin 0 - aoo c0s20 - ao2 sin20], 0 = (boo cos 0 + bol sin 0) cos20. For r - 0 there are at least two singular points: (0, n/2) and (0, 3rt/2). Near (0, rt/2) we write, with ~ = 0 - rt/2, a p p r o x i m a t e l y , ~ = bol~ 2 - boo~ 3,

f = - a l l r,

whereas near (0, 3rt/2), with ~ = 0 - 3rt/2, we a p p r o x i m a t e (9) by

f = allr,

~ = - b o l ~ 2 + boo( ~.

We distinguish between bo~ ~ 0 and bo~ = 0. 4.2.1. bol ~ 0

On r - 0 there are, a p a r t f r o m 0 = rt/2, 3rt/2, singular points in 0 = 0o, 0 _< 0o < ~ and 0 = 0 o + rt with tan 0o = -boo/bo~. Near (0, 00) we write with ~ = 0 - 0 o, a p p r o x i m a t e l y f = allboo - alobol cos Oor + -aoob21 + a°lboob°t - a°Eb°2° bol b2o + b 2, r 2,

~ = bol cos 0o~.

Since all bol 7~ 0, the singular points at (0, n/2) and (0, 3n/2) are second-order saddle nodes with center m a n i f o l d along the 0-axis. If allboo - a~oboo ;~ 0 the singular points at 0 = 00 and 0 = 0o + rt are saddle points if allboo - alobol < 0 and nodes if allboo - alobol > 0. It can then be shown, that when aH boo - alobo~ < 0 the singular point z = u = 0 is o f type HPHP°2 if allbol > 0 and o f type PHPH°,2 if allbol < 0 and when allboo - alob01 > 0 o f type PEPE2,2 if aHbo~ > 0 and o f type EPE_p2,2 if a~lb01 < 0. I f allboo - alobol = 0 the singular points at 0 = 0o and 0 = 0o + n are second-order saddle nodes with center m a n i f o l d n o r m a l to the 0-axis. It can then be shown that the singular point is o f type H-fi-fiE~,2 if aHbo~ > O, bol~p < 0, of type EPpH1,2 if aHbo~ < 0, b01¢ < 0, of type P_HEP_~,zif al~bo~ > 0, bolt/) < 0 and o f type PEHP~,2 if al~bo~ < 0, bo~cp > 0, where ¢ = -aoob21 + ao~boobo~ - a02 b2. If ~ = 0 the system is degenerate, being essentially linear with a line o f singularities t h r o u g h U=Z=0. 4.2.2. bol = 0

On r = 0 there are singular points in 0 = rt/2, 3zt/2 which are third-order saddle points for aHboo < 0 and third-order nodes for al~boo > 0. As a result it can be shown that the singular point is o f type HH°,2 if all boo < 0 and o f type EE2,2 if a11 boo > 0, whereas EP_Ep_22could be also possible in this case; however, this requires a global argument.

918

J.W. REYN and R. E. KOOIJ

I f boo = 0 the system degenerates, with a h y p e r b o l a consisting o f singular points. T h e results o f Section 4.2 are s u m m a r i z e d in Fig. 11. 4.3. Tangentially third-order highly degenerate singular p o i n t s occurring in systems with one transversally nonhyperbolic p o i n t ; (all - bo2 = 0, b02 # 0) In this case (.) m a y be written as = aoo + alo x + aolY + bo2xy + ao2Y 2, = boo + boly + bo2Y 2,

with b02 ~ 0. This system has one transversally nonhyperbolic singular point and in fact only one infinite singular point, being in the direction y -= 0. As a result my can be directly read o f f f r o m the n o t a t i o n o f the singular point. Since for ao2 = 0 the Poincar~ circle consists o f singular points we will assume ao2 # 0. Rewriting (8) for this special case yields z' = - z ( a l o z + boru + aoo z2 + aolZU + ao2U2),

(8a)

U' = booZ 2 + ( - a l o + bol)zu - aolzU 2 - aooz2u - ao2 u~, and the blow up n o w yields i" = r [ - a l o cos 0 + (boo - bo2 ) cos20 sin 0 + bol cos 0 sin20]

+ rZ[-aol cos 0 sin 0 - aoo c0s20 - ao2 sin20],

(9a)

~) = (boo cos20 + bo~ cos 0 sin 0 + bo2 sinXO) cos 0. For r - 0 there are at least two singular points: (0, ~z/2) and (0, 3zt/2). Near (0, zt/2) we write, with ~ = 0 - ~ / 2 , a p p r o x i m a t e l y , f = -ao2 r2 + (alo - bol)r~,

~ = - b o 2 ~ + bol~ 2 - boo~ ~,

whereas near (0, 37r/2), with ~ = 0 - 37t/2, we a p p r o x i m a t e by t: = -ao2 r2 - (alo - bol)r~,

~ = bo2~ - bol~ 2 + boo~ 3.

Both singular points are second-order saddle nodes with center m a n i f o l d n o r m a l to r -= 0. As in Section 4.1 other singular points for r = 0 then 0 = 7r/2, 3rt/2 are the solutions 0± o f boo c0s20 + bol cos 0sin 0 + bo2 sin20 = 0. With A = b2~ - 4boobo2 we distinguish between: (i) A < 0; (ii) A = 0; (iii) A > 0. 4.3.1. A < 0 T h e only real singular points on r - 0 in this case are in 8 = ~t/2, 37t/2. For ao2bo2 > 0 the nodal part o f the saddle node 0 = ~ / 2 is in the region r > 0 as is the saddle part o f the saddle node at O = 3rt/2. For ao2bo2 < 0 the opposite statement can be made. As a result there exists a singular point N21,3 = _PP1,3 for ao2bo2 > 0 and N21,3 = /3_P21,3 for ao2 bo2 < 0.

Infinite singular points

bol:# 0

919

mf=1

a11boo -alo bol
M2,2 u

a11booalobol>O .............

ill,

I //

ml=l

M2,2

a11boo -alo bol=O

9:#0

mf=O z 1

9=0: (*) is essentially linear with line of singular points through z=u=O 2 2 9 = -aoo b01+a01 boobol-ao 2 boo

bol=O

boo 4:0

a11boo<0

M3,2

U,

]I

I z

mf=O

0

M2,2

mf=O

al 1boo >0

M2,2 boo =0: (*) has a hyperbola with singular points Fig. 11. Tangentially second-order highly degenerate singular points occurring in systems with two transversally nonhyperbolic singular points.

920

J.W. REYN and R. E. KOOIJ

4.3.2. A = 0 As for A < 0 the singular points in r = 0, 0 = n/2, 3rr/2 are second-order saddle nodes with the same properties. Moreover, there are singular points in r -- 0, 0 = 00, 00 + rt, where tan 00 = -bol/2bo2. Near (0, 00) we write, with ~ = 0 - 0o, (9a) approximately as

(

i=

-alo+~

1b

)

-4bo2c16 + 2bolc36 r2

b21 +4b22

ol c O s 0 o r + _

+ 4b 2 cos

2

alobol + 2bo2 r~ cos 0o, 2bo2

2,

4bo2 and similarly near (0, 0o + r0. We distinguish between - a l o + ½bo~ ;~ 0 and - a l o + ½bol = 0. 4.3.2.1. - a l o + ½bol ~ 0. If -alo + lbol 7~ 0, the points (0, 00) and (0, 00 + zr) are second-order saddle nodes with center manifold along r - 0. It can be shown that if go2 bo2 > 0 there is a singular point PPHE~,3 for boE(-alo + ½bol ) > 0 and a point EHPP2~,3 for bo2(-alo + ½bol ) < 0, whereas if ao2bo2 < 0 there is a point HEPpppI3 for bo2(-alo + ½bol ) > 0, and a point -fi--fiEHg,3 for bo2(-alo + ½bol) < 0. 4.3.2.2. -alo + ½bol = 0. If - a l o + ½bol = 0, the points (0, 00) and (0, 00 + rt) are fourthorder points if 4fl = -4bo2c~6 + 2bo~c36 = 4(-aoob22 + aolalobo2 - a2oao2) ;~ 0, whereas for fl = 0 the system degenerates into an essentially linear system with a line of singular points through z = u = 0. In (0, 00) and (0, 00 + rt) the fourth-order singularities have three specific directions r = k~, where k = (b21 + 4b22)/2flbo2 cos 0o, 0 and oo. The flow in these directions determines the character of these singular points. If ao2//< 0 the singularity is of type pP-P-fi4~,3 for ao2bo2 > 0 and of type ff-fiPP~,3 for ao2bo2 < 0; if ao2fl > 0 the singularity is of type EHHE,~, 3 for ao2bo2 > 0 and of type HEEH1,3 for ao2bo2 < 0. The results in Sections 4.3.1 and 4.3.2 are summarized in Fig. 12. 4.3.3. A > 0 Apart from the singular points on r - 0 at 0 = rt/2, 37t/2 there are singular points at 0 = 0±, 0 < 0± < rt and 0 = 0± + 7r with tan 0± = (-bol + ~/b2ol - 4boobo2)/2bo2. Near (0, 0±) we write, with ~ = 0 - 0±, (9a) approximately as

i = q~cosO±r + q~cos20±r 2,

~ = q~ cos 0±~,

and similarly near (0, 0_+ + n). Here 1

2

q~: = -alo + lbol :t: ~ / b o l - 4boobo2, 1

q~: = 4--~o22[-4bo2c16 + 2bolc36:1:2c36x/b21 - 4boobo2], 1 [_2bo1(bo21 _ 4boobo2) 4- (2bo2~ + b22 - 4boobo2)~/b21 - 4boobo2]

Infinite singular points

921

u 2 A= b01 -4b00b02<0

mr=2

1 N2,3

2 A= bol -4bo0b02=0

-al0+-~1 bol =~0 .................

ii

I~

"z 1

mf=2

M2,3

_alo+ 1 bo1=0

a02 13<0

mr=0

1 N4,3 u

ao2 13>0 a02 13=0: (*) is essentiallylinear 2 2 13=-a00b02+a01al0b02-al0a02

~

=z 1

m~=0

M4,3

Fig. 12. Tangentially third-order highly degenerate singular points occurring in systems with one transversally nonhyperbolic point; A _< 0.

As for A ~ 0 the singular points at (0, n/2) and (0, 3n/2) are both second-order saddle nodes with center manifold normal to the 8-axis, and for a02 b02 > 0 the nodal part of the saddle node at 8 = n / 2 is in the region r > 0, as is the saddle part of the saddle node at 8 = 3n/2, whereas, for ao2 bo2 < 0 the opposite statement is true. The singular points in (0, 8_+) and (0, 8_+ + n) are also elementary if q~qo ~ 0 and q~q2 ;~ O. Now 4q~q~ = -(16b~o + 2b21 + bo22 + 2A)A < 0 and qgqff = a2o - alobol + boobo2. We distinguish between a2o - alobol + boobo2 ~ 0 and alzo - alobol + boobo2 = 0. We note that a2o - alobol + boobo2 = ( - a l o + ½bol) 2 - ¼A.

922

J.W. REYN and R. E. KOOIJ

4.3.3.1. (-alo + ½bol) 2 - ¼A ~ 0. If b02 ~ 0, then since 1 . tan 0+ - tan 0_ = m x / b ~ l - 4boobo2, bo2 the possibilities for 0± are: (i) 0 < 0 + < 0 _ < n / 2 ; (ii) 0 < 0 _ < n / 2 < 0 ÷ < n ; (iii) n / 2 < 0+ < 0_ < n. As in Section 4.1.3, however, any choice out of these possibilities leads to the same singularities under the same conditions on the coefficients in the differential equations. Apart from the sign of ao2 bo2 which is important for the character of the singular points on r ----0 in 0 = n / 2 and 3n/2, for the singular points at 0 = 0±, 0± + n, we have to consider the cases: (i) qo < 0, q~- < 0; (ii) qo > 0, q~- < 0; (iii) qo > 0, q~- > 0, whereas qo < 0, qg > 0 cannot occur since q~- - qo = -x/b21 - 4boobo2 < 0. Considering all possible combinations of the parameters leads to the following conclusions with regard to the character of the singular point in z = u = 0. When ( - a l o + ½bol) 2 - ¼A > 0, the singular point is of type _PP/-/PEg,~ if ao2bo2 > 0, boz(-alo + ½bol) > 0, of type EP_HPP2~.3 if ao2bo2 > 0, bo2(-a~o + ½bol) < 0, of type HPEP_Pg.3 if ao2bo2 < 0, bo2(-a~o + ½bol) > 0 and of type PPEP_I-Ig,3 if ao2bo2 < 0, bo2(-alo + ½boo < 0. When ( - a l o + ½bo0 2 - ¼A < 0, the singular point is of type PHHHH~[ if ao2bo2 > 0 and o f type H H P H H ~ [ if ao2bo2 < 0. 4.3.3.2. (-alo + ½bol) 2 - ¼A = 0. If a2o - alobo~ + boobo2 = q~qo = O, either qo = 0, q~- < 0 or qo > 0, qg = 0, since q~- - qo = - ~ b 2 1 - 4boobo2 < 0. Considering all possible combinations o f the parameters leads to the following conclusions with respect to the character of the singular point in z = u = 0. Let 1 1 4 , t~ = ~02 (a10c36 - allCl6 ) = b2--~2(-aooaE1 + aloaolall- a2oa02)= ~022]~

then qi- = ~ if qo = 0 and q~ = ~ if q~ = 0. It can then be shown, that if a02fl > 0 that the singular point is o f type M°,3 = P HHHPE°,3 if ao2bo2 > 0, q~ = 0, of type EP_HHHP~, 3 if aoEbo2 > 0, qo = 0, of type HPEP_HH° 3 if aoEb02 < 0, q~- = 0 and o f type H H P E P H ° 3 if ao2bo2 < 0, qo = 0, whereas, if ao2fl < 0' the singular point is o f type SN~3,3 = PPHH°Ia if ao2bo2 > 0, q~ = 0, of type PHHP°3 if aoEbo2 > 0, qo = 0, of type PPHH°~ if ao2b02 < 0, qo = 0 and o f type HHPP3°3 if ao2bo2 < 0, q~ = 0. Finally, it can be remarked that if ~ -- 0, (.) is an essentially linear system with z -- u = 0 on the line of singularities. The results of Section 4.3.3. are summarized in Fig. 13. 4.4. Tangentially third-order highly degenerate singular points occurring in systems with two coinciding transversally nonhyperbolic points; (all -- bo2 = b02 = 0) In this case, (.) may be written as

Jc = aoo + a~oX + ao~Y + ao2Y2 J' = boo + bolY.

Infinite singular points

923

1 (-alo+ ~1 bol )2 -~-A >0 ...............

mt=2

U

(-alo+ 1b01)2-1A <0

1= Z

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

mr=2

M-1

2,3

u 1

\ (-alo+ lb01)2-1A =0

/

a02 I~ <0

~z

mf=l

u

a02 13>0

~

-

mt=l Mo 3,3

ao2 13=0: (*) is essentially linear with z=u=0 on the line of singular points

2 2 I~= "a00bo2+a01alobo2"al 0 a02 Fig. 13. Tangentially third-order highly degenerate singular points occurring in systems with one transversally nonhyperbolic point; A > 0. T h e system has two coinciding transversally nonhyperbolic singular points in the direction y -- 0. Thus my m a y easily be determined. Rewriting (**) for this special case yields z ' = - z ( a ~ o z + aoo z2 -F a o l Z U

U' = boo z2 -/- ( - a l o -t-

bol)zu

-t-

ao2u2),

aloZU 2 -

aooZ2U

-

a02 u3,

924

J.W. REYN and R. E. KOOIJ

and the blow up (9) gives

= r[-am + boo cos 0 sin 0 + bol sin20] cos 0 + r 2 [ - a o t cos 0 sin 0 - aoo cos20 - ao2 sin20], t~ = (boo cos 0 + bol sin 0) cos20. For r --- 0 there are at least two singular points: (0, with ~ = 0 - n/2, approximately, i = -ao2 r2 + (am - bol)r~,

n/2) and (0, 3n/2). Near (0, n/2) we write, ~ = bol~ 2 - boo~ 3,

(10)

whereas, near (0, 3n/2), with ~ = 0 - 3rt/2, we approximate by t: = -ao2 r2 - (alo -

bm)r~,

~ = - b o l ~ 2 + boo~ 3.

(11)

We distinguish between bm ~ 0 and b m = 0. F u r t h e r m o r e , we m a y assume ao2 ;~ 0 since for ao2 = 0 the system (,) is linear.

4.4.1. bol ;~ 0 O n r --- 0 there are, apart f r o m 0 = rt/2, 3n/2 singular points in 0 = 0o, 0 _ 0o < n and 0 = 00 + rr with tan 00 = -(boo~boO. Near (0, 00) we write, with ~ = 0 - 0o, approximately, t: = - a l o cos

Oor +

-aoob21 + aolboobol - ao2b2o r 2,

= bol cos 0o~.

The singular points at r = 0, 0 = rt/2, 3 n / 2 are fourth-order singular points with three specific directions r = k~, where k = ao2/(alo- 2bol) for 0 = n / 2 and k = -ao2/(alo- 2bol) for 0 = 3 n / 2 , whereas k = 0 and k = oo also are directions for both 0 = n/2 and 0 = 3n/2. The flow in these directions determines the character o f these singular points. The singular points in r = 0, 0 = 0o and 0 = 00 + n are saddle points if ambo~ > 0 and nodes if ambol < 0. We, therefore, consider the cases a~o ~ 0 and alo = 0.

4.4.1.1. a~o ;~ 0. It can be shown, that if bol(2bot - am) > 0, atobol < 0 the singular point z= u=0 is o f type PEE2,3, for a l o a 0 2 < 0 and o f type EEP~3 for a l o a 0 2 > 0 , if bol(2box - alo) -> 0, alob01 > 0, o f type P_PHH°,3 for aloa02 < 0 and o f type HH_P/53°,3 for aloa02 > 0, if bol(2bol - alo) < 0, thus alob01 > 0, o f type P_PHP_PH°,3if aloa02 < 0 and o f type H P/3/-/P/3~ 3 if aloao2 > 0.

4.4.1.2. alo = 0. The singular points in r = 0, 0 = 0 o, 0 = 00 + n are n o w second-order saddle nodes if ~0 = - a o o b 2 1 + aolboobo~- ao2b20 ;~ 0(Et ;~ 0). It can be shown that if ~Oao2 > 0(Et < 0) the singular point z = u = 0 is o f type HHEE~, 3 if ao2bo~ > 0 and o f type EEHH~, 3 if ao2bo~ < 0, whereas if ¢a02 < 0(E~ > 0) the point is o f type P_P/54~,3 if ao2bo~ > 0 and o f type _Pppat,3 if ao2b m < 0. If ~o = 0(E 1 = 0) the system is an essentially linear system with a line o f singularities t h r o u g h z = u = 0.

Infinite singular points

4.4.2. bol

=

925

0

On r - 0, there are only the singular points 0 = rt/2 and 37t/2. We m a y assume boo # 0, since for boo = 0 the system is degenerate and has a p a r a b o l a t consisting o f singular points. In order to determine the character o f the singular points we need a further blow up. With r = r/cos tp, ~ -- r/sin tp, (10) becomes, with bol = 0 1

-- (b = (p' = (ao2 COS ¢p -- a l o

sin tp - boo r/sin2~p) sin q~cos tp,

r/ 1 - / / = r/' = e(-a02 cos3cP + alo cos2¢o sin ¢0 - boo r/sin4¢o), r/

and (11) becomes 1

- (b = ~p' = (a02 cos ~p + alo sin ¢p + boor/sin2tp) sin Cacos ~p, 1/ 1//= 1/

r/' = r/(-a02 cos3¢o - a l o COS2tp sin tp + boor/sin4~P).

T a k e a~o > 0, ao2 > 0, boo > 0, then Fig. 14 shows the b e h a v i o u r o f the integral curves in the (0, r / p l a n e for 0 = rt/2 and 3n/2, in the r, 0 plane and finally in the u, z plane to show that the

. X2

-

~)

x 2

q)o=atctan~ x2 %

(p .__X 2

0

q)o=arctana~o 2 ~

_~

2

0

_x 2

3._~ 2

2

- 2'~

,

(_EIHF),3

Fig. 14. t If alo = 0, a21 - 4aooao2 < 0, this parabola is given by aoo + ao~Y + ao2Y 2 = 0, which consists o f two imaginary lines intersecting at the real point at infinity z = u = 0. As in the rest of the paper, nonisolated singular points are not studied in further detail, even if they are isolated in the real plane.

926

J . W . REYN and R. E. KOOIJ

a02=~0

bol~0

alot0

bol(2bol"al0) >0 alO b01
I. / /

~/[[

mf=l 2 M3,3

b01(2b01-al0) 20 al 0 b01>0

"~'~1 / " ~

z~

ml=l

SN0,3

bol(2bo1-alo) <0

] ~

~,

mf=l

SN0,3

alO=0

EleO

E1
\F~

j~

mf=O

M1 4,3

El>0

~

z

mr=0

N13 E1= 0: lineof singularpointsthroughz=u=0

b0,: 0

bOO~0

al0 ,~0

!

/~.~[~'/\~'J/'i

i

m,=0

z

M1 4,3

al0=0 boo=0: (') has a parabolawith singularpoints %2=0: (*) is linear and ('*) has a Poincarecirclewith singularpoints

mr=0 1 J4.3

Fig. 15. Tangentially third-order highly degenerate singular points occurring in systems with t w o coinciding transversally nonhyperbolic singular points.

Infinite singular points

927

singularit._y_yin z = u = 0 is of the type ~ E I H/54~,3. It can then be shown that the singularity is of type PP_EIH,~,3 if aloao2 < 0, ao2boo < 0, of type HiEPP~,3 if aloao2 > 0, ao2boo < 0, of type P_EIHP41,3 if aloao2 > 0, ao2boo > 0 and of type P_H]Ep_I,3 if aloao2 < 0, ao2boo > O. The case alo = 0 may be obtained by a limiting process; at 0 = n / 2 , q~o ~ n/2, at 0 = 3rt/2, ~Po--" -n/2, and the elliptic sector shrinks to a point. For ao2boo < 0 the singularity is of type /5_P4~,3 and for ao2boo > 0 of type _PP),3 • The results in Section 4.4 are collected in Fig. 15. REFERENCES 1. REYN J. W., A Bibliography o f the Qualitative Theory o f Quadratic Systems o f Differential Equations in the Plane, 3rd edn, Report 94-02. Faculty of Technical Mathematics and Informatics, Delft University of Technology (1994). 2. YE YANQIAN et al., Theory of limit cycles, Trans. Am. math. Soc. 66, (1986). 3. CURTZ P., Stabilit6 locale des syst~mes quadratiques, Annls. scient. Ec. norm. sup. 4e S6rie, 13, 293-302 (1980). 4. CURTZ P., Noeuds des syst~mes quadratiques, C. r. Acad. Sci. Paris, S6rie I, 293, 521-524 (1981) 5. CURTZ P., Points ordinaires ou faibles, points explicites des syst~mes quadratiques, C. r. Acad. Sci. Paris, S6rie I, 298, 321-324 (1984). 6. CURTZ P., Varia sur les syst~mes quadratiques, C. r. Acad. Sci. Paris, S6rie I, 300, 475-480 (1985). 7. SIBIRSKII K. S., Introduction to the Algebraic Theory o f Invariants o f Differential Equations. Manchester University Press, Manchester (1988). 8. LATIPOV H. R., On the behaviour of the characteristics of a differential equation in the large on the equator of the Poincar6 sphere, Izd. A kad. Nauk. Uzbek. SSR Tashkent 110-116 (1963). 9. VOROB'EV A. P., Behaviour of integral curves in the neighbourhood of infinity, Vesci Akad. Navuk. B.S.S.R. Set. Fiz.-Tekn. Navuk 2, 20-30 (1961). 10. GUO WEILIE, The topology of nonhomogeneous differential equations in the neighbourhood of the equator, (Chinese, English abstract), J. Beijing Inst. Aero. Astro. 4, 123-140 (1985). 11. GUO WEILIE, The topological structure near the equator for quadratic systems with one or more finite singular points, (Chinese), Coll. Aero. Astro. Eng. (to appear). 12. HU QINXUN & LU YUGI, On the singular points at infinity of second order planar systems, (Chinese, English abstract), J. Beifing Coll. Tech. 16-28 0985). 13. REN YONGTAI, Infinite singular points of a quadratic system with four singular points in the finite part of the plane, (Chinese, English summary), J. math. Res. Expo. 4(4), 37-42 (1984). 14. WANG DONGDA, The infinite singular points of a quadratic system with three singular points in the finite plane, (Chinese), Math. Practice Theory 3, 27-34 (1985). 15. COLL I VICENS B., Estudi qualitatiu d'algunes classes de camps vectorials al pla, (English, with introduction in Catalan), Thesis Universitat Autbnoma de Barcelona, 5-34 (1987). 16. ANDRONOV A. A., LEONTOVICH E. A., GORDON J. J. & MAIER A. G., Qualitative Theory o f Secondorder Dynamic Systems. Wiley, New York (1973).