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Degenerate Hopf Bifurcations in Discontinuous Planar Systems
Journal of Mathematical Analysis and Applications 253, 671᎐690 Ž2001. doi:10.1006rjmaa.2000.7188, available online at http:rrwww.idealibrary.com on
Degenerate Hopf Bifurcations in Discontinuous Planar Systems B. Coll Departament de Matematiques i Informatica, Facultat de Ciencies, Uni¨ ersitat de les ` ` ` Illes Balears, 7071 Palma de Mallorca, Spain
A. Gasull 1 Departament de Matematiques, Edifici Cc, Uni¨ ersitat Autonoma de Barcelona, ` ` 08193 Bellaterra, Barcelona, Spain E-mail: [email protected]
and R. Prohens Departament de Matematiques i Informatica, Facultat de Ciencies, Uni¨ ersitat de les ` ` ` Illes Balears, 7071 Palma de Mallorca, Spain Submitted by U. Kirchgraber Received December 2, 1998
We study the stability of a singular point for planar discontinuous differential equations with a line of discontinuities. This is done, for the most generic cases, by computing some kind of Lyapunov constants. Our computations are based on the so called Ž R, , p, q .-generalized polar coordinates, introduced by Lyapunov, and they are essentially different from the ones used in the smooth case. These Lyapunov constants are also used to generate limit cycles for some concrete examples. 䊚 2001 Academic Press
1. INTRODUCTION In smooth planar differential equations the stability of a nondegenerate critical point with complex eigenvalues is reduced to the computation of the so called Lyapuno¨ constants; see w3x, for instance. Furthermore these 1
Partially supported by DGICYT Grant PB96-1153. 671 0022-247Xr01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
672
COLL, GASULL, AND PROHENS
constants give information about the cyclicity of the point, that is, on the maximum number of small periodic orbits which appear from this critical point via degenerate Hopf bifurcation. The aim of this paper is to obtain some kind of Lyapunov constants for nondegenerate singular points of planar differential equations with a line of discontinuities L, which we will assume is L s Ž x, y ., y s 04 , for sake of simplicity. These type of differential equations appear frequently in applications; see, for instance, w3, 16x. Our first idea was to develop similar methods to the ones used in w2, pp. 449᎐254; 5, 11x for the smooth case. In order to point out the major difficulties that we have found in this extension to the discontinuous case and to explain how we have overcome them, we will give some preliminaries. We consider equations of type
Ž ˙x, ˙y . s
½
Ž Xq Ž x, y . , Yq Ž x, y . . Ž Xy Ž x, y . , Yy Ž x, y . .
if y G 0, if y F 0,
Ž 1.
where Ž Xq, Yq . Žresp. Ž Xy, Yy .. is called the component equation of Ž1. in the upper Žresp. lower. half plane, being X " and Y " real analytical functions in a neighbourhood of Ž0, 0.. We assume that both vector fields can be smoothly extended to an open neighbourhood of the half plane where they are defined. The main property of the flow of a smooth system near a nondegenerate critical point, p, of focus type is that it turns around the point. For systems of type Ž1. the role of such points is taken by four types of singular points that, for short, we will call pseudo-focus and which we will describe in the sequel just assuming the case in which the flow turns around p counterclockwise. See also w10, Chap. 4x. Ži. Points of focus᎐focus type at p g L: both systems Ž X ", Y " . have a critical point at p with complex eigenvalues and their solutions turn around p counterclockwise. Žii., Žiii. Points of focus᎐parabolic Žresp. parabolic᎐focus. type at p g L: the system defined in the upper Žresp. lower. half plane has a critical point of focus type at p while the solutions of the system defined in a neighbourhood of the lower Žresp. upper. half plane have a parabolic contact Ži.e., a second order contact point. with L at p, the solution of which at this contact is locally contained in the upper Žresp. lower. plane. See Fig. 1. Živ. Points of parabolic᎐parabolic type at p g L: the solutions of both systems have a parabolic contact at p with L in such a way that the flow induced by Ž1. turns around p.
BIFURCATIONS IN DISCONTINUOUS SYSTEMS
673
FIG. 1. A singular point of focus᎐parabolic type.
Observe that the parabolic᎐focus type can be reduced to the focus᎐ parabolic case by applying the change of coordinates Ž x, y, t . ª Žyx, yy, t . to Eq. Ž1.. Hence, for short, just the first case will be studied. We point out that the singularity of parabolic᎐parabolic type is also known as a sewed focus Žsee w10, p. 234x.. The main idea for determining the stability of a singular point p Žwhich we will assume is Ž0, 0., again for the sake of simplicity. of Eq. Ž1. consists in starting from Ž r 0 , 0., r 0 ) 0, r 0 small enough, and to evaluate the sign of the first nonzero term of hyŽ hqŽ r 0 .. y r 0 in its r 0-power series expansion, where hq Žresp. hy. is the Poincare ´ return map associated with the flow of the corresponding component equation of Ž1. in y G 04 Žresp. y F 04. defined between both sides of L _ p. When, in the half plane that we study, the point p is of focus type, the ideas are the same as the ones used in the smooth case. The only difficulty appears from the length of the expressions involved. These expression are even larger than in the smooth case. In the smooth case some cancellations related to considering the flow giving a complete turn around the origin appear. In discontinuous planar systems, these cancellations do not appear because we consider just half turns. In any case the maps hq and hy defined above are analytic.
674
COLL, GASULL, AND PROHENS
The case in which the flow has a parabolic contact at p presents more difficulties. If we try to repeat the same ideas as in the focus case, that is, to write the equation in polar coordinates and to study the maps hq and hy, it is not clear if these maps are analytic Žsee Fig. 2 and Section 5.. In order to avoid this problem we use quasihomogeneous polar coordinates with suitable weights; see w6, 15x. By using these new coordinates we can assure that the corresponding return maps are also analytic and a similar treatment to the focus case can be done for this new case. See again Fig. 2. In any case the computations involved are not the standard ones and our approach is essentially different from that of w10, Chap. 4, Sect. 19.4x. At this point and because of the above considerations, it turns out that for a critical point of pseudo-focus type either hyŽ hqŽ r 0 .. y r 0 ' 0 or that there exists a k such that hyŽ hqŽ r 0 .. y r 0 s Vk r 0k q O Ž r 0kq 1 .. In the first case the origin of Ž1. is a center; in the second case it is said that the origin is a weak focus of order k and that its kth Lyapuno¨ constant is Vk . Observe that when an expression for Vk is given, it makes sense only when V1 s V2 s ⭈⭈⭈ s Vky1 s 0. In the case of smooth differential equations, it is well known that this value k with Vk / 0 Žif it exists. is an odd number; see w2, p. 243x. In the case of discontinuous differential equations k does not have to be necessarily odd as will be proved in next results. The main goal of this paper is to obtain the general expressions for the first three Lyapunov constants for a critical point of pseudo-focus type for a differential equation of type Ž1..
FIG. 2. Analyticity of the return map for parabolic contacts.
BIFURCATIONS IN DISCONTINUOUS SYSTEMS
675
2. DEFINITIONS AND MAIN RESULTS Taking complex coordinates z s x q iy, Eq. Ž1. is written as
¡F
˙z s~
¢
q
Ž z, z . s aqq
Fy Ž z, z . s ayq
⬁
Ý Fkq Ž z, z . ks1 ⬁
Ý
if Im Ž z . G 0,
Ž 2. Fky Ž z, z .
if Im Ž z . F 0,
ks1
where Fk" is a complex homogeneous polynomial of degree k in the variables z and z, Fk" Ž z, z . s
Ý
l m f l" mz z ,
Ž 3.
lqmsk
and a "g ⺢. For system Ž1. we put X " Ž x, y . s a "q b "x q c "y q d "x 2 q e "xy q f "y 2 q g "x 3 q ⭈⭈⭈ , Y " Ž x, y . s l "x q m "y q n "x 2 q o "xy q p "y 2 q q "x 3 q r "x 2 y q s "xy 2 q t "y 3 q u "x 4 q ⭈⭈⭈ ,
Ž 4.
the dots being the remaining terms of X " or Y " in their power series expansion at Ž0, 0.. In accordance with previous notation and with the definition of critical points of pseudo-focus type, we will define the following types of component equations for Eq. Ž1.. DEFINITION 1. The component equation Ž Xq, Yq . Žresp. Ž Xy, Yy .. of Eq. Ž1. is of focus type ŽF. if aqs 0 Žresp. ays 0., Ž bqy mq . 2 q 4 lq cq- 0 Žresp. Ž byy my . 2 q 4 ly cy- 0., and lq) 0 Žresp. ly) 0.. Observe that if complex coordinates are used then we have that the component equation of Ž1., Ž2. defined in ImŽ z . G 0 Žresp. ImŽ z . F 0. is of type ŽF. if a "s 0,
2
f 01" f 01" y Ž Im Ž f 10" . . - 0,
and
Im Ž f 10" q f 01" . ) 0.
Ž 5. DEFINITION 2. The component equation Ž Xq, Yq . Žresp. Ž Xy, Yy .. of Eq. Ž1. is of parabolic type ŽP. if aq- 0 Žresp. ay) 0. and l ") 0. Note that, in complex coordinates, this means that the component equation of Ž1., Ž2. defined in ImŽ z . G 0 Žresp. ImŽ z . F 0. is of type ŽP. if aq- 0 Žresp. ay) 0. and ImŽ f 10" q f 01" . ) 0. In agreement with these definitions we have the following main results.
676
COLL, GASULL, AND PROHENS
Let us consider Eq. Ž2., where Fk" is gi¨ en by Ž3..
PROPOSITION A. Define
¡ Re Ž f . ¢'ŽIm Ž f . . y f
[ exp~
"
" 10
2
" 10
" " 01 f 01
¦¥ §.
Ž 6.
Its first Lyapuno¨ quantity at the origin, V1 , is as follows. If Ž0, 0. is a singularity of Ža. Žb. Žc.
focus᎐focus type, then V1 s q yy 1, focus᎐parabolic type, then V1 s qy 1, parabolic᎐parabolic type, then V1 s 0.
The relations between the coefficients of Eqs. Ž1. and Ž2., involved in former proposition, with " are the following: f 10" s Ž b "q m "qŽ l "y c ". i .r2, f 01" s Ž b "y m "qŽ l "q c " . i .r2, and
"s exp Ž b "q m " .
½
'Ž y1. ž Ž b y m "
" 2
. q 4 l "c " / .
5
To go further in the calculations of the Lyapunov constants we make the following assumption. If in, Eq. Ž2., we have any component equation of type ŽF., then we suppose that F1Ž z, z . s Ž i q . z, with g ⺢, for this component equation. In fact what we are assuming is that this focus is written in its real Jordan form. This is a restriction but, as we will see, in this case the computations are tedious enough. THEOREM B. Let us consider Eq. Ž1. or, equi¨ alently Ž2., where Fk" is defined like in Ž3.. Assume that F1q Ž z, z . s Ž i q q . z Ž resp. F1y Ž z, z . s Ž i q y . z . with "g ⺢, when the component equation ˙ z s Fq Ž z, z . Ž resp. yŽ ˙z s F z, z .. in Ž2. is of type ŽF.. Then its first Lyapuno¨ constants at the origin are the following. Ža.
If Ž0, 0. is a singularity of focus᎐focus type, then V1 s e Ž
Žb.
q
q y.
V2 s ey
q
V3 s ey
q
y 1,
y 2 wq 2 Ž . q w2 Ž . e y2 wq 3 Ž . y 2 e
q
q
,
Ž wq2 Ž . .
2
3 q wy 3 Ž . e
q
.
If Ž0, 0. is a singularity of focus᎐parabolic type, then
V1 s e
q
y 1,
y V2 s wq 2 Ž . y 1 ,
2
y V3 s wq 3 Ž . y Ž 1 . .
677
BIFURCATIONS IN DISCONTINUOUS SYSTEMS
Žc.
If Ž0, 0. is a singularity of parabolic᎐parabolic type, then y V2 s q 1 y 1 ,
V1 s 0,
V3 s 0.
where w 2"Ž ., w 3"Ž ., 1" , and " 2 are gi¨ en by the expressions w 2" Ž . s e
"
Ž ye
"
w 3" Ž . s e
"
Ž e2
"
y 1 . Re Ž 1 q " i . ␣ " ,
y 1.
= Re Ž 1 q " i .  "q " ␥ " y Im Ž 1 q " i . ␦ " q ey
"
Ž w 2" Ž . .
2
4
,
with f 20"
␣ "s "  "s ␥ "s "
␦ s
i q " f 30"
2 Ž i q " .
q
f 11" yi q " f 21" 2 "
q
2
4 Ž i q " .
q
Ž f 02" .
f 20" f 11" 2 "
q
q
q
, f 03" 2 Ž y2 i q " .
f 11" f 20" q f 02" f 11" 2 Ž yi q " .
2 f 20" f 02" q Ž f 11" . 4 Ž yi q " .
q
2
q
,
f 02" f 20" 2 Ž y2 i q " .
,
f 11" f 02" 2 Ž y2 i q " .
2
4 Ž y3i q " .
,
2 a "n "y Ž b "q m " . l " 3
y3i q "
2 Ž yi q " .
4"
Ž f 20" .
f 02"
q
f 12"
f 20" f 20" q f 11" f 11" q f 02" f 02"
q
1" s
q
a" l "
.
Remark. The techniques used to prove Theorem B can also be used to obtain Vk for k G 4. In fact, to develop Example C, we compute V4 and V5 , for a particular family of focus᎐focus type. When either q or y is zero or both coincide, for a singularity of focus᎐focus type, the expression for the Lyapunov constants are shorter than in the general case. As an illustration we give V1 , V2 , and V3 in next remark, for the case qs y.
678
COLL, GASULL, AND PROHENS
Remark. Assume that in Theorem B we have qs ys . Then, if Ž0, 0. is a singularity of focus᎐focus type, V1 s e 2 y 1, V2 s 2 ImŽyfq 20 q 1 q 1 y y y q y. q q Ž .. Ž Ž Ž fq q f y yf q f q f , and V s Re f q f y Im f 11 20 11 3 21 21 20 f 11 3 02 3 02 y y .. q f 20 f 11 . Observe that if Eq. Ž1. comes from a smooth vector field, that is, Ž Xq, Yq . s Ž Xy, Yy . s Ž X, Y ., the above result reproduces the well known values of the first three Lyapunov constants for a critical point of focus type, V1 s e 2 y 1, V2 s 0, V3 s 2 ŽReŽ f 21 . y ImŽ f 20 f 11 ... See w7, 9, 13, 14x, for instance. To conclude this section we give in next examples some simple applications of the above theorem. These examples show how to use the above results to generate limit cycles for systems of type Ž1. with singular points of pseudo-focus type. Note that in the smooth case it is known Žsee w2, p. 243x, for instance . that all the even Lyapunov constants associated with a singularity of focus type are zero while that in the non-smooth case it is not true in general. This fact provokes, for instance, that while for quadratic systems the maximum number of limit cycles that bifurcates from the origin is 3 Žusing V1 , V3 , V5 , and V7 ; see w4x., it suffices to use V1 , V2 , V3 , and V4 in system Ž7. to generate also three limit cycles. In fact by a more specific study of system Ž7. we think that six limit cycles can be generated from the origin Žusing V1 , V2 , . . . , V7 .. We remark that although the discontinuous system Ž7. can have twice the number of limit cycles than the smooth case, the free parameters of Ž7. and the ones of the quadratic system ˙ z s Ž i q . z q f 20 z 2 q f 11 zz q f 02 z 2 coincide. EXAMPLE C Žfocus᎐focus case.. Let Eqs. Ž1., Ž2. be given by the expression
˙z s
½
Ž i q . z q f 20 z 2 q f 11 zz q f 02 z 2 ,
if Im Ž z . G 0,
iz,
if Im Ž z . F 0,
Ž 7.
2 15 6y 3x 11 37 where f 20 s y 11 i, f 11 s 12 q 85 z y 4 y q i, f 02 s 48 16 q y y 32 z q 8 2 5 y6 q 3 x q y y 32 z q 8 i, for arbitrary real numbers: , x, y, and z. Then, by choosing - 0, x ) 0, y - 0, and z ) 0 small enough, such that < < < < x < < < y < < < z <, Eq. Ž7. has, at least, four limit cycles bifurcating from the origin.
The fact that ˙ z s 1 q iz q iz and ˙ z s iz induce the same half return map hy in ImŽ z . F 0 allows us to state next result.
679
BIFURCATIONS IN DISCONTINUOUS SYSTEMS
EXAMPLE D Žfocus᎐parabolic case.. Let Eqs. Ž1., Ž2. be given by
˙z s
½
Ž i q . z q f 20 z 2 q f 11 zz q f 02 z 2 ,
if Im Ž z . G 0,
1 q iz q iz,
if Im Ž z . F 0,
where , f 20 , f 11 and f 02 are as in Example C. Then this system has at least four limit cycles bifurcating from the origin. EXAMPLE E Žparabolic᎐parabolic case.. Let Eqs. Ž1., Ž2. be given by the expression
¡y2 q 1 Ž q i . z q 1 Ž y q i . z y i z 2 4 ˙z s~ ¢1 q iz q2 iz,
2
q
i 4
z2,
if Im Ž z . G 0, if Im Ž z . F 0.
Ž 8. Then, by choosing - 0 small enough, Eq. Ž8. has, at least, one limit cycle bifurcating from the origin.
3. PRELIMINARY RESULTS Remember that the Ž R, , p, q .-generalized polar coordinates are x s R CsŽ ., y s R q SnŽ ., where p and q will be fixed afterwards and where CsŽ . and SnŽ . are the solution of the Cauchy problem, p
˙ Ž . s ySn2 py1 Ž . , Cs 2q
Cs Ž 0 . s
(
1 p
˙Sn Ž . s Cs 2 qy1 Ž . , ,
Sn Ž 0 . s 0;
see w6, 8, 15x. By using the Ž R, , p, q .-generalized polar coordinates, each component of Eq. Ž2. is converted into dR d s
Re Ž Ž z q z .
ž
2 qy1
p
q Ž y1 . 2 2Ž qyp. Ž z y z .
2 py1
. ˙z / r Ž 2 2 qy1 R 2 p qy1 .
Im Ž Ž Ž p y q . z q Ž p q q . z . ˙ z . r Ž 2 R pq q .
,
Ž 9.
680
COLL, GASULL, AND PROHENS
where this expression is evaluated on z s R p CsŽ . q iR q SnŽ .. Of course, this change of variables may not make sense for arbitrary values of Ž R, . and p and q. Our first purpose, when we study the focus and the parabolic case, is to find values for p and q for those former change of variables makes sense, i.e., those for which Eq. Ž9. can be written Žin both cases., in ImŽ z . G 0, as an equation of the type dR d
s
q⬁
Ý Tk Ž . R k ,
Ž 10 .
ks1
defined for Ž R, . in the set w0, ␣ . = w0, 2 x, for some real value ␣ ) 0. Once we get Eq. Ž2. written as Eq. Ž10., we will evaluate its solution, RŽ ; s ., RŽ ; s. y s s
q⬁
Ý wk Ž . s k ,
with w k Ž 0 . s 0
for all k G 1 Ž 11 .
ks1
Ži.e., the solution of Ž10. such that when s 0 takes the value s . when s . Hence, by defining hqŽ s . s RŽ ; s . as a ‘‘return’’ function between w0, ␣ . = 04 and w0, ␣ . = 4 , the behaviour of the solution Ž11. near R ' 0 is controlled by the values w k Ž ., k G 1. The next two results, proved in w12x, are given to simplify as much as possible the expressions that allows us to compute functions w k Ž . that appear in Ž11..
LEMMA 1. Ži. The change of ¨ ariables r s ReyH 0 T1Ž . d transforms differential Eq. Ž10. into the differential equation dr d
s
q⬁
Ý Rk Ž . r k ,
Ž 12 .
ks2
where R k Ž . s e Ž ky1.H0 T1Ž . d Tk Ž .. Žii. Let us consider the solution of Eq. Ž12. written as rŽ ; . s q
q⬁
Ý uk Ž . k ,
with u k Ž 0 . s 0
for all k G 2. Ž 13 .
ks2
Then, RŽ ; s. s s q
q⬁
0
Ý wk Ž . s k s e H T Ž . d
ks1
1
sq
q⬁
Ý uk Ž . s k
ks2
is the solution of Ž10., satisfying RŽ0; s . s s; i.e., w 1Ž . s e H0 T1Ž . d y 1 and w k Ž . s e H0 T1Ž . d u k Ž ., for all k G 2.
681
BIFURCATIONS IN DISCONTINUOUS SYSTEMS
The next result, inspired by w2x, allows us to compute the first values of u k Ž . in Ž13.. In the sequel we use the notation f˜s f˜Ž . s H0 f Ž s . ds. PROPOSITION 2. Gi¨ en Eq. Ž12., the functions u i Ž ., i s 2, . . . , 5 in¨ ol¨ ed in its solution, Ž13., are: u 2 Ž . s R˜2 , u 3 Ž . s Ž R˜2 . 2 q R˜3 , u 4 Ž& . s Ž R˜2 . 3 &
2
q 2 R˜2 R˜3 q R˜2 R 3 q R˜4 , and u 5 Ž . s Ž R˜2 . 4 q 3Ž R˜2 . 2 R˜3 q Ž Ž R˜2 . R 3. q &
&
2 R˜2 ŽR˜2 R 3. q 32 Ž R˜3 . 2 q 2 R˜2 R˜4 q 2ŽR 4 R˜2. q R˜5 . 4. THE FOCUS CASE Consider Eq. Ž2.. Let us assume that the component equation in ImŽ z . G 0 is of type ŽF.; that is to say, aqs 0. We will take ordinary Ž R, .-polar coordinates, i.e., p s q s 1, in Ž11.. Hence, if we neglect the plus sign, Eq. Ž9. becomes dR d
s
Re Ž zz˙. rR Im Ž zz˙. rR 2
zsRe i
s
k Ýq⬁ ks 1 Re Ž S k Ž . . R ky1 Ýq⬁ ks 1 Im Ž S k Ž . . R
,
Ž 14 .
where Sk Ž . s zFk Ž z, z .< zse i s eyi Fk Ž e i , eyi .. From the expression of S1Ž . s f 10 q f 01 ey2 i and Ž5. we get that ImŽ S1Ž .. ) 0, which implies that Eq. Ž14. can be written as in Ž10.. In order to evaluate solution Ž11. of Eq. Ž14. when s , we will prove next proposition. PROPOSITION 3. Let the component equation in ImŽ z . G 0 of Ž2. be written in polar coordinates as Ž14.. Let RŽ ; s . be its solution gi¨ en by Ž11.. Then, R Ž ; s . y s s w1Ž . s q O Ž s 2 . , where w 1 Ž . s exp
½'
Re Ž f 10 .
Ž Im Ž f 10 . .
2
y f 01 f 01
5
y 1.
Moreo¨ er, if F1Ž z, z . s Ž i q . z then R Ž , s . y s s w1Ž . s q w 2 Ž . s 2 q w 3 Ž . s 3 q O Ž s 4 . , where w 1Ž . s e y 1, w 2 Ž . s e Žye y 1.RewŽ1 q i . ␣ x, and w 3 Ž . s e Ž e 2 y 1. RewŽ1 q i .  q ␥ x y ImwŽ1 q i . ␦ x4 q ey Ž w 2 Ž .. 2 .
682
COLL, GASULL, AND PROHENS
Proof. First we compute w 1Ž .. Consider the expression Ž14.. Direct substitution shows that w 1Ž . satisfies wX1 Ž . s
Re Ž S1 Ž . . Im Ž S1 Ž . .
Ž w1Ž . q 1. ,
w 1 Ž 0 . s 0.
Then w 1 Ž . s exp
H0
Re Ž f 10 . q Re Ž f 01 . cos 2 q Im Ž f 01 . sin 2 Im Ž f 10 . y Re Ž f 01 . sin 2 q Im Ž f 01 . cos 2
d y 1.
Standard integration techniques give that
H0
Re Ž f 10 . q Re Ž f 01 . cos 2 q Im Ž f 01 . sin 2 Im Ž f 10 . y Re Ž f 01 . sin 2 q Im Ž f 01 . cos 2
d
1 Im Ž f 10 . y Re Ž f 01 . sin 2 q Im Ž f 01 . cos 2 s y ln 2 Im Ž f 10 . q Im Ž f 01 . q
Re Ž f 10 .
'ŽImŽ f
2
10
. . y f 01 f 01
'ŽImŽ f
2
. . y f 01 f 01 tan = arctan q fŽ . , Im Ž f 10 . q Im Ž f 01 . y Re Ž f 01 . tan
ž
10
/
where f Ž . is equal either to 0, if 0 F F r2, or to , if r2 F F . Hence, the expression of w 1Ž . follows. Now we continue computing w 2 Ž . and w 3 Ž . but assuming that f 10 s q i, f 01 s 0. Observe that in this case w 1Ž . s e y 1. Note also that Eq. Ž14. can be written as Ž10. by the expression dR d
s
k R q Ýq⬁ ks 2 R Re Ž S k Ž . .
1q
Ýq⬁ ks 2
R
ky1
Im Ž Sk Ž . .
s
q⬁
Ý Tk Ž . R k ,
ks1
where T1 s , T2 s Re S2 y Im S2 , and T3 s Re S3 y Re S2 Im S2 y Im S3 q ŽIm S2 . 2 . By using Lemma 1Ži. the above equation is converted into dr d where R k s e Ž ky1. Tk , k G 2.
s
q⬁
Ý Rk Ž . r k ,
ks2
683
BIFURCATIONS IN DISCONTINUOUS SYSTEMS
Therefore, from Proposition 2, when we consider its solution in the form Ž13., we must compute
u 2 Ž . s R˜2 Ž . s Re
H0 S Ž . e
2
2
u 3 Ž . s Ž u 2 Ž . . q Re
d y Im
H0
H0 S Ž . e
S3 Ž . q
H0
= e 2 d y Im
2
S3 Ž . q
2
d ,
Ž yS22 Ž . q S2 Ž . S2 Ž . . 1 2
S22 Ž . e 2 d .
Since by Lemma 1Žii., w 2 Ž . s e u 2 Ž . and w 3 Ž . s e u 3 Ž ., direct computations give the final expression of w 2 Ž . and w 3 Ž . in terms of the coefficients of F2 and F3 . As an example we will explain how to calculate u 2 Ž .. From the expression of S2 Ž . s f 20 e i q f 11 eyi q f 02 ey3 i and previous expression of u 2 Ž ., we have u 2 Ž . s Re
ž
ž
H0 S Ž . e 2
s Re Ž 1 q i .
d q i
f 20 e Ž iq . iq
H0 S Ž . e
q
2
f 11 e Žyiq . yi q
d q
/ f 02 e Žy3 iq . y3i q
0
/
.
5. THE PARABOLIC CASE Consider Eq. Ž2.. Let us assume that the component equation in ImŽ z . G 0 of Ž2. is of type ŽP.. Hence, aq/ 0. In the next lemma we argue on the best selection of Ž R, , p, q .-generalized polar coordinates for this case. LEMMA 4. Let the component equation of Ž2. in ImŽ z . G 0 be of type ŽP.. Then, the R-power series expansion of Ž9. is well defined if and only if q s 2 p.
684
COLL, GASULL, AND PROHENS
Proof. If aq/ 0, then the expression Ž9. is written as dR d s
q 2 py1 aq Cs 2 qy1 Ž . R qq 1 q Im Ž fq Ž . Cs Ž . R 2 pq1 q ⭈⭈⭈ 10 q f 01 . Sn q 2 2p yaq q Sn Ž . R q q p Im Ž fq q ⭈⭈⭈ 10 q f 01 . Cs Ž . R
,
where the dots indicate the rest of the terms in the R-power series expansion. We note that all of these terms have order greater than those that appear. Consequently, p and q can be neither q - 2 p nor q ) 2 p, because in these cases the former expression is written as dR d dR d
s s
Cs 2 qy1 Ž . yq Sn Ž . Sn2 py1 Ž . p Cs Ž .
R q O Ž R2 . ,
if q - 2 p,
R q O Ž R2 . ,
if q ) 2 p.
In the case q s 2 p we have dR d
s
q 2 py1 aq Cs 2 qy1 Ž . q Im Ž fq Ž . Cs Ž . 10 q f 01 . Sn q 2 p Ž y2 aq Sn Ž . q Im Ž fq 10 q f 01 . Cs Ž . .
R q O Ž R2 . .
Ž 15 . q. From the fact that aq- 0 and ImŽ fq 10 q f 01 ) 0 we conclude that the two q q. q Ž . Ž real roots of y2 a Sn q Im f 10 q f 01 Cs 2 Ž . s 0 lie in ImŽ z . - 0, which finishes the proof.
As a consequence of the previous lemma, from now on and for simplicity, in the parabolic case we will take Ž R, , 1, 2.-generalized polar coordinates. The next lemma sums up all the previous considerations to the study of the parabolic case. Also, in next lemma, by using a change of variables we get an equivalent expression of Eq. Ž15., written as Eq. Ž10., with a simplest T1Ž .. LEMMA 5. Let the component equation of Ž1. in y G 04 be of type ŽP.. Write Ž1. in the new ¨ ariables x 1 s x q ml y and y 1 s y2l a y. Then, in
BIFURCATIONS IN DISCONTINUOUS SYSTEMS
685
Ž R, , 1, 2.-generalized polar coordinates and remo¨ ing the q sign, it is written as dR d
s
q⬁
Ý Tk Ž . R k
ks1
s
2 Sn Ž . Cs Ž . y Cs 3 Ž . 2 Ž Cs Ž . q Sn Ž . . 2
=
Cs 2 Ž .
Ž Cs 2 Ž . q Sn Ž . .
2
Rq
an y l Ž b q m . 2 al
R 2 q T3 Ž . R 3 q O Ž R 4 . ,
Ž 16 .
where T3 Ž . s
1 4 a l Ž Cs Ž . q Sn Ž . . 2 2
2
3
= Cs 5 Ž . Ž y2 adl 2 q 2 abln y 2 a2 n2 q 2 a2 lq . qCs 3 Ž . Sn Ž . Ž 2 b 2 l 2 y 2 adl 2 q cl 3 q 3bl 2 m q 2 l 2 m2 y2 abln y 2 almn y al 2 o q 2 a2 lq . qCs Ž . Sn2 Ž . Ž cl 3 y bl 2 m q 2 amln y al 2 o . . Proof. In order to simplify further expressions we apply the change of variables suggested in w10x, which does not change the return map on the x-axis x 1 s x q ml y and y 1 s y2l a y, to the component equation of Ž1. in y G 0. Observe that we obtain
˙x s a q Ž b q m . x q ˙y s y2 ax y
2 an l
bm y cl
x2 q
2a
yq
ol y 2 nm l
dl q nm l
x 2 q ⭈⭈⭈ ,
Ž 17 .
xy q ⭈⭈⭈ ,
if we omit the subscripts ‘‘1.’’ If, in addition, we use the Ž R, , 1, 2.-generalized polar coordinates in Eq. Ž17., and we expand the quotient dR d in a R-power series like Ž10., we obtain the expression Ž16.. Next lemma is a technical result which is a key point for the effective computation of the Lyapunov constants in the parabolic case by using the Ž R, , 1, 2.-generalized polar coordinates.
686
COLL, GASULL, AND PROHENS
LEMMA 6. Let SnŽ . and CsŽ . be the generalized trigonometric functions introduced in Section 3. Then Cs m Ž . Sn l Ž .
H ŽCs
2
Ž . q Sn Ž . .
d
n
¡
l
s
Ý Ž y1. i is0
Ž Cs 2 Ž . q Sn Ž . .
lyiynq1
Ž l y i y n q 1 . Cs 2Ž lyiynq1. Ž .
ž /~
q k,
if l y i y n q 1 / y1,
l i
ln
¢
Cs 2 Ž . q Sn Ž .
q k, Cs 2 Ž . if l y i y n q 1 s y1,
for all l, m, and n positi¨ e integers whene¨ er 2 n s m q 2 l q 3, and where k stands for the arbitrary integration constant. Proof. Applying the change of variables x s SnŽ .rCs 2 Ž . to the integral, we get the equivalent expression
H
Ž1 q 2 x2 .
Ž2 nymy2 ly3 .r4
Ž1 q x.
xl
l
dx s
HŽ 1 q x .
yl
x l dx ,
because of the hypothesis. The calculation of last integral is immediate. Hence the proof is finished. In order to evaluate solution Ž11. of Eq. Ž16. when s , we will prove the next proposition. PROPOSITION 7. Let RŽ ; s . be the solution Ž11. of Eq. Ž1. written as Eq. Ž16.. Then w 1Ž . s 0, w 2 Ž . s 23 an y Ž alb q m . l , and w 3 Ž . s Ž 23 an y Ž alb q m . l . 2 . Proof. Using the properties of the Ž R, , 1, 2.-generalized polar coordinates and the expression of T1Ž . given in Lemma 5, standard integration techniques give
e Ž ky1.H0 T1Ž . d s e Ž ky1.r2 H0
d r d logŽCs 2 Ž .qSnŽ .. d 4
s Ž Cs 2 Ž . q Sn Ž . .
y Ž ky1 .r2
.
687
BIFURCATIONS IN DISCONTINUOUS SYSTEMS
Hence, from Lemma 1Ži., Eq. Ž16. is converted into Eq. Ž12., where R2 Ž . s R3 Ž . s
Cs 2 Ž .
an y Ž b q m . l 2 al
Ž Cs 2 Ž . q Sn Ž . .
T3 Ž . Cs Ž . q Sn Ž . 2
and
5r2
.
Now, the idea is to get the solution of Eq. Ž12. in the form Ž13., by means of Proposition 2. Then, from Lemma 1Žii., and using the fact that e H0 T1Ž . d < s s 1, we will have w k Ž . s u k Ž . for all k s 2, 3, . . . , which will finish the proof. From Proposition 2, we must compute u 2 Ž . s R˜2 Ž . s
an y Ž b q m . l 2 al
Cs 2 Ž .
H0
Ž Cs 2 Ž . q Sn Ž . .
5r2
d .
From Lemma 6, we have that u2 Ž . s
an y Ž b q m . l 3al
ž
1y
Cs 3 Ž . Cs 2 Ž . q Sn Ž .
3r2
/
.
Therefore, u2 Ž . s
2 an y Ž b q m . l 3
al
.
To compute u 3 Ž . we need to calculate R˜3 Ž ., but R 3 Ž . is an even expression on SnŽ . and odd on CsŽ .. Hence, R˜3 Ž . s 0. This gives, from Proposition 2, that u 3 Ž . s Ž R˜2 Ž .. 2 , which finishes the proof. Remark 8. About the contact order of the solution, y s y Ž x ., of Eq. Ž1. which passes through Ž0, 0. and L s Ž x, y ., y s 04 , for a component equation of type ŽP. in y G 0, it is not difficult to see that Ž d 2 yrdx 2 .< Ž0, 0. s lra2 and, consequently, that whenever l / 0, this solution has a second order contact with L. If l s 0 then, taking n s 0 and q ) 0, the flow keeps turning around the origin Ž0, 0.. In this case, the contact order becomes four because Ž d 3 yrdx 3 .< Ž0, 0. s 0 and Ž d 4 yrdx 4 .< Ž0, 0. s 6 qra2 . In this situation it is not possible to get Eq. Ž11. by using the change to Ž R, , p, 2 p .-generalized polar coordinates suggested in Lemma 4, but it is not difficult to prove that the suitable change to Ž R, , p, q .-generalized polar coordinates is obtained by taking q s 4 p.
688
COLL, GASULL, AND PROHENS
6. PROOF OF THE MAIN RESULTS Proof of Proposition A and Theorem B. Let us consider Eq. Ž1., or equivalently Ž2., and let us apply to each component equation the suitable change of Ž R, , p, q .-polar coordinates, according Section 4 and Lemma 4. Following the ideas given in Section 3, to prove our results we will compose the map induced by the flow of ˙ z s FqŽ z, z . between s 0 and q s , h , and the map induced by the flow of ˙ z s Fy Ž z, z . between y s and s 2 , h . Their expansions are given by expression Ž11., with h "Ž s . s R "Ž , s ., and are written as h " Ž s . s Ž w 1" Ž . q 1 . s q w 2" Ž . s 2 q w 3" Ž . s 3 q O Ž s 4 . 2 " 3 4 s h1"s q h " 2 s q h3 s q O Ž s . .
We note that, in this last expression, the map hy coincides with the return map induced by the flow of ˙ z s yFyŽ yz, yz . between s 0 and s . Hence the stability of the origin is controlled by the sign of hyŽ hqŽ s .. y s, where q y q y q hy Ž hq Ž s . . y s s Ž hy 1 h1 y 1 . s q Ž h1 h 2 q h 2 Ž h1 . q y q q y q q Ž hy 1 h 3 q 2 h 2 h1 h 2 q h 3 Ž h1 .
3
2
. s2
. s3 q O Ž s4 .
and the three first Lyapunov constants of Ž2. are q V1 s hy 1 h1 y 1,
V2 s 2
V3 s
y q hq 2 q h 2 Ž h1 .
hq 1
q q y q hq 1 h 3 y 2 Ž h 2 . q h 3 Ž h1 .
Ž hq 1 .
2
3
,
5
.
Let us argue about the sign of hq 1 . In the focus case, using Proposition 3, h1 s e ) 0, because w 1Ž . s e y 1. In the parabolic case, using Proposition 7, h1 s 1 ) 0, because w 1Ž . s 0. Hence, we obtain the values Vi , i s 1, 2, and 3, depending on the singularity type of the origin. Whence, the proof follows. Proof of Example C. Lyapunov constants V1 , V2 , and V3 , of the origin of Ž7., follow from Theorem BŽa. by taking into account, among other facts, Ž . that ys 0 and wy i ' 0 for all i G 2, which simplifies their calculation. Ž . In fact, we have Vi s q y 1, V2 s x, i , for all i G 2. Hence, V1 s e and V3 s y. In order to generate four limit cycles we need to know the fourth and fifth Lyapunov constants, V4 and V5 , of the origin of Eq. Ž7.. Their calculation is only interesting if V1 s V2 s V3 s 0. Hence, we assume s x s y s 0. Let us calculate V4 and V5 . It is not diffi-
689
BIFURCATIONS IN DISCONTINUOUS SYSTEMS
cult, following Lemma 5, to get T4 s ReŽ S2 .ŽImŽ S2 .. 2 and T5 s yReŽ S2 .ŽImŽ S2 .. 3. Now, using the fact that s 0, we have R k s Tk if Ž . k G 2. In addition, from Proposition 2, we get the expression of uq 4 and Ž . Ž . uq . Hence, from Proposition 1 ii , using again that s 0, we have that 5 qŽ . Ž . q s u if k G 2. Following the same ideas exposed in the proof of k k Ž . Ž Theorem B we have V4 s q s z and, in the case V s 0 i.e., if 4 4 55 Ž . z s 0., V5 s q s y . 5 384 Hence, the return map associated to the origin of Eq. Ž7. is written as h Ž s; , x, y, z . s Ž e y 1 . s q Ž x q f 2 Ž , x, y, z . . s 2 q Ž y q f 3 Ž , x, y, z . . s 3 q Ž z q f 4 Ž , x, y, z . . s 4 q y
ž
55 384
q f5 Ž , x, y, z . s 5
/
q O Ž s6 . , for small enough values of the variable s, s ) 0, where f i , i s 2, . . . , 5, are analytic real functions satisfying f 2 Ž0, x, y, z . ' 0, f 3 Ž0, 0, y, z . ' 0, f 4Ž0, 0, 0, z . ' 0, and f5 Ž0, 0, 0, 0. s 0. By choosing , x, y, and z small enough and satisfying the conditions of this example, it is easy to find four values of s for which h vanishes. More concretely, if we consider hŽ s; 0, 0, 0, 0., we can get a value of s, s0 , small enough, for which hŽ s0 ; 0, 0, 0, 0. - 0. Because of the continuity of the function h, it is possible to assure that hŽ s0 ; 0, 0, 0, z . - 0, for small enough values of z. Let z1 ) 0 be one such value of z. In this situation, we have that hŽ s; 0, 0, 0, z1 . s z1 s 4 q O Ž s 5 .. Hence, we can have a value of s, s1 , small enough and smaller than s0 , for which hŽ s1; 0, 0, 0, z1 . ) 0. Whence, between s1 and s0 , we obtain the first change of sign of h. Arguing in a similar way, we can get the four changes of sign of h. See also w5x. Proof of Example E. Lyapunov constants V1 , V2 , and V3 , of the origin of Eq. Ž8. follow straightaway from Theorem BŽc.. V4 needs some more computations. They are V1 s 0, V2 s 3 , V3 s 0, and V4 s 301 . Hence, for the choice of the parameter made in the statement of this lemma and arguing as in the proof of Example C, we can create one unstable small limit cycle around the origin as we wanted to prove.
REFERENCES 1. M. A. M. Alwash and N. G. Lloyd, Nonautonomous equations related to polynomial two-dimensional systems, Proc. Roy. Soc. Edinburgh Sect. A 105 Ž1987., 129᎐152. 2. A. A. Andronov, E. A. Leontovich, I. I. Gordon, and A. G. Maıer, ˘ ‘‘Theory of Bifurcations of Dynamic Systems on a Plane,’’ Halsted, New YorkrToronto, 1973.
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COLL, GASULL, AND PROHENS
` Khaıkin, 3. A. A. Andronov, A. A. Vitt, and S. E. ‘‘Theory of Oscillators,’’ Pergamon, ˘ Oxford, 1996. 4. N. N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Amer. Math. Soc. Transl. 100 Ž1954., 397᎐413. 5. T. R. Blows and N. G. Lloyd, The number of limit cycles of certain polynomial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 98 Ž1984., 215᎐239. 6. H. W. Broer, F. Dumortier, S. J. van Strien, and F. Takens, ‘‘Structures in Dynamics,’’ North-Holland, Amsterdam, 1991. 7. A. Cima, A. Gasull, V. Manosa, and F. Manosa, Algebraic properties of the Liapunov and ˜ ˜ period constants, Rocky Mountain J. Math. 27, No. 2 Ž1997., 471᎐501. 8. B. Coll, A. Gasull, and R. Prohens, Differential equations defined by the sum of two quasi-homogeneous vector fields, Canad. J. Math. 49, No. 2 Ž1997., 212᎐231. 9. W. W. Farr, C. Li, I. S. Labouriau, and W. F. Langford, Degenerate Hopf bifurcation formulas and Hilbert’s 16th problem, SIAM J. Math. Anal. 20 Ž1989., 13᎐30. 10. A. F. Filippov, ‘‘Differential Equations with Discontinuous Righthand Sides,’’ Kluwer Academic, Dordrecht, 1988. 11. A. Gasull, A. Guillamon, and V. Manosa, An explicit expression of the first Liapunov and ˜ period constants with applications, J. Math. Anal. Appl. 211 Ž1997., 190᎐212. 12. A. Gasull and R. Prohens, Effective computation of the first Lyapunov quantities for a planar differential equation, Appl. Math. ŽWarsaw . 24, No. 3 Ž1997., 243᎐250. 13. F. Gobber and K.-D. Williamowski, Ljapunov approach to multiple Hopf bifurcation, J. ¨ Math. Anal. Appl. 71 Ž1979., 333᎐350. 14. B. Hassard and Y. H. Wan, Bifurcation formulae derived from center manifold theory, J. Math. Anal. Appl. 63 Ž1978., 297᎐312. 15. A. M. Liapunov, Stability of motion, vol. 30; With a contribution by V. A. Pliss and an introduction by V. P. Basov. Translated from the Russian by Flavian Abramovici and Michael Shimshoni. Mathematics in Science and Engineering, Academic Press, New York, 1966. 16. World Wide Web pages of ‘‘Analysis of Non-Smooth Dynamical System,’’ www.mi.unikoeln.dermirForschungrKuepperrForschung1.htm.
Degenerate Hopf Bifurcations in Discontinuous Planar Systems B. Coll Departament de Matematiques i Informatica, Facultat de Ciencies, Uni¨ ersitat de les ` ` ` Illes Balears, 7071 Palma de Mallorca, Spain
A. Gasull 1 Departament de Matematiques, Edifici Cc, Uni¨ ersitat Autonoma de Barcelona, ` ` 08193 Bellaterra, Barcelona, Spain E-mail: [email protected]
and R. Prohens Departament de Matematiques i Informatica, Facultat de Ciencies, Uni¨ ersitat de les ` ` ` Illes Balears, 7071 Palma de Mallorca, Spain Submitted by U. Kirchgraber Received December 2, 1998
We study the stability of a singular point for planar discontinuous differential equations with a line of discontinuities. This is done, for the most generic cases, by computing some kind of Lyapunov constants. Our computations are based on the so called Ž R, , p, q .-generalized polar coordinates, introduced by Lyapunov, and they are essentially different from the ones used in the smooth case. These Lyapunov constants are also used to generate limit cycles for some concrete examples. 䊚 2001 Academic Press
1. INTRODUCTION In smooth planar differential equations the stability of a nondegenerate critical point with complex eigenvalues is reduced to the computation of the so called Lyapuno¨ constants; see w3x, for instance. Furthermore these 1
Partially supported by DGICYT Grant PB96-1153. 671 0022-247Xr01 $35.00 Copyright 䊚 2001 by Academic Press All rights of reproduction in any form reserved.
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COLL, GASULL, AND PROHENS
constants give information about the cyclicity of the point, that is, on the maximum number of small periodic orbits which appear from this critical point via degenerate Hopf bifurcation. The aim of this paper is to obtain some kind of Lyapunov constants for nondegenerate singular points of planar differential equations with a line of discontinuities L, which we will assume is L s Ž x, y ., y s 04 , for sake of simplicity. These type of differential equations appear frequently in applications; see, for instance, w3, 16x. Our first idea was to develop similar methods to the ones used in w2, pp. 449᎐254; 5, 11x for the smooth case. In order to point out the major difficulties that we have found in this extension to the discontinuous case and to explain how we have overcome them, we will give some preliminaries. We consider equations of type
Ž ˙x, ˙y . s
½
Ž Xq Ž x, y . , Yq Ž x, y . . Ž Xy Ž x, y . , Yy Ž x, y . .
if y G 0, if y F 0,
Ž 1.
where Ž Xq, Yq . Žresp. Ž Xy, Yy .. is called the component equation of Ž1. in the upper Žresp. lower. half plane, being X " and Y " real analytical functions in a neighbourhood of Ž0, 0.. We assume that both vector fields can be smoothly extended to an open neighbourhood of the half plane where they are defined. The main property of the flow of a smooth system near a nondegenerate critical point, p, of focus type is that it turns around the point. For systems of type Ž1. the role of such points is taken by four types of singular points that, for short, we will call pseudo-focus and which we will describe in the sequel just assuming the case in which the flow turns around p counterclockwise. See also w10, Chap. 4x. Ži. Points of focus᎐focus type at p g L: both systems Ž X ", Y " . have a critical point at p with complex eigenvalues and their solutions turn around p counterclockwise. Žii., Žiii. Points of focus᎐parabolic Žresp. parabolic᎐focus. type at p g L: the system defined in the upper Žresp. lower. half plane has a critical point of focus type at p while the solutions of the system defined in a neighbourhood of the lower Žresp. upper. half plane have a parabolic contact Ži.e., a second order contact point. with L at p, the solution of which at this contact is locally contained in the upper Žresp. lower. plane. See Fig. 1. Živ. Points of parabolic᎐parabolic type at p g L: the solutions of both systems have a parabolic contact at p with L in such a way that the flow induced by Ž1. turns around p.
BIFURCATIONS IN DISCONTINUOUS SYSTEMS
673
FIG. 1. A singular point of focus᎐parabolic type.
Observe that the parabolic᎐focus type can be reduced to the focus᎐ parabolic case by applying the change of coordinates Ž x, y, t . ª Žyx, yy, t . to Eq. Ž1.. Hence, for short, just the first case will be studied. We point out that the singularity of parabolic᎐parabolic type is also known as a sewed focus Žsee w10, p. 234x.. The main idea for determining the stability of a singular point p Žwhich we will assume is Ž0, 0., again for the sake of simplicity. of Eq. Ž1. consists in starting from Ž r 0 , 0., r 0 ) 0, r 0 small enough, and to evaluate the sign of the first nonzero term of hyŽ hqŽ r 0 .. y r 0 in its r 0-power series expansion, where hq Žresp. hy. is the Poincare ´ return map associated with the flow of the corresponding component equation of Ž1. in y G 04 Žresp. y F 04. defined between both sides of L _ p. When, in the half plane that we study, the point p is of focus type, the ideas are the same as the ones used in the smooth case. The only difficulty appears from the length of the expressions involved. These expression are even larger than in the smooth case. In the smooth case some cancellations related to considering the flow giving a complete turn around the origin appear. In discontinuous planar systems, these cancellations do not appear because we consider just half turns. In any case the maps hq and hy defined above are analytic.
674
COLL, GASULL, AND PROHENS
The case in which the flow has a parabolic contact at p presents more difficulties. If we try to repeat the same ideas as in the focus case, that is, to write the equation in polar coordinates and to study the maps hq and hy, it is not clear if these maps are analytic Žsee Fig. 2 and Section 5.. In order to avoid this problem we use quasihomogeneous polar coordinates with suitable weights; see w6, 15x. By using these new coordinates we can assure that the corresponding return maps are also analytic and a similar treatment to the focus case can be done for this new case. See again Fig. 2. In any case the computations involved are not the standard ones and our approach is essentially different from that of w10, Chap. 4, Sect. 19.4x. At this point and because of the above considerations, it turns out that for a critical point of pseudo-focus type either hyŽ hqŽ r 0 .. y r 0 ' 0 or that there exists a k such that hyŽ hqŽ r 0 .. y r 0 s Vk r 0k q O Ž r 0kq 1 .. In the first case the origin of Ž1. is a center; in the second case it is said that the origin is a weak focus of order k and that its kth Lyapuno¨ constant is Vk . Observe that when an expression for Vk is given, it makes sense only when V1 s V2 s ⭈⭈⭈ s Vky1 s 0. In the case of smooth differential equations, it is well known that this value k with Vk / 0 Žif it exists. is an odd number; see w2, p. 243x. In the case of discontinuous differential equations k does not have to be necessarily odd as will be proved in next results. The main goal of this paper is to obtain the general expressions for the first three Lyapunov constants for a critical point of pseudo-focus type for a differential equation of type Ž1..
FIG. 2. Analyticity of the return map for parabolic contacts.
BIFURCATIONS IN DISCONTINUOUS SYSTEMS
675
2. DEFINITIONS AND MAIN RESULTS Taking complex coordinates z s x q iy, Eq. Ž1. is written as
¡F
˙z s~
¢
q
Ž z, z . s aqq
Fy Ž z, z . s ayq
⬁
Ý Fkq Ž z, z . ks1 ⬁
Ý
if Im Ž z . G 0,
Ž 2. Fky Ž z, z .
if Im Ž z . F 0,
ks1
where Fk" is a complex homogeneous polynomial of degree k in the variables z and z, Fk" Ž z, z . s
Ý
l m f l" mz z ,
Ž 3.
lqmsk
and a "g ⺢. For system Ž1. we put X " Ž x, y . s a "q b "x q c "y q d "x 2 q e "xy q f "y 2 q g "x 3 q ⭈⭈⭈ , Y " Ž x, y . s l "x q m "y q n "x 2 q o "xy q p "y 2 q q "x 3 q r "x 2 y q s "xy 2 q t "y 3 q u "x 4 q ⭈⭈⭈ ,
Ž 4.
the dots being the remaining terms of X " or Y " in their power series expansion at Ž0, 0.. In accordance with previous notation and with the definition of critical points of pseudo-focus type, we will define the following types of component equations for Eq. Ž1.. DEFINITION 1. The component equation Ž Xq, Yq . Žresp. Ž Xy, Yy .. of Eq. Ž1. is of focus type ŽF. if aqs 0 Žresp. ays 0., Ž bqy mq . 2 q 4 lq cq- 0 Žresp. Ž byy my . 2 q 4 ly cy- 0., and lq) 0 Žresp. ly) 0.. Observe that if complex coordinates are used then we have that the component equation of Ž1., Ž2. defined in ImŽ z . G 0 Žresp. ImŽ z . F 0. is of type ŽF. if a "s 0,
2
f 01" f 01" y Ž Im Ž f 10" . . - 0,
and
Im Ž f 10" q f 01" . ) 0.
Ž 5. DEFINITION 2. The component equation Ž Xq, Yq . Žresp. Ž Xy, Yy .. of Eq. Ž1. is of parabolic type ŽP. if aq- 0 Žresp. ay) 0. and l ") 0. Note that, in complex coordinates, this means that the component equation of Ž1., Ž2. defined in ImŽ z . G 0 Žresp. ImŽ z . F 0. is of type ŽP. if aq- 0 Žresp. ay) 0. and ImŽ f 10" q f 01" . ) 0. In agreement with these definitions we have the following main results.
676
COLL, GASULL, AND PROHENS
Let us consider Eq. Ž2., where Fk" is gi¨ en by Ž3..
PROPOSITION A. Define
¡ Re Ž f . ¢'ŽIm Ž f . . y f
[ exp~
"
" 10
2
" 10
" " 01 f 01
¦¥ §.
Ž 6.
Its first Lyapuno¨ quantity at the origin, V1 , is as follows. If Ž0, 0. is a singularity of Ža. Žb. Žc.
focus᎐focus type, then V1 s q yy 1, focus᎐parabolic type, then V1 s qy 1, parabolic᎐parabolic type, then V1 s 0.
The relations between the coefficients of Eqs. Ž1. and Ž2., involved in former proposition, with " are the following: f 10" s Ž b "q m "qŽ l "y c ". i .r2, f 01" s Ž b "y m "qŽ l "q c " . i .r2, and
"s exp Ž b "q m " .
½
'Ž y1. ž Ž b y m "
" 2
. q 4 l "c " / .
5
To go further in the calculations of the Lyapunov constants we make the following assumption. If in, Eq. Ž2., we have any component equation of type ŽF., then we suppose that F1Ž z, z . s Ž i q . z, with g ⺢, for this component equation. In fact what we are assuming is that this focus is written in its real Jordan form. This is a restriction but, as we will see, in this case the computations are tedious enough. THEOREM B. Let us consider Eq. Ž1. or, equi¨ alently Ž2., where Fk" is defined like in Ž3.. Assume that F1q Ž z, z . s Ž i q q . z Ž resp. F1y Ž z, z . s Ž i q y . z . with "g ⺢, when the component equation ˙ z s Fq Ž z, z . Ž resp. yŽ ˙z s F z, z .. in Ž2. is of type ŽF.. Then its first Lyapuno¨ constants at the origin are the following. Ža.
If Ž0, 0. is a singularity of focus᎐focus type, then V1 s e Ž
Žb.
q
q y.
V2 s ey
q
V3 s ey
q
y 1,
y 2 wq 2 Ž . q w2 Ž . e y2 wq 3 Ž . y 2 e
q
q
,
Ž wq2 Ž . .
2
3 q wy 3 Ž . e
q
.
If Ž0, 0. is a singularity of focus᎐parabolic type, then
V1 s e
q
y 1,
y V2 s wq 2 Ž . y 1 ,
2
y V3 s wq 3 Ž . y Ž 1 . .
677
BIFURCATIONS IN DISCONTINUOUS SYSTEMS
Žc.
If Ž0, 0. is a singularity of parabolic᎐parabolic type, then y V2 s q 1 y 1 ,
V1 s 0,
V3 s 0.
where w 2"Ž ., w 3"Ž ., 1" , and " 2 are gi¨ en by the expressions w 2" Ž . s e
"
Ž ye
"
w 3" Ž . s e
"
Ž e2
"
y 1 . Re Ž 1 q " i . ␣ " ,
y 1.
= Re Ž 1 q " i .  "q " ␥ " y Im Ž 1 q " i . ␦ " q ey
"
Ž w 2" Ž . .
2
4
,
with f 20"
␣ "s "  "s ␥ "s "
␦ s
i q " f 30"
2 Ž i q " .
q
f 11" yi q " f 21" 2 "
q
2
4 Ž i q " .
q
Ž f 02" .
f 20" f 11" 2 "
q
q
q
, f 03" 2 Ž y2 i q " .
f 11" f 20" q f 02" f 11" 2 Ž yi q " .
2 f 20" f 02" q Ž f 11" . 4 Ž yi q " .
q
2
q
,
f 02" f 20" 2 Ž y2 i q " .
,
f 11" f 02" 2 Ž y2 i q " .
2
4 Ž y3i q " .
,
2 a "n "y Ž b "q m " . l " 3
y3i q "
2 Ž yi q " .
4"
Ž f 20" .
f 02"
q
f 12"
f 20" f 20" q f 11" f 11" q f 02" f 02"
q
1" s
q
a" l "
.
Remark. The techniques used to prove Theorem B can also be used to obtain Vk for k G 4. In fact, to develop Example C, we compute V4 and V5 , for a particular family of focus᎐focus type. When either q or y is zero or both coincide, for a singularity of focus᎐focus type, the expression for the Lyapunov constants are shorter than in the general case. As an illustration we give V1 , V2 , and V3 in next remark, for the case qs y.
678
COLL, GASULL, AND PROHENS
Remark. Assume that in Theorem B we have qs ys . Then, if Ž0, 0. is a singularity of focus᎐focus type, V1 s e 2 y 1, V2 s 2 ImŽyfq 20 q 1 q 1 y y y q y. q q Ž .. Ž Ž Ž fq q f y yf q f q f , and V s Re f q f y Im f 11 20 11 3 21 21 20 f 11 3 02 3 02 y y .. q f 20 f 11 . Observe that if Eq. Ž1. comes from a smooth vector field, that is, Ž Xq, Yq . s Ž Xy, Yy . s Ž X, Y ., the above result reproduces the well known values of the first three Lyapunov constants for a critical point of focus type, V1 s e 2 y 1, V2 s 0, V3 s 2 ŽReŽ f 21 . y ImŽ f 20 f 11 ... See w7, 9, 13, 14x, for instance. To conclude this section we give in next examples some simple applications of the above theorem. These examples show how to use the above results to generate limit cycles for systems of type Ž1. with singular points of pseudo-focus type. Note that in the smooth case it is known Žsee w2, p. 243x, for instance . that all the even Lyapunov constants associated with a singularity of focus type are zero while that in the non-smooth case it is not true in general. This fact provokes, for instance, that while for quadratic systems the maximum number of limit cycles that bifurcates from the origin is 3 Žusing V1 , V3 , V5 , and V7 ; see w4x., it suffices to use V1 , V2 , V3 , and V4 in system Ž7. to generate also three limit cycles. In fact by a more specific study of system Ž7. we think that six limit cycles can be generated from the origin Žusing V1 , V2 , . . . , V7 .. We remark that although the discontinuous system Ž7. can have twice the number of limit cycles than the smooth case, the free parameters of Ž7. and the ones of the quadratic system ˙ z s Ž i q . z q f 20 z 2 q f 11 zz q f 02 z 2 coincide. EXAMPLE C Žfocus᎐focus case.. Let Eqs. Ž1., Ž2. be given by the expression
˙z s
½
Ž i q . z q f 20 z 2 q f 11 zz q f 02 z 2 ,
if Im Ž z . G 0,
iz,
if Im Ž z . F 0,
Ž 7.
2 15 6y 3x 11 37 where f 20 s y 11 i, f 11 s 12 q 85 z y 4 y q i, f 02 s 48 16 q y y 32 z q 8 2 5 y6 q 3 x q y y 32 z q 8 i, for arbitrary real numbers: , x, y, and z. Then, by choosing - 0, x ) 0, y - 0, and z ) 0 small enough, such that < < < < x < < < y < < < z <, Eq. Ž7. has, at least, four limit cycles bifurcating from the origin.
The fact that ˙ z s 1 q iz q iz and ˙ z s iz induce the same half return map hy in ImŽ z . F 0 allows us to state next result.
679
BIFURCATIONS IN DISCONTINUOUS SYSTEMS
EXAMPLE D Žfocus᎐parabolic case.. Let Eqs. Ž1., Ž2. be given by
˙z s
½
Ž i q . z q f 20 z 2 q f 11 zz q f 02 z 2 ,
if Im Ž z . G 0,
1 q iz q iz,
if Im Ž z . F 0,
where , f 20 , f 11 and f 02 are as in Example C. Then this system has at least four limit cycles bifurcating from the origin. EXAMPLE E Žparabolic᎐parabolic case.. Let Eqs. Ž1., Ž2. be given by the expression
¡y2 q 1 Ž q i . z q 1 Ž y q i . z y i z 2 4 ˙z s~ ¢1 q iz q2 iz,
2
q
i 4
z2,
if Im Ž z . G 0, if Im Ž z . F 0.
Ž 8. Then, by choosing - 0 small enough, Eq. Ž8. has, at least, one limit cycle bifurcating from the origin.
3. PRELIMINARY RESULTS Remember that the Ž R, , p, q .-generalized polar coordinates are x s R CsŽ ., y s R q SnŽ ., where p and q will be fixed afterwards and where CsŽ . and SnŽ . are the solution of the Cauchy problem, p
˙ Ž . s ySn2 py1 Ž . , Cs 2q
Cs Ž 0 . s
(
1 p
˙Sn Ž . s Cs 2 qy1 Ž . , ,
Sn Ž 0 . s 0;
see w6, 8, 15x. By using the Ž R, , p, q .-generalized polar coordinates, each component of Eq. Ž2. is converted into dR d s
Re Ž Ž z q z .
ž
2 qy1
p
q Ž y1 . 2 2Ž qyp. Ž z y z .
2 py1
. ˙z / r Ž 2 2 qy1 R 2 p qy1 .
Im Ž Ž Ž p y q . z q Ž p q q . z . ˙ z . r Ž 2 R pq q .
,
Ž 9.
680
COLL, GASULL, AND PROHENS
where this expression is evaluated on z s R p CsŽ . q iR q SnŽ .. Of course, this change of variables may not make sense for arbitrary values of Ž R, . and p and q. Our first purpose, when we study the focus and the parabolic case, is to find values for p and q for those former change of variables makes sense, i.e., those for which Eq. Ž9. can be written Žin both cases., in ImŽ z . G 0, as an equation of the type dR d
s
q⬁
Ý Tk Ž . R k ,
Ž 10 .
ks1
defined for Ž R, . in the set w0, ␣ . = w0, 2 x, for some real value ␣ ) 0. Once we get Eq. Ž2. written as Eq. Ž10., we will evaluate its solution, RŽ ; s ., RŽ ; s. y s s
q⬁
Ý wk Ž . s k ,
with w k Ž 0 . s 0
for all k G 1 Ž 11 .
ks1
Ži.e., the solution of Ž10. such that when s 0 takes the value s . when s . Hence, by defining hqŽ s . s RŽ ; s . as a ‘‘return’’ function between w0, ␣ . = 04 and w0, ␣ . = 4 , the behaviour of the solution Ž11. near R ' 0 is controlled by the values w k Ž ., k G 1. The next two results, proved in w12x, are given to simplify as much as possible the expressions that allows us to compute functions w k Ž . that appear in Ž11..
LEMMA 1. Ži. The change of ¨ ariables r s ReyH 0 T1Ž . d transforms differential Eq. Ž10. into the differential equation dr d
s
q⬁
Ý Rk Ž . r k ,
Ž 12 .
ks2
where R k Ž . s e Ž ky1.H0 T1Ž . d Tk Ž .. Žii. Let us consider the solution of Eq. Ž12. written as rŽ ; . s q
q⬁
Ý uk Ž . k ,
with u k Ž 0 . s 0
for all k G 2. Ž 13 .
ks2
Then, RŽ ; s. s s q
q⬁
0
Ý wk Ž . s k s e H T Ž . d
ks1
1
sq
q⬁
Ý uk Ž . s k
ks2
is the solution of Ž10., satisfying RŽ0; s . s s; i.e., w 1Ž . s e H0 T1Ž . d y 1 and w k Ž . s e H0 T1Ž . d u k Ž ., for all k G 2.
681
BIFURCATIONS IN DISCONTINUOUS SYSTEMS
The next result, inspired by w2x, allows us to compute the first values of u k Ž . in Ž13.. In the sequel we use the notation f˜s f˜Ž . s H0 f Ž s . ds. PROPOSITION 2. Gi¨ en Eq. Ž12., the functions u i Ž ., i s 2, . . . , 5 in¨ ol¨ ed in its solution, Ž13., are: u 2 Ž . s R˜2 , u 3 Ž . s Ž R˜2 . 2 q R˜3 , u 4 Ž& . s Ž R˜2 . 3 &
2
q 2 R˜2 R˜3 q R˜2 R 3 q R˜4 , and u 5 Ž . s Ž R˜2 . 4 q 3Ž R˜2 . 2 R˜3 q Ž Ž R˜2 . R 3. q &
&
2 R˜2 ŽR˜2 R 3. q 32 Ž R˜3 . 2 q 2 R˜2 R˜4 q 2ŽR 4 R˜2. q R˜5 . 4. THE FOCUS CASE Consider Eq. Ž2.. Let us assume that the component equation in ImŽ z . G 0 is of type ŽF.; that is to say, aqs 0. We will take ordinary Ž R, .-polar coordinates, i.e., p s q s 1, in Ž11.. Hence, if we neglect the plus sign, Eq. Ž9. becomes dR d
s
Re Ž zz˙. rR Im Ž zz˙. rR 2
zsRe i
s
k Ýq⬁ ks 1 Re Ž S k Ž . . R ky1 Ýq⬁ ks 1 Im Ž S k Ž . . R
,
Ž 14 .
where Sk Ž . s zFk Ž z, z .< zse i s eyi Fk Ž e i , eyi .. From the expression of S1Ž . s f 10 q f 01 ey2 i and Ž5. we get that ImŽ S1Ž .. ) 0, which implies that Eq. Ž14. can be written as in Ž10.. In order to evaluate solution Ž11. of Eq. Ž14. when s , we will prove next proposition. PROPOSITION 3. Let the component equation in ImŽ z . G 0 of Ž2. be written in polar coordinates as Ž14.. Let RŽ ; s . be its solution gi¨ en by Ž11.. Then, R Ž ; s . y s s w1Ž . s q O Ž s 2 . , where w 1 Ž . s exp
½'
Re Ž f 10 .
Ž Im Ž f 10 . .
2
y f 01 f 01
5
y 1.
Moreo¨ er, if F1Ž z, z . s Ž i q . z then R Ž , s . y s s w1Ž . s q w 2 Ž . s 2 q w 3 Ž . s 3 q O Ž s 4 . , where w 1Ž . s e y 1, w 2 Ž . s e Žye y 1.RewŽ1 q i . ␣ x, and w 3 Ž . s e Ž e 2 y 1. RewŽ1 q i .  q ␥ x y ImwŽ1 q i . ␦ x4 q ey Ž w 2 Ž .. 2 .
682
COLL, GASULL, AND PROHENS
Proof. First we compute w 1Ž .. Consider the expression Ž14.. Direct substitution shows that w 1Ž . satisfies wX1 Ž . s
Re Ž S1 Ž . . Im Ž S1 Ž . .
Ž w1Ž . q 1. ,
w 1 Ž 0 . s 0.
Then w 1 Ž . s exp
H0
Re Ž f 10 . q Re Ž f 01 . cos 2 q Im Ž f 01 . sin 2 Im Ž f 10 . y Re Ž f 01 . sin 2 q Im Ž f 01 . cos 2
d y 1.
Standard integration techniques give that
H0
Re Ž f 10 . q Re Ž f 01 . cos 2 q Im Ž f 01 . sin 2 Im Ž f 10 . y Re Ž f 01 . sin 2 q Im Ž f 01 . cos 2
d
1 Im Ž f 10 . y Re Ž f 01 . sin 2 q Im Ž f 01 . cos 2 s y ln 2 Im Ž f 10 . q Im Ž f 01 . q
Re Ž f 10 .
'ŽImŽ f
2
10
. . y f 01 f 01
'ŽImŽ f
2
. . y f 01 f 01 tan = arctan q fŽ . , Im Ž f 10 . q Im Ž f 01 . y Re Ž f 01 . tan
ž
10
/
where f Ž . is equal either to 0, if 0 F F r2, or to , if r2 F F . Hence, the expression of w 1Ž . follows. Now we continue computing w 2 Ž . and w 3 Ž . but assuming that f 10 s q i, f 01 s 0. Observe that in this case w 1Ž . s e y 1. Note also that Eq. Ž14. can be written as Ž10. by the expression dR d
s
k R q Ýq⬁ ks 2 R Re Ž S k Ž . .
1q
Ýq⬁ ks 2
R
ky1
Im Ž Sk Ž . .
s
q⬁
Ý Tk Ž . R k ,
ks1
where T1 s , T2 s Re S2 y Im S2 , and T3 s Re S3 y Re S2 Im S2 y Im S3 q ŽIm S2 . 2 . By using Lemma 1Ži. the above equation is converted into dr d where R k s e Ž ky1. Tk , k G 2.
s
q⬁
Ý Rk Ž . r k ,
ks2
683
BIFURCATIONS IN DISCONTINUOUS SYSTEMS
Therefore, from Proposition 2, when we consider its solution in the form Ž13., we must compute
u 2 Ž . s R˜2 Ž . s Re
H0 S Ž . e
2
2
u 3 Ž . s Ž u 2 Ž . . q Re
d y Im
H0
H0 S Ž . e
S3 Ž . q
H0
= e 2 d y Im
2
S3 Ž . q
2
d ,
Ž yS22 Ž . q S2 Ž . S2 Ž . . 1 2
S22 Ž . e 2 d .
Since by Lemma 1Žii., w 2 Ž . s e u 2 Ž . and w 3 Ž . s e u 3 Ž ., direct computations give the final expression of w 2 Ž . and w 3 Ž . in terms of the coefficients of F2 and F3 . As an example we will explain how to calculate u 2 Ž .. From the expression of S2 Ž . s f 20 e i q f 11 eyi q f 02 ey3 i and previous expression of u 2 Ž ., we have u 2 Ž . s Re
ž
ž
H0 S Ž . e 2
s Re Ž 1 q i .
d q i
f 20 e Ž iq . iq
H0 S Ž . e
q
2
f 11 e Žyiq . yi q
d q
/ f 02 e Žy3 iq . y3i q
0
/
.
5. THE PARABOLIC CASE Consider Eq. Ž2.. Let us assume that the component equation in ImŽ z . G 0 of Ž2. is of type ŽP.. Hence, aq/ 0. In the next lemma we argue on the best selection of Ž R, , p, q .-generalized polar coordinates for this case. LEMMA 4. Let the component equation of Ž2. in ImŽ z . G 0 be of type ŽP.. Then, the R-power series expansion of Ž9. is well defined if and only if q s 2 p.
684
COLL, GASULL, AND PROHENS
Proof. If aq/ 0, then the expression Ž9. is written as dR d s
q 2 py1 aq Cs 2 qy1 Ž . R qq 1 q Im Ž fq Ž . Cs Ž . R 2 pq1 q ⭈⭈⭈ 10 q f 01 . Sn q 2 2p yaq q Sn Ž . R q q p Im Ž fq q ⭈⭈⭈ 10 q f 01 . Cs Ž . R
,
where the dots indicate the rest of the terms in the R-power series expansion. We note that all of these terms have order greater than those that appear. Consequently, p and q can be neither q - 2 p nor q ) 2 p, because in these cases the former expression is written as dR d dR d
s s
Cs 2 qy1 Ž . yq Sn Ž . Sn2 py1 Ž . p Cs Ž .
R q O Ž R2 . ,
if q - 2 p,
R q O Ž R2 . ,
if q ) 2 p.
In the case q s 2 p we have dR d
s
q 2 py1 aq Cs 2 qy1 Ž . q Im Ž fq Ž . Cs Ž . 10 q f 01 . Sn q 2 p Ž y2 aq Sn Ž . q Im Ž fq 10 q f 01 . Cs Ž . .
R q O Ž R2 . .
Ž 15 . q. From the fact that aq- 0 and ImŽ fq 10 q f 01 ) 0 we conclude that the two q q. q Ž . Ž real roots of y2 a Sn q Im f 10 q f 01 Cs 2 Ž . s 0 lie in ImŽ z . - 0, which finishes the proof.
As a consequence of the previous lemma, from now on and for simplicity, in the parabolic case we will take Ž R, , 1, 2.-generalized polar coordinates. The next lemma sums up all the previous considerations to the study of the parabolic case. Also, in next lemma, by using a change of variables we get an equivalent expression of Eq. Ž15., written as Eq. Ž10., with a simplest T1Ž .. LEMMA 5. Let the component equation of Ž1. in y G 04 be of type ŽP.. Write Ž1. in the new ¨ ariables x 1 s x q ml y and y 1 s y2l a y. Then, in
BIFURCATIONS IN DISCONTINUOUS SYSTEMS
685
Ž R, , 1, 2.-generalized polar coordinates and remo¨ ing the q sign, it is written as dR d
s
q⬁
Ý Tk Ž . R k
ks1
s
2 Sn Ž . Cs Ž . y Cs 3 Ž . 2 Ž Cs Ž . q Sn Ž . . 2
=
Cs 2 Ž .
Ž Cs 2 Ž . q Sn Ž . .
2
Rq
an y l Ž b q m . 2 al
R 2 q T3 Ž . R 3 q O Ž R 4 . ,
Ž 16 .
where T3 Ž . s
1 4 a l Ž Cs Ž . q Sn Ž . . 2 2
2
3
= Cs 5 Ž . Ž y2 adl 2 q 2 abln y 2 a2 n2 q 2 a2 lq . qCs 3 Ž . Sn Ž . Ž 2 b 2 l 2 y 2 adl 2 q cl 3 q 3bl 2 m q 2 l 2 m2 y2 abln y 2 almn y al 2 o q 2 a2 lq . qCs Ž . Sn2 Ž . Ž cl 3 y bl 2 m q 2 amln y al 2 o . . Proof. In order to simplify further expressions we apply the change of variables suggested in w10x, which does not change the return map on the x-axis x 1 s x q ml y and y 1 s y2l a y, to the component equation of Ž1. in y G 0. Observe that we obtain
˙x s a q Ž b q m . x q ˙y s y2 ax y
2 an l
bm y cl
x2 q
2a
yq
ol y 2 nm l
dl q nm l
x 2 q ⭈⭈⭈ ,
Ž 17 .
xy q ⭈⭈⭈ ,
if we omit the subscripts ‘‘1.’’ If, in addition, we use the Ž R, , 1, 2.-generalized polar coordinates in Eq. Ž17., and we expand the quotient dR d in a R-power series like Ž10., we obtain the expression Ž16.. Next lemma is a technical result which is a key point for the effective computation of the Lyapunov constants in the parabolic case by using the Ž R, , 1, 2.-generalized polar coordinates.
686
COLL, GASULL, AND PROHENS
LEMMA 6. Let SnŽ . and CsŽ . be the generalized trigonometric functions introduced in Section 3. Then Cs m Ž . Sn l Ž .
H ŽCs
2
Ž . q Sn Ž . .
d
n
¡
l
s
Ý Ž y1. i is0
Ž Cs 2 Ž . q Sn Ž . .
lyiynq1
Ž l y i y n q 1 . Cs 2Ž lyiynq1. Ž .
ž /~
q k,
if l y i y n q 1 / y1,
l i
ln
¢
Cs 2 Ž . q Sn Ž .
q k, Cs 2 Ž . if l y i y n q 1 s y1,
for all l, m, and n positi¨ e integers whene¨ er 2 n s m q 2 l q 3, and where k stands for the arbitrary integration constant. Proof. Applying the change of variables x s SnŽ .rCs 2 Ž . to the integral, we get the equivalent expression
H
Ž1 q 2 x2 .
Ž2 nymy2 ly3 .r4
Ž1 q x.
xl
l
dx s
HŽ 1 q x .
yl
x l dx ,
because of the hypothesis. The calculation of last integral is immediate. Hence the proof is finished. In order to evaluate solution Ž11. of Eq. Ž16. when s , we will prove the next proposition. PROPOSITION 7. Let RŽ ; s . be the solution Ž11. of Eq. Ž1. written as Eq. Ž16.. Then w 1Ž . s 0, w 2 Ž . s 23 an y Ž alb q m . l , and w 3 Ž . s Ž 23 an y Ž alb q m . l . 2 . Proof. Using the properties of the Ž R, , 1, 2.-generalized polar coordinates and the expression of T1Ž . given in Lemma 5, standard integration techniques give
e Ž ky1.H0 T1Ž . d s e Ž ky1.r2 H0
d r d logŽCs 2 Ž .qSnŽ .. d 4
s Ž Cs 2 Ž . q Sn Ž . .
y Ž ky1 .r2
.
687
BIFURCATIONS IN DISCONTINUOUS SYSTEMS
Hence, from Lemma 1Ži., Eq. Ž16. is converted into Eq. Ž12., where R2 Ž . s R3 Ž . s
Cs 2 Ž .
an y Ž b q m . l 2 al
Ž Cs 2 Ž . q Sn Ž . .
T3 Ž . Cs Ž . q Sn Ž . 2
and
5r2
.
Now, the idea is to get the solution of Eq. Ž12. in the form Ž13., by means of Proposition 2. Then, from Lemma 1Žii., and using the fact that e H0 T1Ž . d < s s 1, we will have w k Ž . s u k Ž . for all k s 2, 3, . . . , which will finish the proof. From Proposition 2, we must compute u 2 Ž . s R˜2 Ž . s
an y Ž b q m . l 2 al
Cs 2 Ž .
H0
Ž Cs 2 Ž . q Sn Ž . .
5r2
d .
From Lemma 6, we have that u2 Ž . s
an y Ž b q m . l 3al
ž
1y
Cs 3 Ž . Cs 2 Ž . q Sn Ž .
3r2
/
.
Therefore, u2 Ž . s
2 an y Ž b q m . l 3
al
.
To compute u 3 Ž . we need to calculate R˜3 Ž ., but R 3 Ž . is an even expression on SnŽ . and odd on CsŽ .. Hence, R˜3 Ž . s 0. This gives, from Proposition 2, that u 3 Ž . s Ž R˜2 Ž .. 2 , which finishes the proof. Remark 8. About the contact order of the solution, y s y Ž x ., of Eq. Ž1. which passes through Ž0, 0. and L s Ž x, y ., y s 04 , for a component equation of type ŽP. in y G 0, it is not difficult to see that Ž d 2 yrdx 2 .< Ž0, 0. s lra2 and, consequently, that whenever l / 0, this solution has a second order contact with L. If l s 0 then, taking n s 0 and q ) 0, the flow keeps turning around the origin Ž0, 0.. In this case, the contact order becomes four because Ž d 3 yrdx 3 .< Ž0, 0. s 0 and Ž d 4 yrdx 4 .< Ž0, 0. s 6 qra2 . In this situation it is not possible to get Eq. Ž11. by using the change to Ž R, , p, 2 p .-generalized polar coordinates suggested in Lemma 4, but it is not difficult to prove that the suitable change to Ž R, , p, q .-generalized polar coordinates is obtained by taking q s 4 p.
688
COLL, GASULL, AND PROHENS
6. PROOF OF THE MAIN RESULTS Proof of Proposition A and Theorem B. Let us consider Eq. Ž1., or equivalently Ž2., and let us apply to each component equation the suitable change of Ž R, , p, q .-polar coordinates, according Section 4 and Lemma 4. Following the ideas given in Section 3, to prove our results we will compose the map induced by the flow of ˙ z s FqŽ z, z . between s 0 and q s , h , and the map induced by the flow of ˙ z s Fy Ž z, z . between y s and s 2 , h . Their expansions are given by expression Ž11., with h "Ž s . s R "Ž , s ., and are written as h " Ž s . s Ž w 1" Ž . q 1 . s q w 2" Ž . s 2 q w 3" Ž . s 3 q O Ž s 4 . 2 " 3 4 s h1"s q h " 2 s q h3 s q O Ž s . .
We note that, in this last expression, the map hy coincides with the return map induced by the flow of ˙ z s yFyŽ yz, yz . between s 0 and s . Hence the stability of the origin is controlled by the sign of hyŽ hqŽ s .. y s, where q y q y q hy Ž hq Ž s . . y s s Ž hy 1 h1 y 1 . s q Ž h1 h 2 q h 2 Ž h1 . q y q q y q q Ž hy 1 h 3 q 2 h 2 h1 h 2 q h 3 Ž h1 .
3
2
. s2
. s3 q O Ž s4 .
and the three first Lyapunov constants of Ž2. are q V1 s hy 1 h1 y 1,
V2 s 2
V3 s
y q hq 2 q h 2 Ž h1 .
hq 1
q q y q hq 1 h 3 y 2 Ž h 2 . q h 3 Ž h1 .
Ž hq 1 .
2
3
,
5
.
Let us argue about the sign of hq 1 . In the focus case, using Proposition 3, h1 s e ) 0, because w 1Ž . s e y 1. In the parabolic case, using Proposition 7, h1 s 1 ) 0, because w 1Ž . s 0. Hence, we obtain the values Vi , i s 1, 2, and 3, depending on the singularity type of the origin. Whence, the proof follows. Proof of Example C. Lyapunov constants V1 , V2 , and V3 , of the origin of Ž7., follow from Theorem BŽa. by taking into account, among other facts, Ž . that ys 0 and wy i ' 0 for all i G 2, which simplifies their calculation. Ž . In fact, we have Vi s q y 1, V2 s x, i , for all i G 2. Hence, V1 s e and V3 s y. In order to generate four limit cycles we need to know the fourth and fifth Lyapunov constants, V4 and V5 , of the origin of Eq. Ž7.. Their calculation is only interesting if V1 s V2 s V3 s 0. Hence, we assume s x s y s 0. Let us calculate V4 and V5 . It is not diffi-
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BIFURCATIONS IN DISCONTINUOUS SYSTEMS
cult, following Lemma 5, to get T4 s ReŽ S2 .ŽImŽ S2 .. 2 and T5 s yReŽ S2 .ŽImŽ S2 .. 3. Now, using the fact that s 0, we have R k s Tk if Ž . k G 2. In addition, from Proposition 2, we get the expression of uq 4 and Ž . Ž . uq . Hence, from Proposition 1 ii , using again that s 0, we have that 5 qŽ . Ž . q s u if k G 2. Following the same ideas exposed in the proof of k k Ž . Ž Theorem B we have V4 s q s z and, in the case V s 0 i.e., if 4 4 55 Ž . z s 0., V5 s q s y . 5 384 Hence, the return map associated to the origin of Eq. Ž7. is written as h Ž s; , x, y, z . s Ž e y 1 . s q Ž x q f 2 Ž , x, y, z . . s 2 q Ž y q f 3 Ž , x, y, z . . s 3 q Ž z q f 4 Ž , x, y, z . . s 4 q y
ž
55 384
q f5 Ž , x, y, z . s 5
/
q O Ž s6 . , for small enough values of the variable s, s ) 0, where f i , i s 2, . . . , 5, are analytic real functions satisfying f 2 Ž0, x, y, z . ' 0, f 3 Ž0, 0, y, z . ' 0, f 4Ž0, 0, 0, z . ' 0, and f5 Ž0, 0, 0, 0. s 0. By choosing , x, y, and z small enough and satisfying the conditions of this example, it is easy to find four values of s for which h vanishes. More concretely, if we consider hŽ s; 0, 0, 0, 0., we can get a value of s, s0 , small enough, for which hŽ s0 ; 0, 0, 0, 0. - 0. Because of the continuity of the function h, it is possible to assure that hŽ s0 ; 0, 0, 0, z . - 0, for small enough values of z. Let z1 ) 0 be one such value of z. In this situation, we have that hŽ s; 0, 0, 0, z1 . s z1 s 4 q O Ž s 5 .. Hence, we can have a value of s, s1 , small enough and smaller than s0 , for which hŽ s1; 0, 0, 0, z1 . ) 0. Whence, between s1 and s0 , we obtain the first change of sign of h. Arguing in a similar way, we can get the four changes of sign of h. See also w5x. Proof of Example E. Lyapunov constants V1 , V2 , and V3 , of the origin of Eq. Ž8. follow straightaway from Theorem BŽc.. V4 needs some more computations. They are V1 s 0, V2 s 3 , V3 s 0, and V4 s 301 . Hence, for the choice of the parameter made in the statement of this lemma and arguing as in the proof of Example C, we can create one unstable small limit cycle around the origin as we wanted to prove.
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