Lie Symmetries of Quasihomogeneous Polynomial Planar Vector Fields and Certain Perturbations

On the Integrability of

Quasihomogeneous and Related Planar

Vector Fields

Isaac A. Garcıa∗

Departament de Matematica. Universitat de Lleida.Avda. Jaume II, 69. 25001. Lleida. SPAIN.

E–mail: [email protected]

Abstract

In this work we consider planar quasihomogeneous vector fields and we show,among other qualitative properties, how to calculate all the inverse integratingfactors of such C1 systems. Additionally, we obtain a necessary condition in orderto have analytic inverse integrating factors and first integrals for planar positivelysemi-quasihomogeneous vector fields which is related with the existence of poly-nomial inverse integrating factors and first integrals for the quasihomogeneouscut. Examples are given and their relationship with Kovalevskaya exponents isshown.

1 Introduction

We concentrate our attention to the quasihomogeneous planar differential systems,which are also called similarity invariant systems or weighted homogeneous systems.That is, autonomous differential equations in the real affine plane

dx

dt= x = P (x, y) ,

dy

dt= y = Q(x, y) , (1)

which are invariant under the similarity transformation

(x, y, t) → (αpx, αqy, α−ℓt) (2)

for all α ∈ IR and where p and q are positive integers. In other words, P and Q arep− q quasihomogeneous functions of weighted degrees p+ ℓ and q+ ℓ respectively, i.e.,P (αpx, αqy) = αp+ℓP (x, y) and Q(αpx, αqy) = αq+ℓQ(x, y) for all α ∈ IR. We will saythat system (1) is p− q quasihomogeneous of weighted degree ℓ.

Let us notice that if p is even and q and ℓ are odd, then the p − q quasihomoge-neous systems include some class of time-reversible systems which are invariant underthe symmetry (x, y, t) → (x,−y,−t). Moreover, in the particular case p = q = 1, the

∗The author is partially supported by ACI2000-27.

1

p− q quasihomogeneous systems reduce to an homogeneous system.

Two of the main open problems of the qualitative theory of planar differentialequations are the center-focus problem and the determination of the number of limitcycles and their distribution in the plane. We recall that an isolated periodic orbit inthe set of all periodic orbits is called a limit cycle. Recently, several works has beenshow that an unified method based in the concept of inverse integrating factor can beused to study both problems, see for instance [Chavarriga et.al, 1999] and referencestherein.

One of the best ways to understand the properties of a given planar differentialsystem is through their inverse integrating factors. Existence of inverse integratingfactors gives a lot of information on dynamics, phase space structure and so on, but ingeneral, it is difficult to search inverse integrating factors.

It is known that the existence or non-existence of first integrals of system (1) isrelated to the Kovalevskaya exponents, first called by [Yoshida, 1983]. Although themain theorem of such work seems powerful, it has an important weak point, namely, itgives some conditions for the Kovalevskaya exponents under the existence of a “non-degenerate” first integral. Later, in [Furta, 1996] the author overcame this weak pointalthough the assertion is a little weak.

The paper is organized as follows: In second section we give some qualitative prop-erties of system (1) related with its singular points and the associated monodromyproblem. In the third section we prove one of the main results of the paper, see Theo-rem 4, which states how to calculate all the inverse integrating factors of C1 systems(1) while section 4 is devoted to show some examles. In the section fifth we recallthe concept of Kovalevskaya exponents and, finally the last section tries on anotherfundamental result of the paper (Theorem 9) which gives a necessary condition in or-der to have analytic inverse integrating factors and first integrals for planar positivelysemi-quasihomogeneous vector fields.

2 Some properties of quasihomogeneous vector fields

The point (x0, y0) ∈ lC2 is called a finite singular point of system (1) if P (x0, y0) =Q(x0, y0) = 0. For the subset of real finite singular points of quasihomogeneous systemswe have the next result.

Lemma 1 The polynomial p−q quasihomogeneous system (1) with P and Q coprimesonly has the origin as real finite singular point.

Proof. Let (P (x, y), Q(x, y)) be a quasihomogeneous vector field. Hence taking α = 0into its definition easily follows that P (0, 0) = Q(0, 0) = 0 and the origin is always asingular point.

Assume now that (x0, y0) ∈ IR2 is another singular point different from the ori-gin. In consequence, since P (αpx0, α

qy0) = αp+ℓP (x0, y0) = 0 and Q(αpx0, αqy0) =

αq+ℓQ(x0, y0) = 0, all the points of the curve γ = {(αpx0, αqy0) : α ∈ IR} are sin-

gular points also. But this implies that the polynomials P and Q are not coprimes incontradiction with our hypothesis.

2

There is, in general, common particular solutions of special type for similarityinvariant systems as we will show. See also [Yoshida, 1983] for the particular caseℓ = 1.

Lemma 2 The p− q quasihomogeneous system (1) of weighted degree ℓ = 0 possessesthe particular solution x(t) = c1t

−p/ℓ, y(t) = c2t−q/ℓ where the constant vector (c1, c2)

belongs to the set of real solutions of the algebraic equations

P (c1, c2) +p

ℓc1 = 0 , Q(c1, c2) +

q

ℓc2 = 0 . (3)

Proof. Keeping in mind the p− q quasihomogeneous structure of the system, straight-forward computation gives the result.

A real finite singular point is called monodromic if there are no orbits tending orleaving it with a certain angle. When P and Q are analytic, a monodromic singularpoint is always either a center or a focus, i.e., either the singular point is surroundedby cloded periodic orbits or each trajectory in a neighborhood of the singular point isa spiral winding around it.

Let X = (P,Q) be the analytic vector field associated to a non necessary weightedhomogeneous system (1). A singular point (x0, y0) is called degenerate if the differentialmatrixDX (x0, y0) associated to it is degenerate, that is, the jacobian detDX (x0, y0) =0. Otherwise, we will say that the singular point (x0, y0) is nondegenerate. IfDX (x0, y0)has only one eigenvalue equal to zero then (x0, y0) is an elementary degenerate singu-lar point and, according to [Andronov et al., 1973] it cannot be monodromic. If zerois a double eigenvalue of DX (x0, y0) but DX (x0, y0) is not identically zero then thedegenerate singular point (x0, y0) is called nilpotent. For nilpotent singular points, themonodromy problem, which consists in to determine under what conditions such pointis monodromic, was solved in [Andreev, 1958]. Finally, for degenerate singular points(x0, y0) withDX (x0, y0) identically zero the monodromy problem is poorly-understood.In fact, two different types of behaviour are possible depending on the existence ornot of the socalled characteristic directions of (x0, y0). Let P (x, y) =

∑j≥k Pj(x, y)

and Q(x, y) =∑

j≥k Qk(x, y) where Pj and Qj are homogeneous polynomials of de-gree j and k ≥ 2. Hence (x0, y0) = (0, 0) is a degenerate singular point with as-sociated Jacobian matrix nul. Taking polar coordinates x = r cos θ, y = r sin θ, wesay that θ = θ∗ is a characteristic direction of the origin if cos θ∗Qk(cos θ

∗, sin θ∗) −sin θ∗Pk(cos θ

∗, sin θ∗) = 0. It is well known that if there exists an orbit tending orleaving the origin at a certain angle, this angle is a characteristic direction.

Corollary 3 Assume that the polynomial p−q quasihomogeneous system (1) of weighteddegree ℓ = 0 and P and Q coprimes possesses either a monodromic singular point atthe origin or a limit cycle. Then equations (3) do not have any nontrivial real solution.

Proof. From Lemma 1 it follows that P (0, 0) = Q(0, 0) = 0 and therefore equations(3) have always the trivial solution (0, 0). Let us assume that there exists a real non-trivial solution (c1, c2) = (0, 0) of (3). Hence the particular solution S(t) = (x(t), y(t))described in Lemma 2 goes through the singular point (0, 0) and therefore it cannot bea center of system (1). In addition, since such particular solutions are not spirals thesingular point (0, 0) neither can be a focus. Therefore the origin is not monodromic.

3

On the other hand, let us suppose that system (1) has a limit cycle Γ. Of course,Γ must surround at least one singular point and so, from Lemma 1, Γ only surroundsthe origin. Finally, since S(t) is not bounded we have S ∩ Γ = ∅ which is impossiblefrom the definition of limit cycle.

3 About the integrability of quasihomogeneous sys-tems

Let x = P (x, y), y = Q(x, y) be a C1 vector field defined in an open subset U ⊂ IR2.We call inverse integrating factor for such vector field to a C1(U) solution V (x, y) ofthe linear partial differential equation

P∂V

∂x+Q

∂V

∂y=

(∂P

∂x+

∂Q

∂y

)V . (4)

Observe that 1/V is an integrating factor of the vector field (P,Q) in U\{V = 0}. In[Chavarriga et al., 1999], the authors show that the inverse integrating factor existsand it is unique, except for a multiplicative constant factor, in an open neighborhoodof generic singular points.

If p = q = 1, then the p − q quasihomogeneous vector field (P,Q) reduces to anhomogeneous vector field and it is well known that it possesses the homogeneous inverseintegrating factor V = xQ− yP provided that V ≡ 0. Now we state a theorem whichgeneralizes the above comment to p− q quasihomogeneous vector fields.

Theorem 4 Let x = P (x, y), y = Q(x, y) be a C1 p−q quasihomogeneous vector field.Hence any inverse integrating factor of it must be a p− q quasihomogeneous function.Moreover, if α(u) = Q(1, u)− q

puP (1, u) ≡ 0 then V (x, y) = xm/pf(1, y/xq/p) is a p−qquasihomogeneous inverse integrating factor of weighted degree m, where

f(1, u) = exp

(−∫

β(u)

α(u)du

), (5)

and β(u) = mp P (1, u)− ∂P/∂x(1, u)− ∂Q/∂y(1, u).

Proof. Let V (x, y) be an inverse integrating factor of the p− q quasihomogeneous vec-tor field (P,Q), that is, V satisfies equation (4). Since P , Q and (∂P/∂x+∂Q/∂y) arep − q quasihomogeneous functions of weighted degrees p + ℓ, q + ℓ and ℓ respectively,it is easy to see by straightforward computation that equation (4) is invariant underthe change of variables (x, y) → (αpx, αqy). In consequence their solutions are alsoinvariants, i.e., V (αpx, αqy) = V (x, y) or equivalently V (αpx, αqy) = αmV (x, y).

Let V (x, y) be a p − q quasihomogeneous function of weighted degree m, that isV (αpx, αqy) = αmV (x, y). If we impose that V be an inverse integrating factor of thep− q quasihomogeneous system (1), the following equation is verified

P∂V

∂x+Q

∂V

∂y= div(P,Q)V , (6)

where div(P,Q)(x, y) = (∂P/∂x + ∂Q/∂y)(x, y) is the divergence of the vector field(P,Q). Observe that P , Q, ∂V/∂x, ∂V/∂y, V and div(P,Q) are p − q quasihomoge-neous functions of weighted degrees p+ ℓ, q + ℓ, m− p, m− q, m and ℓ respectively.

4

Now, we make the blow–up (x, y) → (w, u) where w = x and u = y/xq/p into theabove partial differential equation (6). Taking into account the weighted quasihomo-

geneity of the involved functions in such equations we have P (w, uwq/p) = wp+ℓp P (1, u),

Q(w, uwq/p) = wq+ℓp Q(1, u) and div(P,Q)(w, uwq/p) = w

ℓp div(P,Q)(1, u). On the

other hand, by the chain rule we have that ∂/∂x = ∂/∂w − qp

uw∂/∂u and ∂/∂y =

w− qp ∂/∂u. Hence in the new variables (w, u), equation (6) becomes[

Q(1, u)− q

puP (1, u)

]∂V

∂u+ wP (1, u)

∂V

∂w= div(P,Q)(1, u)V ,

where V (w, u) = V (w, uwq/p) = wm/pV (1, u). In consequence ∂V /∂w = mp w

mp −1V (1, u),

∂V /∂u = wm/pdV (1, u)/du and the above partial differential equation reduce to thefollowing first order linear ordinary differential equation

α(u)dV (1, u)

du+ β(u)V (1, u) = 0 , (7)

where the coefficients α(u) and β(u) are the given in the statement of the theorem.Provided that α(u) ≡ 0, the general solution of equation (7) adopts the form (5)

except for a multiplicative arbitrary constant which we take the unity because suchconstant does not affect to our computation. Finally we undoing the blow–up and thetheorem is proved.

Remark 1. We can use Theorem 4 to get a family of inverse integrating factors Vm

of weighted degree m and next compute a first integral H either from H = Vm1/Vm2

with m1 = m2 or from H =∫P/Vm dy + f(x) satisfying ∂H/∂x = −Q/Vm.

4 Some examples

Take the p − q quasihomogeneous vector field x = P (x, y) =∑k

i=0 fp+ℓ−iq(x)yi, y =

Q(x, y) =∑k

i=0 gq+ℓ−iq(x)yi where fj and gj are p− q quasihomogeneous functions of

weighted degree j. The next example leads with the particular case k = 2 and f andg potential functions.

Example 5 The p− q quasihomogeneous system of weighted degree ℓ

x = P (x, y) = a0xp+ℓp + a1x

p+ℓ−qp y + a2x

p+ℓ−2qp y2 ,

y = Q(x, y) = b0xq+ℓp + b1x

ℓp y + b2x

ℓ−qp y2 ,

(8)

has the inverse integrating factor

V (x, y) =

{yx

p+ℓ−2qp [(b1p− a0q)x

2qp + (b2p− a1q)x

qp y − a2qy

2] if b0 = 0 ,

xp+ℓ−q

p [b0px2qp + (b1p− a0q)x

qp y + (b2p− a1q)y

2] if a2 = 0 .

Proof. For system (8) we have α(u) = [b0p+ (b1p− a0q)u+ (b2p− a1q)u2 − a2qu

3]/pand β(u) = [a0(m− ℓ− p)− b1p+(a1(m+ q− ℓ− p)− 2b2p)u+a2(m+2q− ℓ− p)u2]/pfrom Theorem 4. In consequence:

5

• If b0 = 0 then taking m = p + q + ℓ we have f(1, u) = −u[b1p + a0q + (a1q −b2p)u + a2qu

2]. Undoing the blow–up we obtain V (x, y) = xp+q+ℓ

p f(1, y/xqp ) =

yxp+ℓ−2q

p [(b1p−a0q)x2qp +(b2p−a1q)x

qp y−a2qy

2] which is an inverse integratingfactor of system (8).

• If a2 = 0 then, taking m = p+q+ℓ, we have f(1, u) = b0p+(b1p−a0q)u+(b2p−a1q)u

2. In the initial coordinates we obtain V (x, y) = xp+q+ℓ

p f(1, y/xqp ) which

gives, for system (8), the inverse integrating factor V (x, y) = xp+ℓ−q

p [b0px2qp +

(b1p− a0q)xqp y + (b2p− a1q)y

2]. 2

The p−q homogeneous systems with p even and q and ℓ odd are included in the classof time-reversible systems. For the polynomial time–reversible 2n−1 quasihomogeneousvector fields of weighted degree 1 we have the next example.

Example 6 The most general planar polynomial 2n−1 quasihomogeneous of weighteddegree 1 time–reversible system is given by

x = y(ax+ by2) , y = cx+ dy2 , (9)

if n = 1 and byx = y(ax+ by2n) , y = cy2 , (10)

if n = 1. V1(x, y) = −2cx2 + ay2x − 2dxy2 + by4 and V2(x, y) = y2+ac with c = 0

are inverse integrating factors of systems (9) and (10) respectively. In addition, bothsystems (9) and (10) have no limit cycles.

Proof. Since α(u) = [2c+ (2d− a)u2 − bu4]/2 and β(u) = u[−2a− 4d+m(a+ bu2)]/2for system (9), applying Theorem 4 we have

f(1, u) = (2d−a+√∆− 2bu2)

(m−4)(a+2d)+m√

∆

4√

∆ (a− 2d+√∆+2bu2)

−(m−4)(a+2d)+m√

∆

4√

∆ ,

where ∆ := a2 − 4ad+4(2bc+ d2). Taking m = 4 and undoing the blow–up we obtainx2f(1, y/

√x) = −4bV1(x, y). Therefore V1 is a polynomial inverse integrating factor

of system (9).

Now we restrict our attention to the problem of the existence, nature and location inphase plane of limit cycles for system (9). First of all notice that, since V1 is polynomial,it is defined in the whole plane and therefore any limit cycle of the system (if it exists)must be algebraic and contained in the zero level set of V1, see [Giacomini et al., 1996].We can assume that c = 0 since otherwise y = 0 is an invariant straight line throughthe origin of the system and hence it does not have limit cycles. Furthermore, assumethat bc − ad = 0 or equivalently b = (ad)/c. In this particular case we can supposea+2d = 0 because otherwise the components of the system are not coprimes and aftera time rescaling the system becomes linear and does not have limit cycles. Moreover,equations (3) have the nontrivial solution

(c1, c2) =

(2a

c(a+ 2d)2,

−2

a+ 2d

)∈ IR2 ,

and in consequence there is absence of limit cycles from Corollary 3. We continue withthe hypothesis c(bc−ad) = 0 over the parameters of the system. In this case equations

6

(3) have the solution

c1 =1

2c(bc− ad)2[a(−bc+ 2d2)− 2d2{2d+

√∆} − bc{6d+

√∆}] ,

c2 =1

2(bc− ad)[a+ 2d+

√∆] .

If d = 0 then the above solution is nontrivial. Moreover, if ∆ ≥ 0 then (c1, c2) ∈ IR2

and there are no limit cycles in the phase plane from Corollary 3.In summary, if system (9) has a limit cycle then it is algebraic, contained into

the algebraic curve V1(x, y) = 0 with cd(bc − ad) = 0 and ∆ < 0. But if ∆ < 0 thenV1(x, y) = 0 only has an isolated point in IR2 and system (9) does not have limit cycles.

Let us now study system (10). Since α(u) = u2[2nc− a− bu2n+1]/(2n) and β(u) =u[ma− 2n(a+ 2c) +mbu2n+1]/(2n), we have

f(1, u) = uma−2n(a+2c)

a−2nc (a− 2nc+ bu2n+1)2n[a−(m−2)c](1+2n)(a−2nc) .

Finally, if c = 0, taking m = 2 + a/c and undoing the blow–up we obtain V2(x, y) =

x2+a/c

2n f(1, y/x1/2n) = y2+ac as inverse integrating factor for system (10). Obviously

the system cannot have limit cycles since y = 0 is an invariant straight line throughthe origin. 2

5 Kovalevskaya exponents for quasihomogeneous vec-tor fields

Let us consider the p − q quasihomogeneous system (1) of weighted degree ℓ = 0 orequivalently its associated vector field X = (P,Q). We make the following nonau-tonomous change of variables

(x, y) → (u, v) =(tp/ℓx− c1, t

q/ℓy − c2

),

which is a variation of system (1) around the particular solution of Lemma 2. If inthese new variables we add the logarithmic time change τ = log t, then system (1) isexpressed like the following autonomous system

u′ =p

ℓ(u+ c1) + P (u+ c1, v + c2) , v′ =

q

ℓ(v + c2) +Q(u+ c1, v + c2) ,

where the prime denotes differentiation with respect to τ . Moreover, since P and Q areanalytic, we can expand such functions around (c1, c2) and we have P (u+ c1, v+ c2) =P (c1, c2)+ < ∇P (c1, c2), (u, v) > +P (u, v) and Q(u + c1, v + c2) = Q(c1, c2)+ <∇Q(c1, c2), (u, v) > +Q(u, v) where <,> stands for the euclidean inner product andP and Q denote higher order terms. Finally, taking into account that (c1, c2) verifiesfrom definition P (c1, c2) +

pℓ c1 = 0 and Q(c1, c2) +

qℓ c2 = 0, we have(

u′

v′

)= K

(uv

)+

(P (u, v)

Q(u, v)

), (11)

where K := DX (c1, c2)+diag{p/ℓ, q/ℓ} is the Kovalevskaya matrix and represents thelinear part of system (11). The eigenvalues of the Kovalevskaya matrix are called Ko-valevskaya exponents. It is not difficult to prove that λ1 = −1 is always a Kovalevskayaexponent, see [Yoshida, 1983].

7

6 Semi-quasihomogeneous systems

Let us consider a planar system of differential equations

x = X(x, y) , y = Y (x, y) , (12)

defined in lC2 and analytic in a neighbourhood of the origin (0, 0). If system (12)is a p − q quasihomogeneous system of weighted degree ℓ then all the terms in theexpansions of X(x, y) =

∑i,j aijx

iyj and Y (x, y) =∑

i,j bijxiyj verify ip+ jq = p+ ℓ

and ip+ jq = q + ℓ respectively.

Definition 7 (Furta, 1996) The analytic system (12) is called p− q positively semi-quasihomogeneous of weighted degree ℓ if it can be expressed as

x = P (x, y) + P (x, y) , y = Q(x, y) + Q(x, y) , (13)

where (P,Q) is a p − q quasihomogeneous vector field of weighted degree ℓ and allthe terms in the expansions P (x, y) =

∑i,j aijx

iyj and Q(x, y) =∑

i,j bijxiyj satisfy

ip + jq > p + ℓ and ip + jq > q + ℓ respectively. If the above inequalities are reversedthen system (13) is called p− q negatively semi-quasihomogeneous of weighted degreeℓ. Moreover, the truncated system x = P (x, y), y = Q(x, y) is termed the quasihomo-geneous cut.

In [Furta, 1996], the author proves the following result.

Theorem 8 (Furta) Let system (13) be semi-quasihomogeneous. If the Kovalevskayamatrix is diagonalizable and its eigenvalues λ1, λ2 do not satisfay any resonant condi-tion

k1λ1 + k2λ2 = 0 , k1, k2 ∈ IN ∪ {0} , k1 + k2 ≥ 1 ,

then system (13) does not have any polynomial first integral. Moreover, if system (13)is positively semi-quasihomogeneous then there exists no smooth first integrals whichcan be expanded into formal Maclaurin series in a neighbourhood of the origin.

Remark 2. In fact, Furta’s results in [Furta, 1996] are stated for semi-quasihomogeneoussystems in lCn. Since in this work we study the particular planar case n = 2, takinginto account that λ1 = −1, the resonant condition of Theorem 8 reduces to λ2 ∈ lQ.

Let us give some necessary conditions in order to have either an analytic inverseintegrating factor or an analytic first integral for positively semi-quasihomogeneoussystems.

Theorem 9 Assume that a positively semi-quasihomogeneous system (13) has an an-alytic inverse integrating factor (resp. analytic first integral) in a neighbourhood ofthe origin. Then, its quasihomogeneous cut possesses a polynomial inverse integratingfactor (resp. polynomial first integral).

Proof. We only give the proof for the case of having an inverse integrating factor. Theother case is totally simmilar.

Let us define the change of variables u = αpx, v = αqy. Under the action ofthe similarity transformation (2), the positively semi-quasihomogeneous system (13) istransformed into

u = P (u, v) + αp+ℓP (α−pu, α−qv) , v = Q(u, v) + αq+ℓQ(α−pu, α−qv) , (14)

8

where αp+ℓP (α−pu, α−qv) and αq+ℓQ(α−pu, α−qv) are formal power series with re-spect to 1/α without any constant term.

On the other hand, let V (x, y) be an analytic inverse integrating factor in a neigh-bourhood of the origin for system (13). Hence it can be expressed like V (x, y) =∑∞

i=k Vi(x, y) where Vi are homogeneous polynomials of degree i and Vk ≡ 0. Let usdefine ξ = min{p, q}. In the new variables, V becomes

V (u, v) = V (α−pu, α−qv) =

∞∑i=k

Vi(α−pu, α−qv) =

∞∑i=k

α−iξVi(u, v;α)

= α−kξ∞∑i=0

Vk+i(u, v;α)

αiξ.

Hence

W (u, v;α) =∞∑i=0

Vk+i(u, v;α)

αiξ

is an inverse integrating factor of system (14).Finally, system (14) approaches to the quasihomogeneous cut u = P (u, v), v =

Q(u, v) as α → ∞. Moreover W (u, v;α) approaches to Vk(u, v;∞) as α → ∞. HenceVk(u, v;∞) is a polynomial inverse integrating factor of the quasihomogeneous cut andthe theorem is proved.

Remark 3. In [Chavarriga, Giacomini & Gine, 2000], necessary conditions aregiven in order to have polynomial inverse integrating factors for polynomial planarvector fields. However, by using Theorem 4 we determine all the inverse integratingfactors of the quasihomogeneous cut. Therefore it is easy to check if the quasihomoge-neous cut has or does not have a polynomial inverse integrating factor.

Remark 4. Let us consider an analytic vector field

x = P (x, y) =

∞∑k=m

Pk(x, y) , x = Q(x, y) =

∞∑k=m

Qk(x, y) , (15)

where Pk and Qk are homogeneous polynomials of degree k and m ≥ 1. Obviouslysystems (15) are positively semi-quasihomogeneous with p = q = 1 and ℓ = m − 1.In fact, for this particular case the necessary condition given by Theorem 9 is verifiedtrivially as we will see next. Firstly, observe that V = xQm − yPm is always a poly-nomial inverse integrating factor of the homogeneous cut x = Pm(x, y), y = Qm(x, y),provided that V ≡ 0. Moreover, let H(x, y) =

∑∞k=n Hk(x, y) be an analytic first

integral in a neighbourhood U of the origin for system (15). Here Hk are homogeneouspolynomials of degree k. Then, H must verify P∂H/∂x+Q∂H/∂y ≡ 0 on U . Equal-ing to zero the homogeneous polynomials of same degree in the previous expressionPm∂Hn/∂x + Qm∂Hn/∂y ≡ 0 holds on U . Therefore Hn(x, y) is a polynomial firstintegral of the homogeneous cut.

In the next example, let us consider any 2n− 1 positively semi-quasihomogeneoussystem of weighted degree 1 where its quasihomogeneous cut is given by the 2n − 1quasihomogeneous vector field (10) of weighted degree 1 of Example 6.

9

Example 10 For the 2n− 1 positively semi-quasihomogeneous system of weighted de-gree 1

x = y(ax+ by2n) + P (x, y) , y = cy2 + Q(x, y) , (16)

where n ∈ IN\{0, 1} and all the terms in the expansions P (x, y) =∑

i,j aijxiyj and

Q(x, y) =∑

i,j bijxiyj satisfy 2ni + j > p + 1 and 2ni + j > q + 1 respectively, the

following holds:

(i) If c(2cn− a) = 0 and a/c ∈ lQ then system (16) does not have any polynomial firstintegral.

(ii) If bc(2cn− a) = 0 and a/c ∈ IN∪ {−2,−1, 0} then system (16) does not have anyanalytic inverse integrating factor defined in a neighborhood of the origin.

Proof. For the quasihomogeneous cut (P (x, y), Q(x, y)) = (y(ax+by2n), cy2) of system(16), equations (3) have the nontrivial solution(

b/c2n

2cn− a,−1

c

),

when a = 2cn and c = 0. Furthermore the Kovalevskaya matrix associated to suchsolution is

K =

(2n− a

c2bn[c(2n+1)−a]

c2n(2cn−a)

0 −1

),

which eigenvalues are λ1 = −1 and λ2 = 2n − a/c. Therefore, from Theorem 8, weconclude with statement (i) of Example 10.

The results of this paper are applied in the following way. Let us notice that if b = 0then the quasihomogeneous cut (P,Q) of system (16) reduces to an homogeneous vectorfield and therefore V = xQ − yP ≡ 0 is a polynomial inverse integrating factor for itwhen a = c. Otherwise, if b = 0 and a = c then V = x3 is a polynomial inverseintegrating factor.

We continue without loost of generality by assuming b = 0. From Theorem 4 weobtain for the vector field (P,Q) the most general inverse integrating factor

V (x, y) =

xa+c(2−m)

a−2cn ya(m−2n)−4cn

a−2cn

(a− 2cn+ by2n

x

) a+c(2−m)a−2cn

if a = 2cn ,

ym exp(

c(2n+2−m)b

xy2n

)if a = 2cn ,

which is a quasihomogeneous function of degree m. In order to have V polynomial wehave the next possibilities:

• If a = 2cn and c = 0 then V = ax + by2n is a polynomial inverse integratingfactor. Otherwise, i.e., if a = 2cn and c = 0 then a necessary condition in orderto have a polynomial inverse integrating factor is a + c(2 − m) = 0. From thisrelation we have m = 2+a/c and V becomes V = y2+a/c. Then V is polynomialif and only if a/c ∈ IN ∪ {−2,−1, 0}.

• Put a = 2cn. In this case, taking m = 2n+2 the exponential term of V vanishesand moreover V becomes polynomial.

Finally, applying Theorem 9, we have proved statement (ii) of Example 10. 2

10

References

Andreev, A. [1958] “Investigation on the behaviour of the integral curves of a systemof two differential equations in the neighborhood of a singular point”, TranslationAmer. Math. Soc. 8, 187–207.

Andronov, A.A., Leontovich, E.A., Gordon, I.I. and Maier, A.G. [1973]“Theory of bifurcations of dynamic systems on a plane”, John Wiley & Sons, NewYork.

Chavarriga, J., Giacomini, H., Gine, J. and Llibre, J. [1999] “On the integra-bility of two-dimensional flows”, Journal of Differential Equations 157, 163–182.

Chavarriga, J., Giacomini, H. and Gine, J. [2000] “Polynomial inverse integrat-ing factors”, Annals of Differential Equations 16, 320–329.

Furta, S.D. [1996] “On non-integrability of general systems of differential equations”,Z. angew Math. Phys. 47, 112–131.

Giacomini, H., Llibre, J. and Viano, M. [1996] “On the nonexistence, existence,and uniqueness of limit cycles”, Nonlinearity 9, 501–516.

Poincare, H. [1885] “Memoire sur les courbes definies par les equationsdifferentielles”, J. de Mathematiques Pures et Appliquees (4) 1, 167–244; Oeu-vres de Henri Poincare, vol. I, Gauthier-Villars, Paris 1951, 95–114.

Yoshida, H. [1983] “Necessary condition for the existence of algebraic first integrals”,Celestial Mechanics 31, 363–379; 381–399.

11