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Local bifurcation of limit cycles and integrability of a class of nilpotent systems of differential equations
Applied Mathematics and Computation 215 (2009) 314–323
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Local bifurcation of limit cycles and integrability of a class of nilpotent systems of differential equations A. Algaba, C. García, M. Reyes * Department of Mathematics, Facultad de Ciencias Experimentales, Campus del Carmen, University of Huelva, Spain
a r t i c l e
i n f o
Keywords: Limit cycles Center Nilpotent systems
a b s t r a c t We study the analytic system of differential equations in the plane which can be written, in a suitable coordinates system, as _ yÞ _ T¼ ðx;
1 X
Fqpþ2is ;
i¼0
where p; q 2 N; p 6 q; s ¼ ðn þ 1Þp q > 0; n 2 N and Fi ¼ ðP i ; Q i ÞT are quasi-homogeneous vector fields of type t ¼ ðp; qÞ and degree i, with Fqp ¼ ðy; 0ÞT and Q qpþ2s ð1; 0Þ < 0. The origin of this system is a nilpotent and monodromic isolated singular point. We show the Taylor expansion of the return map near the origin for this system, which allow us to generate small amplitude limit cycles bifurcating from the critical point. Also, as an application of the theoretical procedure, we characterize the centers and we generate limit cycles of small amplitude from the origin of several families. Finally, we give a new family integrable analytically which includes the centers of the systems studied. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Probably the most basic and fundamental tool for studying the stability and bifurcations of periodic orbits is the Poincaré map or first return map, defined by Poincaré in 1881, cf. [15]. There are two classes of differential systems whose orbits near an isolated singular point can turn around it (monodromic point): the non-degenerated systems whose linear part has imaginary eigenvalues and the degenerated systems (systems whose matrix of the linear part at the origin is nilpotent or identically null). When the system has an isolated singular point with imaginary eigenvalues, rather than work directly with the return map, it follows the classical method of searching a Liapunov function V of the form Vðx; yÞ ¼ x2 þ y2 þ Oðjx; yj2 Þ defined in a neighborhood of the origin. It is known, see [14], that the function V can be constructed such that its rate of change along _ trajectories be of the form Vðx; yÞ ¼ g2 ðx2 þ y2 Þ þ g4 ðx2 þ y2 Þ2 þ where gj are polynomials in the coefficients of the system. We call g2k the kth focal value and v k ¼ g2k assuming g2j ¼ 0; j 6 k 1, the kth Liapunov constant. It has that a focus is stable or unstable according to whether the first non-zero focal value is negative or positive. Furthermore, if g2 ¼ g4 ¼ . . . ¼ g2k ¼ 0 but g2kþ2 – 0 (focus of order k), no more than k limit cycles can bifurcate from the origin under perturbations of the system, see [14]. And the critical point is a center (a neighborhood of the origin belongs to a continuous of periodic solutions) if all the focal values are zero; in such case, there exists a local analytic first integral defined at the origin. This technique cannot be used for degenerated systems, since, in general, its centers do not have analytic first integral at the origin.
* Corresponding author. E-mail address: [email protected] (M. Reyes). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.04.077
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In this work, we are interested in calculating the multiplicity of a focus, to generate small amplitude limit cycles bifurcating from the origin, and to obtain the centers of some families of nilpotent systems. An analytic system of differential equations in the plane having a nilpotent singular point, in some suitable coordinates, can be written as
x_ ¼ y þ Pðx; yÞ;
y_ ¼ Q ðx; yÞ;
ð1Þ
vwhere Pðx; yÞ and Q ðx; yÞ are analytic functions without constant nor linear terms defined in a certain neighborhood of the origin. The center problem for the system (1) was solved theoretically by Moussu [13] and by Sadovskii [16]. Strozyna and Zoladek [17] characterize the centers which have a local analytic first integral at the origin. Giacomini et al. [10,11] prove that the analytic nilpotent systems with a center can be expressed as limit of systems non-degenerated with a center, and consequently the Poincaré–Liapunov method can be used to find the nilpotent centers. There are only a few families of polynomial differential systems (1) whose centers are known. The centers of the system (1), where Pðx; yÞ ¼ P2nþ1 ðx; yÞ and Q ðx; yÞ ¼ Q 2nþ1 ðx; yÞ are homogeneous polynomials of degree 2n þ 1, have been characterized for n ¼ 1; 2 and 3, see [5] and reference therein. Chavarriga et al. [7] study the centers of (1), with n ¼ 1, having an analytic first integral. They also prove that the nilpotent systems time-reversible under the change of variables ðx; y; tÞ ! ðx; y; tÞ have an analytic first integral. Sadovskii [16] finds the centers of the family (1) for Pðx; yÞ ¼ P 2 ðx; yÞ and Q ðx; yÞ ¼ Q 3 ðx; yÞ. Gasull and Torregrosa [8], using the so-called Cherka’s method, which consists in doing a change of variables that transforms (1) into a Liénard differential equation, calculate the centers of several families of nilpotent systems. In [9], the authors deal with systems of the form (1) with Pðx; yÞ ¼ Pnþ1 ðx; yÞ þ and Q ðx; yÞ ¼ x2n1 þ Q nþ1 ðx; yÞþ Q nþ2 ðx; yÞ þ where, in this case, the vector fields ðP k ; Q k Þ are quasi-homogeneous vector fields of type ð1; nÞ and degree k. In [2,3], Álvarez and Gasull apply the normal form theory to study the center problem for monodromic planar nilpotent singularities and calculate the first two generalized Liapunov constants of (1) and they solve the stability problem of several polynomial families. In this paper, we consider the analytic system of differential equations in the plane whose origin is a nilpotent singular point
x_ ¼ y þ
1 X
Pqpþ2is ðx; yÞ;
y_ ¼
i¼1
1 X
Q qpþ2is ðx; yÞ;
ð2Þ
i¼1
where p; q; n 2 N; p 6 q; s ¼ ðn þ 1Þp q > 0, and Fi ¼ ðPi ; Q i ÞT is a quasi-homogeneous vector field of type ðp; qÞ and degree i with Q ð2nþ1Þpq ð1; 0Þ < 0 (necessary condition of monodromy). That is, according to the degree, Fqp ¼ ðy; 0ÞT is the quasihomogeneous component of minor degree, the second one is Fð2nþ1Þpq which, among others, has the term ð0; x2nþ1 ÞT , and the edges of its Newton’s polygon of the remaining components are parallel to the edge associated to F2ðnþ1Þpq . This class includes, among others, the nilpotent systems which are invariant to the change of variables ðx; yÞ ! ðx; yÞ. In particular, it includes the family
x_ ¼ y þ X 2nþ1 ðx; yÞ;
y_ ¼ Y 2nþ1 ðx; yÞ;
where X 2nþ1 and Y 2nþ1 are homogeneous polynomials of degree 2n þ 1 with Y 2nþ1 ð1; 0Þ < 0 (case p ¼ q ¼ 1; P2n ðx; yÞ ¼ X 2nþ1 ðx; yÞ; Q 2n ðx; yÞ ¼ Y 2nþ1 ðx; yÞ; P2i ¼ Q 2i ¼ 0; i > n in (2)). The main result of the paper is Theorem 2.1 which gives the Taylor expansion of the return map of the system (2). This result allows us to solve theoretically the center problem for the systems (2) (Corollary 2.1) and to generate limit cycles bifurcating from the origin of the system (Corollary 2.2). Finally, as an application, we characterize the centers of several families of (2) and we give the number of small amplitude limit cycles bifurcating from the origin. Applying Proposition 3.1, we prove that the systems obtained have a center at the origin. In particular, we conclude that all have a local analytic first integral at the origin and they can be written in the form
x_ ¼ y þ v y Kðv ; y2 Þ þ yWðv ; y2 Þ;
y_ ¼ v x Kðv ; y2 Þ;
where v ; K; W are analytic functions defined in a neighborhood of O with WðOÞ ¼ 0. Also, we give the cyclicity of a weak focus of these families and we find a system with eight limit cycles.
2. Poincaré map near the origin Recall that a function f of two variables is quasi-homogeneous of type t ¼ ðp; qÞ and degree k if f ðep x; eq yÞ ¼ ek f ðx; yÞ. The vector space of quasi-homogeneous polynomials of type t and degree k will be denoted by Ptk . A vector field F ¼ ðF 1 ; F 2 Þ is said quasi-homogeneous of type t and degree k if F 1 2 Ptkþp and F 2 2 Ptkþq . We will denote Qtk the vector space of quasi-homogeneous polynomial vector fields of type t and degree k. We consider the analytic system of differential equations
_ yÞ _ T¼ ðx;
1 X i¼0
Fqpþ2is ;
ð3Þ
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where p; q 2 N; p 6 q and without common factors, s ¼ ðn þ 1Þp q > 0; n 2 N and Fi ¼ ðP i ; Q i ÞT are quasi-homogeneous vector fields of type t ¼ ðp; qÞ and degree i, with Fqp ¼ ðy; 0ÞT and Q qpþ2s ð1; 0Þ < 0 (without loss of generality, we can assume Q qpþ2s ð1; 0Þ ¼ 1Þ. In this system, this last condition implies that the origin is a monodromic point, see Andreev [4]. Note that if p or q is even, then the origin is a center of (3). Indeed, we assume, for instance, p is even then q will be odd (since p and q have no common factors), in that case P qpþ2is ðx; yÞ ¼ Pqpþ2is ðx; yÞ and Q qpþ2is ðx; yÞ ¼ Q qpþ2is ðx; yÞ since q þ 2is is odd and 2q p þ 2is is even. The system (3) is time-reversible, i.e. has symmetrical phase portrait with regard to a straight line passing through the origin (y ¼ 0, in this case), changing time direction. So, O is a center, since it is monodromic. In what follows, we assume that p and q are odd. We already introduce the generalized polar coordinates. Given any natural number n 2 N, it defines the generalized trigonometric functions, xðhÞ ¼ CsðhÞ; yðhÞ ¼ SnðhÞ, as the unique solution of the Cauchy problem
dx ¼ y; dh
dy ¼ x2nþ1 ; dh
with xð0Þ ¼ 1; yð0Þ ¼ 0. These functions are T-periodic with
rffiffiffiffiffiffiffiffiffiffiffiffi C 1 p 2nþ2 T :¼ 2 n þ 1 C nþ2 2nþ2
and they verify the equality Cs2nþ2 ðhÞ þ ðn þ 1ÞSn2 ðhÞ ¼ 1, see [12] for a proof. We can introduce the generalized polar coordinates, r and h of the real plane ðx; yÞ 2 R2 , as
y ¼ rnþ1 SnðhÞ:
x ¼ rCsðhÞ;
ð4Þ
Furthermore, the following equalities hold
r_ ¼
x2nþ1 x_ þ yy_ ; r 2nþ1
xy_ ðn þ 1Þyx_ h_ ¼ : r nþ2
The return map of (3) is analytic, see [12]. Now we provide a expression of the Taylor expansion of this return map. Theorem 2.1. Let system (3) with p and q odd. The return map of system (3) has the form
PðuÞ ¼ u
Z T 1 X 1 ðunþð2l1Þsþ1 fl Cs3nsþ2lsþ2 ðhÞdhÞð1 þ OðuÞÞ: 2ðn þ 1Þ l¼1 0
where fl 2 R; l P 1, are polynomials in the coefficients of the right-hand sides of (3) (we will call fk the focus quantities of the singular point O of the system (3)). Proof. By making the change (4), after omitting a common factor r n , the system (3) takes the form
r_ ¼ rf ðr; hÞ;
h_ ¼ 1 þ rgðr; hÞ
ð5Þ
with f and g analytic functions and f ð0; hÞ ¼ gð0; hÞ ¼ 0, for all h.We now define the variable u verifying
u2ðnþ1Þ ¼ W rCsðhÞ; r nþ1 SnðhÞ
ð6Þ
being W a C1 -function such that its derivative along the trajectories of the system (3) has the form
_ ¼ x3nsþ2 W
1 X
fl x2ls þ sðx; yÞ
l¼1
where fl ; l P 1, are polynomials in the coefficients of the right-hand sides of (3) and s is a flat function at the origin, see [1]. Let suppose fl ¼ 0 for l ¼ 1; . . . ; m 1 and fm – 0. The expression (6) is valid for r > 0 and for all h. Furthermore, as
u2ðnþ1Þ ¼ W rCsðhÞ; r nþ1 SnðhÞ ¼ r2ðnþ1Þ þ Oðr 2nþ3 ; hÞ
ð7Þ
from inverse function theorem, it has that r ¼ u þ Oðu2 ; hÞ. Next, we express (3) in the new coordinates system ðu; hÞ
1 _ W 2ðn þ 1Þu2nþ1 fm ¼ Cs3nsþ2þ2ms ðhÞu2msþnsþ1 ð1 þ Oðu; hÞÞ; 2ðn þ 1Þ
u_ ¼
h_ ¼ 1 þ Oðu; hÞ
ð8Þ ð9Þ ð10Þ
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whose differential equation associated can be written as
du fm ¼ Cs3nsþ2þ2ms ðhÞu2msþnsþ1 ð1 þ Oðu; hÞÞ: dh 2ðn þ 1Þ
ð11Þ
We write the solution of (11) starting at u ¼ u0 when h ¼ 0 as
uðh; u0 Þ ¼
1 X
ai ðhÞui0 þ sðh; u0 Þ;
ð12Þ
i¼1
with a1 ð0Þ ¼ 1; ai ð0Þ ¼ 0 for i P q2; sð0; u0 Þ ¼ 0 and fðu; hÞ ¼ ðu0 ; 0Þ; u0 > 0g to itself is given by the series
s flat at u0 ¼ 0. Hence the Poincaré return map from the section
Pðu0 Þ ¼ a1 ðTÞu0 þ a2 ðTÞu20 þ :
ð13Þ
By replacing (12) in the equation differential (11) we obtain
a1 ðhÞ 1; ai ðhÞ 0; for i ¼ 2; . . . ; n þ ð2m 1Þs and
anþð2m1Þsþ1 ðTÞ ¼
fm 2ðn þ 1Þ
Z
T
Cs3nsþ2þ2ms ðhÞdh:
0
Remark. The C1 -function W above mentioned in the proof of Theorem 2.1 is not a Liapunov function, since it is not a defined positive function in a neighborhood of the origin. Therefore, it cannot be used for finding limit cycles which bifurcate from the origin. As a consequence, the only significative constant fl is the first one different from zero. It does that the return map differ from the identity map, and it determines the stability of the origin. Also, let notice that the origin is a center if and only if PðuÞ u. So, we have the following results: the first one characterizes the centers of the system (3) and the second result is related to the number of small amplitude limit cycles which can bifurcate from the origin. Corollary 2.1. The origin of (3) with p and q odd is a center if and only if fl ¼ 0, for all l P 1. Proof. As p and q are odd and s ¼ ðn þ 1Þp q, it implies that s is even (odd) if and only if n is even (odd). Thereby, RT 3n s þ 2 þ 2ms is even. So, 0 Cs3nsþ2þ2ms ðhÞdh is a positive value. The result is followed as a consequence. h If we want to find the systems with a center of a polynomial family XðkÞ; k 2 Rm , of systems (3) with p and q odd, we calculate recursively the sets on Rm :
X1 ¼ fk 2 Rm ; f 1 ðkÞ ¼ 0g; Xk ¼ fk 2 Xk1 ; f k ðkÞ ¼ 0g; for k P 2: By Hilbert Basis Theorem, we know that there is a M such that
X1 X2 XM XMþ1 ¼ XMþ2 ¼ : So, the systems Xðk Þ with k 2 XMþ1 have a center at the origin. Also, in such a case, it is said that M is the order of the family XðkÞ. The focus quantities of system (3) can also be used to prove the existence of a certain number of small amplitude limit cycles bifurcating from the nilpotent critical point of a family of systems (3). Next result is used in order to study the degenerate Andronov–Hopf bifurcation, i.e. we analyze the existence of limit cycles which can bifurcate from the origin of XðkÞ under variations of the parameters k. Corollary 2.2. Let XðkÞ be a family of systems (3) with p and q odd, depending on some parameters k 2 Rm . We assume that k enough close to k such that f1 ð kÞ; f 2 ð kÞ; . . . ; k 2 Xr n Xrþ1 (i.e. O is a weak focus of order r of Xðk ÞÞ. If there exists kÞ; f r ð kÞ alternate sign and f r1 ð
0 < jf1 ðkÞj jf2 ðkÞj jfr1 ðkÞj jfr ðkÞj 1 then system Xð kÞ has exactly r limit cycles in a neighborhood of the origin. Proof. By Theorem 2.1, the Taylor expansion of the Poincaré return map of Xð kÞ has the form
PðuÞ ¼ u w1 f1 ðkÞð1 þ uh1 ðuÞÞuj1 w2 f2 ðkÞð1 þ uh2 ðuÞÞuj2 . . . R T 3nsþ2þ2ms 1 where wm ¼ 2ðnþ1Þ Cs ðhÞdh > 0; jm ¼ n þ ð2m 1Þs þ 1 and hm are analytic functions at the origin. 0 Each small limit cycle around the origin corresponds to each positive fixed point of the Poincaré return map of Xð kÞ, i.e. positive zeros of the function
FðuÞ ¼ u PðuÞ ¼
rþ1 X m¼1
ð1 þ uhm ðuÞÞwm fm ðkÞujm þ Oðjrþ1 þ 1Þ:
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m ðuÞÞ, where h m are analytic functions at the origin, it has FðuÞ ¼ ð1 þ uh1 ðuÞÞ By writing 1 þ uhm ðuÞ ¼ ð1 þ uh1 ðuÞÞð1 þ h uj1 F 0 ðuÞ where
F 0 ðuÞ ¼ w1 f1 ðkÞ þ
rþ1 X
m ðuÞÞujm j1 þ Oðj j þ 1Þ: wm fm ðkÞð1 þ uh rþ1 1
m¼2
We must look for positive zeros of F 0 . By differentiating, we have
F 00 ðuÞ ¼
rþ1 X
ð1 þ ug^m ðuÞÞðjm j1 Þwm fm ðkÞujm j1 1 þ Oðjrþ1 j1 Þ;
m¼2
where
m ðuÞÞ þ uðh m ðuÞ þ ug0 ðuÞÞ: ðjm j1 Þð1 þ ug^m ðuÞÞ ¼ ðjm j1 Þð1 þ uh m ~ m ðuÞÞ, where h ~ m are analytic functions at the origin, F 0 has the form ^m ðuÞ ¼ ð1 þ ug^1 ðuÞÞð1 þ h By writing 1 þ ug 0 0 j2 j1 1 ~ F 1 where F 0 ðuÞ ¼ ð1 þ uhm ðuÞÞu
F 1 ðuÞ ¼ ðj2 j1 Þw2 f2 ðkÞ þ
rþ1 X ~ m ðuÞÞujm j2 þ Oðj j þ 1Þ: ðjm j1 Þwm fm ðkÞð1 þ uh rþ1 2 m¼3
Now, the number of positive zeros of F 0 cannot exceed the number of positive zeros of F 1 by more than unity. By continuing this process a further step we obtain a function F 2 such that the number of positive zeros of F 1 cannot exceed the number of positive zeros of F 1 by more than unity. So, the number of positive zeros of F 0 cannot exceed the number of positive zeros of F 2 by more than two. This process stops at the rth step when we obtain a function F r of the form
F r ðuÞ ¼ ðjrþ1 j1 Þwrþ1 frþ1 ðkÞ þ Oð1Þ; which does not have zeros in a neighborhood of origin, since frþ1 ð kÞ is close to frþ1 ðk Þ – 0, by continuity. Therefore, F cannot kÞ; . . . ; f r1 ð kÞ and fr ð kÞ alternate sign and satisfy have more than r positive zeros. Moreover, as f1 ðkÞ; f 2 ð kÞj jf2 ð kÞj jfr1 ð kÞj jfr ð kÞj, we can assure the existence of r limit cycles of small amplitude bifurcating of 0 < jf1 ð the origin of Xðk Þ. h 3. Nilpotent centers and cyclicity of a weak focus of several families of polynomial systems In this section, by applying Corollary 2.1 and 2.2, we characterize the centers and the order of a weak focus of three families of systems (3). First, on the one hand, we give the following result which we will use in order to prove the analytic integrability of the centers of several families of (3). We recall that if the system (3) is monodromic, the existence of an analytic first integral is a sufficient condition so that the origin be a center. Proposition 3.1. The nilpotent systems
x_ ¼ y þ v y Kðv ; y2 Þ þ yWðv ; y2 Þ;
y_ ¼ v x Kðv ; y2 Þ;
ð14Þ
where v ; K; W are analytic functions defined in a neighborhood of the origin with WðOÞ ¼ 0, are integrable analytically in a neighborhood of the origin. Proof. Doing the change of variables u ¼ y2 , denoting dds ¼0 , the system (14) becomes
u0 ¼
2Kðu; v Þ ; 1 þ wðu; v Þ
v 0 ¼ 1:
v ¼ v ðx; yÞ, by redefining the variable time by ds ¼ yv x ð1 þ Wðx; yÞÞdt and by ð15Þ
From Cauchy–Arnold’s Theorem (see Bruno [6], p. 98), the system (15) has got an analytic first integral Hðu; v Þ ¼ cte defined e yÞ ¼ Hðy2 ; v ðx; yÞÞ ¼ cte, which is in a neighborhood of O. Undoing the change of variable, (14) has a first integral of (14), Hðx; analytic in a neighborhood N, since it is a composition of analytic functions. Also, if we denote X the vector field associated to e X ¼ 0, for all ðx; yÞ 2 N n frv X – 0g. So, by continuity, r H e X ¼ 0, for all ðx; yÞ 2 N, that is H e is a local ana(14), it has r H lytic first integral of (14). h Remarks For v ðx; yÞ ¼ x the systems (14) turn out
ðx; y2 Þ; x_ ¼ y þ yW
y_ ¼ Kðx; y2 Þ;
ð16Þ
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319
that is, the family of nilpotent systems time-reversible under the change of variables ðx; y; tÞ ! ðx; y; tÞ.The analytic integrability of the nilpotent systems (16) is one of the main results of Chavarriga et al. [7]. For
v ¼ v 2m ;
Kðy2 ; v Þ ¼ v p2m ; Wðy2 ; v Þ ¼
p X
2mðkþ1Þ2 ak v pk 2m y
k¼0
where p P 0;
v 2m is a homogeneous polynomial of degree 2m and ak arbitrary constants, the systems (14) come given by
8 p > < x_ ¼ y þ ov 2m v p þ P ak v pk y2mðkþ1Þ1 ; 2m 2m oy > :_ y ¼ ovox2m v p2m :
ð17Þ
k¼0
These systems include the nilpotent family integrable analytically given in [5], Lemma 2. On the other hand, we prove the following results which will use in order to obtain a simpler expression of the focus quantities of (3). Proposition 3.2. For all C1 vector fields X and U such that ½X; U ¼ mX with V ¼ X ^ U – 0, it has that
ðm þ div ðUÞÞX ¼ J rV þ div ðXÞU; where ½X; U denotes Lie’s bracket of the fields X and U, and J ¼
0 1
m a C1 -function in a neighborhood of the origin and
1 . 0
Proof. It is easy to prove that for all C1 vector fields X and U, it holds
rV U ¼ ½X; U ^ U þ Vdiv ðUÞ;
rV X ¼ ½X; U ^ X þ Vdiv ðXÞ:
ð18Þ
In our case, rV U ¼ ðm þ div ðUÞÞV and rV X ¼ div ðXÞV. As V – 0, it has that JX and JU are transversals. Therefore, there exist a; b functions of class one in a neighborhood of the origin such that rV ¼ aJX þ bJU. Hence, rV X ¼ bV; rV U ¼ aV. Thus,
V rV ¼ ðaVÞJX þ ðbVÞJU ¼ ðrV UÞJX ðrV XÞJU ¼ ðm þ div ðUÞÞVJX div ðXÞVJU; it follows the result easily.
h
We emphasize the following decomposition which is obtained from Proposition 3.2 with X ¼ Fk 2 Qtk and U ¼ Dt ¼ ðpx; qyÞT where t ¼ ðp; qÞ, and by taking into account the quasi-homogeneous character of Fk and ½Fk ; Dt ¼ kFk . Corollary 3.1 (Conservative-dissipative decomposition). For each t ¼ ðp; qÞ, given Fk ¼ ðP k ; Q k ÞT 2 Qtk , there exist hk 2 Ptkþpþq and lk 2 Ptk such that
Fk ¼
1 ðXhk þ lk Dt Þ; kþpþq
with hk ¼ Fk ^ Dt and
ð19Þ
lk ¼ divðFk Þ, where Dt :¼ ðpx; qyÞT and Xhk :¼
ohk oy
T k ðx; yÞ; oh ðx; yÞ . ox
We now show several applications of our research. We first consider the 11-parameter nilpotent system
x_ ¼ y þ a1 x3 þ a2 x2 y þ a3 xy2 þ a4 y3 þ a6 xy4 þ a7 y5 ;
ð20Þ
y_ ¼ x3 þ b1 x2 y þ b2 xy2 þ b3 y3 þ b5 xy4 þ b6 y5 : _ yÞ _ T ¼ F0 þ F2 þ F4 , with Fi 2 Qti ; i ¼ 0; 2; 4; t ¼ ð1; 1Þ, and This is a subfamily of (3) given by ðx;
F0 ¼
y 0
;
F2 ¼
a1 x3 þ a2 x2 y þ a3 xy2 þ a4 y3 x3 þ b1 x2 y þ b2 xy2 þ b3 y3
!
;
F4 ¼
a6 xy4 þ a7 y5 b5 xy4 þ b6 y5
!
:
The following result characterizes the centers of the systems (20). We note that the focus quantities of (3) have been computed by means of the recursive procedure developed in [1], and they have the form
f1 ¼ a1 g 1 ;
f i ¼ ai g i þ
i1 X
bi;j fj ; i P 2;
j¼1
with ai positive constants and bi;j polynomials in the coefficients of the right-hand sides of (3). Theorem 3.1. The origin of the system (20) is a center if and only if one of the following two series is satisfied: (i) 3a1 þ b1 ¼ 2a2 þ b2 ¼ a3 þ 3b3 ¼ b5 ¼ a6 þ 5b6 ¼ 0 (Hamiltonian system).
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(ii) b1 þ 3a1 ¼ b3 a1 b2 þ 2a21 ¼ a3 þ a1 b2 2a2 þ 6a21 ¼ b6 a1 b5 ¼ a6 þ a1 b5 ¼ 0.Moreover, each one of them has a local analytic first integral. Proof. Taking t ¼ ð1; 2Þ, (that is, the type of the vector field ðy; x3 ÞT , i.e. the quasi-homogeneous principal part of the system _ yÞ _ T ¼ Xh þ lDt , being h the defined (20)) and applying Corollary 3.1, degree to degree, the system (20) comes given by ðx; positive function
hðx; yÞ ¼ and
1 1 1 1 1 1 1 1 ð2y2 þ x4 Þ c1 x3 y c2 x2 y2 c3 xy3 c4 y4 c5 x2 y4 c6 xy5 c 7 y6 4 5 6 7 8 10 11 12
ð21Þ
lðx; yÞ ¼ 15 d1 x2 þ 16 d2 xy þ 17 d3 y2 þ 101 d5 xy3 þ 111 d6 y4 where c1 ¼ b1 2a1 ; d1 ¼ 3a1 þ b1 ; c2 ¼ b2 2a2 ; d2 ¼ 2a2 þ 2b2 ; c3 ¼ b3 2a3 ; d3 ¼ a3 þ 3b3 ; c4 ¼ 2a4 ; c6 ¼ 2a6 ;
d5 ¼ 4c5 ¼ 4b5 ;
d6 ¼ a6 þ 5b6 ; c7 ¼ 2a7 :
The first four focus quantities are
g 1 ¼ d1 ; g 2 ¼ d2 c1 þ 5d3 ; g 3 ¼ d2 25d2 c1 þ 175c1 c2 þ 42c31 þ 375c3 ; g 4 ¼ 4c5 c1 þ 5d6 : In this point, we distinguish two cases depending on the coefficient d2 . If d2 – 0, some necessary conditions for the origin to 1 c1 ð25d2 þ 175c2 þ 42c21 Þ; d6 ¼ 45 c1 c5 . In such a be a center are g 1 ¼ g 2 ¼ g 3 ¼ g 4 ¼ 0, i.e. d1 ¼ 0; d3 ¼ 15 d2 c1 ; c3 ¼ 375 case, it has g 5 ¼ d2 ð3c1 c5 þ 5c6 Þ. Hence, c6 ¼ 35 c1 c5 . Thus, we arrive to the system given by (ii),
x_ ¼ y þ a1 x3 þ a2 x2 y a1 b2 2a2 þ 6a21 xy2 þ a4 y3 a1 b5 xy4 þ a7 y5 ; y_ ¼ x3 3a1 x2 y þ b2 xy2 þ a1 b2 þ 2a21 y3 þ b5 xy4 þ a1 b5 y5 :
ð22Þ
This system belongs to the family (14) given in Proposition 3.1 where
v ¼ 12 x2 þ a1 xy þ 12 a2 a21 y2 ; Kðv ; y2 Þ ¼ 2h ða2 þ b2 Þy2 b5 y4 ;
Uðv ; y2 Þ ¼ a4 þ a2 2a21 2a21 þ b2 y2 þ a7 þ b5 a2 2a21 y4 and, therefore, the origin is a center, since the system has a local analytic first integral and O is a monodromic point. If d2 ¼ 0, from the vanishing of the first four constants above, we have d1 ¼ d3 ¼ 0 and d6 ¼ 45 c1 c5 . In this case, the next focus quantities are g 5 ¼ c5 ð175c1 c2 þ 42c31 þ 375c3 Þ; g 6 0 and g 7 ¼ c5 ð3c1 c5 þ 5c6 Þ. If c5 ¼ 0, the necessary conditions to have a center leads us to the hamiltonian system (i) whose hamiltonian function is (21). We note that the curves hðx; yÞ ¼ cte are closed, therefore, it is a center. And if c5 – 0, g 5 and g 7 must be zero; hence,
c3 ¼
7 c1 ð25c2 þ 6c21 Þ; 375
3 c6 ¼ c1 c5 : 5
In such a case, the system is of the form (22) with d2 ¼ 0. So, O is a center.
h
Theorem 3.2. Under perturbations of the parameters of the system (20), it has: (a) if a2 þ b2 – 0 or b5 – 0, it can bifurcate 0,1,2,3 or 4 limit cycles around the origin. (b) if a2 þ b2 ¼ b5 ¼ 0, it can bifurcate 0,1 or 2 limit cycles around the origin.
Proof. We fix the constants c1 ; d2 ; c2 ; c5 above defined and consider the critical values
d1 ¼ 0;
1 d3 ¼ d2 c1 ; 5
c3 ¼
1 c1 25d2 þ 175c2 þ 42c21 ; 375
4 d6 ¼ c1 c5 ; 5
3 c6 ¼ c1 c5 : 5
Firstly, we assume that a2 þ b2 – 0. From the expression of g 1 ; g 2 ; g 3 ; g 4 and g 5 , applying Corollary 2.2, it deduces the follow ing one: if d1 – d1 there is not limit cycles around the origin. If d1 is near zero and d3 – d3 , then it can exist, at least, 1 limit cycle. If d1 and d3 alternate sign and 0 < jd1 j jd3 j and also c3 is different from c3 , then there are 2 small amplitude limit cycles. If also d3 c3 < 0; jd1 j jd3 j and d6 – d6 , then 3 limit cycles bifurcate of the origin. If we take d1 ; d3 ; c3 and d6 different from d1 ; d3 ; c3 and d6 but near each of them, respectively, and c6 – c6 such that it satisfy the hypothesis of Corollary 2.2 with r ¼ 4, it has that we can bifurcate 4 limit cycles, at least.
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Now, we assume that a2 þ b2 ¼ 0 and b5 – 0. The first constants g i different from zero are
g 1 ¼ d1 ;
g 2 ¼ d3 ;
g 4 ¼ 4c1 c5 þ 5d6 ;
g 5 ¼ c5 175c1 c2 þ 42c31 þ 375c3 ;
g 7 ¼ c5 ð3c1 c5 þ 5c6 Þ:
Thus, if d1 – 0 there is not limit cycles around the origin. If d1 is close to zero and d3 – 0, can exist, at least, 1 limit cycle. If d1 and d3 alternate sign and 0 < jd1 j jd3 j and also d6 is different from d6 , then there are 2 limit cycles of small amplitude. If we 7 c1 ð25c2 þ 6c21 Þ then there now take d6 such that d3 and d6 alternate sign and jd3 j jd6 j and also c3 is different from c3 ¼ 375 exist 3 limit cycles around the origin. If we also choose c3 near c3 such that d6 c3 < 0 and jd6 j jc3 j and take c6 – c6 , then there exist, at least, 4 limit cycles around the origin. Last on, if a2 þ b2 ¼ b5 ¼ 0, that is d2 ¼ c5 ¼ 0, the g i different from zero are
g 1 ¼ d1 ;
g 2 ¼ d3 ;
g 4 ¼ d6 :
Therefore, if d1 – 0 there is not limit cycles near the origin. If d1 is close zero and d3 – 0, it can exist, at least, 1 limit cycle. If d1 and d3 alternate sign and 0 < jd1 j jd3 j and also d6 is different from zero, then 2 small amplitude limit cycles bifurcate from the origin. h We now consider the 9-parameter subfamily of (3) given by
_ yÞ _ T ¼ F2 þ F4 þ F6 ; ðx; with Fi 2
Qti ;
F2 ¼
ð23Þ
i ¼ 2; 4; 6; t ¼ ð1; 3Þ, and
y 0
; F4 ¼
a1 x5 þ a2 x2 y x7 þ b1 x4 y þ b2 xy2
! ;
F6 ¼
!
a3 x7 þ a4 x4 y þ a5 xy2 b6 x9 þ b3 x6 y þ b4 x3 y2 þ b5 y3
with b6 ¼ 0 and b2 ¼ a2 . We already get a lower bound for its cyclicity. Theorem 3.3. Under perturbations of the parameters of the system (23), it has: (a) if 2a4 þ b4 ¼ 0, it can bifurcate 0,1 or 2 limit cycles from the origin. (b) if 2a4 þ b4 – 0 and ð5a2 þ 2ðb1 4a1 Þ2 Þðb1 4a1 Þ ¼ 0, it can bifurcate 0,1,2,3 or 4 limit cycles around the origin. (c) if 2a4 þ b4 – 0 and ð5a2 þ 2ðb1 4a1 Þ2 Þðb1 4a1 Þ – 0, it can bifurcate 0,1,2,3,4,5,6,7 or 8 limit cycles around the origin.
Proof. In this case, we apply Corollary 3.1, degree to degree, with t ¼ ð1; 4Þ (that is, the type of the vector field ðy; x7 ÞT , i.e. _ yÞ _ T ¼ Xh þ lDt , being h the the quasi-homogeneous principal part of the system (23)). The system (23) comes given by ðx; defined positive function
hðx; yÞ ¼ and
1 8 1 1 1 1 1 ðx þ 4y2 Þ c1 x5 y c 2 x2 y 2 c 3 x7 y c 4 x4 y 2 c5 xy3 ; 8 9 10 11 12 13
lðx; yÞ ¼ 19 d1 x4 þ 111 d3 x6 þ 121 d4 x3 y þ 131 d5 y2 where c1 ¼ b1 4a1 ; c5 ¼ b5 4a5
d1 ¼ 5a1 þ b1 ;
c2 ¼ 5a2 ;
c3 ¼ b3 4a3 ;
d3 ¼ 7a3 þ b3 ;
c4 ¼ b4 4a4 ;
d4 ¼ 4a4 þ 2b4 ;
d5 ¼ a5 þ 3b5 :
First, we assume that 2a4 þ b4 ¼ 0, that is d4 ¼ 0. The only g i different from zero are g 1 ¼ d1 ; g 2 ¼ d3 ; g 3 ¼ d5 . Therefore, if d1 – 0 there is a neighborhood of the origin where the system (23) does not have any limit cycle around the origin. If d1 is close to zero and d3 – 0, it can exist, at least, 1 limit cycle. If d1 d3 < 0 with 0 < jd1 j jd3 j and also d5 – 0, then there are 2 limit cycles of small amplitude. We now assume that 2a4 þ b4 – 0 and 5a2 þ 2ðb1 4a1 Þ2 ðb1 4a1 Þ ¼ 0, i.e. d4 – 0 and c2 þ 2c21 c1 ¼ 0. In this case,
g 1 ¼ d1 ;
g 2 ¼ d3 ;
g 3 ¼ d5 þ
12 c1 d4 ; 13
1 g 4 ¼ d4 c3 ; g 5 ¼ d4 c5 þ 4c1 c4 þ d4 : 13
and the remain are zero. So, if d1 – 0 there is not limit cycles around the origin. If d1 is close to zero and d3 – 0, can exist, at least, 1 limit cycle. If d1 and d3 alternate sign and 0 < jd1 j jd3 j and also d5 different from d5 ¼ 12 13 c1 d4 , then there are 2 small amplitude limit 3 limit cycles cycles. If we now take d5 such that d3 and d5 alternate sign and jd3 j jd5 j and also c3 – 0, then there exist 1 d4 then there around the origin. If we also choose c3 close to 0 such that d5 c3 < 0 and jd5 j jc3 j and take c5 – 4c1 c4 þ 13 exist, at least, 4 limit cycles bifurcating from the origin. Lastly, if 2a4 þ b4 – 0 and ð5a2 þ 2ðb1 4a1 Þ2 Þðb1 4a1 Þ – 0, by denoting q ¼ d4 c1 c2 þ 2c21 – 0, the first nine constants g i ; i ¼ 1; . . . ; 9 of (23) are
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g 1 ¼ d1 ; g 2 ¼ d3 ; g 3 ¼ d5 þ 12 c d ; 13 1 4
g 4 ¼ d4 c3 þ 2c1 c2 þ 2c21 ;
1 g 5 ¼ d4 c5 þ 4c1 c4 þ 13 d4 100 c31 c2 þ 2c21 ; 3
g 6 ¼ q c4 þ 12 d4 62 c2 c þ 2c21 ; 3 1 2
24 2 748 g 7 ¼ q d4 5 c2 15 c2 c21 1408 c41 ; 15 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ih pffiffiffiffiffiffiffiffiffiffiffiffiffiffi i g 8 ¼ q 1548c2 4681 þ 5 54049 c21 1548c2 4681 5 54049 c21 ; g 9 ¼ q 381374c2 þ 859813c21 : We can choose d1 ; d3 ; d5 ; c3 ; c5 ; c4 ; d4 and c2 adequately such that g i g iþ1 is negative, g 9 different from zero and
0 < jg 1 j jg 2 j jg 3 j jg 4 j jg 5 j jg 6 j jg 7 j jg 8 j: Applying Corollary 2.2, for r ¼ 1; 2; 3; 4; 5; 6; 7 and 8, there are regions of the parameters where the system has 0,1,2,3,4,5,6,7 or until 8 limit cycles around the origin. h The centers of (23) have been characterized in [1]. These one are included in the class of systems given by Proposition 3.1. Last on, we study the systems
x_ ¼ y þ
2nþ2 ½X 5
ai;n x
2nþ25i 3i1
y
;
y_ ¼ x2nþ1 þ
i¼1
2nþ1 ½X 5
bi;n x2nþ15i y3i :
ð24Þ
i¼1
_ yÞ _ T ¼ F2 þ F6n2 , with with 1 6 n 6 9. (½x means integer part of xÞ. These systems are of the family (3) given by ðx; t Fi 2 Qi ; i ¼ 2; 6n 2; t ¼ ð3; 5Þ. Analogously to the above application, we obtain the cyclicity of the origin of (24). Theorem 3.4. The system (24) with n equal to 1,2,3 or 4, does not have limit cycles of small amplitude surrounding the origin. There are systems inside this family for n ¼ 5 or n ¼ 6 with 0 or 1 limit cycle around the origin. For n ¼ 7 or n ¼ 8 there exist systems inside this family with 0,1,2 or 3 limit cycles around the origin. There exist systems (24) with n ¼ 9 which have 0, 1 or 2 limit cycles around the origin. Proof. Taking t ¼ ð1; n þ 1Þ and by applying Corollary 3.1, degree to degree, the system (24) takes the form
_ yÞ _ T ¼ ðx;
2nþ1 ½X 5 1 1 Xx2nþ2 þðnþ1Þy2 þ di;n x2nþ15i y3i1 Dð1;nþ1Þ þ ci;n Xx2nþ25i y3i 2n þ 2 2n þ 2 þ ð3n 2Þi i¼1
where the new coefficients that appear are
ci;n ¼ ðn þ 1Þai;n bi;n ;
di;n ¼ ð2n þ 2 5iÞai;n þ 3ibi;n
The expressions of the focus quantities different from zero are:
n ¼ 2; g 1 ¼ d1;2 ; n ¼ 3; g 1 ¼ d1;3 ; n ¼ 4; g 1 ¼ d1;4 ; n ¼ 5; g 1 ¼ d1;5 ;
g 2 ¼ c1;5 d2;5
n ¼ 6; g 1 ¼ d1;6 ;
g 2 ¼ c1;6 d2;6 ;
n ¼ 7; g 1 ¼ d1;7 ; g 2 ¼ c1;7 d2;7 þ 35d3;7 ; g 3 ¼ d2;7 33075c3;7 þ 1314c31;7 þ ð2660d2;7 þ 7665c2;7 Þc1;7 ; g 4 ¼ c51;7 d2;7 ; n ¼ 8; g 1 ¼ d1;8 ; g 2 ¼ 3c1;8 d2;8 þ 40d3;8 ; g 3 ¼ d2;8 49600c3;8 þ 2387c31;8 þ ð3960d2;8 þ 13440c2;8 Þc1;8 ; g 4 ¼ c51;8 d2;8 ; n ¼ 9; g 1 ¼ d1;9 ; g 2 ¼ c1;9 d2;9 þ 9d3;9 ; g 3 ¼ d2;9 5103c3;9 þ ð266c21;9 þ 405d2;9 þ 1539c2;9 Þc1;9 : Choosing adequately the constants ci;n and di;n and by applying Corollary 2.2 it follows the result. h
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Acknowledgements This work has been partially supported by Ministerio de Ciencia y Tecnología, Plan Nacional I+ D + I co-financed with FEDER funds, in the frame of the project MTM2007-64193 and by Consejería de Educación y Ciencia de la Junta de Andalucía (FQM276 and EXC/2008). References [1] A. Algaba, C. García, M. Reyes, The center problem for a family of systems of differential equations having a nilpotent singular point, J. Math. Anal. Appl. 340 (2008) 32–43. [2] M.J. Álvarez, A. Gasull, Monodromy and stability for nilpotent critical points, Int. J. Bifur. Chaos Appl. Sci. Eng. 15 (4) (2005) 1253–1265. [3] M.J. Álvarez, A. Gasull, Generating limit cycles from a nilpotent critical point via normal forms, J. Math. Anal. Appl. 318 (1) (2006) 271–287. [4] A. Andreev, Investigation of the behaviour of the integral curves of a system of two differential equations in the neighborhood of a singular point, Trans. Am. Math. Soc. 8 (1958) 187–207. [5] A. Andreev, A.P. Sadovskii, V.A. Tskialyuk, The center-focus problem for a system with homogeneous nonlinearities in the case zero eigenvalues of the linear part, Diff. Equat. 39 (2) (2003) 155–164. [6] A.D. Bruno, Local Methods in Nonlinear Differential Equations, Springer-Verlag, New York, 1989. [7] J. Chavarriga, H. Giacomini, J. Giné, J. Llibre, Local analytic integrability for nilpotent centers, Ergodic Theory Dynam. Syst. 23 (2003) 417–428. [8] A. Gasull, J. Torregrosa, Center problem for several differential equations via Cherka’s method, J. Math. Anal. Appl. 228 (2) (1998) 322–343. [9] A. Gasull, J. Torregrosa, A new algorithm for the computation of the Liapunov constants for some degenerated critical points, Nonlinear Anal. 47 (2001) 4479–4490. [10] H. Giacomini, J. Giné, J. Llibre, The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems, J. Diff. Equat. 227 (2) (2006) 406–426. [11] H. Giacomini, J. Giné, J. Llibre, Corrigendum to: ‘‘The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems [J. Diff. Equat. 227 (2006) 2, 406–426], J. Diff. Equat. 232 (2007) 2, 702. [12] M.A. Liapunov, Stability of Motion, Academic Press, New York, 1966. pp. 123–194. [13] R. Moussu, Symétrie et forme normale des centres et foyers dégénérés, Ergodic Theory Dynam. Syst. 2 (1982) 241–251. [14] V.V. Nemytskii, V.V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, 1960. [15] H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, J. Math. 37 (1881) 375–422. [16] A.P. Sadovskii, Problem of distinguishing a center and a focus for a system with a nonvanishing linear part, Translated Diff. Urav. 12 (7) (1976) 1238– 1246. [17] E. Strozyna, H. Zoladek, The analytic and formal normal form for the nilpotent singularity, J. Diff. Equat. 179 (2002) 479–537.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Local bifurcation of limit cycles and integrability of a class of nilpotent systems of differential equations A. Algaba, C. García, M. Reyes * Department of Mathematics, Facultad de Ciencias Experimentales, Campus del Carmen, University of Huelva, Spain
a r t i c l e
i n f o
Keywords: Limit cycles Center Nilpotent systems
a b s t r a c t We study the analytic system of differential equations in the plane which can be written, in a suitable coordinates system, as _ yÞ _ T¼ ðx;
1 X
Fqpþ2is ;
i¼0
where p; q 2 N; p 6 q; s ¼ ðn þ 1Þp q > 0; n 2 N and Fi ¼ ðP i ; Q i ÞT are quasi-homogeneous vector fields of type t ¼ ðp; qÞ and degree i, with Fqp ¼ ðy; 0ÞT and Q qpþ2s ð1; 0Þ < 0. The origin of this system is a nilpotent and monodromic isolated singular point. We show the Taylor expansion of the return map near the origin for this system, which allow us to generate small amplitude limit cycles bifurcating from the critical point. Also, as an application of the theoretical procedure, we characterize the centers and we generate limit cycles of small amplitude from the origin of several families. Finally, we give a new family integrable analytically which includes the centers of the systems studied. Ó 2009 Elsevier Inc. All rights reserved.
1. Introduction Probably the most basic and fundamental tool for studying the stability and bifurcations of periodic orbits is the Poincaré map or first return map, defined by Poincaré in 1881, cf. [15]. There are two classes of differential systems whose orbits near an isolated singular point can turn around it (monodromic point): the non-degenerated systems whose linear part has imaginary eigenvalues and the degenerated systems (systems whose matrix of the linear part at the origin is nilpotent or identically null). When the system has an isolated singular point with imaginary eigenvalues, rather than work directly with the return map, it follows the classical method of searching a Liapunov function V of the form Vðx; yÞ ¼ x2 þ y2 þ Oðjx; yj2 Þ defined in a neighborhood of the origin. It is known, see [14], that the function V can be constructed such that its rate of change along _ trajectories be of the form Vðx; yÞ ¼ g2 ðx2 þ y2 Þ þ g4 ðx2 þ y2 Þ2 þ where gj are polynomials in the coefficients of the system. We call g2k the kth focal value and v k ¼ g2k assuming g2j ¼ 0; j 6 k 1, the kth Liapunov constant. It has that a focus is stable or unstable according to whether the first non-zero focal value is negative or positive. Furthermore, if g2 ¼ g4 ¼ . . . ¼ g2k ¼ 0 but g2kþ2 – 0 (focus of order k), no more than k limit cycles can bifurcate from the origin under perturbations of the system, see [14]. And the critical point is a center (a neighborhood of the origin belongs to a continuous of periodic solutions) if all the focal values are zero; in such case, there exists a local analytic first integral defined at the origin. This technique cannot be used for degenerated systems, since, in general, its centers do not have analytic first integral at the origin.
* Corresponding author. E-mail address: [email protected] (M. Reyes). 0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.04.077
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In this work, we are interested in calculating the multiplicity of a focus, to generate small amplitude limit cycles bifurcating from the origin, and to obtain the centers of some families of nilpotent systems. An analytic system of differential equations in the plane having a nilpotent singular point, in some suitable coordinates, can be written as
x_ ¼ y þ Pðx; yÞ;
y_ ¼ Q ðx; yÞ;
ð1Þ
vwhere Pðx; yÞ and Q ðx; yÞ are analytic functions without constant nor linear terms defined in a certain neighborhood of the origin. The center problem for the system (1) was solved theoretically by Moussu [13] and by Sadovskii [16]. Strozyna and Zoladek [17] characterize the centers which have a local analytic first integral at the origin. Giacomini et al. [10,11] prove that the analytic nilpotent systems with a center can be expressed as limit of systems non-degenerated with a center, and consequently the Poincaré–Liapunov method can be used to find the nilpotent centers. There are only a few families of polynomial differential systems (1) whose centers are known. The centers of the system (1), where Pðx; yÞ ¼ P2nþ1 ðx; yÞ and Q ðx; yÞ ¼ Q 2nþ1 ðx; yÞ are homogeneous polynomials of degree 2n þ 1, have been characterized for n ¼ 1; 2 and 3, see [5] and reference therein. Chavarriga et al. [7] study the centers of (1), with n ¼ 1, having an analytic first integral. They also prove that the nilpotent systems time-reversible under the change of variables ðx; y; tÞ ! ðx; y; tÞ have an analytic first integral. Sadovskii [16] finds the centers of the family (1) for Pðx; yÞ ¼ P 2 ðx; yÞ and Q ðx; yÞ ¼ Q 3 ðx; yÞ. Gasull and Torregrosa [8], using the so-called Cherka’s method, which consists in doing a change of variables that transforms (1) into a Liénard differential equation, calculate the centers of several families of nilpotent systems. In [9], the authors deal with systems of the form (1) with Pðx; yÞ ¼ Pnþ1 ðx; yÞ þ and Q ðx; yÞ ¼ x2n1 þ Q nþ1 ðx; yÞþ Q nþ2 ðx; yÞ þ where, in this case, the vector fields ðP k ; Q k Þ are quasi-homogeneous vector fields of type ð1; nÞ and degree k. In [2,3], Álvarez and Gasull apply the normal form theory to study the center problem for monodromic planar nilpotent singularities and calculate the first two generalized Liapunov constants of (1) and they solve the stability problem of several polynomial families. In this paper, we consider the analytic system of differential equations in the plane whose origin is a nilpotent singular point
x_ ¼ y þ
1 X
Pqpþ2is ðx; yÞ;
y_ ¼
i¼1
1 X
Q qpþ2is ðx; yÞ;
ð2Þ
i¼1
where p; q; n 2 N; p 6 q; s ¼ ðn þ 1Þp q > 0, and Fi ¼ ðPi ; Q i ÞT is a quasi-homogeneous vector field of type ðp; qÞ and degree i with Q ð2nþ1Þpq ð1; 0Þ < 0 (necessary condition of monodromy). That is, according to the degree, Fqp ¼ ðy; 0ÞT is the quasihomogeneous component of minor degree, the second one is Fð2nþ1Þpq which, among others, has the term ð0; x2nþ1 ÞT , and the edges of its Newton’s polygon of the remaining components are parallel to the edge associated to F2ðnþ1Þpq . This class includes, among others, the nilpotent systems which are invariant to the change of variables ðx; yÞ ! ðx; yÞ. In particular, it includes the family
x_ ¼ y þ X 2nþ1 ðx; yÞ;
y_ ¼ Y 2nþ1 ðx; yÞ;
where X 2nþ1 and Y 2nþ1 are homogeneous polynomials of degree 2n þ 1 with Y 2nþ1 ð1; 0Þ < 0 (case p ¼ q ¼ 1; P2n ðx; yÞ ¼ X 2nþ1 ðx; yÞ; Q 2n ðx; yÞ ¼ Y 2nþ1 ðx; yÞ; P2i ¼ Q 2i ¼ 0; i > n in (2)). The main result of the paper is Theorem 2.1 which gives the Taylor expansion of the return map of the system (2). This result allows us to solve theoretically the center problem for the systems (2) (Corollary 2.1) and to generate limit cycles bifurcating from the origin of the system (Corollary 2.2). Finally, as an application, we characterize the centers of several families of (2) and we give the number of small amplitude limit cycles bifurcating from the origin. Applying Proposition 3.1, we prove that the systems obtained have a center at the origin. In particular, we conclude that all have a local analytic first integral at the origin and they can be written in the form
x_ ¼ y þ v y Kðv ; y2 Þ þ yWðv ; y2 Þ;
y_ ¼ v x Kðv ; y2 Þ;
where v ; K; W are analytic functions defined in a neighborhood of O with WðOÞ ¼ 0. Also, we give the cyclicity of a weak focus of these families and we find a system with eight limit cycles.
2. Poincaré map near the origin Recall that a function f of two variables is quasi-homogeneous of type t ¼ ðp; qÞ and degree k if f ðep x; eq yÞ ¼ ek f ðx; yÞ. The vector space of quasi-homogeneous polynomials of type t and degree k will be denoted by Ptk . A vector field F ¼ ðF 1 ; F 2 Þ is said quasi-homogeneous of type t and degree k if F 1 2 Ptkþp and F 2 2 Ptkþq . We will denote Qtk the vector space of quasi-homogeneous polynomial vector fields of type t and degree k. We consider the analytic system of differential equations
_ yÞ _ T¼ ðx;
1 X i¼0
Fqpþ2is ;
ð3Þ
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where p; q 2 N; p 6 q and without common factors, s ¼ ðn þ 1Þp q > 0; n 2 N and Fi ¼ ðP i ; Q i ÞT are quasi-homogeneous vector fields of type t ¼ ðp; qÞ and degree i, with Fqp ¼ ðy; 0ÞT and Q qpþ2s ð1; 0Þ < 0 (without loss of generality, we can assume Q qpþ2s ð1; 0Þ ¼ 1Þ. In this system, this last condition implies that the origin is a monodromic point, see Andreev [4]. Note that if p or q is even, then the origin is a center of (3). Indeed, we assume, for instance, p is even then q will be odd (since p and q have no common factors), in that case P qpþ2is ðx; yÞ ¼ Pqpþ2is ðx; yÞ and Q qpþ2is ðx; yÞ ¼ Q qpþ2is ðx; yÞ since q þ 2is is odd and 2q p þ 2is is even. The system (3) is time-reversible, i.e. has symmetrical phase portrait with regard to a straight line passing through the origin (y ¼ 0, in this case), changing time direction. So, O is a center, since it is monodromic. In what follows, we assume that p and q are odd. We already introduce the generalized polar coordinates. Given any natural number n 2 N, it defines the generalized trigonometric functions, xðhÞ ¼ CsðhÞ; yðhÞ ¼ SnðhÞ, as the unique solution of the Cauchy problem
dx ¼ y; dh
dy ¼ x2nþ1 ; dh
with xð0Þ ¼ 1; yð0Þ ¼ 0. These functions are T-periodic with
rffiffiffiffiffiffiffiffiffiffiffiffi C 1 p 2nþ2 T :¼ 2 n þ 1 C nþ2 2nþ2
and they verify the equality Cs2nþ2 ðhÞ þ ðn þ 1ÞSn2 ðhÞ ¼ 1, see [12] for a proof. We can introduce the generalized polar coordinates, r and h of the real plane ðx; yÞ 2 R2 , as
y ¼ rnþ1 SnðhÞ:
x ¼ rCsðhÞ;
ð4Þ
Furthermore, the following equalities hold
r_ ¼
x2nþ1 x_ þ yy_ ; r 2nþ1
xy_ ðn þ 1Þyx_ h_ ¼ : r nþ2
The return map of (3) is analytic, see [12]. Now we provide a expression of the Taylor expansion of this return map. Theorem 2.1. Let system (3) with p and q odd. The return map of system (3) has the form
PðuÞ ¼ u
Z T 1 X 1 ðunþð2l1Þsþ1 fl Cs3nsþ2lsþ2 ðhÞdhÞð1 þ OðuÞÞ: 2ðn þ 1Þ l¼1 0
where fl 2 R; l P 1, are polynomials in the coefficients of the right-hand sides of (3) (we will call fk the focus quantities of the singular point O of the system (3)). Proof. By making the change (4), after omitting a common factor r n , the system (3) takes the form
r_ ¼ rf ðr; hÞ;
h_ ¼ 1 þ rgðr; hÞ
ð5Þ
with f and g analytic functions and f ð0; hÞ ¼ gð0; hÞ ¼ 0, for all h.We now define the variable u verifying
u2ðnþ1Þ ¼ W rCsðhÞ; r nþ1 SnðhÞ
ð6Þ
being W a C1 -function such that its derivative along the trajectories of the system (3) has the form
_ ¼ x3nsþ2 W
1 X
fl x2ls þ sðx; yÞ
l¼1
where fl ; l P 1, are polynomials in the coefficients of the right-hand sides of (3) and s is a flat function at the origin, see [1]. Let suppose fl ¼ 0 for l ¼ 1; . . . ; m 1 and fm – 0. The expression (6) is valid for r > 0 and for all h. Furthermore, as
u2ðnþ1Þ ¼ W rCsðhÞ; r nþ1 SnðhÞ ¼ r2ðnþ1Þ þ Oðr 2nþ3 ; hÞ
ð7Þ
from inverse function theorem, it has that r ¼ u þ Oðu2 ; hÞ. Next, we express (3) in the new coordinates system ðu; hÞ
1 _ W 2ðn þ 1Þu2nþ1 fm ¼ Cs3nsþ2þ2ms ðhÞu2msþnsþ1 ð1 þ Oðu; hÞÞ; 2ðn þ 1Þ
u_ ¼
h_ ¼ 1 þ Oðu; hÞ
ð8Þ ð9Þ ð10Þ
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whose differential equation associated can be written as
du fm ¼ Cs3nsþ2þ2ms ðhÞu2msþnsþ1 ð1 þ Oðu; hÞÞ: dh 2ðn þ 1Þ
ð11Þ
We write the solution of (11) starting at u ¼ u0 when h ¼ 0 as
uðh; u0 Þ ¼
1 X
ai ðhÞui0 þ sðh; u0 Þ;
ð12Þ
i¼1
with a1 ð0Þ ¼ 1; ai ð0Þ ¼ 0 for i P q2; sð0; u0 Þ ¼ 0 and fðu; hÞ ¼ ðu0 ; 0Þ; u0 > 0g to itself is given by the series
s flat at u0 ¼ 0. Hence the Poincaré return map from the section
Pðu0 Þ ¼ a1 ðTÞu0 þ a2 ðTÞu20 þ :
ð13Þ
By replacing (12) in the equation differential (11) we obtain
a1 ðhÞ 1; ai ðhÞ 0; for i ¼ 2; . . . ; n þ ð2m 1Þs and
anþð2m1Þsþ1 ðTÞ ¼
fm 2ðn þ 1Þ
Z
T
Cs3nsþ2þ2ms ðhÞdh:
0
Remark. The C1 -function W above mentioned in the proof of Theorem 2.1 is not a Liapunov function, since it is not a defined positive function in a neighborhood of the origin. Therefore, it cannot be used for finding limit cycles which bifurcate from the origin. As a consequence, the only significative constant fl is the first one different from zero. It does that the return map differ from the identity map, and it determines the stability of the origin. Also, let notice that the origin is a center if and only if PðuÞ u. So, we have the following results: the first one characterizes the centers of the system (3) and the second result is related to the number of small amplitude limit cycles which can bifurcate from the origin. Corollary 2.1. The origin of (3) with p and q odd is a center if and only if fl ¼ 0, for all l P 1. Proof. As p and q are odd and s ¼ ðn þ 1Þp q, it implies that s is even (odd) if and only if n is even (odd). Thereby, RT 3n s þ 2 þ 2ms is even. So, 0 Cs3nsþ2þ2ms ðhÞdh is a positive value. The result is followed as a consequence. h If we want to find the systems with a center of a polynomial family XðkÞ; k 2 Rm , of systems (3) with p and q odd, we calculate recursively the sets on Rm :
X1 ¼ fk 2 Rm ; f 1 ðkÞ ¼ 0g; Xk ¼ fk 2 Xk1 ; f k ðkÞ ¼ 0g; for k P 2: By Hilbert Basis Theorem, we know that there is a M such that
X1 X2 XM XMþ1 ¼ XMþ2 ¼ : So, the systems Xðk Þ with k 2 XMþ1 have a center at the origin. Also, in such a case, it is said that M is the order of the family XðkÞ. The focus quantities of system (3) can also be used to prove the existence of a certain number of small amplitude limit cycles bifurcating from the nilpotent critical point of a family of systems (3). Next result is used in order to study the degenerate Andronov–Hopf bifurcation, i.e. we analyze the existence of limit cycles which can bifurcate from the origin of XðkÞ under variations of the parameters k. Corollary 2.2. Let XðkÞ be a family of systems (3) with p and q odd, depending on some parameters k 2 Rm . We assume that k enough close to k such that f1 ð kÞ; f 2 ð kÞ; . . . ; k 2 Xr n Xrþ1 (i.e. O is a weak focus of order r of Xðk ÞÞ. If there exists kÞ; f r ð kÞ alternate sign and f r1 ð
0 < jf1 ðkÞj jf2 ðkÞj jfr1 ðkÞj jfr ðkÞj 1 then system Xð kÞ has exactly r limit cycles in a neighborhood of the origin. Proof. By Theorem 2.1, the Taylor expansion of the Poincaré return map of Xð kÞ has the form
PðuÞ ¼ u w1 f1 ðkÞð1 þ uh1 ðuÞÞuj1 w2 f2 ðkÞð1 þ uh2 ðuÞÞuj2 . . . R T 3nsþ2þ2ms 1 where wm ¼ 2ðnþ1Þ Cs ðhÞdh > 0; jm ¼ n þ ð2m 1Þs þ 1 and hm are analytic functions at the origin. 0 Each small limit cycle around the origin corresponds to each positive fixed point of the Poincaré return map of Xð kÞ, i.e. positive zeros of the function
FðuÞ ¼ u PðuÞ ¼
rþ1 X m¼1
ð1 þ uhm ðuÞÞwm fm ðkÞujm þ Oðjrþ1 þ 1Þ:
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m ðuÞÞ, where h m are analytic functions at the origin, it has FðuÞ ¼ ð1 þ uh1 ðuÞÞ By writing 1 þ uhm ðuÞ ¼ ð1 þ uh1 ðuÞÞð1 þ h uj1 F 0 ðuÞ where
F 0 ðuÞ ¼ w1 f1 ðkÞ þ
rþ1 X
m ðuÞÞujm j1 þ Oðj j þ 1Þ: wm fm ðkÞð1 þ uh rþ1 1
m¼2
We must look for positive zeros of F 0 . By differentiating, we have
F 00 ðuÞ ¼
rþ1 X
ð1 þ ug^m ðuÞÞðjm j1 Þwm fm ðkÞujm j1 1 þ Oðjrþ1 j1 Þ;
m¼2
where
m ðuÞÞ þ uðh m ðuÞ þ ug0 ðuÞÞ: ðjm j1 Þð1 þ ug^m ðuÞÞ ¼ ðjm j1 Þð1 þ uh m ~ m ðuÞÞ, where h ~ m are analytic functions at the origin, F 0 has the form ^m ðuÞ ¼ ð1 þ ug^1 ðuÞÞð1 þ h By writing 1 þ ug 0 0 j2 j1 1 ~ F 1 where F 0 ðuÞ ¼ ð1 þ uhm ðuÞÞu
F 1 ðuÞ ¼ ðj2 j1 Þw2 f2 ðkÞ þ
rþ1 X ~ m ðuÞÞujm j2 þ Oðj j þ 1Þ: ðjm j1 Þwm fm ðkÞð1 þ uh rþ1 2 m¼3
Now, the number of positive zeros of F 0 cannot exceed the number of positive zeros of F 1 by more than unity. By continuing this process a further step we obtain a function F 2 such that the number of positive zeros of F 1 cannot exceed the number of positive zeros of F 1 by more than unity. So, the number of positive zeros of F 0 cannot exceed the number of positive zeros of F 2 by more than two. This process stops at the rth step when we obtain a function F r of the form
F r ðuÞ ¼ ðjrþ1 j1 Þwrþ1 frþ1 ðkÞ þ Oð1Þ; which does not have zeros in a neighborhood of origin, since frþ1 ð kÞ is close to frþ1 ðk Þ – 0, by continuity. Therefore, F cannot kÞ; . . . ; f r1 ð kÞ and fr ð kÞ alternate sign and satisfy have more than r positive zeros. Moreover, as f1 ðkÞ; f 2 ð kÞj jf2 ð kÞj jfr1 ð kÞj jfr ð kÞj, we can assure the existence of r limit cycles of small amplitude bifurcating of 0 < jf1 ð the origin of Xðk Þ. h 3. Nilpotent centers and cyclicity of a weak focus of several families of polynomial systems In this section, by applying Corollary 2.1 and 2.2, we characterize the centers and the order of a weak focus of three families of systems (3). First, on the one hand, we give the following result which we will use in order to prove the analytic integrability of the centers of several families of (3). We recall that if the system (3) is monodromic, the existence of an analytic first integral is a sufficient condition so that the origin be a center. Proposition 3.1. The nilpotent systems
x_ ¼ y þ v y Kðv ; y2 Þ þ yWðv ; y2 Þ;
y_ ¼ v x Kðv ; y2 Þ;
ð14Þ
where v ; K; W are analytic functions defined in a neighborhood of the origin with WðOÞ ¼ 0, are integrable analytically in a neighborhood of the origin. Proof. Doing the change of variables u ¼ y2 , denoting dds ¼0 , the system (14) becomes
u0 ¼
2Kðu; v Þ ; 1 þ wðu; v Þ
v 0 ¼ 1:
v ¼ v ðx; yÞ, by redefining the variable time by ds ¼ yv x ð1 þ Wðx; yÞÞdt and by ð15Þ
From Cauchy–Arnold’s Theorem (see Bruno [6], p. 98), the system (15) has got an analytic first integral Hðu; v Þ ¼ cte defined e yÞ ¼ Hðy2 ; v ðx; yÞÞ ¼ cte, which is in a neighborhood of O. Undoing the change of variable, (14) has a first integral of (14), Hðx; analytic in a neighborhood N, since it is a composition of analytic functions. Also, if we denote X the vector field associated to e X ¼ 0, for all ðx; yÞ 2 N n frv X – 0g. So, by continuity, r H e X ¼ 0, for all ðx; yÞ 2 N, that is H e is a local ana(14), it has r H lytic first integral of (14). h Remarks For v ðx; yÞ ¼ x the systems (14) turn out
ðx; y2 Þ; x_ ¼ y þ yW
y_ ¼ Kðx; y2 Þ;
ð16Þ
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319
that is, the family of nilpotent systems time-reversible under the change of variables ðx; y; tÞ ! ðx; y; tÞ.The analytic integrability of the nilpotent systems (16) is one of the main results of Chavarriga et al. [7]. For
v ¼ v 2m ;
Kðy2 ; v Þ ¼ v p2m ; Wðy2 ; v Þ ¼
p X
2mðkþ1Þ2 ak v pk 2m y
k¼0
where p P 0;
v 2m is a homogeneous polynomial of degree 2m and ak arbitrary constants, the systems (14) come given by
8 p > < x_ ¼ y þ ov 2m v p þ P ak v pk y2mðkþ1Þ1 ; 2m 2m oy > :_ y ¼ ovox2m v p2m :
ð17Þ
k¼0
These systems include the nilpotent family integrable analytically given in [5], Lemma 2. On the other hand, we prove the following results which will use in order to obtain a simpler expression of the focus quantities of (3). Proposition 3.2. For all C1 vector fields X and U such that ½X; U ¼ mX with V ¼ X ^ U – 0, it has that
ðm þ div ðUÞÞX ¼ J rV þ div ðXÞU; where ½X; U denotes Lie’s bracket of the fields X and U, and J ¼
0 1
m a C1 -function in a neighborhood of the origin and
1 . 0
Proof. It is easy to prove that for all C1 vector fields X and U, it holds
rV U ¼ ½X; U ^ U þ Vdiv ðUÞ;
rV X ¼ ½X; U ^ X þ Vdiv ðXÞ:
ð18Þ
In our case, rV U ¼ ðm þ div ðUÞÞV and rV X ¼ div ðXÞV. As V – 0, it has that JX and JU are transversals. Therefore, there exist a; b functions of class one in a neighborhood of the origin such that rV ¼ aJX þ bJU. Hence, rV X ¼ bV; rV U ¼ aV. Thus,
V rV ¼ ðaVÞJX þ ðbVÞJU ¼ ðrV UÞJX ðrV XÞJU ¼ ðm þ div ðUÞÞVJX div ðXÞVJU; it follows the result easily.
h
We emphasize the following decomposition which is obtained from Proposition 3.2 with X ¼ Fk 2 Qtk and U ¼ Dt ¼ ðpx; qyÞT where t ¼ ðp; qÞ, and by taking into account the quasi-homogeneous character of Fk and ½Fk ; Dt ¼ kFk . Corollary 3.1 (Conservative-dissipative decomposition). For each t ¼ ðp; qÞ, given Fk ¼ ðP k ; Q k ÞT 2 Qtk , there exist hk 2 Ptkþpþq and lk 2 Ptk such that
Fk ¼
1 ðXhk þ lk Dt Þ; kþpþq
with hk ¼ Fk ^ Dt and
ð19Þ
lk ¼ divðFk Þ, where Dt :¼ ðpx; qyÞT and Xhk :¼
ohk oy
T k ðx; yÞ; oh ðx; yÞ . ox
We now show several applications of our research. We first consider the 11-parameter nilpotent system
x_ ¼ y þ a1 x3 þ a2 x2 y þ a3 xy2 þ a4 y3 þ a6 xy4 þ a7 y5 ;
ð20Þ
y_ ¼ x3 þ b1 x2 y þ b2 xy2 þ b3 y3 þ b5 xy4 þ b6 y5 : _ yÞ _ T ¼ F0 þ F2 þ F4 , with Fi 2 Qti ; i ¼ 0; 2; 4; t ¼ ð1; 1Þ, and This is a subfamily of (3) given by ðx;
F0 ¼
y 0
;
F2 ¼
a1 x3 þ a2 x2 y þ a3 xy2 þ a4 y3 x3 þ b1 x2 y þ b2 xy2 þ b3 y3
!
;
F4 ¼
a6 xy4 þ a7 y5 b5 xy4 þ b6 y5
!
:
The following result characterizes the centers of the systems (20). We note that the focus quantities of (3) have been computed by means of the recursive procedure developed in [1], and they have the form
f1 ¼ a1 g 1 ;
f i ¼ ai g i þ
i1 X
bi;j fj ; i P 2;
j¼1
with ai positive constants and bi;j polynomials in the coefficients of the right-hand sides of (3). Theorem 3.1. The origin of the system (20) is a center if and only if one of the following two series is satisfied: (i) 3a1 þ b1 ¼ 2a2 þ b2 ¼ a3 þ 3b3 ¼ b5 ¼ a6 þ 5b6 ¼ 0 (Hamiltonian system).
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(ii) b1 þ 3a1 ¼ b3 a1 b2 þ 2a21 ¼ a3 þ a1 b2 2a2 þ 6a21 ¼ b6 a1 b5 ¼ a6 þ a1 b5 ¼ 0.Moreover, each one of them has a local analytic first integral. Proof. Taking t ¼ ð1; 2Þ, (that is, the type of the vector field ðy; x3 ÞT , i.e. the quasi-homogeneous principal part of the system _ yÞ _ T ¼ Xh þ lDt , being h the defined (20)) and applying Corollary 3.1, degree to degree, the system (20) comes given by ðx; positive function
hðx; yÞ ¼ and
1 1 1 1 1 1 1 1 ð2y2 þ x4 Þ c1 x3 y c2 x2 y2 c3 xy3 c4 y4 c5 x2 y4 c6 xy5 c 7 y6 4 5 6 7 8 10 11 12
ð21Þ
lðx; yÞ ¼ 15 d1 x2 þ 16 d2 xy þ 17 d3 y2 þ 101 d5 xy3 þ 111 d6 y4 where c1 ¼ b1 2a1 ; d1 ¼ 3a1 þ b1 ; c2 ¼ b2 2a2 ; d2 ¼ 2a2 þ 2b2 ; c3 ¼ b3 2a3 ; d3 ¼ a3 þ 3b3 ; c4 ¼ 2a4 ; c6 ¼ 2a6 ;
d5 ¼ 4c5 ¼ 4b5 ;
d6 ¼ a6 þ 5b6 ; c7 ¼ 2a7 :
The first four focus quantities are
g 1 ¼ d1 ; g 2 ¼ d2 c1 þ 5d3 ; g 3 ¼ d2 25d2 c1 þ 175c1 c2 þ 42c31 þ 375c3 ; g 4 ¼ 4c5 c1 þ 5d6 : In this point, we distinguish two cases depending on the coefficient d2 . If d2 – 0, some necessary conditions for the origin to 1 c1 ð25d2 þ 175c2 þ 42c21 Þ; d6 ¼ 45 c1 c5 . In such a be a center are g 1 ¼ g 2 ¼ g 3 ¼ g 4 ¼ 0, i.e. d1 ¼ 0; d3 ¼ 15 d2 c1 ; c3 ¼ 375 case, it has g 5 ¼ d2 ð3c1 c5 þ 5c6 Þ. Hence, c6 ¼ 35 c1 c5 . Thus, we arrive to the system given by (ii),
x_ ¼ y þ a1 x3 þ a2 x2 y a1 b2 2a2 þ 6a21 xy2 þ a4 y3 a1 b5 xy4 þ a7 y5 ; y_ ¼ x3 3a1 x2 y þ b2 xy2 þ a1 b2 þ 2a21 y3 þ b5 xy4 þ a1 b5 y5 :
ð22Þ
This system belongs to the family (14) given in Proposition 3.1 where
v ¼ 12 x2 þ a1 xy þ 12 a2 a21 y2 ; Kðv ; y2 Þ ¼ 2h ða2 þ b2 Þy2 b5 y4 ;
Uðv ; y2 Þ ¼ a4 þ a2 2a21 2a21 þ b2 y2 þ a7 þ b5 a2 2a21 y4 and, therefore, the origin is a center, since the system has a local analytic first integral and O is a monodromic point. If d2 ¼ 0, from the vanishing of the first four constants above, we have d1 ¼ d3 ¼ 0 and d6 ¼ 45 c1 c5 . In this case, the next focus quantities are g 5 ¼ c5 ð175c1 c2 þ 42c31 þ 375c3 Þ; g 6 0 and g 7 ¼ c5 ð3c1 c5 þ 5c6 Þ. If c5 ¼ 0, the necessary conditions to have a center leads us to the hamiltonian system (i) whose hamiltonian function is (21). We note that the curves hðx; yÞ ¼ cte are closed, therefore, it is a center. And if c5 – 0, g 5 and g 7 must be zero; hence,
c3 ¼
7 c1 ð25c2 þ 6c21 Þ; 375
3 c6 ¼ c1 c5 : 5
In such a case, the system is of the form (22) with d2 ¼ 0. So, O is a center.
h
Theorem 3.2. Under perturbations of the parameters of the system (20), it has: (a) if a2 þ b2 – 0 or b5 – 0, it can bifurcate 0,1,2,3 or 4 limit cycles around the origin. (b) if a2 þ b2 ¼ b5 ¼ 0, it can bifurcate 0,1 or 2 limit cycles around the origin.
Proof. We fix the constants c1 ; d2 ; c2 ; c5 above defined and consider the critical values
d1 ¼ 0;
1 d3 ¼ d2 c1 ; 5
c3 ¼
1 c1 25d2 þ 175c2 þ 42c21 ; 375
4 d6 ¼ c1 c5 ; 5
3 c6 ¼ c1 c5 : 5
Firstly, we assume that a2 þ b2 – 0. From the expression of g 1 ; g 2 ; g 3 ; g 4 and g 5 , applying Corollary 2.2, it deduces the follow ing one: if d1 – d1 there is not limit cycles around the origin. If d1 is near zero and d3 – d3 , then it can exist, at least, 1 limit cycle. If d1 and d3 alternate sign and 0 < jd1 j jd3 j and also c3 is different from c3 , then there are 2 small amplitude limit cycles. If also d3 c3 < 0; jd1 j jd3 j and d6 – d6 , then 3 limit cycles bifurcate of the origin. If we take d1 ; d3 ; c3 and d6 different from d1 ; d3 ; c3 and d6 but near each of them, respectively, and c6 – c6 such that it satisfy the hypothesis of Corollary 2.2 with r ¼ 4, it has that we can bifurcate 4 limit cycles, at least.
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Now, we assume that a2 þ b2 ¼ 0 and b5 – 0. The first constants g i different from zero are
g 1 ¼ d1 ;
g 2 ¼ d3 ;
g 4 ¼ 4c1 c5 þ 5d6 ;
g 5 ¼ c5 175c1 c2 þ 42c31 þ 375c3 ;
g 7 ¼ c5 ð3c1 c5 þ 5c6 Þ:
Thus, if d1 – 0 there is not limit cycles around the origin. If d1 is close to zero and d3 – 0, can exist, at least, 1 limit cycle. If d1 and d3 alternate sign and 0 < jd1 j jd3 j and also d6 is different from d6 , then there are 2 limit cycles of small amplitude. If we 7 c1 ð25c2 þ 6c21 Þ then there now take d6 such that d3 and d6 alternate sign and jd3 j jd6 j and also c3 is different from c3 ¼ 375 exist 3 limit cycles around the origin. If we also choose c3 near c3 such that d6 c3 < 0 and jd6 j jc3 j and take c6 – c6 , then there exist, at least, 4 limit cycles around the origin. Last on, if a2 þ b2 ¼ b5 ¼ 0, that is d2 ¼ c5 ¼ 0, the g i different from zero are
g 1 ¼ d1 ;
g 2 ¼ d3 ;
g 4 ¼ d6 :
Therefore, if d1 – 0 there is not limit cycles near the origin. If d1 is close zero and d3 – 0, it can exist, at least, 1 limit cycle. If d1 and d3 alternate sign and 0 < jd1 j jd3 j and also d6 is different from zero, then 2 small amplitude limit cycles bifurcate from the origin. h We now consider the 9-parameter subfamily of (3) given by
_ yÞ _ T ¼ F2 þ F4 þ F6 ; ðx; with Fi 2
Qti ;
F2 ¼
ð23Þ
i ¼ 2; 4; 6; t ¼ ð1; 3Þ, and
y 0
; F4 ¼
a1 x5 þ a2 x2 y x7 þ b1 x4 y þ b2 xy2
! ;
F6 ¼
!
a3 x7 þ a4 x4 y þ a5 xy2 b6 x9 þ b3 x6 y þ b4 x3 y2 þ b5 y3
with b6 ¼ 0 and b2 ¼ a2 . We already get a lower bound for its cyclicity. Theorem 3.3. Under perturbations of the parameters of the system (23), it has: (a) if 2a4 þ b4 ¼ 0, it can bifurcate 0,1 or 2 limit cycles from the origin. (b) if 2a4 þ b4 – 0 and ð5a2 þ 2ðb1 4a1 Þ2 Þðb1 4a1 Þ ¼ 0, it can bifurcate 0,1,2,3 or 4 limit cycles around the origin. (c) if 2a4 þ b4 – 0 and ð5a2 þ 2ðb1 4a1 Þ2 Þðb1 4a1 Þ – 0, it can bifurcate 0,1,2,3,4,5,6,7 or 8 limit cycles around the origin.
Proof. In this case, we apply Corollary 3.1, degree to degree, with t ¼ ð1; 4Þ (that is, the type of the vector field ðy; x7 ÞT , i.e. _ yÞ _ T ¼ Xh þ lDt , being h the the quasi-homogeneous principal part of the system (23)). The system (23) comes given by ðx; defined positive function
hðx; yÞ ¼ and
1 8 1 1 1 1 1 ðx þ 4y2 Þ c1 x5 y c 2 x2 y 2 c 3 x7 y c 4 x4 y 2 c5 xy3 ; 8 9 10 11 12 13
lðx; yÞ ¼ 19 d1 x4 þ 111 d3 x6 þ 121 d4 x3 y þ 131 d5 y2 where c1 ¼ b1 4a1 ; c5 ¼ b5 4a5
d1 ¼ 5a1 þ b1 ;
c2 ¼ 5a2 ;
c3 ¼ b3 4a3 ;
d3 ¼ 7a3 þ b3 ;
c4 ¼ b4 4a4 ;
d4 ¼ 4a4 þ 2b4 ;
d5 ¼ a5 þ 3b5 :
First, we assume that 2a4 þ b4 ¼ 0, that is d4 ¼ 0. The only g i different from zero are g 1 ¼ d1 ; g 2 ¼ d3 ; g 3 ¼ d5 . Therefore, if d1 – 0 there is a neighborhood of the origin where the system (23) does not have any limit cycle around the origin. If d1 is close to zero and d3 – 0, it can exist, at least, 1 limit cycle. If d1 d3 < 0 with 0 < jd1 j jd3 j and also d5 – 0, then there are 2 limit cycles of small amplitude. We now assume that 2a4 þ b4 – 0 and 5a2 þ 2ðb1 4a1 Þ2 ðb1 4a1 Þ ¼ 0, i.e. d4 – 0 and c2 þ 2c21 c1 ¼ 0. In this case,
g 1 ¼ d1 ;
g 2 ¼ d3 ;
g 3 ¼ d5 þ
12 c1 d4 ; 13
1 g 4 ¼ d4 c3 ; g 5 ¼ d4 c5 þ 4c1 c4 þ d4 : 13
and the remain are zero. So, if d1 – 0 there is not limit cycles around the origin. If d1 is close to zero and d3 – 0, can exist, at least, 1 limit cycle. If d1 and d3 alternate sign and 0 < jd1 j jd3 j and also d5 different from d5 ¼ 12 13 c1 d4 , then there are 2 small amplitude limit 3 limit cycles cycles. If we now take d5 such that d3 and d5 alternate sign and jd3 j jd5 j and also c3 – 0, then there exist 1 d4 then there around the origin. If we also choose c3 close to 0 such that d5 c3 < 0 and jd5 j jc3 j and take c5 – 4c1 c4 þ 13 exist, at least, 4 limit cycles bifurcating from the origin. Lastly, if 2a4 þ b4 – 0 and ð5a2 þ 2ðb1 4a1 Þ2 Þðb1 4a1 Þ – 0, by denoting q ¼ d4 c1 c2 þ 2c21 – 0, the first nine constants g i ; i ¼ 1; . . . ; 9 of (23) are
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g 1 ¼ d1 ; g 2 ¼ d3 ; g 3 ¼ d5 þ 12 c d ; 13 1 4
g 4 ¼ d4 c3 þ 2c1 c2 þ 2c21 ;
1 g 5 ¼ d4 c5 þ 4c1 c4 þ 13 d4 100 c31 c2 þ 2c21 ; 3
g 6 ¼ q c4 þ 12 d4 62 c2 c þ 2c21 ; 3 1 2
24 2 748 g 7 ¼ q d4 5 c2 15 c2 c21 1408 c41 ; 15 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ih pffiffiffiffiffiffiffiffiffiffiffiffiffiffi i g 8 ¼ q 1548c2 4681 þ 5 54049 c21 1548c2 4681 5 54049 c21 ; g 9 ¼ q 381374c2 þ 859813c21 : We can choose d1 ; d3 ; d5 ; c3 ; c5 ; c4 ; d4 and c2 adequately such that g i g iþ1 is negative, g 9 different from zero and
0 < jg 1 j jg 2 j jg 3 j jg 4 j jg 5 j jg 6 j jg 7 j jg 8 j: Applying Corollary 2.2, for r ¼ 1; 2; 3; 4; 5; 6; 7 and 8, there are regions of the parameters where the system has 0,1,2,3,4,5,6,7 or until 8 limit cycles around the origin. h The centers of (23) have been characterized in [1]. These one are included in the class of systems given by Proposition 3.1. Last on, we study the systems
x_ ¼ y þ
2nþ2 ½X 5
ai;n x
2nþ25i 3i1
y
;
y_ ¼ x2nþ1 þ
i¼1
2nþ1 ½X 5
bi;n x2nþ15i y3i :
ð24Þ
i¼1
_ yÞ _ T ¼ F2 þ F6n2 , with with 1 6 n 6 9. (½x means integer part of xÞ. These systems are of the family (3) given by ðx; t Fi 2 Qi ; i ¼ 2; 6n 2; t ¼ ð3; 5Þ. Analogously to the above application, we obtain the cyclicity of the origin of (24). Theorem 3.4. The system (24) with n equal to 1,2,3 or 4, does not have limit cycles of small amplitude surrounding the origin. There are systems inside this family for n ¼ 5 or n ¼ 6 with 0 or 1 limit cycle around the origin. For n ¼ 7 or n ¼ 8 there exist systems inside this family with 0,1,2 or 3 limit cycles around the origin. There exist systems (24) with n ¼ 9 which have 0, 1 or 2 limit cycles around the origin. Proof. Taking t ¼ ð1; n þ 1Þ and by applying Corollary 3.1, degree to degree, the system (24) takes the form
_ yÞ _ T ¼ ðx;
2nþ1 ½X 5 1 1 Xx2nþ2 þðnþ1Þy2 þ di;n x2nþ15i y3i1 Dð1;nþ1Þ þ ci;n Xx2nþ25i y3i 2n þ 2 2n þ 2 þ ð3n 2Þi i¼1
where the new coefficients that appear are
ci;n ¼ ðn þ 1Þai;n bi;n ;
di;n ¼ ð2n þ 2 5iÞai;n þ 3ibi;n
The expressions of the focus quantities different from zero are:
n ¼ 2; g 1 ¼ d1;2 ; n ¼ 3; g 1 ¼ d1;3 ; n ¼ 4; g 1 ¼ d1;4 ; n ¼ 5; g 1 ¼ d1;5 ;
g 2 ¼ c1;5 d2;5
n ¼ 6; g 1 ¼ d1;6 ;
g 2 ¼ c1;6 d2;6 ;
n ¼ 7; g 1 ¼ d1;7 ; g 2 ¼ c1;7 d2;7 þ 35d3;7 ; g 3 ¼ d2;7 33075c3;7 þ 1314c31;7 þ ð2660d2;7 þ 7665c2;7 Þc1;7 ; g 4 ¼ c51;7 d2;7 ; n ¼ 8; g 1 ¼ d1;8 ; g 2 ¼ 3c1;8 d2;8 þ 40d3;8 ; g 3 ¼ d2;8 49600c3;8 þ 2387c31;8 þ ð3960d2;8 þ 13440c2;8 Þc1;8 ; g 4 ¼ c51;8 d2;8 ; n ¼ 9; g 1 ¼ d1;9 ; g 2 ¼ c1;9 d2;9 þ 9d3;9 ; g 3 ¼ d2;9 5103c3;9 þ ð266c21;9 þ 405d2;9 þ 1539c2;9 Þc1;9 : Choosing adequately the constants ci;n and di;n and by applying Corollary 2.2 it follows the result. h
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Acknowledgements This work has been partially supported by Ministerio de Ciencia y Tecnología, Plan Nacional I+ D + I co-financed with FEDER funds, in the frame of the project MTM2007-64193 and by Consejería de Educación y Ciencia de la Junta de Andalucía (FQM276 and EXC/2008). References [1] A. Algaba, C. García, M. Reyes, The center problem for a family of systems of differential equations having a nilpotent singular point, J. Math. Anal. Appl. 340 (2008) 32–43. [2] M.J. Álvarez, A. Gasull, Monodromy and stability for nilpotent critical points, Int. J. Bifur. Chaos Appl. Sci. Eng. 15 (4) (2005) 1253–1265. [3] M.J. Álvarez, A. Gasull, Generating limit cycles from a nilpotent critical point via normal forms, J. Math. Anal. Appl. 318 (1) (2006) 271–287. [4] A. Andreev, Investigation of the behaviour of the integral curves of a system of two differential equations in the neighborhood of a singular point, Trans. Am. Math. Soc. 8 (1958) 187–207. [5] A. Andreev, A.P. Sadovskii, V.A. Tskialyuk, The center-focus problem for a system with homogeneous nonlinearities in the case zero eigenvalues of the linear part, Diff. Equat. 39 (2) (2003) 155–164. [6] A.D. Bruno, Local Methods in Nonlinear Differential Equations, Springer-Verlag, New York, 1989. [7] J. Chavarriga, H. Giacomini, J. Giné, J. Llibre, Local analytic integrability for nilpotent centers, Ergodic Theory Dynam. Syst. 23 (2003) 417–428. [8] A. Gasull, J. Torregrosa, Center problem for several differential equations via Cherka’s method, J. Math. Anal. Appl. 228 (2) (1998) 322–343. [9] A. Gasull, J. Torregrosa, A new algorithm for the computation of the Liapunov constants for some degenerated critical points, Nonlinear Anal. 47 (2001) 4479–4490. [10] H. Giacomini, J. Giné, J. Llibre, The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems, J. Diff. Equat. 227 (2) (2006) 406–426. [11] H. Giacomini, J. Giné, J. Llibre, Corrigendum to: ‘‘The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems [J. Diff. Equat. 227 (2006) 2, 406–426], J. Diff. Equat. 232 (2007) 2, 702. [12] M.A. Liapunov, Stability of Motion, Academic Press, New York, 1966. pp. 123–194. [13] R. Moussu, Symétrie et forme normale des centres et foyers dégénérés, Ergodic Theory Dynam. Syst. 2 (1982) 241–251. [14] V.V. Nemytskii, V.V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, 1960. [15] H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, J. Math. 37 (1881) 375–422. [16] A.P. Sadovskii, Problem of distinguishing a center and a focus for a system with a nonvanishing linear part, Translated Diff. Urav. 12 (7) (1976) 1238– 1246. [17] E. Strozyna, H. Zoladek, The analytic and formal normal form for the nilpotent singularity, J. Diff. Equat. 179 (2002) 479–537.