## Abstract

We study the number of limit cycles of polynomial differential systems of the form
where *g*_{1},*f*_{1},*g*_{2} and *f*_{2} are polynomials of a given degree. Note that when *g*_{1}(*x*)=*f*_{1}(*x*)=0, we obtain the generalized polynomial Liénard differential systems. We provide an accurate upper bound of the maximum number of limit cycles that the above system can have bifurcating from the periodic orbits of the linear centre , using the averaging theory of first and second order.

## 1. Introduction

The second part of the 16th Hilbert problem wants to find an upper bound on the maximum number of limit cycles that a polynomial vector field of a fixed degree can have. In this paper, we will try to give a partial answer to this problem for the class of polynomial differential systems given by
1.1Note that when *g*_{1}(*x*)=*f*_{1}(*x*)=0 coincides with generalized polynomial Liénard differential systems. The classical polynomial Liénard differential systems are
1.2where *f*(*x*) is a polynomial in the variable *x* of degree *n*. For these systems, Lins *et al.* (1977) stated the conjecture that if *f*(*x*) has degree *n*≥1, then system (1.2) has at most [*n*/2] limit cycles. They proved this conjecture for *n*=1,2. Recently, the conjecture has also been proved for *n*=3, see Li & Llibre (2012). For *n*≥5, Dumortier *et al.* (2007) and De Maesschalck & Dumortier (2011) showed that the conjecture is not true for *n*≥5. In short, at this moment, the conjecture is only open for *n*=4.

Many of the results on the limit cycles of polynomial differential systems have been obtained by considering limit cycles that bifurcate from a single degenerate singular point (i.e. from a Hopf bifurcation), which are called *small amplitude limit cycles*, see Lloyd (1988). There are partial results concerning the maximum number of small-amplitude limit cycles for Liénard polynomial differential systems. The number of small-amplitude limit cycles gives a lower bound for the maximum number of limit cycles that a polynomial differential system can have.

There are many results concerning the existence of small-amplitude limit cycles for the following generalization of the classical Liénard polynomial differential system (1.2):
1.3where *f*(*x*) and *g*(*x*) are polynomials in the variable *x* of degrees *n* and *m*, respectively. We denote by *H*(*m*,*n*) the maximum number of limit cycles that system (1.3) can have. This number is usually called the *Hilbert number* for system (1.3).

In 1928, Liénard (1928) proved that if

*m*=1 and is a continuous odd function, which has a unique root at*x*=*a*and is monotone increasing for*x*≥*a*, then equations (1.3) have a unique limit cycle.In 1973, Rychkov (1975) proved that if

*m*=1 and*F*(*x*) is an odd polynomial of degree five, then equations (1.3) have at most two limit cycles.In 1977, Lins

*et al.*(1977) proved that*H*(1,1)=0 and*H*(1,2)=1.In 1998, Coppel (1998) proved that

*H*(2,1)=1.Dumortier & Rousseau (1990) and Dumortier & Li (1997) proved that

*H*(3,1)=1.In 1997, Dumortier (1997) proved that

*H*(2,2)=1.In 2012, Li & Llibre (2012) proved that

*H*(1,3)=1.

Up to now and as far as we know, only for these five cases (iii)–(vii) has the Hilbert number for system (1.3) been determined.

The maximum number of small-amplitude limit cycles for system (1.3) is denoted by . Blows & Lloyd (1984), Lloyd & Lynch (1988) and Lynch (1995) have used inductive arguments in order to prove the following results:

— if

*g*is odd, then ;— if

*f*is even, then , whatever*g*is;— if

*f*is odd, then ; and— if

*g*(*x*)=*x*+*g*_{e}(*x*), where*g*_{e}is even, then .

Christopher & Lynch (1999), Lynch (1998, 1999) and Lynch & Christopher (1999) have developed a new algebraic method for determining the Liapunov quantities of system (1.3) and proved some other bounds for for different *m* and *n*:

— ;

— ;

— , for all 1<

*m*≤50;— , for all 1<

*m*≤50; and— ,

*k*=6,7,8,9 and .

In 1998, Gasull & Torregrosa (1998) obtained upper bounds for , , and .

In 2006, Yu & Han (2006) gave some accurate values of , for *n*=4, *m*=10,11,12,13; *n*=5, *m*=6,7,8,9; *n*=6, *m*=5,6; see also Llibre *et al.* (2010) for a table with all the specific values.

In 2010, Llibre *et al.* (2010) computed the maximum number of limit cycles of system (1.3) that bifurcate from the periodic orbits of the linear centre , , using the averaging theory of order *k*, for *k*=1,2,3.

In this paper, first we consider the system
1.4where *g*_{11}, *f*_{11}, *g*_{21} and *f*_{21} have degree *k*, *l*, *m* and *n*, respectively, and *ε* is a small parameter.

### Theorem 1.1

*For |ε| sufficiently small, the maximum number of limit cycles of the generalized Liénard polynomial differential system (1.4) bifurcating from the periodic orbits of the linear centre* *using the averaging theory of first order is
*1.5

The proof of theorem 1.1 is given in §3.

Now we consider the system
1.6where *g*_{11} and *g*_{12} have degree *k*; *f*_{11} and *f*_{12} have degree *l*; *g*_{21} and *g*_{22} have degree *m*; and *f*_{21}, *f*_{22} have degree *n*. Furthermore, *ε* is a small parameter.

### Theorem 1.2

*For |ε| sufficiently small, the maximum number of limit cycles of the generalized Liénard polynomial differential systems (1.6) bifurcating from the periodic orbits of the linear centre* *using the averaging theory of second order is* *, where
*1.7*with* *.*

The proof of theorem 1.2 is given in §4.

The results that we shall use from the averaging theory of first and second order for computing limit cycles are presented in §2.

## 2. The averaging theory of first and second order

The averaging theory for studying specifically limit cycles up to first order in *ε* was developed many years ago, and can be found in Verhulst (1991), Guckenheimer & Holmes (1990) and Buică & Llibre (2004). The averaging theory for computing limit cycles up to second order in *ε* was developed in Llibre (2002–2003) and Buică & Llibre (2004). It is summarized as follows.

Consider the differential system
2.1where , are continuous functions, *T*-periodic in the first variable, and *D* is an open subset of . Assume that the following conditions hold.

*F*_{1}(*t*,⋅)⊂*C*^{2}(*D*),*F*_{2}(*t*,⋅)⊂*C*^{1}(*D*), for all ,*F*_{1},*F*_{2},*R*, are locally Lipschitz with respect to*x*, and*R*is twice differentiable with respect to*ε*.We define for

*k*=1,2 as whereFor

*V*⊂*D*, an open and bounded set and for each*ε*∈(−*ε*_{f},*ε*_{f})\{0}, there exists*a*_{ε}∈*V*such that*F*_{10}(*a*_{ε})+*εF*_{20}(*a*_{ε})=0 and*d*_{B}(*F*_{10}+*εF*_{20},*V*,*a*_{ε})≠0.

Then, for |*ε*|>0 sufficiently small, there exists a *T*-periodic solution *ϕ*(⋅,*ε*) of the system such that *ϕ*(0,*a*_{ε})→*a*_{ε} when *ε*→0.

The expression *d*_{B}(*F*_{10}+*εF*_{20},*V*,*a*_{ε})≠0 means that the Brouwer degree of the function at the fixed point *a*_{ε} is not zero. A sufficient condition in order that this inequality holds is that the Jacobian of the function *F*_{10}+*εF*_{20} at *a*_{ε} is not zero.

If *F*_{10} is not identically zero, then the zeros of *F*_{10}+*εF*_{20} are mainly the zeros of *F*_{10} for *ε* sufficiently small. In this case, the previous result provides the *averaging theory of first order*.

If *F*_{10} is identically zero and *F*_{20} is not identically zero, then the zeros of *F*_{10}+*εF*_{20} are mainly the zeros of *F*_{20} for *ε* sufficiently small. In this case, the previous result provides the *averaging theory of second order*.

## 3. Proof of theorem 1.1

We shall need the first-order averaging theory to prove theorem 1.1. We write system (1.4) in polar coordinates (*r*,*θ*) where
In this way, system (1.4) will become written in the standard form for applying the averaging theory. If we write
3.1then, system (1.4) becomes
3.2Now taking *θ* as the new independent variable, system (3.2) becomes
and
Now using the expressions for the integrals in appendix A (note that *α*_{k+1}=(2*k*+1)*α*_{k}), we get
3.3Then, the polynomial *F*_{10}(*r*) has at most *λ*_{1} (see (1.5)) positive roots, and we can choose the coefficients *a*_{2i,2} and *b*_{2i+1,1} in such a way that *F*_{10}(*r*) has exactly *λ*_{1} simple positive roots. Hence, theorem 1.1 is proved.

## 4. Proof of theorem 1.2

We write *f*_{11}, *f*_{21}, *g*_{11} and *g*_{21} as in (3.1), and
Then, system (1.4) in polar coordinates (*r*,*θ*) with *r*>0 becomes
4.1Taking *θ* as the new independent variable, system (4.1) becomes
where
4.2where
and
In order to compute *F*_{20}(*r*), we need that *F*_{10} be identically zero. Then from (3.3),
4.3First we compute
and . To do it, we rewrite
Then, taking into account that
and using the integrals of appendix A, we obtain
where
Again, using the integrals of appendix A, we conclude that
where *P*_{1}(*r*^{2}) is equal to
with
Then, *P*_{1}(*r*^{2}) is a polynomial in the variable *r*^{2} of degree *λ*_{2}, see (1.7). In the notation of (4.2), we have that
where *P*_{2} is a polynomial in the variable *r*^{2} of degree *λ*_{1}.

Furthermore,
4.4Using relation (4.3), we get
where the third and fourth (respectively, fifth and sixth) lines come from taking together the second and fourth (respectively, third and sixth) lines of (4.4). We write , where *P*_{3} is equal to
where
Then, *P*_{3}(*r*^{2}) is a polynomial in the variable *r*^{2} of degree *λ*_{2}, see (1.7). Then,
Then, to find the real positive roots of *F*_{20}, we must find the zeros of a polynomial in *r*^{2} of degree *λ*_{3}. This yields that *F*_{20} has at most *λ*_{3} real positive roots. Moreover, we can choose the coefficients *a*_{i,1},*a*_{i,2},*b*_{i,1},*b*_{i,2},*c*_{i,1},*c*_{i,2},*d*_{i,1},*d*_{i,2} in such a way that *F*_{20} has exactly *λ*_{3} real positive roots. Hence, the theorem is proved.

## Acknowledgements

J.L. has been supported by the grants MICINN/FEDER MTM 2009-03437, CIRIT 2009SGR 410 and ICREA Academia. C.V. is supported by the grant AGAUR PIV-DGR-210, by FCT through PTDC/MAT/117106/210 and by CAMGDS, Lisbon.

## Appendix A. Formulae

In this appendix, we recall some formulae that will be used during the paper, for more details, see Abramowitz & Stegun (1964). For *i*≥0, we have
where *C*_{i,j} and *K*_{i,l} are non-zero constants.

- Received December 22, 2011.
- Accepted March 12, 2012.

- This journal is © 2012 The Royal Society