## Abstract

We study the simultaneous existence of centres for two families of planar

## 1. Introduction

The second part of Hilbert’s 16th problem deals with the existence of a uniform upper bound on the number of limit cycles *H*(*n*) of a planar polynomial differential system
*n*, where *H*(1)=0. For *n*≥2, the problem remains open. Only lower bounds for *H*(*n*) with *n*≥2 are known and our objective is to improve these lower bounds. An efficient method to do this is to perturb symmetric Hamiltonian systems. Symmetric Hamiltonian systems are Hamiltonian systems with certain symmetries that allow the existence of a great number of centres whose perturbations can produce a large number of limit cycles.

A generalization of these symmetric Hamiltonian systems are the *q* centres. After the survey, some new particular cases are studied in detail.

Let *G* be a finite group of transformations acting on *G*-*equivariant function* if, for all *g*∈*G* and all *x*, *ϕ*(*gx*)=*gϕ*(*x*). Given a *G*-equivariant function *ϕ*, the vector field d*x*/d*t*=*ϕ*(*x*) is called a *G*-*equivariant vector field*.

All the actions considered below are orthogonal two-dimensional real *q*≥2. If *q*>2, the representation is irreducible, while, for *q*=2, one has a direct sum of two one-dimensional antipodal representations. *z*=*x*+*iy* and

The following result characterizes a

### Theorem 1.1.

*The vector field* (*1.2*) *is a* *-equivariant complex vector field if and only if the complex function* *has the form
**where g*_{ℓ} *and h*_{ℓ} *are polynomials in |z|*^{2} *with complex coefficients. Moreover, system* (*1.2*) *is Hamiltonian if and only if*

The *q*=4 there appears the term

Several authors have studied the *n*=5 for *q* varying from 2 to 6, and with the help of numerical analysis it was proved that at least 24 limit cycles can bifurcate for such systems; see [2,5]. In fact, the cases *q*=2 and *q*=3 were studied separately in [6,7], where at least 15 and 23 limit cycles were found, respectively.

In [7–13],

The limit cycles of

Recently, the limit cycles for

Observe that there exist *q*≠2 which are not

The importance of *H*(*n*). In other words, the phase portraits of *H*(*n*)≥*k*_{1}*n*^{2} for some constant *k*_{1}. In [33], perturbing some *H*(*n*) grows at least as *k*_{2}. In fact, in [38], it was conjectured that *H*(*n*) is *O*(*n*^{3}) as

In this paper, we study the number of simultaneous centres in planar differential systems. Up to now, the simultaneity of centres was investigated only for a very few particular families. For instance, the existence of two simultaneous centres was studied in [39,40] for quadratic systems, and in [41,42] for some particular cubic systems. The simultaneity of centres in planar differential systems is important because perturbations of such systems give a great number of bifurcations of limit cycles; see [30,43,44].

In [11,12], the authors studied *X*_{i} and *Y* _{i} are homogeneous polynomials of degree *i* having a focus–centre singular point at points (−1,0) and (1,0). In those works, the authors gave the necessary and sufficient conditions to have a centre and an isochronous centre at these singular points. Note that when we have a centre at one of the singular points then automatically we have a centre at the other because the system is *a*,*b*) and (−*a*,−*b*) with *ab*≠0 arbitrary. We will see that these three additional centres appear simultaneously.

In [12], the existence of two centres and isochronous centres is characterized at points (−1,0) and (1,0) for the *X*_{i} and *Y* _{i} are homogeneous polynomials of degree *i*. However, in [12], only a particular case is studied because the general case is computationally unfeasible. The particular case studied in [12] has *x*=0 as an invariant straight line, which implies that the origin cannot be a centre. In this paper, we study the existence of more centres for such systems by also studying when they appear simultaneously. As before, we will give a condition providing that the system admits a centre at the origin and two additional centres at points (*a*,*b*) and (−*a*,−*b*) with *ab*≠0 arbitrary.

## 2. Definitions and preliminary results

In this section, we introduce some definitions and preliminary results which will be used throughout the paper.

By a linear change of coordinates and a time rescaling, system (1.1) with a weak focus can be written in the form
*X* and *Y* are polynomials without constant and linear terms. We denote by *H* defined in a neighbourhood of the origin which is constant along the trajectories, i.e.
*R* which is not identically zero is an *integrating factor* of system (2.1) if
*first integral* *H* associated with this integrating factor *R* is given by
*H* must satisfy ∂*H*/∂*x*=−*RQ*. A function *V* which is not identically zero is an *inverse integrating factor* of system (2.1) if it satisfies
*V* defines the integrating factor *R*=1/*V* where it does not vanish. The next results characterize when system (2.1) has a centre at the origin; see, for instance, [1].

### Theorem 2.1.

*System* (*2.1*) *has a centre at the origin if and only if there exists a local analytic first integral of the form H*(*x,y*)=*x*^{2}+*y*^{2}+*F*(*x,y*) *defined in a neighbourhood of the origin, where F starts with terms of order higher than* 2.

The next theorem is known as Reeb’s criterion for the classical centre problem; see [45,46].

### Theorem 2.2.

*Let p be a focus or a centre of system* (*2.1*). *Then p is a centre if and only if there is a non-zero analytic integrating factor V defined in a neighbourhood of p with V* (*p*)≠0.

For system (2.1), it is possible to construct a formal first integral of the form *H*(*x*,*y*)=*x*^{2}+*y*^{2}+⋯ , such that *V* _{i} are polynomials in the coefficients of system (2.1) called the *Poincaré–Liapunov constants*. These constants are the obstructions to the existence of a first integral for system (2.1). Hence if system (2.1) has a first integral, then *V* _{i}=0 for all *i*≥1. Consequently, the simultaneous vanishing of all the Poincaré–Liapunov constants provides sufficient conditions to have a centre at the origin of system (2.1). We define the ideal generated by these constants by *λ* are the parameters of system (2.1). This ideal is called the *Bautin ideal*, and the affine variety *centre variety* of system (2.1).

The Hilbert basis theorem ensures the existence of a positive value *k* such that

Finally, we recall some results from the Darboux theory on integrability for polynomial differential systems; for more details, see [1] or ch. 8 of [49] and references therein. An *invariant algebraic curve* *f*(*x*,*y*)=0 of system (2.1) is given by a polynomial *f*(*x*,*y*) satisfying
*K* is called the *cofactor* of the invariant algebraic curve, which is a polynomial of degree at most *n*−1. A *Darboux first integral* of system (2.1) is a first integral of the form *f*_{i} are invariant algebraic curves of system (2.1) and *Darboux integrating factor* of system (2.1) is an integrating factor of the form *f*_{1},*f*_{2},…,*f*_{k} are *K*_{1},*K*_{2},…,*K*_{k}, then if there exist *i*=1,…,*k* such that *i*=1,…,*k*, satisfying

A *time-reversible system* is a system which has a line through the origin such that this line is a symmetry axis of the phase portrait. More specifically, if this line is given by the straight line through the origin with slope *α*/2 the system is invariant under the symmetry (*x*,*y*,*t*)→(*x*,−*y*,−*t*). If we know that a singular point on this line is a centre or a focus, the presence of this time-reversible symmetry prevents this singularity being a focus; consequently, it must be a centre.

## 3. Simultaneity of centres for a Z 2 -equivariant cubic system

Liu & Li [11] studied what is called the bi-centre problem for a *c*_{i,j} and *d*_{i,j} are real parameters. Theorem 7 from [11] suggests splitting the centre variety of system (3.1) into 11 families. In what follows the existence of more centres for such a system that appears simultaneously is studied. First we impose the existence of a singular point at (*a*,*b*) with *ab*≠0 arbitrary, and, after that, we impose that this point is a focus–centre singular point with purely complex eigenvalues and we get

### Theorem 3.1.

*System* (*3.1*) *with the additional pair of focus–centre singular points* (*a,b*) *and* (−*a*,−*b*) *becomes*
*Moreover, the above system has five singular points of focus–centre type that simultaneously become centres if a*^{2}+*b*^{2}−1=0.

### Proof.

The real solution of system (3.2) is
*a*)/2,*b*/2), (−(1+*a*)/2,−*b*/2), ((*a*−1)/2,*b*/2), (−(*a*−1)/2,−*b*/2), (−*a*,−*b*) and (*a*,*b*). It is easy to see that the singular points (0,0), (−1,0), (1,0), (−*a*,−*b*) and (*a*,*b*) are focus–centre singular points, the first two because system (3.1) already had them. As we have assumed that (*a*,*b*) is a focus–centre singular point, by symmetry, we obtain that (−*a*,−*b*) is also a focus–centre singular point. The eigenvalues at the origin of system (3.3) are purely complex; consequently, the origin is also a focus–centre singular point of system (3.3). Moreover, the divergence of system (3.3) is
*a*^{2}+*b*^{2}−1=0 system (3.3) is Hamiltonian and all its focus–centre singular points simultaneously become centres because system (3.3) has a polynomial first integral. ▪

## 4. Simultaneity of centres for a Z 2 -equivariant quintic system

Recently, in [12], the authors studied the bi-centre problem for a *c*_{i,j} and *d*_{i,j} are real parameters. In theorem 4.2 in [12] four different families of centres are given that provide the centre variety of system (4.1), but only in the particular case *c*_{41}=−1 and *c*_{05}=0. Under these last conditions, the origin cannot be a centre because line *x*=0 is an invariant straight line of system (4.1). In the following, the existence of more centres for the general system (4.1) and the simultaneity in their appearance is studied. Assuming that (*a*,*b*) with *ab*≠0 is a singularity of focus–centre type, we obtain
*c*_{41}=−5/4 and *d*_{01}=0. To compute the focal values at point (1,0), we first move this point to the origin, applying the transformation *u*=*x*−1 and *v*=*y*. Computing these focal values and decomposing the ideal generated by the Poincaré–Liapunov constants we can establish the following theorem.

### Theorem 4.1.

*System* (*4.1*) *with c*_{41}=−5/4 *and d*_{01}=0 *has a centre at points* (−1,0) *and* (1,0) *if and only if one of the following conditions holds*.

(a)

*d*_{05}*=d*_{23}*=c*_{14}*=c*_{32}*=0,*(b)

*c*_{32}*−d*_{23}*=c*_{14}*−d*_{05}*=2d*_{32}*+1=4c*_{05}*+1=4c*_{23}*−4d*_{14}*+3=0,*(c)

*c*_{32}*+d*_{23}*=c*_{23}*+2d*_{14}*=c*_{14}*+5d*_{05}*=2d*_{32}*−5=0,*(d)

*c*_{14}*+d*_{23}*+2d*_{05}*=c*_{23}*−c*_{05}*+d*_{32}*−d*_{14}*+1=c*_{32}*+3d*_{05}*=2d*_{32}*d*_{05}*−d*_{23}*−2d*_{05}*= c*_{05}*d*_{23}*+d*_{23}*d*_{14}*−c*_{05}*d*_{05}*+3d*_{14}*d*_{05}*−d*_{23}*−4d*_{05}*=2c*_{05}*d*_{32}*+2d*_{32}*d*_{14}*+2c*_{23}*−5c*_{05}*−d*_{14}*=0.*

### Proof.

We computed the first eight non-zero focal values using the method described in §2. Their expressions are extremely long and we only write the first two here.

Case (a). Under condition (a) of theorem 4.1, system (4.1) with *c*_{41}=−5/4, *d*_{01}=0, and with point (1,0) at the origin takes the form
*u*,*v*,*t*)→(*u*,−*v*,−*t*), hence it is a time-reversible system and it has a centre at the points (1,0) and (−1,0).

Case (b). System (4.1) with *c*_{41}=−5/4 and *d*_{01}=0 under the conditions of statement (b) of theorem 4.1 and with point (1,0) at the origin takes the form
*V* =(1−2*u*+*u*^{2}+*v*^{2})^{3}, hence by Reeb’s theorem (see [46]) system (4.3) has a centre at points (1,0) and (−1,0). However, if we go back to the origin the inverse integrating factor takes the form *V* =(*x*^{2}+*y*^{2})^{3} and Reeb’s theorem cannot be applied to this point.

Case (c). Under the conditions of statement (c) of theorem 4.1, system (4.1) with *c*_{41}=−5/4, *d*_{01}=0, and with point (1,0) at the origin takes the form

Case (d). Under the conditions of statement (c) of theorem 4.1, system (4.1) with *c*_{41}=−5/4, *d*_{01}=0, and with point (1,0) at the origin takes the form
*V* =(1−2*u*+*u*^{2}+*v*^{2})^{5/2−d32}, hence by Reeb’s theorem system (4.3) has a centre at points (1,0) and (−1,0). However, if we go back to the origin the inverse integrating factor takes the form *V* =(*x*^{2}+*y*^{2})^{5/2−d32} and Reeb’s theorem cannot be applied at (0,0). ▪

Now we will investigate to what extent conditions (a)–(d) from theorem 4.1 describing centres at (−1,0) and (1,0) are compatible with the requirement that the system also admits a centre at the origin.

### Theorem 4.2.

*Any system* (*4.1*) *with c*_{41}=−5/4 *and d*_{01}=0 *satisfying one of the conditions* (*a*), (*b*) *and* (*d*) *of theorem 4.1 always has a centre at the origin. Moreover, satisfying condition* (*b*) *of theorem 4.1 has a centre at the origin if and only if* 3*d*_{05}+*d*_{23}=0.

### Proof.

For case (a), system (4.1) takes the form

For case (b), although the system has an inverse integrating factor given by *V* =(*x*^{2}+*y*^{2})^{3}, we have that *V* (0,0)=0 and Reeb’s theorem cannot be applied at the origin. In fact, if we construct the first integral associated with this inverse integrating factor we obtain a first integral which is not analytic at the origin. Moreover, computing the first focal value at the origin we obtain *V* _{4}=3*d*_{05}+*d*_{23}. If this focal value vanishes, we get the first integral

Case (c) is Hamiltonian and, consequently, all its focus–centre singular points are centres because in this case system (4.1) has a polynomial first integral.

Finally, case (d) also has the inverse integrating factor *V* =(*x*^{2}+*y*^{2})^{5/2−d32} with *V* (0,0)=0, and Reeb’s theorem cannot also be applied at the origin. But the associated first integral with *V* is

The next result follows when the families of centres given in theorem 4.1 have a centre at the singular point (*a*,*b*).

### Theorem 4.3.

*Any system* (*4.1*) *with c*_{41}=−5/4 *and d*_{01}=0 *satisfying one of the conditions* (*a*), (*b*), (*c*) *or* (*d*) *of theorem 4.1 has additional centres at the singular point* (*a,b*) *and* (−*a*,−*b*) *with ab*≠0 *if, and only if, for case* (*a*) −1+*a*^{4}+5*a*^{2}*b*^{2}=0, *for cases* (*b*), (*c*) *and* (*d*) *always*.

### Proof.

We take system (4.1) and impose the conditions of statement (a). Next we impose that the system has the singular point (*a*,*b*) and system (4.1) becomes
*a*,*b*) to the origin by applying the transformation *u*=*x*−*a* and *v*=*y*−*b*, and compute the focal values at this point. The first two non-zero focal values are
*V* _{1} and *V* _{2} with real roots are −1+*a*^{4}+5*a*^{2}*b*^{2}, which are *a*^{4}+5*a*^{2}*b*^{2}=0 system (4.5) becomes Hamiltonian.

For cases (b) and (d), we have that *V* =(*x*^{2}+*y*^{2})^{3} and *V* =(*x*^{2}+*y*^{2})^{5/2−d32} are inverse integrating factors, respectively. Therefore, at point (*a*,*b*) with *ab*≠0, we have the inverse integrating factor *V* =((*u*+*a*)^{2}+(*v*+*b*)^{2})^{3} and *V* =((*u*+*a*)^{2}+(*v*+*b*)^{2})^{5/2−d32}, both with *V* (0,0)≠0, respectively. Consequently, applying Reeb’s theorem (see [46]) in both cases, we have a centre at points (*a*,*b*).

Finally, case (c) is Hamiltonian and any focus–centre singular point must be a centre. ▪

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## Authors' contributions

J.G., J.L. and C.V. conceived the mathematical structure of the paper, deduced the new results and wrote the paper. All authors gave final approval for publication.

## Competing interests

We have no competing interests.

## Funding

J.G. is partially supported by a MINE CO/FEDER grant (no. MTM 2017-84383-P) and an AGAUR (Generalitat de Catalunya) grant (no. 2017SGR-1276). J.L. is partially supported by a FEDER-MINECO grant (no. MTM2016-77278-P), a MINECO grant (no. MTM2013-40998-P) and an AGAUR grant (no. 2014SGR-568). C.V. is supported by Portuguese national funds through FCT - Fundação para a Ciência e a Tecnologia within project PEst-OE/EEI/LA0009/2013 (CAMGSD).

## Acknowledgements

We thank the reviewers for their comments and suggestions, which helped us to improve this paper.

- Received November 20, 2017.
- Accepted April 13, 2018.

- © 2018 The Author(s)

Published by the Royal Society. All rights reserved.