Simultaneity of centres in ℤ_q-equivariant systems

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

in function of its degree n, where $n=max(\mathrm{deg}P,\mathrm{deg}Q)$ (see, for instance, [1,2]). It is well known that linear polynomial differential systems have no limit cycles, so 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 ${\mathbb{Z}}_q$-equivariant systems defined below using a cyclic group ${\mathbb{Z}}_q$. First, we give a survey on the existing results related to the local and global bifurcations of limit cycles for such systems perturbing their centres. The existence of a ${\mathbb{Z}}_q$-symmetry implies that, in analysing the number of limit cycles which can bifurcate from one centre, we are simultaneously analysing this problem for q centres. After the survey, some new particular cases are studied in detail.

Let G be a finite group of transformations acting on ${\mathbb{R}}^n$. A function $\varphi :{\mathbb{R}}^n\to {\mathbb{R}}^n$ is a G-equivariant function if, for all g∈G and all x, ϕ(gx)=gϕ(x). Given a G-equivariant function ϕ, the vector field dx/dt=ϕ(x) is called a G-equivariant vector field.

All the actions considered below are orthogonal two-dimensional real ${\mathbb{Z}}_q$-representations, where ${\mathbb{Z}}_q$ stands for a finite cyclic group of order q≥2. If q>2, the representation is irreducible, while, for q=2, one has a direct sum of two one-dimensional antipodal representations. ${\mathbb{Z}}_2$-equivariant vector fields are also called ${\mathbb{Z}}_2$-symmetric. Obviously, a phase portrait of a ${\mathbb{Z}}_2$-symmetric field in ${\mathbb{R}}^2$ is symmetric with respect to the origin. Its characterization is very straightforward in complex variables. After the change of variables z=x+iy and $\overline{z}=x-\mathrm{i}y$, system (1.1) takes the form

where $F(z,\overline{z})=P(x,y)+\mathrm{i}Q(x,y)$ with $x=(z+\overline{z})/2$ and $y=(z-\overline{z})/(2i)$.

The following result characterizes a ${\mathbb{Z}}_q$-equivariant complex vector field and is proved in [3]; see also [2,4].

Theorem 1.1.

The vector field (1.2) is a ${\mathbb{Z}}_q$-equivariant complex vector field if and only if the complex function $F(z,\overline{z})$ 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 $\mathrm{\partial }F/\mathrm{\partial }z+\mathrm{\partial }\overline{F}/\mathrm{\partial }\overline{z}=0$.

The ${\mathbb{Z}}_5$-equivariant complex vector fields of degree 5 have been described in [2]. However, in this classification, there are some mistakes. For instance, for q=4 there appears the term $A_5{\overline{z}}^5$, which is incompatible with the form of $F(z,\overline{z})$. This mistake is also repeated in [4].

Several authors have studied the ${\mathbb{Z}}_q$-equivariant systems in order to classify their centres and to compute the number of limit cycles which can bifurcate from these centres under convenient perturbations. For instance in [2], a method was given to control the parameters in order to obtain as many limit cycles as possible. The method was applied to ${\mathbb{Z}}_q$-equivariant perturbed polynomial Hamiltonian systems of degree 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], ${\mathbb{Z}}_2$-equivariant systems have been studied. In particular the centres, isochronous centres and local critical periods of ${\mathbb{Z}}_2$-equivariant cubic systems have been studied in [7–9,11,12]. In fact, the study of the number of limit cycles which can bifurcate from the centres of ${\mathbb{Z}}_2$-equivariant cubic systems has provided the highest lower bound on the number of limit cycles for cubic systems (see [14] and references therein). The centre problem and the bifurcation of limit cycles for ${\mathbb{Z}}_2$-equivariant quintic systems have been studied in [10–12]. Finally, the study of ${\mathbb{Z}}_2$-equivariant Liénard systems was started in [13].

The limit cycles of ${\mathbb{Z}}_3$-equivariant near-Hamiltonian systems have been considered in [15,16]. In [17–22], ${\mathbb{Z}}_4$-equivariant systems have been analysed, with special attention paid to ${\mathbb{Z}}_4$-equivariant quintic systems. However, ${\mathbb{Z}}_4$-equivariant cubic systems were treated in [17], and the small limit cycles of the ${\mathbb{Z}}_4$-equivariant near-Hamiltonian systems have been studied in [21]. Finally, the study of ${\mathbb{Z}}_6$-equivariant quintic systems began in [23,24], and the bifurcations of limit cycles in the ${\mathbb{Z}}_8$-equivariant planar vector field of degree 7 have been considered in [25].

Recently, the limit cycles for ${\mathbb{Z}}_{2n}$-equivariant systems without an infinite singular point have been discussed in [26]. As a classification of the phase portraits is not the goal of the present paper, we do not mention the works devoted to this topic for some ${\mathbb{Z}}_q$-equivariant systems.

${\mathbb{Z}}_2$-symmetric systems are of special interest. Recently, the centre problem for such systems was solved in [27]. The bifurcations of the limit cycles for ${\mathbb{Z}}_2$-symmetric cubic systems have been studied in [28], and for ${\mathbb{Z}}_2$-symmetric Liénard systems in [29]. The first natural question that arises is: what is the relation between ${\mathbb{Z}}_q$-equivariant systems and ${\mathbb{Z}}_2$-symmetric systems?

Observe that there exist ${\mathbb{Z}}_q$-equivariant systems with q≠2 which are not ${\mathbb{Z}}_2$-symmetric. For instance, the differential system

is ${\mathbb{Z}}_4$-equivariant but is not ${\mathbb{Z}}_2$-symmetric.

The importance of ${\mathbb{Z}}_q$-equivariant systems in the context of this paper is due to the fact that if an equilibrium is a centre, then any point belonging to its ${\mathbb{Z}}_q$-orbit is also a centre; hence bifurcations of limit cycles related to ${\mathbb{Z}}_q$-symmetric systems come in orbits, which is crucial for increasing the lower bounds for the Hilbert numbers H(n). In other words, the phase portraits of ${\mathbb{Z}}_q$-equivariant systems are invariant under a certain rotation angle, which allows a large number of centres and consequently of limit cycles. It was shown in [30–32] that H(n)≥k₁n² for some constant k₁. In [33], perturbing some ${\mathbb{Z}}_q$-equivariant systems, it was proved that H(n) grows at least as $k_2{n}^2\log n$. Some small improvements to this bound have been given in [34–37], increasing, in some cases, the values of the constant k₂. In fact, in [38], it was conjectured that H(n) is O(n³) as $n\to \mathrm{\infty }$.

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 ${\mathbb{Z}}_2$-equivariant cubic systems of the form

where 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 ${\mathbb{Z}}_2$-symmetric. The necessary and sufficient conditions, providing these centres are isochronous, were given in [12]. In this paper, we will study the conditions providing multiple centres in system (1.3). More specifically, we will give conditions providing that the system admits a centre at the origin and two more centres at points (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 ${\mathbb{Z}}_2$-equivariant quintic system of the form

where 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

where X and Y are polynomials without constant and linear terms. We denote by $\mathcal{X}=P\mathrm{\partial }/\mathrm{\partial }x+Q\mathrm{\partial }/\mathrm{\partial }y$ the vector field associated with system (2.1). A first integral of system (2.1) is a non-constant function H defined in a neighbourhood of the origin which is constant along the trajectories, i.e. A function R which is not identically zero is an integrating factor of system (2.1) if A first integral H associated with this integrating factor R is given by where this 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 This function 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)=x²+y²+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)=x²+y²+⋯ , such that $\dot{H}=\mathcal{X}H=\sum _{i=1}^{\mathrm{\infty }}{V}_i{(x^2+y^2)}^{2i}$, where the 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 $\mathcal{B}=⟨V_1,V_2,\dots ⟩\subset \mathbb{C}[\mathrm{\lambda }]$, where λ are the parameters of system (2.1). This ideal is called the Bautin ideal, and the affine variety $V(\mathcal{B})$ is the centre variety of system (2.1).

The Hilbert basis theorem ensures the existence of a positive value k such that $\mathcal{B}={\mathcal{B}}_k=⟨V_1,V_2,\dots V_k⟩$. In fact, we always have that $V(\mathcal{B})\subset V({\mathcal{B}}_k)$. The opposite inclusion is satisfied if any point of each component of the irreducible decomposition of $V({\mathcal{B}}_k)$ corresponds to a system having a centre at the origin. To find the irreducible component of $V({\mathcal{B}}_k)$, we use the routine minAssGTZ [47] of the computer algebra system Singular [48].

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

where 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 $H=f_1^{{\alpha }_1}\cdots f_k^{{\alpha }_k}$, where f_i are invariant algebraic curves of system (2.1) and ${\alpha }_i\in \mathbb{C}$. A Darboux integrating factor of system (2.1) is an integrating factor of the form $R=f_1^{{\beta }_1}\cdots f_k^{{\beta }_k}$, with ${\beta }_i\in \mathbb{C}$. Assume that the cofactors of invariant curves f₁,f₂,…,f_k are K₁,K₂,…,K_k, then if there exist ${\alpha }_i\in \mathbb{C}$ for i=1,…,k such that $\sum _{i=1}^k{\alpha }_i{K}_i=0$ then $H=f_1^{{\alpha }_1}\cdots f_k^{{\alpha }_k}$ is a Darboux first integral of system (2.1). Moreover, if there exist ${\beta }_i\in \mathbb{C}$ for i=1,…,k, satisfying $\sum _{i=1}^k{\alpha }_i{K}_i+\mathrm{\partial }P/\mathrm{\partial }x+\mathrm{\partial }Q/\mathrm{\partial }y=0$, then $R=f_1^{{\beta }_1}\cdots f_k^{{\beta }_k}$ is a Darboux integrating factor of system (2.1).

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 $\tan (\alpha /2)$, then after a rotation of 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 ${\mathbb{Z}}_2$-equivariant cubic system

Liu & Li [11] studied what is called the bi-centre problem for a ${\mathbb{Z}}_2$-equivariant cubic system of the form (1.3), and they found the necessary and sufficient conditions for the existence of two centres at points (−1,0) and (1,0). Assume that (−1,0) and (1,0) are focus–centre singularities for (1.3). Then (1.3) takes the following form:

where 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 The unique real solution of this algebraic system of equations gives the following result.

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 a²+b²−1=0.

Proof.

The real solution of system (3.2) is

Plugging these values into system (3.1), we obtain system (3.3). System (3.3) has the finite singular points (0,0), (−1,0), (1,0), ((1+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 Therefore, if a²+b²−1=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 ${\mathbb{Z}}_2$-equivariant quintic system

Recently, in [12], the authors studied the bi-centre problem for a ${\mathbb{Z}}_2$-equivariant quintic system of the form (1.4), and they gave the necessary and sufficient conditions for the existence of two centres at points (−1,0) and (1,0). Assuming that system (1.4) admits focus–centre singularities at these points, it takes the form

where 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 c₄₁=−1 and c₀₅=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 The unique real solution of this algebraic system of equations is However, we do not introduce all these values in system (4.1), because we have to compute the Poincaré–Liapunov constants or focal values at point (1,0), and with these substitutions the computations become harder. Hence we only impose the simple conditions c₄₁=−5/4 and d₀₁=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 c₄₁=−5/4 and d₀₁=0 has a centre at points (−1,0) and (1,0) if and only if one of the following conditions holds.

• (a) d₀₅=d₂₃=c₁₄=c₃₂=0,

• (b) c₃₂−d₂₃=c₁₄−d₀₅=2d₃₂+1=4c₀₅+1=4c₂₃−4d₁₄+3=0,

• (c) c₃₂+d₂₃=c₂₃+2d₁₄=c₁₄+5d₀₅=2d₃₂−5=0,

• (d) c₁₄+d₂₃+2d₀₅=c₂₃−c₀₅+d₃₂−d₁₄+1=c₃₂+3d₀₅=2d₃₂d₀₅−d₂₃−2d₀₅= c₀₅d₂₃+d₂₃d₁₄−c₀₅d₀₅+3d₁₄d₀₅−d₂₃−4d₀₅=2c₀₅d₃₂+2d₃₂d₁₄+2c₂₃−5c₀₅−d₁₄=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.

Next, we compute the irreducible decomposition of the variety $V({\mathcal{B}}_8)=V(⟨V_1,V_2,V_3,V_4,V_5,V_6,V_{17},V_8⟩)$ of these Poincaré–Liapunov constants using the routine minAssGTZ of the computer algebra system Singular over the field of rational numbers and we obtain the families given in the statement of the theorem. Now we prove the sufficiency.

Case (a). Under condition (a) of theorem 4.1, system (4.1) with c₄₁=−5/4, d₀₁=0, and with point (1,0) at the origin takes the form

This system respects the symmetry (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 c₄₁=−5/4 and d₀₁=0 under the conditions of statement (b) of theorem 4.1 and with point (1,0) at the origin takes the form

The above system has an inverse integrating factor of the form V =(1−2u+u²+v²)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 =(x²+y²)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 c₄₁=−5/4, d₀₁=0, and with point (1,0) at the origin takes the form

System (4.5) is Hamiltonian and then it has a centre at the singular points (1,0) and (−1,0).

Case (d). Under the conditions of statement (c) of theorem 4.1, system (4.1) with c₄₁=−5/4, d₀₁=0, and with point (1,0) at the origin takes the form

This system has an inverse integrating factor of the form V =(1−2u+u²+v²)^{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 =(x²+y²)^{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 c₄₁=−5/4 and d₀₁=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 3d₀₅+d₂₃=0.

Proof.

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

Therefore, it is also time reversible and consequently has a centre at (0,0).

For case (b), although the system has an inverse integrating factor given by V =(x²+y²)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=3d₀₅+d₂₃. If this focal value vanishes, we get the first integral

or the first integral where Then the first integral is well defined at the origin. So the origin is a centre.

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 =(x²+y²)^{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

where $f=1+2d_{32}+3x^4-2d_{32}{x}^4+4(3+d_{32}-2d_{32}^2)x^2{y}^2+4d_{05}(3-4(d_{32}-1)d_{32})x{y}^3-(2d_{32}-3)(1-c_{23}-d_{32}-2c_{23}{d}_{32}-2d_{32}^2)y^4$. Hence this first integral or its inverse is always analytic at the origin. Therefore, the origin of system (4.1) is a centre. ▪

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 c₄₁=−5/4 and d₀₁=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+a⁴+5a²b²=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

The next step is to move point (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 and Then we have that the unique common factors of V ₁ and V ₂ with real roots are −1+a⁴+5a²b², which are $b=\pm \sqrt{1-a^4}/(\sqrt{5}a)$. Furthermore, when −1+a⁴+5a²b²=0 system (4.5) becomes Hamiltonian.

For cases (b) and (d), we have that V =(x²+y²)³ and V =(x²+y²)^{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. ▪

Data accessibility

This work does not have any experimental data.

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).