Bi-center problem for some classes of Z2-equivariant systems.

UNIVERSIDADE DE SÃO PAULO

Instituto de Ciências Matemáticas e de Computação

ISSN 0103-2577

_______________________________

BI-CENTER PROBLEM FOR SOME CLASSES OF Z2-EQUIVARIANT SYSTEMS

V.G. ROMANOVSKI W. FERNANDES

R. OLIVEIRA

Nº 422

_______________________________

NOTAS DO ICMC

SÉRIE MATEMÁTICA

São Carlos – SP

Set./2016

SYSTEMS

V.G. ROMANOVSKI^1,2,3, W. FERNANDES⁴, R. OLIVEIRA⁴

Abstract. We investigate the simultaneous existence of two centers (bi-center) for two families of planar Z2-equivariant differential systems. First we present necessary and suf- ficient conditions for the existence of an isochronous bi-center for a planar Z2-equivariant cubic system having two centers at the points (−1,0) and (1,0), completing the study done by Liu and Li (2011). Next, we give conditions for the existence of a bi-center and study its isochronicity for a planarZ2-equivariant quintic system having two weak foci or centers at the points (−1,0) and (1,0). We also give an example of a cubic system with three isochronous centers.

1. Introduction

In a real planar analytic differential system a singular point with pure imaginary eigenvalues of the matrix of the linear approximation can be either a focus or a center. The problem to distinguish between a center or a focus is called thecenter-focus problem. If the singular point is a center the next arising problem is to determine whether the center is isochronous.

Although the center-focus problem have been studied during more than hundred years by many authors it is unresolved even for planar systems with cubic nonlinearities. For polynomial quadratic systems it is solved in [11, 17, 18, 36], were the authors presented necessary and sufficient conditions for the existence of a center. For polynomial cubic systems only particular families were investigated, see e.g. [1, 4, 21, 22, 33, 34, 35, 37] and references therein. There are also some works on the center problem for families of polynomial systems of higher degrees, see e.g. [12, 13, 14] and references given there.

The studies of isochronicity of polynomial systems goes back to Loud [27], who found the necessary and sufficient conditions for isochronicity of the quadratic system. Latter on, the isochronicity problem was solved for the linear center perturbed by homogeneous polynomials of degree three [29] and degree five [31]. There are also many works devoted to the investigation of particular families of polynomial systems, see e.g. [2, 3, 6, 28, 33] and references therein.

The existence of two simultaneous centers in planar differential systems was investigated only for very few particular families of systems. Kirnitskaya and Sibirskii in [19] and Li in [20]

studied the bi-center problem for quadratic systems. They presented necessary and sufficient conditions for a planar quadratic differential system to have two centers simultaneously. Conti [7] investigated a particular family of cubic systems obtaining the first cubic system possessing a bi-center. Chen, Lu and Wang [5] studied other particular family of cubic systems called Kukles system, characterizing when such systems have a bi-center.

Recently Liu and Li [24] studied theZ2-equivariant cubic system of the form (1.1) x˙ =X₁(x, y) +X₃(x, y), y˙=Y₁(x, y) +Y₃(x, y),

1

whereX_i(x, y), Y_i(x, y), (i= 1,3), are homogeneous polynomials of degreei, in the variables xand yand having two weak foci or centers at the points (−1,0) and (1,0). They presented necessary and sufficient conditions for system (1.1) to have a bi-center at these points. Du [10]

also studied a system havingZ₂-equivariant symmetry. He obtained necessary and sufficient conditions for the existence of a bi-center and an isochronous bi-center for a particular family of Z₂-equivariant system of degree seven.

The main purpose of this paper is to find conditions for the existence of bi-centers and isochronous bi-centers for two families ofZ₂-equivariant systems. This paper has three main results: Theorem 3.1 which gives the necessary and sufficient conditions for the existence of an isochronous bi-center in system (1.1) at the points (−1,0) and (1,0), Theorem 4.2 and Theorem 4.4, which characterize the existence of a bi–center and an isochronous bi–center, respectively, in the Z2-equivariant quintic system of the form

(1.2) x˙ =X₁(x, y) +X₅(x, y) =X(x, y), y˙=Y₁(x, y) +Y₅(x, y) =Y(x, y),

where Xi(x, y), Yi(x, y), (i= 1,5) are homogeneous polynomials of degree iin the variables x and y and (1.2) has two weak foci or centers at the points (−1,0) and (1,0). The reason for choosing homogeneous polynimials of degree five in (1.2) (rather than polynomials of degree four) is to assure the existence of Z₂-equivariant symmetry, which can appear if the polynomials defining the system have only odd degree monomials. So, if we replace in (1.2) X5(x, y) andY5(x, y) with homogeneous polynomials of degree four, then the system can not have a bi-center at (−1,0) and (1,0).

2. Preliminaries

In this section we introduce some definitions and basic results which are used in this paper.

2.1. The center and integrability problems. Consider a real polynomial differential sys- tem on R² having a weak focus or an elementary center at the point (x₀, y₀). Moving the point (x0, y0) to the origin and applying a linear change of coordinates and a time rescaling we can write the system in the form

(2.1) x˙ =−y+

n

X

p+q≥2

a_p,qx^py^q =P(x, y), y˙ =x+

n

X

p+q≥2

b_p,qx^py^q =Q(x, y), whereap,q, bp,q∈R. We will denote byXits corresponding vector field

X=P ∂

∂x +Q ∂

∂y.

A local first integral of system (2.1) is a nonconstant differentiable function Φ from a neigh- bourhood of the origin inR² intoRwhich is constant on trajectories of (2.1), equivalently,

XΦ =PΦx+QΦy ≡0.

A formal first integral of system (2.1) is a formal power series Φ in the variables x and y satisfying XΦ≡0.

The next theorem due to Poincar´e and Lyapunov ([23, 30]) characterizes when a system of the form (2.1) posses a center at the origin.

Theorem 2.1. System (2.1)has a center at the origin if and only if it admits a local analytic first integral of the form

(2.2) Φ(x, y) =x²+y²+· · ·.

Moreover, the existence of a formal first integral Φ of the form (2.2) implies the existence of a local analytic first integral of the same form.

For system (2.1) it is always possible to find a formal power series of the form Ψ = x²+y²+P

k+j≥3uk,jx^ky^j,such that

XΨ =v₁(xy)²+v₂(x²y)²+v₃(x³y)²+· · ·+v_k(x^ky)²+· · ·,

wherev_kare polynomials in the parametersap,qandbp,qof system (2.1) calledfocus quantities.

The polynomialsv_krepresent obstacles for the existence of the first integral of the form (2.2), that is, system (2.1) admits a first integral of the form (2.2) if and only ifvk= 0, for allk≥1.

Thus, the simultaneous vanishing of all focus quantities provide conditions which characterize when a system of the form (2.1) has a center at the origin. The ideal defined by the focus quantities, B = hv₁, v₂, . . .i ⊂ C[a, b], where a and b represents all the parameters a_p,q and b_p,q, respectively, of system (2.1) is called Bautin ideal. The affine variety VC = V(B), is called the center variety of system (2.1).

By the Hilbert Basis Theorem there exist a positive integer k such that B = B_k = hv₁, ..., vki. Note that the inclusion VC = V(B) ⊂ V(B_k) holds for any k ≥ 1. The op- posite inclusion is verified finding the irreducible decomposition of V(B_k) and then checking that any point of each component of the decomposition corresponds to a system having a center at the origin. To find the irreducible decomposition ofV(B_k) we performed computa- tions with the routineminAssGTZ[9] (which is based on the algorithm of [16]) of the computer algebra system Singular[8].

We now recall few results from the Darboux theory of integrability that allow to find first integrals of polynomial systems (see e.g. [25, 26, 33] for more details).

Consider the general polynomial differential system

(2.3) x˙ =U(x, y), y˙=V(x, y),

where x, y ∈R,U and V are polynomials in the variables x, y without constant terms that have no common factor, andm=max(deg(U), deg(V)).

A Darboux factor of system (2.3) is a polynomialf(x, y) satisfying

∂f

∂xU +∂f

∂yV =Kf,

whereK(x, y) is a polynomial of degree at mostm−1, called thecofactor of f.

A Darboux first integral of system (2.3) is a first integral of system (2.3) of the form H= f₁^α1· · ·f_k^αk, wheref₁, f₂, ..., f_kare Darboux factors of system (2.3) andα_i ∈R, 1≤i≤k. If a first integral of system (2.3) cannot be found, then we turn our attention to possible existence of an integrating factor. A Darboux integrating factor of system (2.3) is an integrating factor of the formµ=f₁^β1···f_k^βk, wheref1, f2, ..., f_kare Darboux factors of system (2.3) andβi∈R, 1≤i≤k.

Suppose that f1, f2, ..., fk are Darboux factors of system (2.3) with respective cofactors K₁, K₂, ..., K_k. Then it is easy to see that

1. If there exist constants α_i ∈R, 1≤i≤k, satisfying (2.4)

k

X

i=1

α_iK_i = 0,

then H=f₁^α1 · · ·f_k^αk is a Darboux first integral of (2.3).

2. If there exist constants βi ∈R, 1≤i≤k, satisfying (2.5)

k

X

i=1

β_iK_i+∂U

∂x + ∂V

∂y = 0,

then µ=f₁^β1 · · ·f_k^βk is a Darboux integrating factor of system (2.3).

Finally, we recall that a differential system possess atime-reversible symmetrywith respect to a lineLif its phase portrait is invariant after reflecting with respect to the line and reversing the sense of all orbits (reversing of time). If we know that a singular point on the line L is a center or a focus, the presence of time-reversible symmetry with respect to L prevents this singularity to be a focus, consequently forcing it to be a center. When the line Lis the x-axis, system (2.3) posses time–reversible symmetry if and only if obviously,

(2.6) U(x,−y) =−U(x, y), V(x,−y) =V(x, y).

2.2. Isochronicity and linearizability problems. If the singular point at the origin of (2.1) is known to be a center we say that this center is isochronous if all periodic solutions of (2.1) in a neighbourhood of the origin have the same period. System (2.1) is said to be linearizable if there is an analytic change of coordinates

(2.7) x₁ =x+ X

m+n≥2

c_m,nx^myⁿ, y₁=y+ X

m+n≥2

d_m,nx^myⁿ, that reduces (2.1) to the canonical linear center,

(2.8) x˙₁ =−y₁, y˙₁=x₁.

The following theorem which goes back to Poincar´e and Lyapunov shows that there is an intimate relation between linearizability and isochronicity. A proof can be found e.g. in [33].

Theorem 2.2. The origin of system (2.1)is an isochronous center if and only if the system is linearizable.

It follows from Theorem 2.2 that solving the isochronicity problem is equivalent to solving the linearizability problem, but the investigation of the latter problem is computationally much simpler.

Differentiating with respect totboth sides of each equation of (2.7) we obtain

(2.9)

˙

x1 = ˙x+



 X

m+n≥2

mcm,nx^m−1yⁿ



x˙+



 X

m+n≥2

ncm,nx^myⁿ⁻¹



y,˙

˙ y1 = ˙y+



 X

m+n≥2

mdm,nx^m−1yⁿ



x˙+



 X

m+n≥2

ndm,nx^myⁿ⁻¹



y.˙

Thus, the substitution (2.7) linearizes system (2.1) if it holds that (2.10)

X

m+n≥2

dm,nx^myⁿ+

n

X

p+q≥2

ap,qx^py^q+



 X

m+n≥2

mcm,nx^m−1yⁿ







−y+

n

X

p+q≥2

ap,qx^py^q





+



 X

m+n≥2

ncm,nx^myⁿ⁻¹







x+

n

X

p+q≥2

bp,qx^py^q



≡0,

− X

m+n≥2

c_m,nx^myⁿ+

n

X

p+q≥2

b_p,qx^py^q+



 X

m+n≥2

md_m,nx^m−1yⁿ







−y+

n

X

p+q≥2

a_p,qx^py^q





+



 X

m+n≥2

nd_m,nx^myⁿ⁻¹







x+

n

X

p+q≥2

b_p,qx^py^q



≡0.

The left hand sides of two equations in (2.10) can be written in the form P

k,l≥2h^(k,l)₁ x^ky^l, and P

k,l≥2h^(k,l)₂ x^ky^l, respectively, where h^(k,l)₁ and h^(k,l)₂ are polynomials in the parameters a_p,q, b_p,q,p+q ≥2, of system (2.1) and c_m,n,d_m,n m+n≥2, of (2.7). It is clear that both equations in (2.10) are satisfied if and only if h^(k,l)_i = 0,i= 1,2, for allk, l∈N0,k+l≥2.

Using the computer algebra systemMathematicawe can construct a linearizing transfor- mation (2.7). The process starts from solving the polynomial system h^(k,l)_i = 0,i= 1,2 such that k+l = 2. At this step we are able to determine the coefficients c_m,n,d_m,n,m+n= 2 of (2.7) in terms of the parametersap,q’s andbp,q’s of system (2.1). The next step is to solve the polynomial system h^(k,l)_i = 0, i= 1,2 such that k+l = 3, determining the coefficients c_m,n,d_m,n,m+n= 3, of (2.7) in terms of the parametersa_p,q’s andb_p,q’s of system (2.1). In general case the polynomial system h^(k,l)_i = 0,i= 1,2 such that k+l= 3, cannot be solved.

However dropping two suitable equations we obtain a system that has a solution. We denote the two dropped polynomials on the left hand sides of these two equations by i1 and j1.

Proceeding step-by-step we obtain that the polynomial systemh^(k,l)_i = 0,i= 1,2,k+l=r, can not be solved whenr=k+lis odd number. Dropping on each step two suitable equations (and denoting by i_(r−1)/2 and j_(r−1)/2 the corresponding polynomials), we obtain a system that has a solution.

The polynomials i_k and j_k obtained at each odd step of the process are polynomials in the parameters a_p,q and b_p,q of system (2.1) called the linearizability quantities. They represent obstacles for the existence of a linearizing change of coordinates (2.7), that is, system (2.1) admits a linearizing change of coordinates (2.7) if and only if i_k = j_k = 0, for all k > 1. Thus, the simultaneous vanishing of all linearizability quantities provide conditions which characterize when a system of the form (2.1) is linearizable (equivalently it has an isochronous center at the origin). The ideal defined by the linearizability quantities, L=hi₁, j1, i2, j2, ...i ⊂C[a, b], is called linearizability ideal and its affine variety,VL=V(L) is called thelinearizability variety.

By the Hilbert Basis Theorem there exists a positive integer k such that L = L_k = hi₁, j₁, ..., i_k, j_ki. Computing the irreducible decomposition of the variety V(L_k) (using the

routineminAssGTZ of the computer algebra system Singular) we find necessary conditions for the existence of the linearizable change of coordinates of the form (2.7). To prove the sufficiency of the obtained conditions we check that any point of each component of V(L_k) corresponds to a linearizable system.

To find a linearizing change of coordinates we use the method of Darboux linearization.

To construct Darboux linearization it is convenient to complexify the real system (2.1). To this end, we introduce the complex variable z =x+iy, obtaining from (2.1) the system of the form

(2.11) z˙=i(z−X(z, z)),

where i = √

−1, X = (P +iQ)/i and P and Q are evaluated at ((z+z)/2,(z −z)/2i).

Adjoining to equation (2.11) its complex conjugate, ˙z=−i(z−X(z, z)), and replacing zby w, we obtain the pair of equations

(2.12) z˙=i(z−X(z, w)), w˙ =−i(w−Y(z, w)),

whereY(z, z) =X(z, z). System (2.12) is the complexification of the real system (2.1).

The linearizability problem for the complex system (2.12) is to decide whether it can be transformed to the linear system ˙z₁ =iz₁, ˙w₁ =−iw₁ by an analytic change of coordinates of the form

(2.13) z₁ =z+Z₁(z, w), w₁=w+W₁(z, w).

Remark 2.3. If the system

(2.14) z˙=z−X(z, w), w˙ =−w+Y(z, w),

is transformed into the linear system ˙z1 =z1, ˙w1=−w₁by transformation (2.13), then (2.12) is reduced to ˙z₁ =iz₁, ˙w₁ =−iw₁by the same transformation. Conversely, if (2.13) linearizes (2.12), then it also linearizes (2.14). Therefore systems (2.12) and (2.14) are equivalent with regard to the problem of linearizability.

A Darboux linearization of system (2.14) is an analytic change of coordinates, z₁ = Z1(z, w), w1 =W1(z, w), such that

Z₁(z, w) =

m

Y

j=0

f_j^αj(z, w) =z+Z₁⁰(z, w),

W1(z, w) =

n

Y

j=0

g_j^βj(z, w) =w+W₁⁰(z, w),

wheref_j, g_j ∈C[z, w],α_j, β_j ∈C, and Z₁⁰ andW₁⁰ have neither constant nor linear terms.

A system is said to be Darboux linearizable if it admits a Darboux linearization. The next theorem provides a way to construct a Darboux linearization using Darboux factors (see e.g.

[33] for a proof).

Theorem 2.4. System (2.14) is Darboux linearizable if and only if there exits s+ 1 ≥ 1 Darboux factors f₀, ..., f_s with corresponding cofactors K₀, ..., K_s and t+ 1 ≥ 1 Darboux factors g0, ..., gt with corresponding cofactors L0, ..., Lt with the following properties:

a. f0(z, w) =z+· · · but fj(0,0) = 1for j ≥1;

b. g₀(z, w) =w+· · · butg_j(0,0) = 1 for j≥1; and

c. there ares+t constants α₁, ..., α_s, β₁, ..., β_t∈Csuch that

(2.15) K₀+α₁K₁+· · ·+α_sK_s = 1 and L₀+β₁L₁+· · ·+β_tL_t=−1.

The Darboux linearization is then given by

z₁ =H₁(z, w) =f₀f₁^α1 · · ·f_s^αs, w₁ =H₂(z, w) =g₀g₁^β1· · ·g^β_t^t. 3. Isochronicity and linearizability problems for system (1.1)

In [24] Liu and Li studied the bi-center problem for aZ2-equivariant cubic system of the form (1.1) and found the necessary and sufficient conditions for existence of a bi-center at the points (−1,0) and (1,0). After a change of coordinates, they obtained from (1.1) the following standard form of the system (see Theorem 7 of [24]):

(3.1)

˙

x=−(a₁+ 1)y+a1x²y+a2xy²+a3y³,

˙ y=−1

2x−a₄y+1

2x³+a₄x²y+a₅xy²+a₆y³.

The following eleven necessary and sufficient conditions for the existence of a bi-center at the points (−1,0) and (1,0) for system (3.1) are given in Theorem 11 of [24]:

(1) a4 = 0,a1 =−a₅,a6 =−a₂/3;

(2) a₄ = 0,a₁+a₅6= 0, a₂=a₆ = 0;

(3) a1+a56= 0,a6 =−(a₂+2a1a2−2a₄+2a2a5−2a₄a5)/3, 2(1+a1)(a1+a5)²−a²₄(1+2a1+ 2a₅) = 0, 3(a₁+a₅)(−a₃+ 2(1 +a₁)(1 +a₅))−2a₄(2a₄(1 +a₅) +a₂(2 +a₁+a₅)) = 0;

(4) a3 = 2(1 +a1)(1 +a5),a6 =−(a₂+ 2a1a2−2a4+ 2a2a5−2a4a5), 2(1 +a5)a4+ (2 + a₁+a₅)a₂ = 0;

(5) a4 6= 0,a1 =−(2−3a²₄)/3,a2=a4,a3=a24(1−a²₄+a5),a6=a4(1−a²₄);

(6) a₄ 6= 0, a₁ = −(8−5a²₄)/8, a₂ = a₄/2, a₅ = −(8 +a²₄)/8, a₃ = −5a⁴₄/32, a₆ = a4(2−a²₄)/4;

(7) a₄ 6= 0, a₁ = −(32 + 15a²₄)/32, a₂ = a₄/4, a₃ = a²₄(64 + 15a²₄)/512, a₅ = −(96 + 17a²₄)/32, a6 =−3a₄(4 +a²₄)/16;

(8) a₄ 6= 0, a₁ =−(50 + 21a²₄)/50, a₂ =a₄/5, a₃ =a²₄/1250(250 + 63a²₄), a₅ =−(200 + 39a²₄)/50, a6 =−a₄/25(35 + 9a²₄);

(9) a₄ 6= 0,a₁ =−(9 + 4a²₄)/9,a₂= 0, a₃ = 0,a₆ = 2a₄/3(1 +a₅);

(10) a4 6= 0,a1 =−(8 + 3a²₄)/8,a2 =−a₄/2,a3 = 3a²₄(4 +a²₄+ 4a5)/16, a6 =a4(4−a²₄+ 8a₅)/8;

(11) a4 6= 0, a1 =−(32 + 15a²₄)/32, a2 =−a₄/4, a3 =a²₄/512(832 + 495a²₄), a5 = (160 + 111a²₄)/32, a₆ =a₄(76 + 45a²₄)/16.

To complete the study of bi-centers started by Liu and Li in [24] we investigate the existence of isochronous bi–centers for system (3.1) and obtain the following result.

Theorem 3.1. System (3.1) has an isochronous bi-center at the points (1,0) and (−1,0) if and only if one of the following conditions is satisfied:

(i) a1 =−3/2, a2 = 0, a3= 1/2, a4= 0, a5 =−3/2, a6 = 0;

(ii) a₁ =−3, a₂= 0, a₃ = 0, a₄ = 0, a₅=−9, a₆= 0.

Proof. To compute the linearizability quantities we move the singular point (1,0) to the origin.

Applying the translation u=x−1, v=y, we obtain from (3.1) the system (3.2)

˙

u=−v+a1(2 +u)uv+a2(1 +u)v²+a3v³,

˙

v=u+3 2u²+1

2u³+a₄(2 +u)uv+a₅(1 +u)v²+a₆v³.

Using the computer algebra system Mathematica we computed the first seven pairs of the linearizability quantities for system (3.2) using the procedure described in Subsection 2.2.

Their expressions are very large, so we present here only the first pair:

i₁ = 18 + 4a²₁+ 10a²₂+ 9a₃−2a₂a₄+ 4a²₄+ 12a₅−10a₁a₅+ 4a²₅, j₁ =a₂+ 2a₁a₂−2a₄+ 2a₂a₅−2a₄a₅+ 3a₆.

The reader can easily compute the others quantities using any available computer algebra system.

The next computational step is to compute the irreducible decomposition of the variety V(L₇) = V(hi₁, j₁, ..., i₇, j₇i). For this purpose we use the routine minAssGTZ of the com- puter algebra systemSingular. Since an isochronous center must be a center, to make the computations easier we investigated the existence of an isochronous center using the eleven cases found by Liu and Li in [24] and given above. Thus, our proof is split in eleven cases corresponding to these eleven conditions.

Case(1): Computation withminAssGTZ of Singularshows that I =hL₇, a4, a1+a5, a6+a2/3i, is a primary ideal whose minimal associate prime is

I₁=ha₄, a²₆+ 1,2a₅−3, a₁+a₅,−2a₄a₅+a₂−2a₄+ 3a₆, fi, wheref = 40

9 a²₄a²₅+76

9 a²₄a5− 40

3 a4a5a6+40

9 a²₄+ 2a²₅−38

3 a4a6+ 10a²₆+a3+ 4 3a5+ 2.

Analysing this variety we see that the corresponding polynomial system has only complex solutions, so V(I) is the empty set in R⁶. Thus system (3.1) can not have an isochronous center if condition (1) of Theorem 11 of [24] holds.

Case(2): The irreducible decomposition of the variety of the ideal hL₇, a₂, a₄, a₆i

computed using the routine minAssGTZ over the field of rational numbers consists of the varieties of the following two ideals

I₁ =ha₆,2a₅+ 3, a₄, 4 9a²₁+4

9a²₄−10

9 a₁a₅+4

9a²₅+a₃+ 4

3a₅+ 2, a₂,2a₁+ 3i, I2 =ha₆, a5+ 9, a4, 4

9a²₁+4

9a²₄− 10

9 a1a5+4

9a²₅+a3+4

3a5+ 2, a2, a1+ 3i.

The varieties of the ideals I₁ and I₂ give conditions (i) and (ii) of this theorem. Thus, (i) and (ii) are the necessary conditions for existence of an isochronous bi-center at the points (−1,0) and (1,0) for system (3.1). Now we need to show that these two conditions are also sufficient for existence of an isochronous bi-center. To do so, we look for the Darboux linearizations of the corresponding systems.

Condition (i): In this case system (3.1) becomes

(3.3) x˙ =y(1

2− 3 2x²+1

2y²), y˙=x(−1 2+1

2x²− 3 2y²).

Translating the point (1,0) to the origin, using the substitution u=x−1, v =y, and then the complexification z=u+iv,w=u−iv, we obtain the system

(3.4) z˙=z(1 +3

2z+1

2z²), w˙ =w(−1−3 2w−1

2w²).

This system is a particular case of the system studied in [15] (namely, it satisfies condition (1) of Theorem 4 of [15]) where it was shown that this system is linearizable. Thus, system (3.4) is linearizable at the origin. Consequently, system (3.3) has an isochronous bi–center at the points (1,0) and (−1,0).

Condition (ii): The conditions of this case yield the system (3.5) x˙ =y(2−3x²), y˙ =x(−1

2 +1

2x²−9y²).

As in the previous case, after apply the translation,u=x−1,v=y, and the complexification, z=u+iv,w=u−iv, to system (3.5) we obtain the system

(3.6)

˙

z=z+33z²

8 − 15zw 4 +9w²

8 +25z³

16 −9z²w

16 −21zw²

16 +13w³ 16 ,

˙

w=−w−9z²

8 +15zw

4 −33w²

8 − 13z³

16 +21z²w

16 +9zw²

16 −25w³ 16 . This system has the following Darboux factors

l1=z+ 3z²

8 −3zw 4 +3w²

8 + z³

16 +3z²w

16 +3zw² 16 +w³

16, l2=w+3z²

8 + 3zw 4 +3w²

8 +z³

16 +3z²w

16 +3zw² 16 + w³

16, l₃= 1 + 3z+ 3w+3z²

4 + 3zw 2 +3w²

4 , with respective cofactors

k₁= 1

4(4 + 18z−18w+ 9z²−9w²), k2= 1

4(−4 + 18z−18w+ 9z²−9w²), k3= 3

2(2z−2w+z²−w²).

It is easy to verify that conditions (2.15) are satisfied with f₀ =l₁,f₁ =l₃, g₀ =l₂,g₁ =l₃ and constants α₁ =β₁ =−3

2. Thus, the Darboux linearization for system (3.6) is given by the following analytic change of coordinates

z₁ = 16z+ 6z²+z³+ 12zw+ 3z²w+ 6w²+ 3zw²+w³ 2(4 + 12z+ 3z²+ 12w+ 6zw+ 3w²)^3/2 , w₁ = 6z²+z³+ 16w+ 12zw+ 3z²w+ 6w²+ 3zw²+w³

2(4 + 12z+ 3z²+ 12w+ 6zw+ 3w²)^3/2 .

Thus system (3.6) is linearizable and therefore system (3.5) has an isochronous bi–center at the points (1,0) and (−1,0).

Case(3): The irreducible decomposition of the variety of the ideal

(3.7) hL₇, f₁, f₂, f₃i,

where

f1 =3a6+ (a2+ 2a1a2−2a4+ 2a2a5−2a4a5), f₂ =2(1 +a₁)(a₁+a₅)²−a²₄(1 + 2a₁+ 2a₅),

f₃ =3(a₁+a₅)(−a₃+ 2(1 +a₁)(1 +a₅))−2a₄(2a₄(1 +a₅) +a₂(2 +a₁+a₅)), computed using the routine minAssGTZ over the field of rational numbers consists of the seven components presented in Appendix I. Analysing the components we see that the corresponding polynomial systems have only complex solutions, so the varieties are empty sets inR⁶. Thus, system (3.1) cannot have an isochronous center if condition (3) of Theorem 11 of [24] holds.

The remaining eight cases (Cases (4),(5),...,(11)) are analogous to Case (3). All the varieties are empty sets inR⁶. Therefore system (3.1) can not have an isochronous center if conditions (4), (5),..., (11) of Theorem 11 of [24] holds.

We note that for both conditions (i) and (ii) we obtain concrete systems (without param- eters). Figures 1 and 2 present the behaviour of the vector field in a neighbourhood of the origin of such systems. From the pictures we can suppose that the singular points at the origin also are centers. We prove the existence of such centers at the origin and that these centers are also isochronous.

-2 -1 0 1 2

-2 -1 0 1 2

Figure 1. Condition (i).

-2 -1 0 1 2

-2 -1 0 1 2

Figure 2. Condition (ii).

Proposition 3.2. If conditions (i) or (ii) of Theorem 3.1 holds system (3.1)has a center at the origin.

Proof. Under condition (i) of Theorem 3.1 system (3.1) becomes system (3.3), namely

˙ x=y(1

2− 3 2x²+1

2y²), y˙=x(−1 2+1

2x²− 3 2y²).

By a time rescaling and a linear change of coordinates system (3.3) becomes (3.8) x˙1=−y₁−3x²₁y1+y₁³, y˙1 =x1+x³₁−3x1y₁².

System (3.8) has Darboux factors f₁ =x²₁+y₁² and f₂ = 1 + 2x²₁−2y²₁, with respective cofactors k₁ =−4x₁y₁ and k₂ =−8x₁y₁.It is easy to verify that α₁ = 1 andα₂ = 1/2 is a solution of equation (2.4) with k1, k2 given above. So we obtain that

H= x²+y² p1 + 2x²₁−2y²₁,

is an analytic Darboux first integral of system (3.8). Thus, the origin is a center for (3.8), and, hence, the origin of (3.3) is a center as well.

Under condition (ii) of Theorem 3.1 system (3.1) becomes system (3.5), namely

˙

x=y(2−3x²), y˙ =x(−1 2 +1

2x²−9y²).

By a time rescaling and a linear change of coordinates system (3.5) becomes (3.9) x˙₁ =−y₁+ 3x²₁y₁, y˙₁ =x₁−2x³₁+ 9x₁y²₁.

System (3.9) has Darboux factorsf1 = 9x²₁−27y₁²−4,f2 =x1−1/√

3 andf3=x1+ 1/√ 3, which allow to construct the first integral

H= 9x²₁−27y₁²−4 (−1 + 3x²₁)³ .

Thus, the origin is a center of (3.9). Therefore the origin is a center of (3.5).

Remark 3.3. In the proof of Proposition 3.2 it is possible to obtain first integrals using the results of [24], but we have presented a direct proof. In [24] the authors gave examples of Hamiltonian systems (1.1) having more than two centers. Here we have presented the examples of non-Hamiltonian systems with 3 centers.

It turns out all three centers are isochronous.

Proposition 3.4. If condition (i) or (ii) of Theorem 3.1 holds, then system (3.1) has an isochronous center at the origin.

Proof. As showed in the proof of Proposition 3.2, by a linear change of coordinates system (3.3) is transformed to system (3.8) and system (3.5) is transformed to system (3.9).

System (3.8) corresponds to system (S₁^∗) in Table II of [28], and it was shown in [28] that it possess an isochronous center at the origin. Thus, system (3.3) also has an isochronous center at origin. System (3.9) corresponds to system (S₃^∗) in Table II of [28] so it has an isochronous center at the origin. Hence, system (3.5) has an isochronous center at origin as

well.

Therefore under the conditions of Theorem 3.1 system (3.1) has three isochronous centers at points (−1,0), (0,0) and (1,0). We are unaware about other examples of cubic systems with 3 isochronous centers.

4. The center and isochronocity problems for system(1.2)

In this section we investigate the existence of a bi-center and an isochronous bi-center for another class ofZ2-invariant system, namely, for the quintic system (1.2).

4.1. Canonical form of bi-centers for system (1.2). To apply the methods described in the preliminaries to study system (1.2) we look for a canonical form of the system for which the computations are simpler.

Suppose that system (1.2) satisfies the following conditions (4.1) X(1,0) =Y(1,0) = 0, ∂X(1,0)

∂x = ∂Y(1,0)

∂y = 0, ∂X(1,0)

∂y =−1, ∂Y(1,0)

∂x = 1, which mean that the point (1,0) is a singular point of (1.2), and its linearization at the point (1,0) is

(4.2) dx

dt =−y, dy

dt =x−1.

According to Lemma 6 in [24], if system (1.2) has two weak foci or centers at (−1,0) and (1,0) then the transformation

x1 =x+

∂Y(1,0)

∂y /∂Y(1,0)

∂x

y, y1=

r/∂Y(1,0)

∂x

y, t1=rt, where

r= s

∂X(1,0)

∂x

∂Y(1,0)

∂y −∂X(1,0)

∂y

∂Y(1,0)

∂x ,

carries system (1.2) into a system whose linear part at the point (1,0) is given by (4.2) and the conditions (4.1) holds. Moreover, this linear transformation preserves the distance between the origin and the two singular points. Consequently we have the following result.

Proposition 4.1. Suppose that system (1.2) has two weak foci or centers that could be arranged at the points (−1,0) and (1,0). Then there is a change of coordinates that maps the orbits of system (1.2)to the orbits of the following system

(4.3)

˙

x=−(a₁+ 1)y+a1x⁴y+a2x³y²+a3x²y³+a4xy⁴+a5y⁵,

˙ y=−1

4x−a₆y+1

4x⁵+a₆x⁴y+a₇x³y²+a₈x²y³+a₉xy⁴+a₁₀y⁵, where a_i∈R, i= 1, ...,10.

System (4.3) is a Z₂-equivariant quintic system, so the existence of a bi-center or an isochronous bi-center at the points (1,0) and (−1,0) for such system follows from the existence of a center or an isochronous center at the point (1,0).

To compute the focus and linearizability quantities we have to move the singular point (1,0) to the origin. Applying the transformationu =x−1,v =y, we obtain from (4.3) the

system

(4.4)

˙

u=−v+ 4a₁uv+a₂v²+ 6a₁u²v+ 3a₂uv²+a₃v³+ 4a₁u³v+ 3a₂u²v²+ 2a₃uv³+ +a4v⁴+a1u⁴v+a2u³v²+a3u²v³+a4uv⁴+a5v⁵,

˙

v=u+5u²

2 + 4a6uv+a7v²+5u³

2 + 6a6u²v+ 3a7uv²+a8v³+5u⁴

4 + 4a6u³v+

+ 3a7u²v²+ 2a8uv³+a9v⁴+u⁵

4 +a6u⁴v+a7u³v²+a8u²v³+a9uv⁴+a10v⁵. Our goal is to find systems with bi-centers and isochronous bi-centers within the family (4.3). Unfortunately, because this system has ten parameters the computations described in Section 2 become infeasible for the whole family (4.3). So we restrict our study to a subcase.

From now on we assume that system (4.3) posses the y-axis as an invariant curve, i.e, we fixed two parameters of the original system (4.3), a1 =−1 and a5 = 0. Thus, we look for necessary and sufficient conditions for system

(4.5)

˙

x=−x⁴y+a2x³y²+a3x²y³+a4xy⁴,

˙ y=−1

4x−a₆y+1

4x⁵+a₆x⁴y+a₇x³y²+a₈x²y³+a₉xy⁴+a₁₀y⁵,

to have a bi-center (respectively, an isochronous bi-center) at the points (−1,0) and (1,0), or, equivalently, for the system

(4.6)

˙

u=−v−4uv+a₂v²−6u²v+ 3a₂uv²+a₃v³−4u³v+ 3a₂u²v²+ 2a₃uv³+a₄v⁴+

−u⁴v+a2u³v²+a3u²v³+a4uv⁴,

˙

v =u+5u²

2 + 4a₆uv+a₇v²+5u³

2 + 6a₆u²v+ 3a₇uv²+a₈v³+5u⁴

4 + 4a₆u³v+

+ 3a₇u²v²+ 2a₈uv³+a₉v⁴+u⁵

4 +a₆u⁴v+a₇u³v²+a₈u²v³+a₉uv⁴+a₁₀v⁵, to have a center (respectively, an isochronous center) at the origin.

4.2. Bi-center conditions for the Z2-equivariant system (4.5). In this subsection we find conditions for existence of a bi-center for theZ₂-equivariant system (4.5).

Theorem 4.2. System (4.5) has a bi-center at the points (−1,0) and (1,0) if one of the following conditions holds:

(1) a6 = 0, a8 = 1

3(a2−2a2a7), a9 = 1

2(a3−a3a7), a10= 1

5(3a4−2a4a7);

(2) a2 =−4a₆, a4 = 4a3a6, a8= 4a6a7, a10= 4a6a9; (3) a₄ = 4(a₃a₆−4a₂a²₆−16a³₆), a₈ = 1

3(a₂+ 4a₆−2a₂a₇+ 4a₆a₇), a₉ = 1

6(3a₃−4a₂a₆−16a²₆−3a₃a₇−4a₂a₆a₇−16a²₆a₇), a10= 2a6(−a₃+ 4a2a6+ 16a²₆)(−1 +a7);

(4) a₇ =−1, a₈=a₂, a₉=a₃, a₁₀=a₄.

Proof. Using the computer algebra systemMathematicawe computed the first nine nonzero focus quantities for system (4.6) using the procedure described in Subsection 2.1. Their

expressions are very large, so we present here only the first two:

v₁=−a₂−4a₆+ 2a₂a₇−4a₆a₇+ 3a₈,

v2=60a10+ 75a2−40a2a3−36a4+ 300a6+ 80a²₂a6−16a3a6+ 256a2a²₆−256a³₆

−154a₂a₇+ 40a₂a₃a₇+ 24a₄a₇+ 284a₆a₇+ 64a₃a₆a₇+ 320a₂a²₆a₇−256a³₆a₇ + 148a2a²₇+ 544a6a⁻₇280a2a³₇+ 560a6a³₇−225a8−80a2a6a8+ 64a²₆a8+ 12a7a8

−420a²₇a₈+ 80a₂a₉+ 80a₆a₉.

The reader can easily compute the others quantities using any available computer algebra system.

The next computational step is to compute the irreducible decomposition of the variety V(B₉) = V(hv₁, ..., v₉i). For this purpose we use the routine minAssGTZ of the computer algebra system Singular. Due to the complexity of the focus quantities the computations become infeasible over the field of rational numbers. To be able to complete our compu- tations we computed in the field of finite characteristic 32003 and then lifted the resulting ideals to the ring of polynomials with rational coefficients using the rational reconstruction algorithm of [38] and the computational procedure of [32]. The irreducible decomposition of the variety of the ideal B₉ computed using the routine minAssGTZ of the computer algebra system Singular over a field of characteristic 32003 consists of 4 irreducible components.

After the rational reconstruction we obtain the 4 conditions in the statement of this theorem, which are necessary conditions for existence of a bi-center at the points (−1,0) and (1,0) for system (4.5).

Now we show that they are also sufficient.

Case(1): Under conditions (1) of Theorem 4.2 system (4.5) becomes

(4.7)

˙

x=−xy(x³−a2x²y−a3xy²−a4y³),

˙ y=−x

4 +x⁵

4 +a₇x³y²+

a₂−2a₂a₇ 3

x²y³ +

a3−a3a7

2

xy⁴+

3a4−2a4a7

5

y⁵.

System (4.7) has the Darboux factor l1 =x with the cofactork1 =−y(x³−a2x²y−a3xy²− a₄y³). It is easy to verify thatβ₁= 2a₇−4 is a solution of equation (2.5). Thus,µ=x^2a7−4 is an integrating factor of system (4.7) and computing we obtain the first integral

H(x, y) =x^2a7−3(−(15a₇+ 15)x+ (15a₇−15)x⁵+ (60a²₇−60)x³y²

+ (40a₂−40a₂a²₇)x²y³+ (30a₃−30a₃a²₇)xy⁴+ (24a₄−24a₄a²₇)y⁵),

which is analytic in a neighbourhood of the point (1,0). Thus, the points (−1,0) and (1,0) are centers of (4.7).

Case(2): Under conditions (2) of Theorem 4.2 system (4.5) becomes (4.8)

˙

x=−xy(x+ 4a₆y)(x²−a₃y²),

˙ y=1

4(x+ 4a6y)(−1 +x⁴+ 4a7x²y²+ 4a9y⁴).

Note that polynomials on the right hand sides of equation (4.8) have the common factor x+ 4a6y, so after the reparametrization of time we obtain the system

(4.9)

˙

x=−xy(x²−a₃y²) =P(x, y),

˙ y=1

4(−1 +x⁴+ 4a7x²y²+ 4a9y⁴) =Q(x, y).

It is easy to verify that the polynomials P(x, y) and Q(x.y) satisfy equation (2.6). So, system (4.9) posses time-reversible symmetry with respect to the x-axis. Thus, the singular points (−1,0) and (1,0) are centers for (4.8).

Case(3): In this case the corresponding system (4.5) is written as (4.10)

˙

x=−xy(x+ 4a6y)(x²−a2xy−4a6xy−a3y²+ 4a2a6y²+ 16a²₆y²),

˙ y=1

12(x+ 4a₆y)(−3 + 3x⁴+ 12a₇x²y²+ 4a₂xy³+ 16a₆xy³−8a₂a₇xy³−32a₆a₇xy³+ + 6a₃y⁴−24a₂a₆y⁴−96a²₆y⁴−6a₃a₇y⁴+ 24a₂a₆a₇y⁴+ 96a²₆a₇y⁴).

Applying the reparametrization of time we obtain the system

(4.11)

˙

x=−xy(x²−(a2+ 4a6)xy+ (−a₃+ 4a2a6+ 16a²₆)y²),

˙ y=1

12(−3 + 3x⁴+ 12a₇x²y²+ (4a₂+ 16a₆−8a₂a₇−32a₆a₇)xy³ + (6a3−24a2a6−96a²₆−6a3a7+ 24a2a6a7+ 96a²₆a7)y⁴).

System (4.11) has the Darboux factor l1 = x with cofactor k1 = −y(x² −a2xy−4a6xy− a₃y²+ 4a₂a₆y²+ 16a²₆y²). It is easy to verify thatβ₁= 2a₇−3 is a solution of equation (2.5).

Thus,µ=x^2a7−3 is an integrating factor of system (4.11) and computing we obtain the first integral

H(x, y) =x^2a7−2(−3−3a7 + (3a7−3)x⁴+ (12a²₇−12)x²y² + (8a₂+ 32a₆−8a₂a²₇−32a₆a²₇)xy³

+ (6a₃−24a₂a₆−96a²₆−6a₃a²₇+ 24a₂a₆a²₇+ 96a²₆a²₇)y⁴),

which is analytic in a neighbourhood of the point (1,0). Thus the points (−1,0) and (1,0) are centers of (4.10).

Case(4): For this case, system (4.6) becomes

(4.12)

˙

u=−v−4uv+a2v²−6u²v+ 3a2uv²+a3v³−4u³v+ 3a2u²v²+ 2a3uv³+a4v⁴+

−u⁴v+a₂u³v²+a₃u²v³+a₄uv⁴,

˙

v=u+ 5u²

2 + 4a6uv−v²+5u³

2 + 6a6u²v−3uv²+a2v³+5u⁴

4 + 4a6u³v

−3u²v²+ 2a2uv³+a3v⁴+u⁵

4 +a6u⁴v−u³v²+a2u²v³+a3uv⁴+a4v⁵.