Centers of Projective Vector Fields of Spatial Quasi-homogeneous Systems with Weight (m, M, N) and Degree 2 on the Sphere

In this paper we study the centers of projective vector fields Q T of three-dimensional quasi-homogeneous differential system dx/dt = Q(x) with the weight (m, m, n) and degree 2 on the unit sphere S 2. We seek the sufficient and necessary conditions under which Q T has at least one center on S 2. Moreover, we provide the exact number and the positions of the centers of Q T. First we give the complete classification of systems dx/dt = Q(x) and then, using the induced systems of Q T on the local charts of S 2 , we determine the conditions for the existence of centers. The results of this paper provide a convenient criterion to find out all the centers of Q T on S 2 with Q being the quasi-homogeneous polynomial vector field of weight (m, m, n) and degree 2.

The three-dimensional polynomial differential systems occur as models or at least as simplifications of models in many domains in science.For example, the population models in biology.In recent years, the qualitative theory of three-dimensional polynomial differential systems has been and still is receiving intensive attention [1,2,5,7,9,10,13,15].
Just as the author of [16] point out that, the study of three-dimensional polynomial differential system is much more difficult than that of planar polynomial system.For example, it is an arduous task to determine the global topological structure of the Lorenz system ẋ1 = σ(x 2 − x 1 ), ẋ2 = ρx 1 − y − xz, ẋ3 = −βz + xy (σ, β, ρ ≥ 0), although this system has a simply form, see [14].
An efficient method for studying the qualitative behavior of orbits of system (1) is to project the system to the unit sphere S 2 .In what follows we will adopt the notations used in [6] to introduce some basic theory of the projective system on S 2 .
For any y ∈ S 2 , we define a curve as S(y) = {(r α 1 y 1 , r α 2 y 2 , r α 3 y 3 )|r > 0}.The orbit Γ of system (4) on S 2 can be regarded as the projection of W Γ along the family of curves {S(y)|y ∈ S 2 }.In this sense, we call Q T (y) the projective vector field of Q(y) on S 2 and call (4) the projective system of (1).
To study the behavior of orbits of system (4), we will use the local charts of S 2 .Denoted by
In the literature many authors study the projective vector field of system (1) with degree two (d = 2).Most of them consider the homogeneous case, i.e., α 1 = α 2 = α 3 = 1.For instance, Camacho in [1] investigates the projective vector fields of homogeneous polynomial system of degree two.The classification of projective vector fields without periodic orbits on S 2 is given.Wu in [15] corrects some mistakes of [1] and provide several properties of homogeneous vector fields of degree two.Llibre and Pessoa in [10] study the homogeneous polynomial vector fields of degree two, it was shown that if the vector field on S 2 has finitely many invariant circles, then every invariant circle is a great circle.[11] deals with the phase portraits for quadratic homogeneous polynomial vector fields on S 2 , they verify that if the vector field has at least a non-hyperbolic singularity, then it has no limit cycles.They also give necessary and sufficient conditions for determining whether a singularity of (4) on S 2 is a center.Pereira and Pessoa in [12] classify all the centers of a certain class of quadratic reversible polynomial vector fields on S 2 .
Under the homogeneity assumption we know that whenever x(t) is a solution of system (1), then so is x = λx(λ d−1 t).But this conclusion is not true for the quasi-homogeneous system.Recently, the authors of [6] study the projective vector field of a three-dimensional quasi-homogeneous system with weight (1, 1, α) (α > 1) and degree d = 2.Some interesting qualitative behaviours are determined according to the parameters of the systems.Another meaningful work about the spatial quasi-homogeneous systems is [7].In that paper the authors generalize the results of [2,13] by studying the limit set of trajectories of three-dimensional quasi-homogeneous systems.They also point out, by a counterexample, the mistake of [2].
However, to the best of our knowledge, there is no paper dealing with the center of the projective vector field of spatial quasi-homogeneous.Motivated by this fact, in the present paper we study the sufficient and necessary conditions for the projective vector field Q T of the system (1) with the weight (m, m, n) and degree 2 to have at least one center on S 2 .We would like to emphasize that, in the above mentioned papers dealing with homogeneous systems, many authors concern on the periodic orbits of system (4), see [1,6,11].This is because the periodic behavior of system (4) provide a threshold to investigate the periodic and spirally behaviors of the spatial system.Our work provides a criterion for the projective vector field associated to system (1) to have a family of periodic orbits.
This paper is organized as follows.In Section 2, we prove some properties and establish the canonical forms of quasi-homogeneous polynomial system (1) with weight (m, m, n) and degree 2. In Sections 3, 4, and 5, we are going to seek the sufficient and necessary conditions under which the projective system (4) has at least one center on S 2 , where Section 3 (resp.Section 4 and Section 5) deals with the case that n = 1 (resp.m > 1, n > 1 and m = 1).

Properties and canonical forms for quasi-homogeneous systems with
weight (m, m, n) and degree d = 2 The first goal of this section is to derive some properties of the three-dimensional quasihomogeneous polynomial vector field with weight (m, m, n) and degree d = 2.The results obtained will be used in the next sections.
Proposition 2.1.Assume that Q is a quasi-homogeneous polynomial vector field with weight (m, m, n) and degree d = 2. Then Proof.Firstly, it follows from the quasi-homogeneity property of Q that Secondly, by the expression of Q T (y) we have Since ψ(y) = Dψ • ȳ, it follows that ψ(y), ψ(y) = ȳ, y .Hence The proof is finished.
This prove the first and the second equation of ( 5).If a (3) The equation (11) is satisfied if and only if This proves the third equation of (5).
We deduce from (12 This proves the first and the second equation of (6).If a (3) The equation ( 13) is satisfied if and only if This proves the third equation of (6).
Thus we obtain the third equation of (7).
Proof.Firstly, in the case that m = 1, the canonical forms ( 15) and ( 16) follow from the result of [6] directly.
The systems ( 15) and ( 16) are considered in [6].It is shown that the projective system of system (15) has no closed orbits on S 2 .But the authors do not give the conditions for projective systems of system ( 16) to acquire at least one center.The purpose of the rest of this paper is to find the sufficient and necessary conditions for all the projective systems (4) of the systems in Theorem 2.4 to have at least one center.

Center of the quasi-homogeneous systems with weight (m, m, 1)
In this section we deal with the canonical forms of ( 17) and ( 18).The main results of this section are the following two theorems.Theorem 3.1.Suppose that Q T is the projective vector field of system (17), then the following statements hold.
Moreover, if J(c 1 , c 2 ) > 0 (resp.J(c 1 , c 2 ) < 0), then at the equator Q T has a unique center at E (resp.−E).If the condition (i) (i ∈ {1, 2, 3}) of (B) holds, then Q T has a unique center at y i on S 2 ∩ H + 3 and has a unique center at −y i on S 2 ∩ H − 3 which satisfies φ 3 + (y i ) = (x * i , y * i , 1), where Theorem 3.2.The projective vector field Q T of system (18) with m > 2 has at least one center on S 2 if and only if Furthermore, if (22) is satisfied then Q T has exactly two centers at the points where λ = λ 0 is the unique positive solution of equation 3.1.Quasi-homogeneous systems with weight (2, 2, 1).In this subsection we assume that Q T is the projective vector field of system (17).We will firstly study the centers on the H + 3 ∩ S 2 .The centers on H − 3 ∩ S 2 can be obtained by the symmetry (see Proposition 2.1).By straightforward calculations we find that the induced system of where . By direct computation we obtain that on l + , writing x = (p 1 λ, p2 λ, 1), we have 3 ) = 0.This means that f (λ) ≡ 0. Therefore, l + is an invariant straight line of W + 3 .Let S be the great circle containing the points (p 1 , p2 , 0) and (0, 0, ±1).Clearly, p ∈ S. By Proposition 2.2, the half-great circle S ∩ H + 3 is an integral curve of the vector field Q T .Thus p can no be a center of Q T .
Next consider the critical point (0, 0, ±1) of Q T with η = 0. We need the following result.
By applying the Proposition 2.1, we can conclude that the relations (26) are also the sufficient and necessary conditions for (0, 0, −1) to be a center of Q T on S 2 .
Let us consider the case η = 1.Proposition 3.6.Assume that η = 1, then Q T has at least a center on S 2 ∩ H + 3 if and only if one of the conditions of Theorem 3.1.(B)is satisfied.Moreover, if the condition (i) (i ∈ {1, 2, 3}) of Theorem 3.1.(B)holds, then Q T has a unique center y i on S 2 ∩ H + 3 and has a unique center −y i on S 2 ∩ H − 3 which satisfy φ 3 + (y i ) = (x * i , y * i , 1), where x * i and y * i are defined in (21).Proof.Suppose that p ∈ H + 3 ∩ S 2 is a singularity of Q T , and let (x 0 , y 0 , 1) = φ + 3 (p).It is easy to see that (x 0 , y 0 ) is a singularity of system (24).By taking the transformation u = x1 − x 0 , v = x2 − y 0 , we change system (24) to In what follows we will consider the singularity (0, 0) of system (27).
One can check directly that system (27) satisfies the condition (1) of Lemma 3.4.Thus the point (0, 0) is a center of system (27) if and only if the following two equalities hold where x 0 and y 0 are the isolated solutions of the following equations By equations ( 28) and (30), we get 3a 2 y 0 = ( b2 −2ā 1 )x 0 .We will now split our discussion into two cases.
Next assume that 6a The equations (28) and (30) have a unique solution Substituting into (31) yields Moreover, it follows that 2 ) < 0. This complete the proof.
Next we are going to study the singularities of Q T at the equator.Proof.The conclusion follows directly from Proposition 3.8.Q T has centers on the equator if and only if J(c 1 , c 2 ) = 0.Moreover, if J(c 1 , c 2 ) > 0 (resp.J(c 1 , c 2 ) < 0), then Q T has a unique center at E 1 (resp.E 2 ).Proof.By Proposition 3.7, the equator contains centers only if c 2 1 +c 2 2 = 0.In what follows we will split our discussion into three cases.
When η = 1, on the straight line x2 = −c 1 /c 2 we have dx 2 dτ = x3 3 , meaning that the orbits of system (37) pass through the straight line x2 = −c 1 /c 2 from the left to the right on the upper half-plane and from the right to the left on the lower half-plane.On the other hand, noting also that the direction of vector field (37) at the x2 axis is upward on the right hand side of the critical point (−c 1 /c 2 , 0) and is downward on the left hand side of the critical point (−c 1 /c 2 , 0).See Figure 2. We conclude that there is no closed orbit around the singularity (−c 1 /c 2 , 0).So the singularity (−c 1 /c 2 , 0) is not a center.Next we study the singular point where η = 0, 1 and dτ = (2y . The characteristic equation of the linear approximation system of (38) at the critical point (c 1 /c 2 , 0) is In the same way, we can easily verify that the critical point (c We make the transformation φ 1 − : H − 1 ∩ S 2 → Π − 1 and then we obtain the induced system of Q T on Π − 1 , which is system (38).
Using analog arguments we conclude that E 2 is a center of Q T on S 2 if and only if c 2 J(c 1 , c 2 ) > 0.
Analogously, the singularity The case that c 1 < 0 can be studied in a similar way, we omit the discussion for the sake of brevity.
In conclusion, the vector field Q T has a center at the equator if and only if and it is different from zero except for c 1 = c 2 = 0.It follows from Proposition 3.5 that Q T has exactly three centers at (0, 0, 1), (0, 0, −1), and E or −E.

3.2.
Quasi-homogeneous systems with weight (m, m, 1).Throughout this subsection we suppose that Q T is the projective vector field of system (18).
4. Center of the quasi-homogeneous systems with weight (2, 2, This section is devoted to derive the sufficient and necessary conditions for the projective vector field Q T of systems ( 19) and ( 20) to possess at least one center.The main results of this section are the following two theorems.Theorem 4.1.Suppose that Q T is the projective vector field of system (19), then Q T has at least one center on S 2 if and only if one of the following conditions holds: (1) (ii) Suppose that (2) is satisfied.If c 2 < 0, c 2 + c 3 > 0, then Q T has exactly four centers at ±(0, 0, 1) and ±D; If c 2 < 0, c 2 + c 3 ≤ 0, then Q T has exactly two centers at ±(0, 0, 1); If c 2 + c 3 > 0, c 2 > 0, then Q T has exactly two centers at ±D.Here Theorem 4.2.The projective vector field Q T of system (20) has no centers on S 2 .
To prove our results, we need the following lemma, which is a part of the Nilpotent Singular Points Theorem.The readers are referred to [3] for the complete result.Lemma 4.3.Let (0, 0) be the isolate singularity of system where A and B are analytic in a neighborhood of (0, 0) and also j 1 A(0, 0) = j 1 B(0, 0) = 0. Let y = f (x) be the solution y + A(x, y) = 0 in a neighborhood of the point (0, 0).And let n ≥ 1 and ab = 0, then we have (i) if m is even and m < 2n + 1, then the origin of system (44) is a cusp.(ii) if m is even and m > 2n + 1, then the origin of system (44) is a saddle-node.

Let us consider firstly system (19).
Proof of Theorem 4.1.The induced systems of 2 ).( 45) System (45) has an isolated singularity if and only if c 1 + c 2 + c 3 = 0. Assume that c 1 + c 2 + c 3 = 0, then system (45) has a unique isolated singularity at the point The characteristic equation of the linear approximation system of (45) at the singularity Hence it is easy to see that E is not a center of system (45).This mean that Q T has no centers on S 2 ∩ H + 3 .By the symmetry of system (19), we know that Q T has also no centers on S 2 ∩ H − 3 .By direct computation, we have , 0 and −G 0 , respectively; If c 1 = c 2 = 0, then Q T has exactly two singularities at (1, 0, 0) and (−1, 0, 0), respectively; (iii) if c 2 2 − 4c 1 c 3 > 0, then Q T has exactly four singularities ±G 1 and ±G 2 at the equator, where ±G i (i = 1, 2) are defined in (43).
Suppose that c 1 > 0. The induced system of Q T giving by φ 2 + : Let The singularity G i (i = 0, 1, 2) is a center of Q T if and only if the origin is a center of system (47).If c 2 2 − 4c 1 c 3 = 0, then c 2 + 2c 1 x0 = 0. Obviously, in this case the origin of system (47) is not a center if 1 − x0 = 0. Thus we assume that 1 − x0 = 0, which means that, by the time rescaling, system (47) can be transformed to the same form as (44).By the result of Lemma 4.3 we know that the origin of system (47) is a cusp.
If c 2 2 − 4c 1 c 3 > 0, then c 2 + 2c 1 x0 = 0.By Lemma 3.4, the origin is a center of system (47) if and only if (c 2 + 2c 1 x0 )(1 − x0 ) < 0. Therefore, using the explicit expression of x 0 , G i (and hence In other words, ±G 1 are centers of Q T if and only if and ±G 2 are centers of Q T if and only if Similarly, if c 1 < 0, then by the induced system of Q T on Π − 2 , we conclude that ±G 1 are centers of Q T if and only if Finally consider the case that c 1 = 0.If c 2 = 0, then Q T has two pairs of singularities at the equator: ±(1, 0, 0) and ±D.Using the procedure as above, we conclude that ±D are centers of Q T if and only if c 2 + c 3 > 0.
To study the singularities ±(1, 0, 0), we use the induced system of Q T giving by φ 1 + : And φ 1 + (1, 0, 0) = (0, 0, 0).If c 2 = 0, then by the result of Lemma 3.4 we know that the origin is a center of system (48) if and only if c 2 < 0. If c 2 = 0, then ±(1, 0, 0) is the unique pair of singularities of Q T at the equator.By Lemma 4.3, the origin is not a center of system (48), meaning that Q T has no centers in this case.
We are now in the position to prove the result for system (20).
Proof of Theorem 4.2.The induced system of System (49) has an isolated singularity if and only if c 1 + c 2 > 0. And if c 1 + c 2 > 0, then system (45) has a unique isolated singularity at By direct computation, we obtain that the eigenvalues of the linear approximation of system of (49) at that singularity are Hence it is not a center of system (45), meaning that Q T has no centers on S 2 ∩ H + 3 .By the symmetry of (49), we know that Q T also has no centers on S 2 ∩ H − 3 .At the equator, we have , 0) and −E respectively.
Assume firstly that c 1 > 0. Then E ∈ H + 2 ∩ S 2 .The induced system of Q T giving by And The singularity E is a center of Q T if and only if the origin is a center of system (51). where Before proving the above results, we give some necessary information about the projective vector field Q T .Firstly, we have 3 ), y 3 − y 2 p 4 (y 2 , y 3 ), q 4 (y 2 , y 3 )), with y 2 2 + y 2 3 = 1, where p 4 (y 2 , y 3 ) and q 4 (y 2 , y 3 ) are two polynomials of degree not more than 4. Therefore, Q T has no singularities at S 1 := {(y 1 , y 2 , y 3 ) ∈ S 2 |y 1 = 0} if a 2 = 0. Suppose that a 2 = 0, then the first component of Q T is identically zero.This means that S 1 is invariant under the vector field Q T .In any case, Q T has no centers at S 1 .
Next let us study the singularity of Q T on S 2 \ S 1 .By the symmetry (see Proposition 2.1), it is enough to study the singularity on Suppose that (x 0 , y 0 ) is a singularity of system (52), i.e., Taking the transformation u = x2 − x 0 , v = x3 − y 0 , we change system (52) to where (55) Thus, the point (x 0 , y 0 ) is a center of system (52) if and only if system (54) has a center at the origin.
By regarding x 0 and y 0 as the parameters of system (54), we have the following result.
Lemma 5.4.The origin is a center of system (54) if and only if (x 0 , y 0 ) satisfies one of the following conditions: (1) a 3 = 0, D(x 0 ) = 0 and α It is easy to see that system (54) has a center at the origin only if (x 0 , y 0 ) satisfies the relations (53) and If we regard the equality (56) as an equation in the variable x 0 , then (56) has solution if and only if one of the following three conditions is satisfied: If a 3 = 0, then we use (56) to reduce the power of x 0 in (53) and we obtain that Finally, if a 3 = 2B 3 + C 2 = B 2 + C 1 = 0, then a + ā = 0 holds automatically and the function G reduces to G 0 .Remark 5.5.We would like to point out here that if a 3 = 2B 3 + C 2 = B 2 + C 1 = 0, then the divergence of system (54) is identically zero.