New Family of Centers of Planar Polynomial Differential Systems of Arbitrary Even Degree

The problem of distinguishing between a focus and a center is one of the classical problems in the qualitative theory of planar differential systems. In this paper, we provide a new family of centers of polynomial differential systems of arbitrary even degree. Moreover, we classify the global phase portraits in the Poincaré disc of the centers of this family having degree 2, 4, and 6.


Introduction and Statement of the Main Results
Let P (x, y) and Q(x, y) be two real polynomials. In this work, we deal with polynomial differential systems in R 2 of the forṁ x = P (x, y),ẏ = Q(x, y), (1) where the dot denotes derivative with respect to an independent real variable t, usually called the time. The degree of the polynomial differential system (1) is the maximum of the degrees of the polynomials P (x, y) and Q(x, y). The origin O = (0, 0) of R 2 is a singular point for system (1) if P (0, 0) = Q(0, 0) = 0.
When all the orbits of system (1) in a neighborhood U \ {O} of the singular point O are periodic, we say that the origin O is a center.
If all the orbits of system (1) in a neighborhood U \ {O} of the singular point O spiral to O when t → +∞ or when t → −∞, we say that the origin is a focus.
The center-focus problem consists in distinguishing when the singular point O is either a center or a focus. The center-focus problem started with Poincaré [13] and Dulac [1], and in the present days many questions about them remain open. More recent results on the center-focus problem can be found in [3][4][5][6][7]9] and in their references.
In this paper, we consider the planar polynomial differential systems of the forṁ of degree 2k depending of k parameters r i for i = 1, 2, . . . , k such that 0 < r 1 < r 2 < · · · < r k . We denote the vector field of this system by X.
It is easy to show that the function satisfies the equality ∂V ∂xẋ Therefore V is an inverse integrating factor of system (for more details see for instance [2].) By multiplying the vector field X by the integrating factor, 1/V system (2) becomes a Hamiltonian system. If we compute the Hamiltonian H of that system for k = 1 we obtain H (x, y) = e −2x (x 2 + y 2 − r 2 1 ), for k = 2 we have and for k = 3 we get An important property of systems (2), which will help for characterizing their phase portraits, is that all the circles f i (x, y) = x 2 + y 2 − r 2 i = 0 for i = 1, 2, . . . , k are invariant algebraic curves of system (2), i.e., they are formed by orbits of systems (2), because they satisfy that where K i is the polynomial 2y k j =1,j =i (x 2 + y 2 − r 2 j ) (see Chapter 8 of [2] for additional information on the invariant algebraic curves.) In this paper, we prove that polynomial differential systems (2) provide a new family of centers of degree 2k for all k = 1, 2, . . .. Moreover, we classify the global phase portraits of systems (2) in the Poincaré disc for k = 1, 2, 3.

Theorem 1
For k = 1, 2, ... the polynomial differential systems (2) have a unique singular point in the interior of the circle x 2 + y 2 = r 2 1 and this singular point is a center.
Theorem 1 is proved in Section 3.

Theorem 2
For k = 1 the polynomial differential systems (2) have a phase portrait in the Poincaré disc topologically equivalent to the phase portrait of Fig. 1.

Theorem 3
For k = 2 the polynomial differential systems (2) have a phase portrait in the Poincaré disc topologically equivalent to one of the three phase portraits of Fig. 3.

Theorem 4
For k = 3 the polynomial differential systems (2) have a phase portrait in the Poincaré disc topologically equivalent to one of the seven phase portraits of Fig. 5.
In Section 2, we recall basic definitions and results for proving our theorems.

Poincaré Compactification
In this section, we summarize some basic results about the Poincaré compactification, which was done by Poincaré in [13]. He provided a tool for studying the behavior of a planar polynomial differential system near the infinity. (For more details on the Poincaré compactification, see Chapter 5 of [2].) be a polynomial vector field of degree d. We consider the Poincaré sphere S 2 = {y = (y 1 , y 2 , y 3 ) ∈ R 3 : y 2 1 + y 2 2 + y 2 3 = 1}; its tangent plane to the point (0, 0, 1) is identified with R 2 . Now we consider the central projection f : R 2 → S 2 of the vector field X, which sends every point x ∈ R 2 to the two intersection points of the straight line passing through the point x and the origin of coordinates with the sphere S 2 . We note that the equator S 1 = {y ∈ S 2 : y 3 = 0} of the sphere is in bijection with the infinity of R 2 . The differential Df sends the vector field X on R 2 into a vector field X defined on S 2 \ S 1 , which is formed by two symmetric copies of X with respect to the origin of coordinates.
We can extend the vector field X analytically to a vector field on S 2 multiplying X by y d 3 . This new vector field is denoted by p(X) and it is called the Poincaré compactification of the polynomial vector field X on R 2 . The dynamics of p(X) near S 1 corresponds with the dynamics of X in the neighborhood of the infinity. Since S 2 is a curved surface, for working with the vector field p(X) on S 2 , we need the expressions of this vector field in the local charts (U i , φ i ) and (V i , ψ i ), where U i = {y ∈ S 2 : y i > 0}, V i = {y ∈ S 2 : y i < 0}, φ i : U i −→ R 2 and ψ i : V i −→ R 2 for i = 1, 2, 3, with φ i (y) = −ψ i (y) = (y m /y i , y n /y i ) for m < n and m, n = i. In the local chart (U 1 , φ 1 ), the expression of p(X) iṡ In (U 2 , φ 2 ), the expression of p(X) iṡ The expressions for p(X) in the local chart The points of S 1 in any local chart have its v coordinate equal to zero. We note that the equator S 1 is invariant by the vector field p(X). The infinite singular points of X are the singular points of p(X) which lie in S 1 . Note that if y ∈ S 1 is an infinite singular point, then −y is also an infinite singular point and these two points have the same stability if the degree of vector field is odd. Such stability change to the opposite if the degree of the vector field is even.
The image of the northern hemisphere of S 2 onto the plane y 3 = 0 under the projection π(y 1 , y 2 , y 3 ) = (y 1 , y 2 ) is called the Poincaré disc which is denoted by D. The integral curves of S 2 are symmetric with respect to the origin, therefore it is sufficient to investigate the flow of p(X) only in the closed northern hemisphere. In order to draw the phase portrait on the Poincaré disc, it is needed to project by π the phase portrait of p(X) on the northern hemisphere of S 2 .
We note that the points (u, 0) are the points at infinity in the local charts U i and V i with i = 1, 2. Moreover, we remark that for studying the infinite singularities it is sufficient to study them on the local chart U 1 , and to check if the origin of the local chart U 2 is or not a singularity.

Topological Equivalence of Two Polynomial Vector Fields
Let X 1 and X 2 be two polynomial vector fields on R 2 . We say that they are topologically equivalent if there exists a homeomorphism on the Poincaré disc D which preserves the infinity S 1 and sends the orbits of π(p(X 1 )) to orbits of π(p(X 2 )), preserving or reversing the orientation of all the orbits.
A separatrix of the Poincaré compactification π(p(X)) is one of following orbits: all the orbits at the infinity S 1 , the finite singular points, the limit cycles, and the two orbits at the boundary of a hyperbolic sector at a finite or an infinite singular point (see for more details on the separatrices [8,10]).
The set of all separatrices of π(p(X)), which we denote by X , is a closed set (see [10]). A canonical region of π(p(X)) is an open connected component of D \ X . The union of the set X with an orbit of each canonical region form the separatrix configuration of π(p(X)) and is denoted by X . We denote the number of separatrices of a phase portrait in the Poincaré disc by S, and its number of canonical regions by R.
Two separatrix configurations X 1 and X 2 are topologically equivalent if there is a homeomorphism h : According to the following theorem which was proved by Markus [8], Neumann [10] and Peixoto [11], it is sufficient to investigate the separatrix configuration of a polynomial differential system, for determining its global phase portrait.
Theorem 5 Two Poincaré compactified polynomial vector fields π(p(X 1 )) and π(p(X 2 )) with finitely many separatrices are topologically equivalent if and only if their separatrix configurations X 1 and X 2 are topologically equivalent.

Proof of Theorem 1
It is easy to see that all singular points of system (2) and these imply that the function f (x) for all k has at least one zero in the interval (−r 1 , r 1 ).
If k is even and x ∈ (0, r 1 ), then f (x) is strictly decreasing and, if x ∈ (−r 1 , 0) then f (x) is positive. Hence for k even, the equation f (x) = 0 has only one root in the interval (−r 1 , r 1 ). By similar argument, we can easily show that if k is odd, then the equation f (x) = 0 has exactly one root in the interval (−r 1 , r 1 ). Thus, system (2) has a unique singular point inside the circle x 2 + y 2 = r 2 1 . The Jacobian matrix of the system at any singular point (x, 0) is as follows tr and det represent the trace and determinant of a matrix, respectively. Let (x, 0) be the singular point inside the disc of radius r 1 . If k is even, then x ∈ (0, r 1 ) and det(M) > 0. If k is odd, then x ∈ (−r 1 , 0) and det(M) > 0. Hence, the singular point (x, 0) is either a focus or a center, because its eigenvalues are purely imaginary. Since system (2)

Infinite Singular Points
Here, we study the infinite singular points of system (2) for all k. The Poincaré compactifi- It is obvious that there is no singular point in this local chart. The expression for p(X) in the local chart (U 2 , φ 2 ) has the forṁ So the unique infinite singular point in U 2 is the origin which is a hyperbolic stable node. Since the degree of system (2) is even, the origin of the chart V 2 is a hyperbolic unstable node.

Proofs of Theorems 2, 3, and 4
In general system (2) has two important properties that we use for drawing its phase portrait. These properties are: (i) Since system (2) has the inverse integrating factor (3), its corresponding first integral is defined in the whole plane except perhaps on the circles x 2 + y 2 = r 2 i . Therefore system (2) cannot have any focus as a singular point. (ii) System (2) is invariant by the change (x, y, t) → (x, −y, −t). Thus, the phase portrait of this system is symmetric with respect to the x-axis.
Proof of Theorem 2 System (2) with k = 1 has the two finite singular points P ± = ⎛ ⎜ ⎝ The Jacobian matrix at the point P ± is Therefore P + is a hyperbolic saddle, and P − is a center. By using the symmetry (x, y, t) → (x, −y, −t), the first integral (4), and the result of Section 4.1, it follows that the global phase portrait of system (2) for k = 1 in the Poincaré disc is topologically equivalent to the phase portrait of Fig. 1.

Proof of Theorem 3
For finding the finite singular points (x, y) of system (2) with k = 2, we must take y = 0, and find the real zeros of the equation f (x) = −x + (x 2 − r 2 1 )(x 2 − Fig. 1 The phase portrait in the Poincaré disc of system (2) for k = 1 r 2 2 ) = 0. In other words, it is enough to find the fixed points of the polynomial g(x) = (x 2 − r 2 1 )(x 2 − r 2 2 ). Since g(0) = r 2 1 r 2 2 > 0 and function g has four real roots ±r 1 , ±r 2 , we have exactly one of the following three cases: (i) f has two simple positive roots (see Fig. 2a).
(ii) f has one double negative and two simple positive roots (see Fig. 2b).
(iii) f has two simple negative and two simple positive roots (see Fig. 2c).
The Jacobian matrix at every singular point (x, 0) of system (2) with k = 2 is It is easy to see that tr M = 0 and det M = −f (x).
In case (i) the polynomial f (x) has only two simple positive roots x = a and x = b satisfying 0 < a < r 1 < r 2 < b, −f (a) > 0 and −f (b) < 0 (see Fig. 2a). Therefore in this case system (2) with k = 2 has two singular points (a, 0) and (b, 0), which are a center and a hyperbolic saddle, respectively.
For in case (ii) the polynomial f (x) has one negative double root x = a, and two simple positive roots Fig. 2b). Thus in this case, system (2) with k = 2 has three singular points (a, 0), (b, 0), and (c, 0), where (a, 0) is a nilpotent singular point, and (b, 0) and (c, 0) are a center and a hyperbolic saddle, respectively. Here, for determining the local phase portrait of the nilpotent singular point (a, 0), we use the index theory. Based on the Poincaré-Hopf theorem, for every vector field on S 2 with finitely many singular points, the sum of their (topological) indices is two (see for instance [2]). By applying this theorem to the Poincaré sphere with the Poincaré compactification of our Fig. 2 The graphics for all different cases of fixed points of g(x) when k = 2 system, it is easy to see that the index of the singular point (a, 0) is zero. Since the flow of a Hamiltonian system preserve the area, and the unique nilpotent singular points with index zero are the saddle-nodes and the cusps (see Theorem 3.5 of [2]), it follows that the singular point (a, 0) is a cusp.
Hence, by using the symmetry (x, y, t) → (x, −y, −t), the first integral (5), and that at infinity, we have a pair of nodes at the origins of the local charts U 2 and V 2 , the first stable and the second unstable (see Section 4.1), it follows that the global phase portrait of system (2) for k = 2 for each of three cases (i), (ii), and (iii) in the Poincaré disc is topologically equivalent to the one of the phase portrait (a), (b), and (c) of Fig. 3, respectively.
Proof of Theorem 4 Similar to the proof of Theorem 3, for finding the finite singular points (x, y) of system (2) with k = 3, we must have y = 0 and x must be a real zero of the equation Hence, it is enough to find Fig. 3 The phase portraits in the Poincaré disc of system (2) for k = 2 the fixed points of the polynomial function g(x) = (x 2 − r 2 1 )(x 2 − r 2 2 )(x 2 − r 2 3 ). Since g(0) = −r 2 1 r 2 2 r 2 3 < 0 and the polynomial g(x) has six real roots ±r 1 , ±r 2 and ±r 3 , we have exactly one of the following nine cases for the roots of the polynomial f (x).
(i) One simple negative and one simple positive roots (see Fig. 4a).
(ii) One simple negative, one double positive, and one simple positive roots (see Fig. 4b). (iii) One simple negative and three simple positive roots (see Fig. 4c). (iv) Three simple negative and three simple positive roots (see Fig. 4d).
(v) Three simple negative and one simple positive roots (see Fig. 4e). (vi) One double negative, one simple negative, and one simple positive roots (see Fig. 4f). (vii) One double negative, one simple negative, one double positive, and one simple positive roots (see Fig. 4g). (viii) One double negative, one simple negative, and three simple positive roots (see Fig. 4h). (ix) Three simple negative, one double positive, and one simple positive roots (see Fig. 4i).
The three invariant algebraic curves x 2 + y 2 = r 2 i for i = 1, 2, 3, play an important role in drawing the phase portraits for system (2) with k = 3. Actually, if there is one singular point inside and one singular point outside of an invariant algebraic curve, then these two singular points do not have any connection, i.e., there are no orbits going from one to the other.

Case (i):
In this case, we have two singular points (a, 0) and (b, 0) where −r 3 < −r 2 < −r 1 < a < 0 < r 1 < r 2 < r 3 < b. By computing the Jacobian matrix in each singular point, we can conclude that (a, 0) is a center and (b, 0) is a hyperbolic saddle. The symmetry (x, y, t) → (x, −y, −t) and the first integral (6) together with the result of Section 4.1 force to the system to have a phase portrait topologically equivalent to the phase portrait of Fig. 5a. Case (ii): Then system (2) has three singular points (a, 0), (b, 0), and (c, 0), where −r 3 < −r 2 < −r 1 < a < 0 < r 1 < b < r 2 < r 3 < c. Again, by computing the Jacobian matrix in each singular point we have (a, 0) is a center and (c, 0) is a hyperbolic saddle. Using the index theory as it is done in case (ii) for k = 2, we can conclude that (b, 0) is a nilpotent cusp. Since the cusp (b, 0) is the only singular point between the two invariant algebraic curves x 2 + y 2 = r 2 1 and x 2 + y 2 = r 2 2 , it implies the existence of a cuspidal loop which surrounds the center (a, 0) and the invariant algebraic curve x 2 + y 2 = r 2 1 . By using the symmetry (x, y, t) → (x, −y, −t) and the first integral (6), we also obtain a homoclinic loop passing through (c, 0) and surrounding all the finite singular points and all the three invariant algebraic curves. By taking into account the result of Section 4.1, the phase portrait of system (2) in this case is topologically equivalent to the phase portrait of Fig. 5b. Case (iii): Then the system has four singular points (a, 0), (b, 0), (c, 0), and (d, 0), where −r 3 < −r 2 < −r 1 < a < 0 < r 1 < b < c < r 2 < r 3 < d. By computing the Jacobian matrix in each of these singular points, we have that (a, 0) and (c, 0) are centers, and (b, 0) and (d, 0) are hyperbolic saddles. Due to the fact that the singular point (b, 0) is located between the two centers (a, 0) and (c, 0), and inside the invariant algebraic curve x 2 + y 2 = r 2 2 , the point (b, 0) is a saddle having two homoclinic loops. The left homoclinic loop surrounds the center (a, 0) and the invariant algebraic  (d, 0) is a hyperbolic saddle on the right side of r 3 . Two of its separatrics form a homoclinic loop which surrounds the other three singular points and the three algebraic invariant circles. The symmetry (x, y, t) → (x, −y, −t), the first integral (6) and the result of Section 4.1 show that the phase portrait of system (2) in this case is topologically equivalent to the phase portrait of Fig. 5c. Case (iv): We have six singular points (a, 0), (b, 0), (c, 0), (d, 0), (e, 0), and (f, 0), where −r 3 < a < b < −r 2 < −r 1 < c < 0 < r 1 < d < e < r 2 < r 3 < f . The singular points (a, 0), (c, 0), and (e, 0) are centers and (b, 0), (d, 0), and (f, 0) are hyperbolic saddles. With a similar discussion as to the one of the previous case, we obtain that the phase portrait of the system is topologically equivalent to the phase portrait of Fig. 5d. Case (v): Doing a similar discussion to the case (iii), we obtain that the phase portrait of the system is topologically equivalent to the phase portrait of Fig. 5c. Case (vi): Working in a similar way to the case (ii), we obtain that the phase portrait of the system is topologically equivalent to the phase portrait of Fig. 5b.

Case (vii):
We have four singular points (a, 0), (b, 0), (c, 0), and (d, 0), where −r 3 < a < −r 2 < −r 1 < b < 0 < r 1 < c < r 2 < r 3 < d. By obtaining the Jacobian matrix in each singular point, we get that (b, 0) is a center and (d, 0) is a hyperbolic saddle. By using the index theory and Corollary 2 in chapter 3 of [12], it follows that the singular point (b, 0) inside the invariant algebraic curve x 2 +y 2 = r 2 1 , and the singular point (a, 0) inside the invariant algebraic curve x 2 +y 2 = r 2 3 , are cusps. Using the invariant algebraic curves together with the symmetry (x, y, t) → (x, −y, −t) and the first integral (6), we obtain that the phase portrait of the system is topologically equivalent to the phase portrait of Fig. 5(e). Case (viii): Again in this case using similar arguments to previous cases, we conclude that there are five singular points (a, 0), (b, 0), (c, 0), (d, 0), and (e, 0), where −r 3 < a < −r 2 < −r 1 < b < 0 < r 1 < c < d < r 2 < r 3 < e. The singular points (b, 0) and (d, 0) are centers, (e, 0) and (c, 0) are hyperbolic saddles and (a, 0) is a cusp. In this case the phase portrait is topologically equivalent to the one of Fig. 5f. Case (ix): The phase portrait of system (2) in this case is topologically equivalent to the phase portrait that it is shown in Fig. 5g. Similar to the case (viii), we have five singular points (a, 0), (b, 0), (c, 0), (d, 0), and (e, 0), where −r 3 < a < b < −r 2 < −r 1 < c < 0 < r 1 < d < r 2 < r 3  Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.