Global Phase Portraits of Quadratic Systems with a Complex Ellipse as Invariant Algebraic Curve

In this paper, we study a new class of quadratic systems and classify all its phase portraits. More precisely, we characterize the class of all quadratic polynomial differential systems in the plane having a complex ellipse x2 + y2 + 1 = 0 as invariant algebraic curve. We provide all the different topological phase portraits that this class exhibits in the Poincaré disc.


Introduction and Statement of the Main Results
A planar polynomial differential system is a differential system of the forṁ x = P (x, y), y = Q(x, y), (1.1) where P and Q are real polynomials. The polynomial differential system (1.1) has degree n, if n is the maximum of the degrees of the polynomials P and Q. A polynomial differential system of degree 2 is called simply a quadratic system. The dot in system (1.1) denotes the derivative with respect to the independent variable t.
In this paper, we want to study a new class of quadratic systems and classify all its phase portraits. More precisely we analyze the class of all quadratic polynomial differential systems having a complex invariant ellipse, that after an affine transformation it can be written as The family of quadratic systems having a real invariant ellipse has been studied by Qin [21] in 1958.
Our first result is to provide a normal form for all quadratic polynomial differential systems having a complex invariant ellipse x 2 + y 2 + 1 = 0 as an invariant algebraic curve. Proposition 1.1 A planar polynomial differential system of degree 2 having a complex invariant ellipse x 2 + y 2 +1 = 0 as an invariant algebraic curve, after an affine change of coordinates, can be written asẋ The proof of Proposition 1.1 is the same as the proof of the normal form of the quadratic system with a real invariant ellipse, see for more details the normal form given in system (12) of [13].
In the next theorem we present the topological classification of all the phase portraits in the Poincaré disc of planar polynomial differential systems of degree 2 having a complex invariant ellipse as an invariant algebraic curve. For a definition of the Poincaré compactification, the Poincaré disc and of topological equivalent phase portraits of a polynomial differential system in the Poincaré disc, see Section 2.
It is known that if a quadratic system has a limit cycle this surrounds a focus, see for more details [4]. Due to this fact we shall see that such quadratic systems have no limit cycles. Theorem 1.2 Given a planar polynomial differential system of degree 2 having a complex ellipse x 2 + y 2 + 1 = 0 as an invariant algebraic curve its phase portrait is topological equivalent to one of the L phase portraits of Figure 1. Theorem 1.2 is proved in Section 3.

Poincaré Compactification
The planar polynomial vector field of degree n associated to the polynomial differential system (1.1) of degree n is The Poincaré compactified vector field p(X ) associated to X is an analytic vector field on S 2 constructed as follows (see, for more details Chapter 5 of [7]). Let S 2 = {y = (y 1 , y 2 , y 3 ) ∈ R 3 : y 2 1 + y 2 2 + y 2 3 = 1} (the Poincaré sphere) and T y S 2 be the tangent plane to S 2 at point y. The tangent plane T (0,0,1) S 2 is identified with the plane R 2 where we have the polynomial vector field X . Consider the central projections f ± : T (0,0,1) S 2 → S 2 . These two maps define two copies of X , one X + in the northern hemisphere and the other X − in the southern hemisphere. Denote by X = X + ∪ X − the vector field defined on S 2 except on its equator S 1 = {y ∈ S 2 : y 3 = 0}. The infinity of R 2 is identified with the equator S 1 . The extension of X to a vector field P (X ) on S 2 (including S 1 ) is the unique analytic extension of y n−1 3 X to S 2 . Knowing the behaviour of p(X ) in a neighborhood of S 1 , we know the behaviour of X near the infinity. The Poincaré compactification is such that S 1 is invariant under the flow of p(X ).
The Poincaré disc, denoted by D, is the projection of the closed northern hemisphere of S 2 on y 3 = 0 under (y 1 , y 2 , y 3 ) → (y 1 , y 2 ).
Two polynomial vector fields X and Y on R 2 are topologically equivalent if there is a homeomorphism on S 2 carrying orbits of the flow induced by p(X ) into orbits of the flow induced by p(Y), preserving the infinity S 1 and preserving or reversing simultaneously the sense of all orbits.
We consider the six local charts U i = {y ∈ S 2 : y i > 0}, and V i = {y ∈ S 2 : y i < 0} where i = 1, 2, 3; and the diffeomorphisms F i : U i → R 2 and G i : V i → R 2 for i = 1, 2, 3 (which are the inverses of the central projections from the planes tangent at the points (1, 0, 0), (−1, 0, 0), (0, 1, 0), (0, −1, 0), (0, 0, 1) and (0, 0, −1) respectively in order to study the sphere S 2 which is a differentiable manifold). If we denote by (u, v) the value of F i (y) or G i (y) for any i = 1, 2, 3 (so (u, v) represents different things according to the local charts under consideration), then we get for p(X ) the following expressions: . In what follows we omit the factor Δ(u, v) by rescaling the vector field p(X ). Thus we obtain a polynomial vector field in each local chart. The expression for V i is the same as that for U i except for a multiplicative factor (−1) n−1 . In these coordinates for i = 1, 2, v = 0 always denotes the points of S 1 .

Separatrices and Canonical Regions
Let p(X ) be the Poincaré compactification in the Poincaré disc D of system (1.1), and let Φ be its analytic flow. Following Markus [17] and Neumann [18] we denote by (U, Φ) the flow of a differential system on an invariant set U ⊂ D under the flow Φ. Two flows (U, Φ) and (V, Ψ) are topologically equivalent if and only if there exists a homeomorphism h : U → V which sends orbits of the flow Φ into orbits of the flow Ψ either preserving or reversing the orientation of all the orbits.
A flow (U, Φ) is parallel if it is topologically equivalent to one of the following flows: (i) The flow defined in R 2 by the differential systemẋ = 1,ẏ = 0, called strip flow.
The separatrices of a vector field p(X ) in the Poincaré disc D are: (I) all the orbits of p(X ) which are in the boundary S 1 of the Poincaré disc (i.e. at the infinity of R 2 ), (II) all the finite singular points of p(X ), (III) all the limit cycles of p(X ), and (IV) all the separatrices of the hyperbolic sectors of the finite and infinite singular points of p(X ).
The vector fields p(X ), coming from polynomial vector fields (1.1) of R 2 having finitely many singular points finite and infinite, have finitely many separatrices. For more details see for instance [12]. For a proof of this proposition see [18] or [12]. For a proof of this result see [17,18,20]. Theorem 2.2 implies that in order to classify the phase portraits in the Poincaré disc of a planar polynomial differential system having finitely many separatrices finite and infinite, it is enough to describe their separatrix configuration. This is what we have done in Figure 1.

Phase Portraits
It is clear that system (1.2) with α = 0, after a rescaling of the time τ = (ax + by + c)t becomeṡ x = y, y = −x, whose phase portrait is formed by all the invariant circles centered at the origin of coordinates, providing the first two phase portraits of Figure 1.
In what follows we shall study the phase portraits of system (1.2) with α = 0. Doing the rescaling of the time τ = αt, and renaming a/α, b/α, c/α again by a, b, c, respectively we have the quadratic systemẋ =x 2 + y 2 + 1 + y(ax + by + c),  The way for studying the phase portraits of systems (3.1) is the following. First we shall characterize the finite equilibria of those systems together with their local phase portraits. After we do the same for the infinite equilibria, and finally using this information on the equilibria and the existence of the invariant complex ellipse x 2 + y 2 + 1 = 0 we shall provide the classification of all the phase portraits of systems (3.1).

The Finite Singular Points
The finite singular points of system (3.1) are characterized in the next result. (c) if b = −1 and c 2 > 4(b + 1), then Proof The proof follows easily studying the real solutions of the system x 2 + y 2 + 1 + y(ax + by + c) = 0, −x(ax + by + c) = 0.
We recall that c ≥ 0. Now in the half plane (b, c) we define the regions the curves From now on we only consider the remaining cases: L 1 , L 2 , R 1 and R 2 .
For definitions of elliptic, parabolic and hyperbolic sectors, cusp, and hyperbolic, semihyperbolic and nilpotent singular points see [7].
We introduce some notation that will be used in the following result. If a = 0, using the fact that system (3.1) is reversible with respect to the y-axis, and the eigenvalues of the Jacobian matrix are ± i √ 1 + c 2 we get that U is a center. When a = 0, U is a hyperbolic node if a 2 ≥ 4c 2 (1 + c 2 ). Using that c ≥ 0, it is unstable if a < 0 and stable if a > 0, see for more details Theorem 2.15 of [7] where are described the local phase portraits of the hyperbolic singular points. If a 2 < 4c 2 (1 + c 2 ), since a = 0, then it is a hyperbolic focus, unstable if a < 0 and stable if a > 0.
On the curve L 2 , we have b > −1 and c = 2 √ b + 1. Clearly system (3.1) has the unique singular point V . The Jacobian matrix of system (3.1) at V is Note that b + 2 = 0 because b > −1. The eigenvalues of the Jacobian matrix of system (3.1) at V are 0 and −a/ √ 1 + b. If a = 0 then it is a semi-hyperbolic point. We will use Theorem 2.19 of [7] for studying the local phase portraits of semi-hyperbolic singular points. Doing the change of variables x = X, y = Y − 1 √ b+1 the equilibrium V goes to the origin. Now the linear change of variables ⎛ transforms the linear part of the system into its real Jordan form. In these new variables we get the system Using Theorem 2.19 of [7] we get that V is a semi-hyperbolic saddle-node. When a = 0 the V is a nilpotent singular point. We will use Theorem 3.5 of [7] for studying the local phase portraits of the nilpotent singular points. Again doing the change of variables x = X, y = Y − 1 √ b+1 , and after the change x = v, y = u, and the rescaling of the independent variable τ = 2+b √ 1+b t, we get the systeṁ By Theorem 3.5 of [7] the singular point V is a cusp. On the region R 1 . System (3.1) has two singular points W + and W − . We use the notation Δ in (3.2). Note that Δ > 0. The Jacobian matrix of system (3.1) at W + is 3) The determinant of J W + is equal to D W + given in (3.2). Here D W + > 0. The trace of J W + , T W + is given in (3.2). Note that −c − Δ < 0. So, sign(T W + ) = sign(a). By Remark 3.2, if a = 0 the system is reversible with respect to the y-axis and so W + is a center. If Dis + ≤ 0 (see (3.2)) then W + is a hyperbolic node (stable if a < 0 and unstable if a > 0), and if Dis + < 0 then it is a hyperbolic focus (stable if a < 0 and unstable if a > 0). The Jacobian matrix of system (3.1) at W − is The determinant is equal to D W − given in (3.2). Here, again D W − > 0. The trace of J W − is T W − given in (3.2). Notice that since Δ > c then sign(T W − ) = −sign(a). If a = 0 the system is reversible with respect to the y-axis and so W − is a center. If Dis + ≤ 0 (see (3.2)) then W − is a hyperbolic node (stable if a > 0 and unstable if a < 0), and if Dis + < 0 then it is a hyperbolic focus (stable if a > 0 and unstable if a < 0). On the region R 2 the study can be done in a similar way to the study in the region R 1 . Again, system (3.1) has the two singular points W + and W − . The Jacobian matrix at W + , J W + is given in (3.3) and its determinant, D W + is negative. So, W + is a hyperbolic saddle. The Jacobian matrix at W − , J W − is given in (3.4) and its determinant, D W − is positive. The trace of J W − is T W − . So, if a = 0 then W − is a center, and the other cases follow directly from the hyperbolic singular theorem, see Theorem 2.15 of [7]. Proof First we study the infinite singular points of the local chart U 1 . System (3.1) in the local chart U 1 using (2.1) iṡ

The Infinite Singular Points
there is a unique infinite singular point of (3.5) which is u = −a/ (1 + b), v = 0. The eigenvalues of the Jacobian matrix at this singular point are −1 and −(a 2 + (1 + b) 2 )/(1 + b), which implies that this singular point is a hyperbolic saddle if b < −1, and a hyperbolic stable node if b > −1. Since the origin of the local chart U 2 is not a singular point, the statement (a) follows.
If b = −1, from the study done for system (3.5) there are no infinite singular points in the local chart U 1 . System (3.1) in the local chart U 2 using (2.2) iṡ If a = 0, then the infinity is filled by singular points.
If a = 0, the origin is a singular point of (3.6). It is easy to get that the eigenvalues of the Jacobian matrix at the origin are 0 and a, which implies that the origin is a semi-hyperbolic singular point. Using Theorem 2.19 of [7] we get that if c = 0 then the origin is a semi-hyperbolic saddle-node. If c = 0 and a < 0 the origin is a semi-hyperbolic stable node, and if c = 0 and a > 0 the origin is a semi-hyperbolic unstable node.
Proof of Theorem 1.2 Using Propositions 3.4 and 3.5, and the existence of some first integrals, we shall obtain the global phase portraits of system (3.1) in Poincaré disc described in Figure 1.
For P 1 if a = 0 the phase portrait is given in C of Figure 1, and if a = 0 then it is given in the D of Figure 1. We note that the phase portrait C follows from the local study of the finite and infinite singular points done in Propositions 3.4 and 3.5 together with the Bendixson-Poincaré Theorem, see for instance [7]. The phase portrait D follows from the fact that on P 1 the system has the first integral H = y 2 /(1 + x 2 ) and all the orbits satisfy H = constant.
For L 1 if a = 0 the phase portrait is given in E of Figure 1, and if a = 0 then it is given in F of Figure 1. The phase portrait E follows from the fact that on L 1 the system has the first integral H = (c − y) 2 /(1 − c 2 + x 2 + 2cy). The phase portrait F follows from the local study of the finite and infinite singular points done in Propositions 3.4 and 3.5 together with the Bendixson-Poincaré Theorem.
For L 2 if a = 0 the phase portrait is given in G of Figure 1, and if a = 0 then we get the phase portrait H of Figure 1. Both phase portraits follow from the local study of the finite and infinite singular points done in Propositions 3.4 and 3.5 together with the Bendixson-Poincaré Theorem.
For R 1 if a = 0 the phase portrait is given in I of Figure 1, and if a = 0 then we get the phase portrait H of Figure 1. The phase portrait I follows from the fact that on R 1 the system has the first integral H = (c + by) 2 (1 + x 2 + y 2 ) b . The phase portrait H follows from the local study of the finite and infinite singular points done in Propositions 3.4 and 3.5 together with the Bendixson-Poincaré Theorem.
For R 2 if a = 0 the phase portrait is given in K of Figure 1, and if a = 0 then it is given in L of Figure 1. The phase portrait K follows from the fact that on R 2 the system has the first integral H = (c + by) 2 (1 + x 2 + y 2 ) b . The phase portrait L follows from the local study of the finite and infinite singular points done in Propositions 3.4 and 3.5 together with the Bendixson-Poincaré Theorem.
For R 3 the phase portrait is given in C of Figure 1. This completes the proof of Theorem 1.2.