Phase portraits of the quadratic polynomial Liénard differential systems

Abstract We classify the global phase portraits in the Poincaré disc of the quadratic polynomial Liénard differential systems \dot{x}=y, \quad \dot{y}=(ax+b)y+cx^2+dx+e,where (x, y) ∈ ℝ2 are the variables and a,b,c,d,e are real parameters.


Introduction
A quadratic polynomial differential system is a system of the forṁ where P and Q are polynomials in the variables x and y, and the maximum of the degrees of P and Q is two.
On the other hand a polynomial Liénard differential system is a system of the formẋ = y,ẏ = f (x)y + g(x), where f and g are polynomials in the variable x. The polynomial Liénard differential systems and their applications also have been analysed by many authors these recent years. Thus some authors studied their limit cycles (see for instance [14,15,17,21,26,27,39,41]), or their algebraic limit cycles (see [28,31,37]), or their invariant algebraic curves (see [7,8,49]), or their canard limit cycles (see [43]), or the shape of their limit cycles (see [46]), or the period function of their centres (see [47]), or their integrability (see [9,30]), or a kind of a generalized Liénard system (see [18]).
Roughly speaking the Poincaré disc D 2 is the closed unit disc centred at the origin of coordinates of R 2 , where its interior is identified with R 2 and its boundary S 1 is identified with the infinity of R 2 , i.e. in the plane we can go to or come from infinity in as many directions as points for the circle S 1 . So a polynomial differential system in R 2 (i.e. in the interior of D 2 ) can be extended analytically to the whole D 2 . In this way we can study the dynamics of the differential system in a neighbourhood of infinity. For details on the Poincaré disc see § 3, and chapter 5 of [20].
Up to now the phase portraits in the Poincaré disc of the quadratic polynomial Liénard differential systems have not been studied, their study is the goal of this paper. More precisely, our objective is to classify the different topological phase portraits in the Poincaré disc of the systemṡ where (x, y) ∈ R 2 are the variables and a, b, c, d, e are real parameters. We denote by X = (y, (ax + b)y + cx 2 + dx + e) the vector field defined by system (1.1). We observe that since we are interested in the quadratic polynomial Liénard differential systems we must assume that the parameters satisfy a 2 + c 2 = 0 and a 2 + b 2 = 0 in order to avoid the non-quadratic systems and in order to have systems of Liénard type respectively. Moreover we need that c 2 + d 2 + e 2 = 0, otherwise y = 0 is a straight line filled of equilibria and the system can be reduced to a linear one.
Two phase portraits in the Poincaré disc D 2 are topologically equivalent if there exists a homeomorphism h : D 2 → D 2 which sends orbits of one of the phase portraits into orbits of the other phase portrait, preserving or reversing the orientation of all the orbits.
Our main result is the following one.
Theorem 1. A quadratic polynomial Liénard differential system (1.1) has a phase portrait in the Poincaré disc topologically equivalent to one of the phase portraits of figures 3, 4, 7, 9, 10, 11 and 12. That is, there are 17 different topological phase portraits in the Poincaré disc for system (1.1).
In order to prove theorem 1 we will make use of the normal forms of system (1.1) in § 2, which simplify in somehow the envolved calculations. Afterthat in § 3 we analyse the local phase portraits of the infinite singular points. Finally in § 4 we study the local phase portraits of the finite singular points, and the proof of theorem 1 is given in § 4.
For studying the local phase portraits at the finite and infinite singular points of the compactified quadratic polynomial Liénard differential systems we use notations and results presented in chapters 2, 3 and 5 of [20]. For classifying the global phase portraits of the quadratic polynomial Liénard differential systems in the plane R 2 extended to infinity we follow the notations and results on Poincaré disc in chapter 5 in [20], and with the result due to Markus [33], Neumann [35] and Peixoto [38], which guarantees that we only need to classify all the different configurations of separatrices of the compactified quadratic polynomial Liénard differential systems, in order to obtain their topologically different phase portraits in the Poincaré disc.

Infinite singular points
In this section we study the infinite singular points of the quadratic polynomial Liénard differential systems using the notation and results of chapter 5 in [20].

Infinite singular points in the local charts U 1 and V 1
From equation (5.2) in [20], we obtain that the expression of the Poincaré compactification p(X ) of system (1.1) in the local chart U 1 iṡ   At the singular point (−c/a, 0) the Jacobian matrix has trace a and determinant 0. From § 1.5 in [20] we know that (−c/a, 0) is a semi-hyperbolic singular point, and using theorem 2.19 in [20] we obtain that it is a saddle-node such that when c > 0 it has in v > 0 the unstable parabolic sector, and in v < 0 there are two hyperbolic sectors, recall that the infinity v = 0 is invariant. When c < 0 the sectors of the saddle-node interchange their localization with respect the line of the infinity.
Since the local phase portraits in the infinite singular points of the local chart V 1 of the Poincaré sphere are the symmetric phase portrait with respect to the centre of the sphere reversing the orientation of the orbits, we have that the infinite singular point (−c/a, 0) in V 1 when c > 0 has in v > 0 the two hyperbolic sectors, and in v < 0 the parabolic sector. When c < 0 the sectors of the saddle-node interchange their localization with respect the line of the infinity.
When c = 0 the origin is the unique infinite singular point and it is a semihyperbolic singular point. Again, by theorem 2.19 of [20] we obtain that (0, 0) is either a saddle if ad < 0, or an unstable node if ad > 0, or a saddle-node if d = 0 and e = 0.

The origin of the local charts U 2 and V 2
Once we have studied the infinite singular points in the local charts U 1 and V 1 , it only remains to study if the origins of the local charts U 2 and V 2 are infinite singular points.
Again using the results stated in [20], chapter 5, we obtain the expression of the Poincaré compactification p(X ) of system (1.1) in the local chart U 2 , i.e.  Proof. The Jacobian matrix of system (3.3) at the singular point (0, 0) has trace and determinant equal to zero, but it is not the zero matrix. So according to the definitions of § 1.5 in [20] the origin of system (3.3) is a nilpotent singular point, and we can study its local phase portrait using theorem 3.5 of [20]. Doing that we get that if a = 0 then the local phase portrait at the origin is formed by two sectors one elliptic and one hyperbolic, of course separating these two sectors we can consider two parabolic sectors. When a = 0 we have that c = 0, and then the origin is a stable node if c > 0, and an unstable node if c < 0.
In order to know the position of the invariant straight line of the infinity v = 0 with respect to the elliptic and hyperbolic sectors when a = 0, we need to do the changes of variable known as blow up's, see for more details chapter 3 of [20] or [1]. Doing such changes we get that the elliptic sector is in v > 0 and the hyperbolic sector is in v < 0. Moreover there are no parabolic sectors in v < 0.
In short we have completed the description of the local phase portraits at the infinite singular points.

Finite singular points for each normal form
In this section we study the local behaviour of the finite singular points for each normal form of system (1.1) presented in table 1. To analyse the local phase portraits  of every finite singular point we use the results of [20]. Furthermore, we aim to obtain all the possible global phase portraits for each Poincaré disc with a different configuration of all local phase portraits of finite and infinite singular points. In order to get this we use the following important remark for all the six cases that we have to analyse in this section.  [20] in the semi-hyperbolic case we get that p is a saddlenode. For the case nilpotent we apply theorem 3.5 of [20] and conclude that p is a cusp. These local phase portraits are shown in figure 5. Using remark 5 the correspondent global phase portraits are shown in figure 5.
Phase portraits of the quadratic polynomial Liénard differential systems 209  When A 2 − 4C > 0 there are two singular points, that is, p ± . Since Δ + < 0 we have that p + is a saddle, and p − is either a focus, or a node. If In all the above phase portraits there exists at most one limit cycle for the Liénard differential system that we are analysing, this is a known result which can be found in [12].
Hopf Bifurcation: In Case (i) with A 2 − 4C > 0 and T − = 0 as the singular point p − is a weak focus it, eventually, could be a centre, but this is not the case. In order to prove this we compute the eigenvalues of the Jacobian matrix of system (4.1) at the singular point p − . The real part of the eigenvalues is −A + 2B − α, where α 2 = A 2 − 4C, and we assume α > 0 to facilitate the calculations. The possible situation where the weak focus p − could be a centre is when the real part of its eigenvalues is zero, i.e. α = −A + 2B. With some effort we can check that the singular point p − is always an unstable focus, and never a centre, because    Figure 3 the first non-zero Lyapunov constant is equal π/4β 5 > 0, where β > 0 and is given by The result proved in [12] ensures that the Liénard differential system that we are analysing has at most one periodic orbit. Using a result proved in [25] (theorem 3.3), when B 1 = (A + √ A 2 − 4C)/2 we have a Hopf bifurcation, and an unstable limit cycle bifurcates from p − when B < B 1 (see figure 7(b)). This limit cycle ends in a loop formed with the sadle p + and a homoclinic orbit to it (see figure 7(c)).
Case (ii). Then system (1.1) becomeṡ Phase portraits of the quadratic polynomial Liénard differential systems 211  The finite singular points of system (4.2) are p ± = ((−D ± √ D 2 − 4E)/2, 0), and the Jacobian matrix at p ± is which has trace T = 1 and determinant Δ = √ D 2 − 4E for p − , and Δ = − √ D 2 − 4E for p + . First we observe the necessary condition D 2 − 4E > 0 for the existence of the equilibrium points p ± . If D 2 − 4E = 0 we get a unique singular point which is semi-hyperbolic. Applying theorem 2.19 of [20] at this semi-hyperbolic singular point we get that this singularity is a saddle-node.
If D 2 − 4E > 0 we obtain two singular points, i.e. p ± . In this case p + is a saddle because T = 1 and Δ = − √ D 2 − 4E < 0, and the other equilibrium point p − is an unstable node if 0 < D 2 − 4E 1/16, or an unstable focus if D 2 − 4E > 1/16. Since the divergence of system (4.2) is constant equal to 1, this system cannot have periodic orbits or homoclinic loops (by Bendixson theorem, see for instance theorem 7.10 of [17]). Then from the local phase portraits of figure 8 we obtain the correspondent global phase portraits as shown in figure 9.
All these global phase portraits in figure 9 are achievable with the respective parameters shown in table 3.
In Case (i) the singular point is a node and in Case (ii) the singular point is a focus.

In both cases the singular point is stable if H > G and unstable if H < G.
For all these local phase portraits and the correspondent global phase portraits see figure 10 when G > 0, figure 11 when G < 0 and when G = 0 the local and the correspondent global phase portrait are topologically equivalent to those ones of figure 3. It is easy to check that the quadratic differential system (4.3) has no invariant straight lines. So by a result of Sotomayor and Paterlini [44] it follows that the two finite separatrices of the saddles at the origins of the local charts U 1 and V 1 of figure 11(a) cannot connect, otherwise the connection must be through an invariant straight line. Therefore the global phase portrait of figure 11(b) is the unique possible. All these global phase portraits are achievable with the respective parameters shown in table 4.
In Case (i) the singular point (I, 0) is a stable node if I −2, or an unstable node if I 2. In Case (ii) the singular point (I, 0) is a stable focus if −2 < I < 0, or an