This is a preprint of: “Global phase portraits of the key pitchfork bifurcation”, Shimin Li, Jaume Llibre, SCIENTIA SINICA Mathematica, vol. 49(9), 1201–1208, 2019. DOI: [10.1360/SCM-2018-0266] GLOBAL PHASE PORTRAITS OF THE KEY PITCHFORK BIFURCATION SHIMIN LI1 AND JAUME LLIBRE2 Abstract. This paper deals with the following quadratic polynomial differ- ential systems dx dy = y 2 − y − x, = x2 − µx − y, dt dt with parameter µ ∈ R, which is the key example of [30] for studying the pitchfork bifurcation of a singular point. We classify the global phase portraits in the Poincaré disc of these systems when µ varies. 1. Introduction and statement of the main results Global phase portraits are an invaluable tool in studying the long dynamical behaviour of differential systems. They reveals information such as whether an attractor, a repellor or a limit cycle is present for a given parameter value. Hence the global phase portraits analysis is the one of most important problems in the qualitative theory of differential systems. The possibilities of topological distinct phase portraits for a general polynomial differential system are huge, it is expected that the quadratic polynomial differential systems have more than 2000 topological distinct phase portraits. Most of known results about global phase portraits of differential systems are mainly deal with special differential systems, see [1, 2, 3, 4, 6, 9, 11, 18, 27, 29, 31] for quadratic polynomial differential systems, see [5, 7, 10, 13, 14, 33, 34] for cubic polynomial differential systems, see [8, 25, 24, 26] for quartic polynomial differential systems, see [12, 15, 16, 17] for Liénard differential systems, see [22, 23, 28] for Hamiltonian differential systems. In [20] the authors introduce how to use the computer program P 4 for drawing phase portraits in a Poincaré disc. In a recent paper [30] Rajapakse and Smale considered the following quadratic polynomial differential systems dx dy (1) = y 2 − y − x, = x2 − µx − y, dt dt as a key example to describe the pitchfork bifurcation, where µ ∈ R. They showed that symmetry is a dispensable condition for the existence of pitchfork bifurcation. Let X1 and X2 be two vector fields defined on open subsets ∆1 and ∆2 of R2 , respectively. We say that two vector fields X1 and X2 are topologically equivalent 2010 Mathematics Subject Classification. 34C05, 34C23. Key words and phrases. Phase portrait; Pitchfork bifurcation; Poincaré compactification. 1 2 SHIMIN LI AND JAUME LLIBRE when there exists a homeomorphism h : ∆1 → ∆2 which sends orbits of X1 to orbits of X2 preserving or reversing the orientation. In this paper we provide the topological classification of the phase portraits of systems (1) in the Poincaré disc. Theorem 1. The global phase portrait of a differential system (1) is topologically equivalent to Figure 1.1 if µ 6 1, Figures 1.2 if 1 < µ < µ∗ , Figure 1.3 if µ > µ∗ and Figure 1.4 if µ = µ∗ , where µ∗ ∈ (10.4722, 10.4723). 1.1 S = 10, R = 3 1.2 S = 16, R = 5 1.3 S = 16, R = 5 1.4 S = 15, R = 4 Figure 1. Topological phase portraits of differential systems (1). S and R denote the number of separatricies and canonical regions, respectively. Note from Figure 1 that the pitchfork bifurcation at the origin of coordinates takes place at µ = 1, where the stable node bifurcates into two stable foci and one saddle. The layout of this paper is as follows. In section 2 we analyze the equilibria of systems (1) in the Poincaré compactification. In section 3 we prove Theorem 1. GLOBAL PHASE PORTRAITS OF THE KEY PITCHFORK BIFURCATION 3 2. Poincaré compactification 2.1. Poincaré compactification. For a given polynomial differential system dx dy (2) = P (x, y), = Q(x, y), dt dt of degree d = max{deg(P ), deg(Q)}. Let X = (P, Q) be the vector field associated system (2). We call S2 = {s = (s1 , s2 , s3 ) ∈ R3 : s21 + s22 + s23 = 1} the Poincaré sphere. The Poincaré compactified vector field p(X ) corresponding to X is an analytic vector field induced on S2 as follows, for more details see Chapter 5 of [21]. First we take R2 as the plane in R3 defined by (x, y, 1) ∈ R3 , and then project each point (x, y, 1) in two points of the Poincaré sphere S2 using the straight line through (x, y, 1) and the origin (0, 0, 0). It is obvious that the equator S1 = {s ∈ S2 , s3 = 0} corresponds to the infinity of R2 . So we have two copies of the vector field X on the Poincaré sphere S2 , one in the open northern hemisphere S− = {s ∈ S2 : s3 > 0}, and the other in the open southern hemisphere S+ = {s ∈ S2 : s3 < 0}. This vector field on S2 \ S1 can be extended to a vector field p(X ) defined in the whole S2 multiplying it by sd3 . For studying the Poincaré sphere we use the following six local charts (3) Ui = {s ∈ S2 : si > 0}, Vi = {s ∈ S2 : si < 0}, for i = 1, 2, 3. The expression of p(X ) in the local charts U1 and V1 are given by [ ( ) ( )] ( ) du d u 1 u 1 dv d+1 u 1 (4) = v − uP , − uQ , , = −v Q , ; dt v v v v dt v v with v > 0 and v < 0, respectively. The expression of p(X ) in the local charts U2 and V2 are given by [ ( ) ( )] ( ) du 1 u 1 u dv 1 u (5) = v d − uP , +Q , , = −v d+1 P , ; dt v v v v dt v v with v > 0 and v < 0, respectively. The expression of p(X ) in the local charts U3 and V3 are just du dv (6) = P (u, v), = Q(u, v). dt dt For studying the phase portrait of a differential system (2), we just need to study its Poincaré compactification p(X ) restricted to the closed northern hemisphere. We do the orthogonal projection π(s1 , s2 , s3 ) = (s1 , s2 ) of the closed northern hemi- sphere onto the Poincaré disc D2 = {s21 + s22 6 1} for drawing the phase portrait. It is obvious that the finite equilibria of system (2) are the equilibria in the interior of D2 , and they can be studied using U3 . The infinite equilibria of systems (2) are the equilibria of p(X ) in the boundary of D2 . Note that for studying the infinite equilibria it suffices to look the ones at the local charts U1 |v=0 and V1 |v=0 , and at the origin of the local charts U2 and V2 . 4 SHIMIN LI AND JAUME LLIBRE 1 u 2.2. In charts U1 and V1 . Doing the change of variables x = , y = , systems v v (1) become du dv = 1 − v − u3 + µu2 v, = v 2 (v − u2 + µuv). dt dt Let v = 0, then we obtain the unique infinite equilibrium (1, 0) in the chart U1 , which is a stable node. Since the degree of systems (1) is 2, so we can deduce that there is also a unique unstable node infinite equilibrium (1, 0) in chart V1 by symmetry. u 1 2.3. In charts U2 and V2 . Doing the change of variables x = , y = , systems v v (1) become du dv (7) = 1 − µv − u3 + u2 v, = v(v − u2 + uv). dt dt Therefore the origin of U2 and V2 are not equilibria. 2.4. In charts U3 and V3 . In order to determine the number of finite equilibrium of systems (1), we introduce the following result, see for example [32]. Lemma 2. For a general quartic polynomial (8) a0 x4 + a1 x3 + a2 x2 + a3 x + a4 , (a0 ̸= 0). We define the following parameters: E = 8a20 a3 + a31 − 4a0 a1 a2 , D2 = 3a21 − 8a0 a2 , D3 = 16a20 a2 a4 − 18a20 a23 − a0 a32 + 14a0 a1 a2 a3 − 6a0 a21 a4 + a21 a22 − 3a31 a3 , (9) D4 = 256a30 a34 − 27a20 a43 − 192a20 a1 a3 a24 − 27a41 a24 − 6a0 a21 a23 a4 + a21 a22 a23 −4a0 a32 a23 + 18a31 a2 a3 a4 + 144a0 a21 a2 a24 − 80a0 a1 a22 a3 a4 + 18a0 a1 a2 a33 −4a21 a32 a4 − 4a31 a33 + 16a0 a42 a4 − 128a20 a22 a24 + 144a20 a2 a23 a4 . The following statements hold. (i) If D4 > 0, D3 > 0, D2 > 0, then (8) has four simple real zeros. (ii) If D4 < 0, then (8) has two simple real zeros. (iii) If D4 = 0, D3 = 0, D2 > 0, E ̸= 0, then (8) has a simple zero and a triple zero. The finite equilibria of systems (1) must satisfy the following quartic polynomial (10) f (y) = y(µ − 1 − (µ − 1)y − 2y 2 + y 3 ). By direct computation we obtain E = −8, D2 = 4(1 + 2µ), D3 = (µ − 1)3 (4µ2 + 5µ + 23), (11) D4 = 2(µ − 1)(2µ2 + 3µ + 7). GLOBAL PHASE PORTRAITS OF THE KEY PITCHFORK BIFURCATION 5 If µ < 1, then D4 < 0. Therefore systems (1) have two simple finite equilibria E1 = (0, 0) and E2 . If µ = 1, then D2 > 0 and D3 = D4 = 0. Hence systems (1) have a simple finite equilibrium E1 = (0, 0) and a triple finite equilibrium E2 . If µ > 1, then D2 , D3 and D4 are positive. So systems (1) have four simple finite equilibria Ei , i = 1, 2, 3, 4. We state the well known Bendixson’s Theorem as follows, see for instance The- orem 7.10 of [21]. Theorem 3 (Bendixson’s Theorem). Assume that the divergence function of d- ifferential system has constant sign in a simply connected region R, and is not identically zero on any subregion of R. Then differential system does not have a periodic orbit which lies entirely in R. According to Bendixson’s Theorem, we can obtain a preliminary result for sys- tems (1). Proposition 4. Systems (1) have no periodic orbits, no homoclinic loops as the one of Figure 2.1. l5 E2 E4 s2,3 s2,2 s1,4 E2 Γ s1,2 s2,1 s2,4 l1 E1 s1,1 E3 s1,3 l2 l3 l4 2.1 µ ∈ R 2.2 µ > 1 Figure 2. 2.1. The homoclinic loop of the saddle E2 . 2.2. Local dynamics of the finite equilibria of systems (1) for µ > 1. l1 , l2 , l4 are the straight lines through the equilibria E1 and E3 , E2 , E4 , respectively. l3 is the straight line y = x. l5 is the straight line through equilibria E2 and E3 . Proof. Since the divergence of systems (1) is −2, it follows that these systems have no periodic orbits by Theorem 3. Assume that systems (1) have a homoclinic loop Γ containing equilibrium E2 , see Figure 2.1. From Green’s formula we have ∫ ∫∫ ( ) ∂P ∂Q (12) P dy − Qdx = + dxdy. Γ int(Γ) ∂x ∂y 6 SHIMIN LI AND JAUME LLIBRE Note that Γ is an orbit of systems (1), P dy − Qdx = 0 holds everywhere along Γ. Hence the left side of (12) is zero. Recall that the divergence of systems (1) is -2, so the right side of (12) negative, which giving a contradiction.  In order to study the types and stabilities of finite equilibria of systems (1), we introduce the Poincaré-Hopf Theorem and the Berlinskii Theorem, see Theorem 6.30 of [21] and Theorem 7 of [19], respectively. Theorem 5 (Poincaré-Hopf Theorem). For every tangent vector field on S2 with a finite number of equilibria, the sum of their topological indices is 2. Theorem 6 (Berlinskii Theorem). Consider the quadratic polynomial differential systems (2). Suppose that there are four equilibrium points. If the quadrilateral with vertices at these points is convex then two opposite equilibrium points are saddles and the other two are antisaddles (nodes, foci, or centers). But if the quadrilateral is not convex then either the three exterior vertices are saddles and the interior vertex an antisaddle, or the exterior vertices are antisaddles and the interior vertex a saddle. The types and stabilities of the infinite equilibria of systems (1) can be stated as follows: Proposition 7. Consider systems (1) we have (i) If µ < 0, then systems (1) have two finite equilibria: E1 is a hyperbolic stable focus, and E2 is a hyperbolic saddle. (ii) If 0 6 µ < 1, then systems (1) have two finite equilibria: E1 is a hyperbolic stable node, and E2 is a hyperbolic saddle. (iii) If µ = 1, then systems (1) have two finite equilibria: E1 is a semi-hyperbolic stable node, and E2 = (2, 2) is a hyperbolic saddle. (iv) If µ > 1, then systems (1) have four finite equilibria: E1 and E2 are hyper- bolic saddles, E3 and E4 are hyperbolic stable foci, see Figure 2.2. Proof. First we consider the finite equilibrium E1 = (0, 0). The Jacobian matrix of systems (1) at the equilibrium E1 = (0, 0) is ( ) −1 −1 (13) . −µ −1 √ √ The eigenvalues of the Jacobian matrix (13) are −1− µ and −1+ µ. It is obvious that E1 is a hyperbolic saddle when µ > 1, a stable hyperbolic node for 0 6 µ < 1, and a hyperbolic stable focus if µ < 0. For the case µ = 1 doing the change of variables X = x − y, Y = x + y, T = −t, then systems (1) become dX dY X2 Y2 (14) = XY, = 2Y − − . dT dT 2 2 According to statement (ii) of Theorem 2.19 of [21], we know that the origin of systems (14) is an unstable topological node. Note that we have reverse the time, hence the origin of systems (1) for µ = 1 is a semi-hyperbolic stable node. Second we study the other finite equilibria. For the case µ = 1, it is obvious that E2 = (2, 2) is a hyperbolic saddle. For the case µ < 1. Since the two infinity equilibria are hyperbolic nodes and the finite equilibrium E1 = (0, 0) is a node GLOBAL PHASE PORTRAITS OF THE KEY PITCHFORK BIFURCATION 7 or a focus, the other finite equilibrium E2 should be a hyperbolic saddle applying Theorem 5 to the Poincaré sphere. For the case µ > 1. According to Theorem 5 again, the sum of the indices of finite equilibria Ei , i = 1, 2, 3, 4 is 0 because the two infinity equilibria are hyperbolic nodes. Thus we can conclude that the quadrilateral with vertices at these finite equilibria is convex by Theorem 6, and then E2 is a hyperbolic saddle, E3 and E4 are hyperbolic antisaddles, i.e. they can be nodes, foci or centers. The equilibria E3 and E4 cannot be centers because systems (1) have no periodic orbits by Proposition 4. In the following we prove that E3 and E4 are stable foci. The eigenvalues of the Jacobian matrix of systems (1) at a point (x, y) are √ (15) λ± = −1 ± (2x − µ)(2y − 1). Let µ = 2, then we know that E3 and E4 are stable foci. Since (2x − µ)(2y − 1) with µ > 1 never vanish in the equilibria E3 and E4 of systems (1) and taking into account that the sum of the eigenvalues at any equilibrium point is −2, we can deduce that E3 and E4 are stable foci for all µ > 1.  3. Proof of Theorem 1 Let φ(t, p) be an orbit of an analytic vector field. If this orbit is defined for all t > 0 we denote its ω−limit set as ω(p) or ω(φ). If this orbit is defined for all t 6 0 its α−limit set is defined by α(p) or α(φ). Before prove our main result we state the well known Poincaré-Bendixson Theo- rem and the Markus-Neumann-Peixoto Theorem, see Corollary 1.30 and Theorem 1.43 of [21]. Theorem 8 (Poincaré-Bendixson Theorem). Let φ(t, p) be an integral curve of an analytic vector field X in R2 defined for all t > 0, such that γp+ = {φ(t, p) : t > 0} is contained in a compact set K. Assume that the vector field X has a finite number of equilibrium points in ω(p). Then one of the following statement holds. (i) If ω(p) contains only regular points, then ω(p) is a limit cycle. (ii) If ω(p) contains regular and equilibrium points, then ω(p) is formed by a finite number of orbits γ1 , γ2 , · · · , γn and a finite number of equilibrium points p1 , p2 , · · · , pn such that α(γi ) = pi , ω(γi ) = pi+1 for i = 1, 2, · · · , n − 1, α(γn ) = pn and ω(γn ) = p1 . Such kind of ω−limit sets are called graphics. Possibly some of the equilibrium points γi can be identified. (iii) If ω(p) does not contain regular points, then ω(p) is a equilibrium point. (iv) Similar results for the α−limit set. Let X be a polynomial vector field, and let p(X ) be its Poincaré compactification. Assume that p(X ) has finitely many equilibria. Then the set Σ of all separatrices of p(X ) in the Poincaré disc are all its infinite orbits, its finite equilibria, its limit cycles and its graphics. It is known that Σ is a closed set in the Poincaré disc D2 . Each open component of D2 \ Σ is called a canonical region. Then a separatrix skeleton S of X or of p(X ) is the union of Σ plus an orbit of each canonical region. We say that two separatrix skeletons S1 and S2 corresponding to two polynomial vector fields X1 and X2 are equivalent if there exists a homeomorphism h : S1 → S2 . 8 SHIMIN LI AND JAUME LLIBRE Theorem 9 (Markus-Neumann-Peixoto Theorem). Assume that p(X1 ) and p(X2 ) are two Poincaré compactifications of two polynomial vector fields X1 and X2 with finitely many separatrices. Then their phase portraits in the Poincaré disc are topologically equivalent if and only if their separatrix skeletons are equivalent. In order to obtain the position of the local separatrices of the saddle E1 and E2 (see Figure 2.2), we introduce the following result which is given in [19]. Lemma 10. On any straight line which is not composed of paths the total number of equilibrium points and contacts is at most two. If there are two such points, P1 and P2 , then the paths intersecting the segment ∞P1 cross in the same sense as the paths intersecting P2 ∞ and in the opposite sense to the paths intersecting P1 P2 . Proof of Theorem 1. From subsections 2.2 and 2.3, systems (1) have two infinite equilibria, a stable node and an unstable node for all values of µ ∈ R. Recall that systems (1) have no periodic orbits and no homoclinic loops by Proposition 4. (I) For the case µ < 0. There are two finite equilibria: E1 = (0, 0) is a hyperbolic stable focus and E2 is a hyperbolic saddle by statements (i) of Proposition 7. Since there are no limit cycles and no graphics, the α− and ω−limit sets of the separatrices of the saddle E2 it must be equilibrium points by Theorem 8. We know that the two unstable separatrices of the saddle E2 cannot go together to the stable focus E1 , or to the stable node at infinity, because one of the stable separatrix of the saddle E2 would not have its α−limit set. So one of the unstable separatrices of the saddle E2 goes to the stable focus E1 and the other goes to the stable node at infinity. Therefore the two stable separatrices of E2 have their α−limit set at the unstable node at infinity. From the above analysis, the phase portrait of systems (1) for µ > 1 is topologically equivalent to the one of Figure 1.1. (II) For the case µ ∈ [0, 1]. There are two finite equilibria: E1 = (0, 0) is a hyperbolic stable node if µ ∈ [0, 1), or is a semi-hyperbolic stable node if µ = 1, and E2 is a hyperbolic saddle by statements (ii) and (iii) of Proposition 7. Similar with the proof of case (I) we obtain the phase portrait of systems (1) for µ ∈ [0, 1] is topologically equivalent to the one of Figure 1.1. (III) For the case µ > 1. There are four finite equilibria Ei , i = 1, 2, 3, 4 by the statement (iv) of Proposition 7, see Figure 2.2. Ei for i = 1, 2 are hyperbolic saddles, which have two unstable separatrices si,1 and si,2 and two stable separatrices si,3 and si,4 . Ei for i = 3, 4 are hyperbolic stable foci. We divide the proof of Theorem 1 for µ > 1 into five steps. Step 1. The flow of systems (1) on the straight line l1 (see its definition in Figure 2.2) can be determined as follows. According to the orientation of the flow at infinity we can deduce the flows of systems (1) on the segment ∞E1 is upward, see Figure 2.2. Then we can determine the flow of systems (1) on the segment E1 E3 and E3 ∞ by Lemma 10. The flow of systems (1) on the straight lines l2 , l4 and l5 can be deduced similarly. Recall that l3 : y = x. Since the inner product of the vectors (−1, 1) and (y 2 − y − x, x2 − µx − y) on the straight line y = x is equal to (1 − µ)x, the flows of systems (1) on the line y = x is described in Figure 2.2. GLOBAL PHASE PORTRAITS OF THE KEY PITCHFORK BIFURCATION 9 Thus we can obtain the position of the local separatrices of the saddles E1 and E2 as they are indicated in Figure 2.2. Let µ = 2 we know that the finite equilibria E2 and E3 are located in the region y < x, and E4 is contained in the region y > x. Taking into account the direction of the flows on the straight line l3 and that the quadrilateral with vertices at these equilibria Ei , i = 1, 2, 3, 4 is convex (due to Berlinskii Theorem), then we can deduce that for µ > 1 the equilibria E2 and E3 always are located in the region y < x, while E4 is always contained in the region y > x. Step 2. Since systems (1) have no homoclinic loops (see Proposition 4), and taking into account that the sense of the flow of systems (1) on the straight lines l3 and l4 , we know that the α−limit set of separatrix s1,3 of the saddle E1 is the unstable node at infinity. Similarly we can deduce the α−limit set of the separatrix s1,4 of the saddle E1 is the unstable node at infinity, and the ω−limit set of the separatrix s2,2 of the saddle E2 is the stable node at infinity. Step 3. Note that the separatrix s1,4 of the saddle E1 must intersect the straight line l4 in the segment ∞E4 . It is easy to know using the Poincaré-Bendixson Theorem that the ω−limit set of the separatrix s1,2 of the saddle E1 is the stable focus E4 because the systems have no periodic orbits (see Proposition 4), and the α−limit set of the separatrix s2,3 of the saddle E2 is the unstable node at infinity. Step 4. We claim using the Poincaré-Bendixson Theorem that the ω−limit set of the separatrix s2,1 of the saddle E2 is the stable focus E3 because the systems have no periodic orbits (see Proposition 4). Otherwise it can only be the stable node at infinity, this situation cannot occurs because then the α−limit set of the separatrix s2,4 of the saddle E2 cannot exist. Step 5. The remaining separatrices are s1,1 and s2,4 . The ω−limit set of the separatrix s1,1 of the saddle E1 can be either the stable focus E3 or the stable node at infinity. Subcase (III.1) If the ω−limit set of the separatrix s1,1 of the saddle E1 is the stable focus E3 , then the α−limit set of the separatrix s2,4 of the saddle E2 is the unstable node at infinity. In this case the phase portrait of systems (1) for µ > 1 is topologically equivalent to the one of Figure 1.2. In fact it is not difficult to check that for µ > 1 but close to 1, then the phase portrait of a differential systems (1) is topologically equivalent to Figure 1.2. Subcase (III.2) If the ω−limit set of the separatrix s1,1 of the saddle E1 is the stable node at infinity, then the α−limit set of the separatrix s2,4 of the saddle E2 is also the unstable node at infinity. In this case the phase portrait of systems (1) for µ > 1 is topologically equivalent to the one of Figure 1.3. If µ = 20 it is not difficult to verify that the phase portrait of a differential systems (1) is topologically equivalent to Figure 1.3. Subcase (III.3) From the above subcases (III.1) and (III.2), we know that the separatrix s1,1 of the saddle E1 and the separatrix s2,4 of the saddle E2 must connect for some value µ∗ of the parameter µ by continuity. In this case the phase portrait of a systems (1) is topologically equivalent to the one of Figure 1.4. Moreover, numerically we find that µ∗ ∈ (10.4722, 10.4723).  10 SHIMIN LI AND JAUME LLIBRE 4. Acknowledgements The first author is partially supported by the Natural Science Foundation of Guangdong Province (2017A030313010), Science and Technology Program of Guang zhou (No. 201707010426, 20180401350). The second author is partially supported by the MINECO-FEDER grant MTM 2016-77278-P, the AGAUR grant 2017 SGR1617, and the projecte MDM-2014-0445 (BGSMath). References [1] J.C. Artés, J. 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Lan, Phase portraits, Hopf bifurcations and limit cycles of the Holling-Tanner models for predator-prey interactions, Nonlinear Analysis: Real World Applications 12 (2011) 1961–1973. 1 School of Mathematics and Statistics, Guangdong University of Finance and Eco- nomics, Guangzhou, 510320, P.R. China E-mail address: lism1983@126.com 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bel- laterra, Barcelona, Catalonia, Spain E-mail address: jllibre@mat.uab.cat