This is a preprint of: “Phase portraits and bifurcation diagram of the Gray-Scott model”, Ting Chen, Shimin Li, Jaume Llibre, J. Math. Anal. Appl., vol. 496, 124840, 2021. DOI: [10.1016/j.jmaa.2020.124840] PHASE PORTRAITS AND BIFURCATION DIAGRAM OF THE GRAY-SCOTT MODEL TING CHEN1,2 , JAUME LLIBRE2,∗ AND SHIMIN LI3 Abstract. We give a complete classification of the phase portraits in the Poincaré disk for the cubic polynomial systems ẋ = 1 − x − axy 2 , ẏ = −by + axy 2 , in R2 according with the values of its two parameters a and b. These differ- ential systems correspond to the Gray-Scott model. Moreover we provide the bifurcation diagram in the parameter plane (a, b) of these systems. The Gray-Scott model for studying the autocatalysis has been ana- lyzed by several authors these recent years. The original partial differen- tial equations can be simplified to ordinary differential equations, more precisely to a family cubic polynomial differential systems depending on two parameters a and b. Here we characterize the global dynamics of these cubic systems, taking into account their behavior near the infinity using the Poincaré compactification. Moreover we provide the bifurca- tion diagram of the phase portraits in the parameter plane (a, b). 1. Introduction and statement of the main results The Gray-Scott model [13, 18] is a cubic autocatalysis system that exhibits many interesting patterns, see for instance the papers [3, 10, 13, 16, 18] and the refer- ences quoted there. This model has been studied with slightly distinct differential equations, here we consider the model given by the differential equations ∂U = 1 − U − aU V 2 + DU ∆U, ∂t (1) ∂V = −bV + aU V 2 + DV ∆V, ∂t where U = U (u, v, t) and V = V (u, v, t) are the concentrations of an inhibitor and an activator, (u, v) ∈ R2 , a and b are positive constants, DU > DV are the diffusivities, ∆ is the Laplace operator, and t is the time. For more details on the Gray-Scott model see the previous mentioned references. 0 Corresponding author 2010 Mathematics Subject Classification. Primary: 34C07, 34C08. Key words and phrases. Gray-Scott model, cubic polynomial system, phase portraits, bifurca- tion diagram, Hopf bifurcation. 1 http://www.gsd.uab.cat 2 TING CHEN, JAUME LLIBRE AND SHIMIN LI In order to obtain information on the dynamics of the partial differential equa- tions (1) some authors have studied the ordinary differential systems ẋ =1 − x − axy 2 , (2) ẏ = − by + axy 2 , obtained from systems (1) taking DU = DV ≡ 0. If we put the keywords “Gray- Scott model” in MathSciNet we obtain 161 references related with this model. As far as we know any of these references studied the complete dynamics of the differential systems (2). The objective of this paper is to obtain all the phase portraits of the cubic polynomial differential systems (2) for all the values of the parameters a, b ∈ R in the Poincaré disk, i.e. including the dynamics of systems (2) at infinity and consequently in its neighborhood. See Section 2 for a summary on the Poincaré compactification. These recent years many authors have studied the phase portraits of different classes of planar polynomial differential systems, we only quote few of them [2, 4, 5, 12, 6, 7, 8, 9, 14, 15, 17]. From this point of view our paper contributes to study a new class, the class defined by the cubic polynomial differential systems (2). Assume that a is not zero, otherwise the differential systems (2) become linear. Hence we study the phase portraits of systems (2) and provide their bifurcation diagram in the parameter plane (a, b) ∈ R2 \ {(0, b) : b ∈ R}. To do this we will use the Poincaré compactification of the polynomial differential systems. Here we say that two differential systems (2) in the Poincaré disk D2 are topologically equivalent if there exists a homeomorphism h : D2 → D2 which sends orbits to orbits preserving or reversing the direction of all orbits. For the polynomial differential systems in the Poincaré disk it is known that the separatrices are all the infinite orbits, all the finite singular points, the separatrices of the hyperbolic sectors of the finite and infinite singular points, and the limit cycles. If Σ denotes the set of all separatrices in the Poincaré disk D2 , Σ is a closed set and the components of D2 \ Σ are called the canonical regions, for more details see [11]. We denote by s, r and LC the number of separatrices, canonical regions and limit cycles, respectively. Our main result is the following theorem. Theorem 1. The phase portraits of the planar cubic polynomial differential sys- tems (2) are topologically equivalent to the following ones of Figures 1, 2 and 3: 1.1 if 0 < a < f1 (b) and b < 0; 1.2 if a = f1 (b) and b < 0; 1.3 if a > f1 (b) and b < 0; 1.4 if b = 0 and a > 0; 1.5 if a > f1 (b), a > f2 (b) and a > f5 (b), or a = f2 (b) and b > b3 ; 1.6 if a = f5 (b) and b1 < b ≤ b3 ; 1.7 if f1 (b) < a < f5 (b), a > f4 (b) and a ≥ f2 (b); 1.8 if a = f4 (b) and 2 < b ≤ b2 ; 1.9 if a = f4 (b) and b2 < b < b4 ; 1.10 if f2 (b) < a < f4 (b), or a = f2 (b) and 4 < b < b2 ; 1.11 if f3 (b) < a < f2 (b), a < f4 (b) and 4 < b < b2 ; 1.12 if a = f3 (b) and 4 < b < b4 ; PHASE PORTRAITS OF THE GRAY-SCOTT MODEL 3 (1.1) s=17, r=4; (1.2) s=21, r=6; (1.3) s=23, r=6; (1.4) s=17, r=4; (1.5) s=23, r=6; (1.6) s=22, r=5; (1.7) s=23, r=6; (1.8) s=22, r=6; (1.9) s=23, r=7, LC=1; (1.10) s=24, r=7, LC=1; (1.11) s=25, r=8, LC=2; (1.12) s=24, r=7, LC=1. Figure 1. Topological phase portraits of the cases 1.1-1.12 in The- orem 1. 4 TING CHEN, JAUME LLIBRE AND SHIMIN LI (1.13) s=22, r=6; (1.14) s=24, r=7, LC=1; (1.15) s=23, r=6, LC=1; (1.16) s=24, r=7, LC=1; (1.17) s=23, r=6; (1.18) s=21, r=6; (1.19) s=20, r=5; (1.20) s=21, r=6; (1.21) s=20, r=5; (1.22) s=21, r=6; (1.23) s=17, r=4; (1.24) s=25, r=8. Figure 2. Topological phase portraits of the cases 1.13-1.24 in Theorem 1. PHASE PORTRAITS OF THE GRAY-SCOTT MODEL 5 (1.25) s=17, r=4; (1.26) s=21, r=4; (1.27) s=22, r=5, LC=1; (1.28) s=21, r=5; (1.29) s=21, r=5; (1.30) s=23, r=6; (1.31) s=24, r=7, LC=1; (1.32) s=22, r=6; (1.33) s=23, r=6. Figure 3. Topological phase portraits of the cases 1.25-1.33 in Theorem 1. 1.13 if a = f4 (b) and b ≥ b4 ; 1.14 if f4 (b) < a < f2 (b) and a < f6 (b); 1.15 if a = f6 (b) and b > b3 ; 1.16 if f6 (b) < a < f2 (b) and b > b3 ; 1.17 if f1 (b) < a < f2 (b), a < f3 (b), a < f4 (b) and b > 2, or a = f2 (b) and 2 < b ≤ 4; 1.18 if a = f1 (b) and 0 < b < b1 ; 1.19 if a = f1 (b) and b = b1 ; 1.20 if a = f1 (b) and b1 < b < 2; 1.21 if a = 16 and b = 2; 1.22 if a = f1 (b) and b > 2; 1.23 if 0 < a < f1 (b) and b > 0; 1.24 if a < 0 and b > 0; 6 TING CHEN, JAUME LLIBRE AND SHIMIN LI 1.25 if a < 0 and b = 0; 1.26 if a ≤ f2 (b) and −1 ≤ b < 0; 1.27 if f2 (b) < a < f7 (b) and −1 ≤ b < 0; 1.28 if a = f7 (b) and −1 ≤ b < 0; 1.29 if f7 (b) < a < 0 and −1 ≤ b < 0; 1.30 if a ≤ f2 (b) and b < −1; 1.31 if f2 (b) < a < f7 (b) and b < −1; 1.31 if a = f7 (b) and b < −1; 1.33 if f7 (b) < a < 0 and b < −1; where f1 (b) = 4b2 , f2 (b) = b4 /(b − 1), fi (b) (i = 3, 4, · · · , 7) are convenient functions, b1 ∈ (1.4202, 1.4204), b2 ∈ (4.4, 4.403), b3 ∈ (4.413, 4.414) and b4 ∈ (6.24, 6.25). Moreover the corresponding bifurcation diagrams are shown in Fig- ures 4, 5 and 6 (see Section 4 for the explanation of the bifurcation diagrams). In fact, the curve a = f1 (b) corresponds to a saddle-node bifurcation, the curve a = f2 (b) corresponds to a Hopf bifurcation, the curve a = f3 (b) corresponds to a semi-stable limit cycle, the curve a = f4 (b) corresponds to a homoclinic orbit, the curves a = f5 (b) and a = f6 (b) correspond to connection of two separatrices, and the curve a = f7 (b) corresponds to heteroclinic connection. a a=f1(b) 1.3 1.4 1.2 1.1 0 b Figure 4. The bifurcation diagram of systems (2) when a > 0 and b ≤ 0, where f1 (b) = 4b2 . In this paper we analyze different types of equilibria, hyperbolic, semi-hyperbolic, nilpotent, degenerate and Hopf. For studying the local phase portraits of the semi- hyperbolic equilibria we use Theorem 2.19 of [11], for the nilpotent equilibria we use Theorem 3.5 of [11], while for the degenerate cases we apply blow-ups for studying the local phase portrait at the equilibrium points, for more details on the changes of variables called blow-ups see [1]. And for the Hopf bifurcation we need to compute the Liapunov constants at the singularities. For more information on the Liapunov constants see Chapter 4 of [11]. To complete a global phase portrait of a differential PHASE PORTRAITS OF THE GRAY-SCOTT MODEL 7 a a=f2(b) a=f6(b) 1.16 1.4 1.5 1.15 1.6 1.7 1.14 1.13 a=f5(b) 1.8 1.11 1.9 a=f4(b) 1.12 a=f1(b) 1.10 1.20 1.19 1.21 a=f3(b) 1.18 1.23 1.22 1.17 0 b1 2 4 b2 b3 6 b4 b Figure 5. The bifurcation diagram of systems (2) when a > 0 and b ≥ 0. f1 (b) = 4b2 , f2 (b) = b4 /(b − 1), fi (b) (i = 3, 4, · · · , 6) are convenient functions, b1 ∈ (1.4202, 1.4204), b2 ∈ (4.4, 4.403), b3 ∈ (4.413, 4.414) and b4 ∈ (6.24, 6.25). a -1 b 0 1.29 1.33 1.27 1.28 1.25 1.32 1.26 1.31 1.24 1.30 a=f7(b) a=f2(b) Figure 6. The bifurcation diagram of systems (2) when a < 0. f2 (b) = b4 /(b − 1) and f7 (b) is a convenient function. system it is important the characterization of its separatrices, and the continuity of phase portraits with respect to the parameters is also essential. 8 TING CHEN, JAUME LLIBRE AND SHIMIN LI 2. Poincaré compactification In this section we present the Poincaré compactification for describing the phase portraits of the cubic polynomial differential systems (2). See more details in the Chapter 5 of [11]. The sphere S2 is the set of points (s1 , s2 , s3 ) ∈ R3 with s21 + s22 + s23 = 1, here called the Poincaré sphere. Consider a polynomial vector field (3) X = (x˙1 , x˙2 ) = (P (x1 , x2 ), Q(x1 , x2 )) of degree d in R2 identified with the plane x3 = 1 of R3 . We can analyze the Poincaré sphere by projecting each point x ∈ R2 identified with the point (x1 , x2 , 1) ∈ R3 into the Poincaré sphere using a straight line through x and the origin of R3 . Then we obtain a vector field X ′ formed by two copies of X: one on the northern hemisphere {(s1 , s2 , s3 ) ∈ S2 : s3 > 0} and another on the southern hemisphere {(s1 , s2 , s3 ) ∈ S2 : s3 < 0}. Note that the equator S1 = {(s1 , s2 , s3 ) ∈ S2 : s3 = 0} corresponds to the infinity of R2 . This vector field X ′ on S2 \ S1 can be extended to a vector field p(X), which is called the Poincaré compactification of the vector field X on the whole sphere S2 multiplying the vector field X ′ by xd3 . What we need for working with the vector field p(X) on the Poincaré sphere are the local charts (4) Ui = {s ∈ S2 : si > 0}, Vi = {s ∈ S2 : si < 0}, where s = (s1 , s2 , s3 ), with the corresponding local diffeomorphisms (5) ϕi (s) : Ui → R2 , ψi (s) : Vi → R2 , such the ϕi (s) = −ψi (s) = (sm /si , sn /si ) for m < n and m, n 6= i, for i = 1, 2, 3. In the local chart U1 the expression for the corresponding vector field on S2 is        1 u 1 u 1 u (6) u̇ = v d − uP , +Q , , v̇ = −v d+1 P , , v v v v v v the expression in the local chart U2 is        d u 1 u 1 d+1 u 1 (7) u̇ = v − uP , − uQ , , v̇ = −v Q , , v v v v v v and the expression in the local chart U3 is (8) u̇ = P (u, v), v̇ = Q(u, v). The expressions for the charts Ui multiplied by (−1)d−1 , provide the expression for the charts Vi , for i = 1, 2, 3. To study the vector field X it is enough to study its Poincaré compactification p(X) restricted to the northern hemisphere plus S1 . To draw the phase portraits we will consider the orthogonal projection π(s1 , s2 , s3 ) = (s1 , s2 ) of the closed northern hemisphere onto the closed unit disc centered at the origin of coordinates in the plane x3 = 0. This closed disc D2 is the Poincaré disk. Using U3 , finite singular points of X, which are the singular points of its com- pactification in S2 \ S1 . Infinite singular points of X are the singular points of the corresponding vector field in the Poincaré disk lying on S1 . If s ∈ S1 is an infinite PHASE PORTRAITS OF THE GRAY-SCOTT MODEL 9 v u (a) Systems (11) (b) Systems (10) (c) Systems (9) Figure 7. Blow-up of the origin of U1 when a > 0. singular point, then −s ∈ S1 is another infinite singular point, and the local be- havior of one is that of the other multiplied by (−1)d−1 . Note that for studying the infinite singular points it suffices to look the ones at U1 |v=0 and at the origin of U2 . We note that the coordinates (u, v) means different things in every local charts, but in the local charts Ui and Vi for i = 1, 2 the infinite points have always the coordinate v = 0. We note that the local phase portraits of a node and a focus are topological equivalent. 3. Phase portraits of systems (2) We remark that the straight line y = 0 of systems (2) is invariant by the flow of these systems. That is, it is formed by orbits of systems (2). This simplifies the study of the phase portraits of systems (2). 3.1. The infinite singular points. For the study of the infinite singular points of systems (2) we use the Poincaré compactification, and we first restrict our attention to the local chart U1 . In this chart systems (2) become u̇ =u(au + au2 + v 2 − bv 2 − v 3 ), (9) v̇ =v(au2 + v 2 − v 3 ). On the infinity, i.e. on v = 0, there are two singular points on U1 , namely A′ = (0, 0) and B ′ = (−1, 0). The linear part of systems (9) is   au(2 + 3u) 0 . 0 au2 Hence the singular point B ′ is an attracting node or a repelling node when a < 0 or a > 0, respectively. At the origin A′ of U1 the linear part is identically zero. We need to do blow-up’s in order to determine its the local phase portrait. Doing the blow-up (u, v) 7→ (u, w) with w = v/u we obtain the systems u̇ =u2 (a + au + uw2 − buw2 − u2 w3 ), (10) ẇ =uw(−a + buw2 ). 10 TING CHEN, JAUME LLIBRE AND SHIMIN LI  (a) Systems (11) (b) Systems (9) Figure 8. Blow-up of the origin of U1 when a < 0. Doing a rescaling of the time we eliminate the common factor u between u̇ and ẇ, and we get the systems u̇ =u(a + au + uw2 − buw2 − u2 w3 ), (11) ẇ =w(−a + buw2 ). When u = 0 the unique singular point of systems (11) is the origin. The eigenvalues of the linear part of systems (11) at the origin are ±a, hence the origin is a saddle. Going back through the changes of variables until systems (9), we find that A′ is a saddle-node point as it is shown in Figures 7 and 8, that is a singular point whose neighborhood consists of one parabolic sector and two hyperbolic sectors. Now we check if the origin of the local chart U2 is a singular point. In U2 we use (7) to get u̇ = − au − au2 + (b − 1)uv 2 + v 3 = −au + B(u, v), (12) v̇ = − v(au − bv 2 ) = A(u, v). The origin C ′ of systems (12) is a singular point, and it is a semi-hyperbolic singular point. Assume a < 0 then we get that v3 (13) u = f (v) = + O(v 5 ) a is the solution of the equation −au + B(u, v) = 0 in a neighborhood of the origin. Substituting the variable u from (13) into A(u, v), we have g(v) = A(f (v), v) = bv 3 − v 4 + O(v 5 ). From Theorem 2.19 of [11], we obtain that there always exists an invariant analytic curve, called the strong unstable manifold, tangent at 0 to the u-axis, on which the vector field is analytically conjugate to u̇ = −au, it represents repelling behavior if a < 0. Moreover, if b < 0 then C ′ is a saddle, see Figure 9(a). If b > 0 it is a repelling node, see Figure 9(b). If b = 0 it is a saddle-node, see Figure 9(c). PHASE PORTRAITS OF THE GRAY-SCOTT MODEL 11     (a) b < 0 (b) b > 0 (c) b = 0 Figure 9. The origin of systems (12) when a < 0.    (a) b < 0 (b) b > 0 (c) b = 0 Figure 10. The origin of systems (12) when a > 0. Similar to the case a < 0, if a > 0 by changing the sign of the time t 7→ −t, systems (12) become e v), u̇ =au + au2 + (1 − b)uv 2 − v 3 = au + B(u, (14) e v). v̇ =v(au − bv 2 ) = A(u, e v) = Then we choose the solution u = f (v) = v 3 /a+O(v 5 ) of the equation au+ B(u, e 0 in a neighborhood of the origin. Substituting the variable u into A(u, v), we obtain e (v), v) = −bv 3 + v 4 + O(v 5 ). g(v) = A(f Due to the change t 7→ −t, we know that the trajectories of systems (12) are traveled in the converse direction to the ones of systems (14). Hence we have that C ′ is an attracting node when b < 0 (see Figure 10(a)), a saddle when b > 0 (see Figure 10(b)), and a saddle-node when b = 0 (see Figure 10(c)). For systems (2) we have found six equilibria in the infinite region, two (A′ and B ) are in U1 , one (C ′ ) is in U2 and their diametrically opposite (A′′ , B ′′ and C ′′ ) ′ are in V1 and V2 . Note that for systems (2) the degree is 3 so the flow in the charts V1 and V2 have the same sense than in the charts U1 and in U2 , respectively. 3.2. The finite singular points. Having determined the local phase portraits at the infinite singular points of systems (2), we now compute the phase portraits at their finite singular points. We separate the study of the finite singular points 12 TING CHEN, JAUME LLIBRE AND SHIMIN LI Figure 11. The number of separatrices and of canonical regions of systems (2) when a > 0 and b = 0. in three cases: (i) either a(a − 4b2 ) < 0 or b = 0, (ii) a = 4b2 6= 0, and (iii) a(a − 4b2 ) > 0 and b 6= 0. Case (i). If either a(a − 4b2 ) < 0 or b = 0, that is either 0 < a < 4b2 , or b = 0. Then systems (2) have the unique finite singular point (1, 0). Subcase (i.a). Assuming b = 0 the linear part of systems (2) is   −1 − ay 2 −2axy (15) M= . ay 2 2axy Thus (1, 0) is a semi-hyperbolic singular point of systems (2). We move it to the origin doing the translation x 7→ x + 1, and in order to apply Theorem 2.19 of [11] we reverse the sign of the time t 7→ −t, and we get the systems ẋ =x + a(x + 1)y 2 = x + D(x, y), (16) ẏ = − a(x + 1)y 2 = C(x, y). Then x = f (y) = −ay 2 + O(y 3 ) is the solution of the equation x + D(x, y) = 0 in a neighborhood of the origin of systems (16). Therefore we have g(y) = C(f (y), y) = ay 2 + O(y 3 ). From Theorem 2.19 of [11] we obtain that the singular point (1, 0) of systems (2) (or the origin of systems (16)) is a saddle-node. In this case taking into account the local phase portraits at the infinite and finite singular points together with the fact that the straight line y = 0 is invariant systems (2) have 17 separatrices and 4 canonical regions in their phase portraits when a > 0, see Figure 11. Then we obtain the phase portraits 1.4 of Figure 1 and 1.25 of Figure 3 for a > 0 and a < 0, respectively. Subcase (i.b). If 0 < a < 4b2 since the linear part of systems (2) is given in (15), we get that (1, 0) is a saddle when b < 0. Therefore the phase portrait of systems (2) is topologically equivalent to the phase portrait 1.1 of Figure 1. When b > 0 the singular point (1, 0) is an attracting node. Therefore the only possible phase portrait in this case is the 1.23 of Figure 2. PHASE PORTRAITS OF THE GRAY-SCOTT MODEL 13 Case (ii). If a = 4b2 6= 0 systems (2) have two finite singular points (1, 0) and (1/2, 1/(2b)). As in the subcase (i.b) we have that (1, 0) is an attracting node or a saddle when b > 0 or b < 0, respectively. Now we study the local phase portrait of the singular point (1/2, 1/(2b)). The eigenvalues of (15) at (1/2, 1/(2b)) are 0 and −2 + b. Subcase (ii.a) If b = 2 the singular point (1/2, 1/4) is a nilpotent singular point. We translate this singular point to the origin doing the change of variables x 7→ x+ 1/2, y 7→ y + 1/4, and we have the systems ẋ = − 2(x + 2y + 4xy + 4y 2 + 8xy 2 ) = H(x, y), (17) ẏ =(1 + 4y)(x + 2y + 4xy) = x + G(x, y). Therefore x = f (y) = −2y + O(y 2 ) is the solution of the equation x + G(x, y) = 0 in a neighborhood of the origin of systems (17). Then we obtain H(y) = H(f (y), y) = −8y 2 + O(y 3 ) and   ∂G ∂H + (f (y), y) = −8y + O(y 2 ). ∂y ∂x From Theorem 3.5 of [11] we obtain that the singular point (1/2, 1/4) of systems (2) (or the origin of systems (17)) is a cusp. Hence in this subcase we have the phase portrait 1.21 of Figure 2. Subcase (ii.b) Assuming b 6= 2 the point (1/2, 1/(2b)) is a semi-hyperbolic singular point. We move the point (1/2, 1/(2b)) to the origin doing the translation x 7→ x + 1/2, y 7→ y + 1/(2b), and in order to apply Theorem 2.19 of [11] we reverse the sign of the time t 7→ −t, and we obtain the systems ẋ =2(x + by + 2bxy + b2 y 2 + 2b2 xy 2 ) = 2x + F (x, y), (18) ẏ = − (1 + 2by)(x + by + 2bxy) = E(x, y), Then x = f (y) = −by + O(y 2 ) is the solution of the equation 2x + F (x, y) = 0 in a neighborhood of the origin of systems (18). Hence we have g(y) = E(f (y), y) = b2 y 2 + O(y 3 ). From Theorem 2.19 of [11] it follows that the singular point (1/2, 1/(2b)) of systems (2) (or the origin of systems (18)) is a saddle-node. In summary we obtain the phase portrait 1.2 of Figure 1 when b < 0, the phase portrait 1.18 of Figure 2 when 0 < b < b1 , the phase portrait 1.20 of Figure 2 when b1 < b < 2, and the phase portrait 1.22 of Figure 2 when b > 2. By the continuity of the phase portraits moving the parameter b we obtain the phase portrait 1.19 when (a, b) = (4b21 , b1 ), in this phase portrait two separatrices have connected. It is difficult to get the value of b1 , but b1 belongs to the interval (1.4202,1.4204). 14 TING CHEN, JAUME LLIBRE AND SHIMIN LI Case: (iii). If a(a − 4b2 ) > 0, then either a > 4b2 or a < 0, additionally we can assume b 6= 0. The finite singular points of systems (2) different from (1, 0) are p p ! 1 a(a − 4b2 ) a − a(a − 4b2 ) p1 = (x1 , y1 ) = + , , 2 2a 2ab p p ! 1 a(a − 4b2 ) a + a(a − 4b2 ) p2 = (x2 , y2 ) = − , . 2 2a 2ab Similarly to the above case we get that (1, 0) is an attracting node or a saddle when b > 0 or b < 0, respectively. Next we consider two subcases and some subsubcases in order to study the local phase portraits of the singular points p1 and p2 . Subcase (iii.a). Assuming a > 4b2 we have that the determinant and the trace of (15) at p1 are p p a − 4b2 − a(a − 4b2 ) −a + 2b3 + a(a − 4b2 ) (19) Det[M ] = , Tr[M ] = . p1 2b p1 2b2 Similarly we obtain p p a − 4b2 + a(a − 4b2 ) −a + 2b3 − a(a − 4b2 ) (20) Det[M ] = , Tr[M ] = . p2 2b p2 2b2 Now we shall study the local phase portraits at the singular points p1 and p2 . For this we need to know the sign of Tr[M ]|p1 or Tr[M ]|p2 . Hence we give the Remarks 2 and 3. Remark 2. From Tr[M ]|p1 < 0 we have either b < 0 and b4 /(b − 1) < a ≤ 0, or b < 0 and a ≥ 4b2 , or 0 < b ≤ 1 and a ≥ 4b2 , or 1 < b < 2 and 4b2 ≤ a < b4 /(b−1). From Tr[M ]|p1 = 0 we have a = b4 /(b − 1) and a ≥ 2b3 . From Tr[M ]|p1 > 0 we have either b ≤ 0 and a < b4 /(b − 1), or b > 0 and a ≤ 0, or 1 < b ≤ 2 and a > b4 /(b − 1), or b > 2 and a ≥ 4b2 . Remark 3. From Tr[M ]|p2 < 0 we have either b < 0 and a ≤ 0, or b < 0 and a ≥ 4b2 , or 0 < b < 1 and a < b4 /(b − 1), or 0 < b < 2 and a ≥ 4b2 , or b ≥ 2 and a > b4 /(b − 1). From Tr[M ]|p2 = 0 we have a = b4 /(b − 1) and a ≤ 2b3 . From Tr[M ]|p2 > 0 we have either 0 < b < 1 and b4 /(b − 1) < a ≤ 0, or 1 ≤ b and a ≤ 0, or b > 2 and 4b2 ≤ a < b4 /(b − 1). Subsubcase (iii.a.1). If b > 0 we have 1 > x1 > x2 > 0 and y2 > y1 > 0. On the other hand, we get Det[M ]|p1 < 0, because (a − 4b2 )2 − a(a − 4b2 ) = 4b2 (4b2 − a) < 0. Hence the singular point p1 is a saddle. From (20) we obtain that Det[M ]|p2 is positive. Then we need to compute the discriminant   N1 (21) ∆1 = (Tr[M ])2 − 4Det[M ] = 4 p2 2b where p (22) N1 = a2 − 2ab2 − 6ab3 + 16b5 + 2b6 + (a − 6b3 ) a(a − 4b2 ). PHASE PORTRAITS OF THE GRAY-SCOTT MODEL 15 a a=a1 a=f2(b) a=a2 a=f1(b) 0 2 4 b Figure 12. The roots a1,2 of N1 when a > 0. f1 (b) = 4b2 , f2 (b) = b4 /(b − 1). When 0 < b < 1 it is easy to check that N1 > 0, because it has no real roots. In addition we have Tr[M ]|p2 < 0, hence the singular point p2 is an attracting node, and we get the phase portrait 1.5 of Figure 1. If b ≥ 1 the equation N1 = 0 has two roots p b3 (8 + 7b + 3b2 ) ± 2 2(−1 + b)b7 (2 + b)2 (23) a1,2 = . 1 + 2b + b2 The graphics of a1,2 , which help us to analyze the singular points p1,2 , see Figure 12. Since b6 (8 + 7b + 3b2 )2 − 8(−1 + b)b7 (2 + b)2 = b6 (1 + b)2 (8 + b)2 > 0, it is easy to get that 0 < 4b2 ≤ a2 ≤ a1 . Then we have N1 ≥ 0 when 4b2 < a ≤ a2 or a ≥ a1 . We have Tr[M ]|p2 < 0 when a ≥ a1 , or 1 ≤ b < 2 and 4b2 < a ≤ a2 . Hence the singular point p2 is an attracting node. In this subsubcase the phase portraits are topologically equivalent to 1.5-1.7 of Figure 1. From the phase portraits 1.5 and 1.7 it follows by the continuity of the phase portraits with respect to the parameters the existence of the phase portrait 1.6. Hence we obtain the bifurcation curve a = f5 (b) on which we have the phase portrait 1.6. Note that the intersection of the curves a = f5 (b) and a = f1 (b) is the point (4b21 , b1 ). We have Tr[M ]|p2 > 0 when b > 2 and 4b2 < a ≤ a2 . Hence the singular point p2 is a repelling node. In this case the phase portraits of systems (2) are topologically equivalent to 1.17 of Figure 2. On the other hand, we have N1 < 0 when a2 < a < a1 . Thus the singular point p2 is a stable focus when Tr[M ]|p2 < 0, that is 1 < b < 2, a2 < a < a1 and a 6= b4 /(b − 1), or b > 2 and b4 /(b − 1) < a < a1 . The phase portraits are topologically equivalent to 1.5-1.7 of Figure 1. The singular point p2 is an unstable focus when Tr[M ]|p2 > 0, that is 2 < b and a2 < a < b4 /(b − 1). The phase portrait is topologically equivalent to 1.17 of Figure 2. When a = b4 /(b − 1) and b 6= 2 we have either Tr[M ]|p1 = 0, or Tr[M ]|p2 = 0. Indeed, if from Remarks 2 and 3 we have Tr[M ]|p1 = Tr[M ]|p2 = 0 if and only if 16 TING CHEN, JAUME LLIBRE AND SHIMIN LI 2 (a, b) = (0, 0) or (a, b) = (16, 2), which √ is in contradiction with a > 4b . Simplifying 4 a2 < b /(b − 1) < a1 , we have −1 + 5 < b < 2 or 2 < b. Assume that a = b4 /(b − 1) and 2 < b, we have the singular points p′1 = ((b − 1)/b, 1/b2) and p′2 = (1/b, (b − 1)/b2 ) where 1 > (b − 1)/b > 1/b > 0 and (b − 1)/b2 > 1/b2 . From Tr[M ]|p′2 = 0 and ∆1 = −4(b − 2)b < 0, we get that p′2 must be a center or a focus. We move p′2 to the origin doing the translation x 7→ x + 1/b, y 7→ y + (b − 1)/b2 , and we get the systems b3 2 b4 ẋ = − bx − 2by − 2b2 xy − y − xy 2 , b−1 b−1 (24) b3 2 b4 ẏ =(b − 1)x + by + 2b2 xy + y + xy 2 . b−1 b−1 Doing the change of variable (x, y, t) 7→ (ξ, η, τ ) where p p b − (b − 2)bi b + (b − 2)bi ξ= x+ y, p 4b p 4b (b − 2)b − 3bi (b − 2)b + 3bi η= x− y, p 4b 4b √ τ = (b − 2)bt, i = −1, systems (24) become p p dξ (3b − 4) (b − 2)b 2 (b − 2)b b (b − 2)b 2 =−η− ξ − ξη + η dτ 2(b − 2)(b − 1) b−1 2(b − 1) p p b3 (b − 2)b 3 2b3 2 b2 (b − 2)b 2 + ξ + ξ η+ ξη , (b − 2)(b − 1) b−1 b−1 (25) p dη b2 (3b − 4) b (b − 2)b b2 =ξ − ξ2 − ξη + η2 dτ 2(b − 2)(b − 1) b−1 2(b − 1) p b4 2b3 (b − 2)b 2 b3 ξ3 + ξ η+ ξη 2 . (b − 2)(b − 1) (b − 2)(b − 1) b−1 We present some basic formulas for computing the Liapunov constants of the general differential systems n X ẋ =δx − βy + Pk (x, y), k=2 (26) n X ẏ =βx + δy + Qk (x, y), k=2 where Pk (x, y), Qk (x, y) are homogeneous polynomials of degree k in x and y. Introducing the polar coordinate transformation, x = ρ cos θ and y = ρ sin θ, system (26) can be written as the differential equation P dρ δρ + nk=2 Υk (θ)ρk (27) = Pn , dθ β + k=2 Θk (θ)ρk−1 where Υk (θ) = cos θPk (cos θ, sin θ) + sin θQk (cos θ, sin θ), Θk (θ) = cos θQk (cos θ, sin θ) − sin θPk (cos θ, sin θ), PHASE PORTRAITS OF THE GRAY-SCOTT MODEL 17 in which Pk and Qk are polynomials in sin θ and cos θ. The general solution of (27) in a neighborhood of the origin can be expressed as X (28) Π(ρ, θ) = vk (θ)ρk , |ρ| ≪ 1, k≥1 and we define the displacement function X (29) d(ρ) = Π(ρ, 2π) − ρ = (v1 − 1)ρ + v2k+1 ρ2k+1 , k≥1 where vk (v2k ≡ 0) is called the kth-order Liapunov constant at the origin of systems (26). The number of fixed points of d(ρ) (or zeros of d(ρ)) corresponds to the number of limit cycles of systems (29). If the displacement function (29) satisfies v1 = 1, v3 = · · · = v2k−1 = 0 and v2k+1 6= 0, any perturbation of (26) have at most k limit cycles bifurcating at the origin. Then we compute the first and the third Liapunov constants of systems (25) at the origin and we obtain v1 = 1 and p (4 − b)b2 (b − 2)bπ (30) v3 = . 2(b − 2)2 If v3 = 0, namely b = 4, the fifth Liapunov constant is p √ b4 (b − 2)b(192 − 1228b + 1417b2 − 567b3 + 72b4 )π 256 2π v5 = = > 0. 12(b − 2)4 (b − 1) 3 So the origin of systems (25) is an unstable weak focus of second order. The phase portrait is topologically equivalent to 1.16 of Figure 2. From [19, Theorem 2] and since the derivative ∂v3 √ = 4 2π 6= 0, ∂b b=4 then there exist exactly two small amplitude limit cycles bifurcating from the singu- lar point p′2 of systems (2) (or from the origin of systems (24)) by small perturbation of b = 4, see Figure 13. We obtain the phase portrait 1.11 of Figure 1. In order to obtain two small amplitude limit cycles, it requires to find a set of explicit per- turbation value b, which is not an easy task. We give the convenient function f3 (b) to describe the region, whose existence is given by continuity for proving from the phase portrait 1.11 to 1.17. When a = f3 (b), systems (2) have only one semi-stable (externally unstable and internally stable) limit cycle. Hence, the phase portrait is topologically equivalent to 1.12 of Figure 1. If 2 < b < 4 the third Liapunov constant v3 > 0 from (30), we have that the singular point p′2 is an unstable weak focus. Furthermore we obtain a phase portrait which is topologically equivalent to the phase portrait 1.17 of Figure 2. There exists exactly one small amplitude limit cycle bifurcating from the singular point p′2 of systems (2) by small perturbation of b. In fact, in the neighborhood of the curve a = b4 /(b − 1) the singularity p′2 is an unstable focus when a < b4 /(b − 1) and a stable focus when b4 /(b − 1) < a. The stability of the focus reverses, and a Hopf bifurcation occurs, there must bifurcate one limit cycle from this singular point. Hence we obtain the phase portrait 1.10 of Figure 1. There exists a convenient function f4 (b), its existence is given by continuity between the phase portraits 1.7 to 1.10, i.e. at the curve a = f4 (b) systems (2) have a homoclinic loop, and 18 TING CHEN, JAUME LLIBRE AND SHIMIN LI Figure 13. Two small amplitude limit cycles bifurcated from the singular point p′2 of systems (2). topologically their phase portrait is the 1.8 of Figure 1. We get that the coordinate of the point corresponding to the intersection point of the curves a = f2 (b) and a = f4 (b) is (b42 /(b2 − 1), b2 ), and b2 belongs to (4.4, 4.403). Similarly, if b > 4 the singular point p′2 is a stable weak focus. The phase portraits are topologically equivalent to 1.5-1.7 of Figure 1. And there exists exactly one small amplitude limit cycle bifurcating from the singular point p′2 of systems (2). Hence we obtain the phase portraits 1.10 of Figure 1 and 1.14-1.16 of Figure 2. From the phase portraits 1.14 and 1.16, we obtain the bifurcation curve a = f6 (b) which corresponds to the phase portrait 1.15. In addition, it is difficult to obtain the coordinates of the intersection point (a, b3 ) between the curves a = f5 (b) and a = f6 (b). But we can get that b3 ∈ (4.413, 4.414). √ Now we assume a = b4 /(b − 1) and 5 − 1 < b < 2, and we have that the singular points p′1 = (1/b, (b−1)/b2) and p′2 = ((b−1)/b, 1/b2) where 1 > 1/b > (b−1)/b > 0 and 1/b2 > (b − 1)/b2 . The trace of M at p′2 is Tr[M ]|p′2 = (b − 2)b/(b − 1) < 0 and the discriminant is b(8 − 8b + b3) < 0. Hence the singular point p′2 is a stable focus. Then the phase portraits are topologically equivalent to 1.5-1.7 of Figure 1. By the continuity of the phase portraits with respect to parameters from the phase portrait 1.11 to 1.14, we obtain the bifurcation curve a = f4 (b) with b2 < b < b4 which describes the phase portrait 1.9, i.e. systems (2) have a homoclinic loop with a limit cycle inside the loop. The intersection point of the curves a = f3 (b) and a = f4 (b) is (f4 (b4 ), b4 ), and b4 belongs to the interval (6.24, 6.25). On the PHASE PORTRAITS OF THE GRAY-SCOTT MODEL 19 other hand, from the phase portrait 1.14 to 1.17 we obtain the phase portrait 1.13, i.e. systems (2) have a homoclinic loop in the bifurcation curve a = f4 (b) with b ≥ b4 . Subsubcase (iii.a.2). If b < 0 we have 1 > x1 > x2 > 0 and y2 < y1 < 0. From (20) we have Det[M ]|p2 < 0. Therefore the singular point p2 is a saddle. Since Det[M ]|p1 > 0 from (19) we obtain that the discriminant at p1 is   2 N2 ∆2 = (Tr[M ]) − 4Det[M ] = 4, p1 2b where p (31) N2 = a2 − 2ab2 − 6ab3 + 16b5 + 2b6 + (6b3 − a) a(a − 4b2 ). When b = −1 we obtain p p N2 = a2 + 4a − 14 − a3 (a − 4) − 6 a(a − 4) > 0. If b 6= −1 solving N2 = 0 for a we obtain the same solutions a1,2 given in (23), see Figure 14. We have 49 lim a1 = − and lim a2 = −∞. b→−1 8 b→−1 It is easy to check that a2 ≤ a1 ≤ 0 when b < 0. Hence we have N2 ≥ 0 and a -8 -6 -4 -2 0 b a=a1 a=a2 a=f2(b) Figure 14. The roots a1,2 of N2 = 0 when a < 0. f2 (b) = b4 /(b − 1). Tr[M ]|p1 < 0, because a > 4b2 . Furthermore we get that the singular point p1 is an attracting node. The only possible phase portrait in this case is 1.3 of Figure 1. Subcase (iii.b). Assume a < 0. Subsubcase (iii.b.1). If b > 0 we have x1 < 0, x2 > 1, y1 > 1/b and y2 < 0. Consequently Det[M ]|p1 < 0 and Det[M ]|p2 < 0, because (32) (a − 4b2 )2 − a(a − 4b2 ) = 4b2 (4b2 − a) > 0. Hence the singular points p1,2 are saddles, and we obtain the phase portrait 1.24 of Figure 2. Subsubcase (iii.b.2). If b < 0 we have x1 < 0, x2 > 1, y1 < 1/b and y2 > 0. From (32) it is easy to check that Det[M ]|p1 > 0 and Det[M ]|p2 > 0. And we have Tr[M ]|p2 < 0 from Remark 3. Hence the singular point p2 must be stable. From 20 TING CHEN, JAUME LLIBRE AND SHIMIN LI (22) and (31) we need to determine the signs of Ni for i = 1, 2 for characterizing the local phase portraits at the singular points p1 and p2 . Assume b = −1. We obtain that N1 = 0 has no solution, and N2 has one solution a = −49/8. 1) If a = −1/2 we have N1 < 0 and N2 < 0. So the singular point p′2 = (2, 1) is a stable focus. On the other hand, we have Tr[M ]|p′1 = 0. Then the singular point p′1 = (−1, −2) is a center or a focus. From (30) we have that the third Liapunov constant of p′1 is √ 5 3π v3 = > 0. 18 Thus the singular point p′1 is an unstable weak focus. Then the phase portrait is 1.26 of Figure 3. Furthermore the Liapunov constant is independent of b, hence there exists exactly one small amplitude limit cycle bifurcating from p′1 of systems (2) under small perturbations. The phase portrait is topologically equivalent to 1.27 of Figure 3. 2) If 0 > a > −49/8 and a 6= −1/2, we get N1 < 0 and N2 < 0. Hence the singular point p2 is a stable focus. By Remark 2 the trace of M at p′1 is positive or negative when −49/8 < a < −1/2 or −1/2 < a < 0, respectively. Then p1 is an unstable focus or a stable focus. Hence we obtain the phase portraits 1.26 of Figure 3 or 1.29 of Figure 3. By the continuity of the phase portraits, we give the point (f7 (−1), −1) to describe one case, whose existence is given by continuity going from the phase portraits 1.27 to 1.29, systems (2) have a heteroclinic orbit. Then the phase portrait in this case turn out to be topologically equivalent to 1.28 of Figure 3. 3) If a ≤ −49/8 we obtain N1 < 0 and N2 > 0. We have Tr[M ]|p1 > 0 from Remark 2. So the singular points p1 and p2 are a repelling node and a stable focus, respectively. The node has the same topology with the focus. Therefore the phase portrait is topologically equivalent to 1.26 of Figure 3. Assume b 6= −1. For determining the signs of Ni for i = 1, 2, we solve Ni = 0 for a, and we obtain the same solutions a2 ≤ a1 , see (23). On the other hand, from N1 = 0 we have p (33) a2 − 2ab2 − 6ab3 + 16b5 + 2b6 = (6b3 − a) a(a − 4b2 ). Thus we obtain (34) ∆′ = (a2 − 2ab2 − 6ab3 + 16b5 + 2b6 )(6b3 − a) ≥ 0, when N1 = 0. Taking a = a1 into the above inequality, we get −8 ≤ b ≤ −2. If b = −8 we have a = a1 = 0, which is not possible because a < 0. If a = a2 then, from inequality (34) we get −2 ≤ b < −1 or −1 < b < 0. Similarly, we have ∆′ ≤ 0 when N2 = 0. Taking a = a1 into this inequality ∆ ≤ 0, we have b < −8 or −2 ≤ b < −1. If a = a2 we obtain b ≤ −2 from ∆′ ≤ 0. ′ Hence we have Table 1. Now we analyze the local phase portraits of the singular points p1 and p2 by the sign of N2 and N1 , respectively. Assume −1 < b < 0. PHASE PORTRAITS OF THE GRAY-SCOTT MODEL 21 Table 1. The solutions of N1 = 0 and N2 = 0 when b < 0. The value of b The solutions of N1 = 0 The solutions of N2 = 0 b ∈ (−1, 0) a = a2 a = a1 b = −1 no solution a = − 49 8 b ∈ (−2, −1) a = a2 a = a1 b = −2 a = −48 a = −48 b ∈ (−8, −2) a = a1 a = a2 b = −8 a=0 a = 0 or a = − 147456 49 b ∈ (−∞, −8) no solution a = a1,2 We get that N1 = 0 has one solution a = a2 , and N2 = 0 has one solution a = a1 from Table 1. 1) If a1 < a < 0 and a = b4 /(b − 1), we get N1 < 0 and N2 < 0. Hence we have that p′2 = ((b − 1)/b, 1/b2) is a stable focus. The trace of M at p′1 = (1/b, (b − 1)/b2 ) is Tr[M ]|p′1 = 0. From (30) we have the third Liapunov constant 0 < v3 at p′1 . Hence the singular point p′1 is an unstable weak focus. Therefore we obtain that the phase portrait is topologically equivalent to 1.26 of Figure 3. Furthermore the Liapunov constant is independent of b, hence there exists exactly one small amplitude limit cycle bifurcating from p′1 of systems (2) under small perturbations. The phase portrait is topologically equivalent to 1.27 of Figure 3. 2) If a1 < a < 0 and a 6= b4 /(b − 1), we have N1 < 0 and N2 < 0. Hence the singular point p2 is a stable focus. From Remark 2 we get that Tr[M ]|p1 > 0 when a1 < a < b4 /(b − 1), and Tr[M ]|p1 < 0 when b4 /(b − 1) < a < 0. Hence the singular point p1 is an unstable focus or a stable focus. The phase portrait is topologically equivalent to 1.26 of Figure 3 or 1.29 of Figure 3. By continuity from the phase portraits 1.27 to 1.28, we obtain the curve a = f7 (b) where systems (2) have a heteroclinic loop. Then the phase portrait in this case turn out to be topologically equivalent to 1.28 of Figure 3. 3) If a2 < a ≤ a1 we have N1 < 0 and N2 ≥ 0. So p2 is a stable focus. From Remark 2 we have Tr[M ]|p1 > 0. Hence the singular points p1 is a repelling node. The phase portrait is topologically equivalent to 1.26 of Figure 3. 4) If a ≤ a2 we have N1 ≥ 0 and N2 > 0. So we obtain that p2 is an attracting node. We have Tr[M ]|p1 > 0 from Remark 2. Hence the singular points p1 is a repelling node. Consequently the phase portrait is topologically equivalent to 1.26 of Figure 3. Assume −8 < b < −1. From Table 1, we get that N1 = 0 and N2 = 0 has one solution, respectively. 1) If a1 < a < 0 and a = b4 /(b−1), we have N1 < 0 and N2 < 0. Hence the singular point p′2 = ((b−1)/b, 1/b2) is a stable focus. The trace of M at p′1 = (1/b, (b−1)/b2) is Tr[M ]|p′1 = 0. Furthermore from (30) we get that the singular points p′1 is an unstable weak focus. The phase portrait is topologically equivalent to 1.30 of Figure 3. Furthermore the Liapunov constant is independent of b, hence systems (2) have 22 TING CHEN, JAUME LLIBRE AND SHIMIN LI exactly one small amplitude limit cycle bifurcating from p′1 under small pertubation. In this case the only possible phase portrait is topologically equivalent to 1.31 of Figure 3. 2) If a1 < a < 0 and a 6= b4 /(b − 1), we have N1 < 0 and N2 < 0. Then the singular point p2 is a stable focus. And from Remark 2 we have Tr[M ]|p1 > 0 or Tr[M ]|p1 < 0 when a1 < a < b4 /(b − 1) or b4 /(b − 1) < a < 0, respectively. So the singular point p1 is an unstable focus or a stable focus. Therefore the phase portraits are equivalent to 1.30 or 1.33 of Figure 3. When a = f7 (b) systems (2) have a heteroclinic loop. In this case the phase portrait is topologically equivalent to 1.32 of Figure 3. 3) If a2 < a ≤ a1 and a = b4 /(b−1), we obtain N1 ≥ 0 and N2 < 0. From Remark 2 we get that Tr[M ]|p′1 = 0. It is easy check that p′1 is an unstable weak focus and p′2 is an attracting node. Consequently we obtain the phase portrait 1.30 of Figure 3. Furthermore the Liapunov constant is independent of b, hence there exists exactly one small amplitude limit cycle bifurcating from p′1 of systems (2). We obtain that the phase portrait is topologically equivalent to 1.31 of Figure 3. 4) If −2 ≤ b < −1 and a2 < a ≤ a1 , we have N1 < 0 and N2 ≥ 0. So p2 is a stable focus. From Remark 2 we have Tr[M ]|p1 > 0. Hence the singular point p1 is a repelling node. If b < −2, a2 < a < b4 /(b − 1) and a < a1 , we have N1 > 0 and N2 < 0. We have Tr[M ]|p1 > 0 from Remark 2. Therefore p1 is an unstable focus and p2 is an attracting node. Then we easily get that the phase portraits are topologically equivalent to 1.30 of Figure 3. 5) If b4 /(b − 1) < a < a1 we get that N1 > 0 and N2 < 0. It is easy to check that Tr[M ]|p1 < 0. So p1 is a stable focus and p2 is an attracting node. Therefore the phase portrait is topologically equivalent to 1.32 or 1.33 shown in Figure 3. 6) If a ≤ a2 we get N1 > 0 and N2 ≥ 0. From Remark 2, we obtain Tr[M ]|p1 > 0. Hence p1 and p2 are a repelling node and an attracting node. The phase portrait is topologically equivalent to 1.30 of Figure 3. √ Remark 4. Assume b < 0. If a1 ≤ b4 /(b − 1) then √ −1 − 5 ≤ b < −1 or −1 < b < 0. If a2 < b4 /(b − 1) < a1 then b < −1 − 5. On the other hand, there is no parameter b satisfying b4 /(b − 1) ≤ a2 . Assume that b ≤ −8. If b < −8 we obtain that N1 = 0 has no solution, and N2 = 0 has two solutions a = a1,2 . If b = −8, N1 = 0 has no solution, and N2 = 0 has one solution. 1) If a1 ≤ a < 0 we have N1 > 0 and N2 ≥ 0. On the other hand, from Remark 4 we have b4 /(b − 1) < a < 0. Then we have Tr[M ]|p1 < 0 from Remark 2. Hence p1 and p2 are two attracting nodes. The phase portrait is topologically equivalent to 1.33 of Figure 3. 2) If a2 < a < a1 and a = b4 /(b−1) (let a1 = 0 and a2 = −147456/49 when b = −8), we get that N1 ≥ 0 and N2 < 0. Hence p′2 = ((b − 1)/b, 1/b2) is an attracting node. The trace of M at p′1 = (1/b, (b − 1)/b2) is Tr[M ]|p′1 = 0. Furthermore from (30) we obtain that p′1 is an unstable weak focus. It is easy to see that the phase portrait is topologically equivalent to the phase portrait 1.30 of Figure 3. Furthermore PHASE PORTRAITS OF THE GRAY-SCOTT MODEL 23 the Liapunov constant is independent of b, hence there exists exactly one small amplitude limit cycle bifurcating from p′1 of systems (2). We obtain that the phase portrait is topologically equivalent to 1.31 of Figure 3. 3) If a2 < a < b4 /(b − 1) we obtain N1 > 0 and N2 < 0. It is easy to get that Tr[M ]|p1 > 0. Then p1 is an unstable focus and p2 is an attracting node. The phase portrait is topologically equivalent to 1.30 of Figure 3. 4) If b4 /(b − 1) < a < a1 we get that N1 > 0 and N2 < 0. From Remark 2 we have Tr[M ]|p1 < 0. Thus p1 is a stable focus and p2 is an attracting node. The phase portrait is topologically equivalent to the phase portrait 1.33 of Figure 3. When a = f7 (b) systems (2) have a heteroclinic loop. The phase portrait is topologically equivalent to 1.32 of Figure 3. 5) If a ≤ a2 we get N1 > 0 and N2 ≥ 0. Then p2 is an attracting node. On the other hand, we have Tr[M ]|p1 > 0 from Remark 2. Hence p1 is a repelling node. Thus the phase portraits in this case turn out to be topologically equivalent to 1.30 of Figure 3. This completes the classification of the phase portraits of systems (2) in the parameter plane (a, b) ∈ R2 \ {(0, b) : b ∈ R}. From systems (2), if a = 0 we obtain linear systems (35) ẋ = 1 − x, ẏ = −by. If b 6= 0 systems (35) have one finite singular point (1, 0). It is an unstable node when b > 0, or a saddle when b < 0. If b = 0 systems (35) have the straight line x = 1 filled of singular points. Then we give the bifurcation diagram of the finite singular points, see Figure 15. Here, we denote by S, N , SN and AS the saddle, the node, the saddle-node and the anti-saddle, i.e. a node or focus. a=f2(b) a=f1(b) a  AS=weak focus 2      1 stable N 1 stable N 1 stable N+ 1 SN+ 1 cusp 1 stable N 1 stable N 0 2 AS+1 S 0 2 b 1 SN 2 S+ 1 stable N 1 AS+1 S+ 1 unstable weak focus a=f2(b) Figure 15. The bifurcation diagram of finite singular points of systems (2) where f1 (b) = 4b2 , f2 (b) = b4 /(b − 1). 24 TING CHEN, JAUME LLIBRE AND SHIMIN LI C' C' C' C' B'' S3 B'' B'' B'' S3 S2=S5 A'' A' A'' S10 A' A'' S2=S5 A'' S1=S6 A' S8 S4 S1=S6 A' S8 S4 S7 B' S7 S9 S11 B' S4 B' S9 B' C'' C'' C'' C'' (a) (b) (c) (d) Figure 16. Some separatrices of the phase portraits. (a) phase portrait 1.1, (b) phase portrait 1.2, (c) phase portrait 1.3, (d) phase portrait 1.4. 4. Bifurcation diagrams Now we explain the bifurcation diagrams of the phase portraits of systems (2) in the parameter plane (a, b) ∈ R2 \ {(0, b) : b ∈ R}. We note that the bifurcation lines for the phase portraits are b = 0 and a = fi (b) (i = 1, · · · , 7). The bifurcation diagrams are described in Figures 4, 5 and 6. From the following analysis it would be at most 20 different separatrices, which are not at infinity, connecting with the infinite, finite equilibrium points or with the limit cycle and the homoclinic loop, see Table 2. Table 2. The separatrices, which are not at infinity, connecting the infinite, the finite equilibrium points, or with the limit cycle and the homoclinic loop. Equilibria Type of the equilibria Separatrices (1, 0) saddle or saddle-node S1 , S2 , S3 ,S4 p1 saddle, saddle-node, cusp or focus S13 , S14 , S15 , S16 , L3 p2 saddle, saddle-node or focus S7 , S8 , S9 , S10 , H1 , L1 , L2 A′ saddle-node S5 A′′ saddle-node S6 C′ saddle or saddle-node S12 C ′′ saddle or saddle-node S11 When the parameter (a, b) in Figure 4 provides the phase portrait 1.1, see Figure 16(a), systems (2) have six infinite equilibria, A′ and B ′ in U1 , C ′ in U2 and their diametrically opposite, i.e. A′′ , B ′′ and C ′′ , in V1 and V2 . A′ and A′′ are saddle- nodes, B ′ and B ′′ are repelling nodes, C ′ and C ′′ are attracting nodes. And systems (2) have only one finite equilibrium (1, 0), which is a saddle. The saddle has four separatrices Si (i = 1, 2, 3, 4), S1 connects (1, 0) and the infinite equilibrium A′′ , S2 connects (1, 0) and the infinite equilibrium A′ , S3 connects (1, 0) and the infinite equilibrium C ′ , and S4 connects (1, 0) and the infinite equilibrium C ′′ . The saddle- node A′ has the separatrix S5 = S2 , and A′′ has the separatrix S6 = S1 , these separatrices are not at infinity. PHASE PORTRAITS OF THE GRAY-SCOTT MODEL 25 C' C' C' C' S12 S15 B'' S15 B'' S15 S16 B'' S15 S16 B'' S14 L1 S16 S12=S16 S12 S13 S12 A'' S6 A' A'' A' A'' A' A'' S5 S11 B' B' B' C'' C'' C'' (a) (b) (c) (d) Figure 17. Some separatrices of the phase portraits. (a) phase portrait 1.5, (b) phase portrait 1.6, (c) phase portrait 1.7, (d) phase portrait 1.14. On the bifurcation curve a = f1 (b) with b < 0, we get the phase portrait 1.2, see Figure 16(b). This phase portrait has the finite equilibrium (1/2, 1/(2b)), which is a saddle-node. The saddle-node has three separatrices Si (i = 7, 8, 9), S7 connects (1/2, 1/(2b)) and the infinite equilibrium C ′′ , S8 connects (1/2, 1/(2b)) and the infinite equilibrium A′′ , and S9 connects (1/2, 1/(2b)) and the infinite equilibrium B ′ . The infinite equilibria keep their type. When the parameter (a, b) in Figure 4 provides the phase portrait 1.3, see Figure 16(c), the saddle-node p(1/2, 1/(2b)) of the phase pportrait 1.2 splits into the attract- ing node p1 =p(1/2 + a(a − 4b2 )/(2a), p (a − a(a − 4b2 ))/(2ab)) and the saddle 2 2 p2 = (1/2 − a(a − 4b )/(2a), (a + a(a − 4b ))/(2ab)). Then it generates the separatrix S10 , which connects p1 and p2 . On the bifurcation straight line b = 0 with 0 < a we have the phase portrait 1.4, see Figure 16(d). The node p1 of the phase portrait 1.3 moves to (1, 0), p2 moves to the infinite equilibrium C ′′ in the local chart V2 , and the separatrices Si (i = 4, 7, 8, 9) disappear. C ′ and C ′′ become saddle-nodes. The saddle-node C ′ has one separatrix S11 , they are not at infinite. Hence systems (2) have only one finite equilibrium (1, 0). This completes the description of the bifurcation diagram of Figure 4. We start the analysis of the bifurcation diagram described in Figure 5. The phase portrait 1.4 has six infinite equilibria A′ , A′′ , B ′ , B ′′ , C ′ and C ′′ . A′ , A′′ , C ′ and C ′′ are saddle-nodes, and B ′ and B ′′ are repelling nodes. There is only one finite equilibrium (1, 0), which is a saddle-node. The saddle-node (1, 0) has three separatrices Si (i = 1, 2, 3). A′ has one separatrix S5 = S2 , A′′ has one separatrix S6 = S1 , and C ′′ has one separatrix S11 , these separatrices are not at infinity. When (a, b) in Figure 5 provides the phase portrait 1.5, see Figure 17(a), the finite saddle-node of the phase portrait 1.4 splits into two finite equilibria the same (1, 0) (an attracting node) and the saddle p1 . The saddle-node C ′ splits into a saddle at C ′ and an attracting node or focus at p2 , which are connected by the separatrix S12 . The saddle p1 has four separatrices Si (i = 13, 14, 15, 16), S13 connects p1 and (1, 0), S14 connects p1 and the infinite equilibrium A′ , S15 connects p1 and p2 , S16 connects p1 and the infinite equilibrium B ′′ . The saddle-node A′ has one separatrix 26 TING CHEN, JAUME LLIBRE AND SHIMIN LI C' C' C' C' S12 H1 S12 B'' S15 B'' B'' S16 B'' S15 S15 S12= S16 L1 S16 L1 S13 S14 A'' A' A'' A' A'' A' A'' S6 A' S5 B' B' B' S11 B' C'' C'' C'' C'' (a) (b) (c) (d) Figure 18. Some separatrices of the phase portraits. (a) phase portrait 1.15, (b) phase portrait 1.16, (c) phase portrait 1.13, (d) phase portrait 1.17. S5 , A′′ has one separatrix S6 , and the saddle C ′′ has one separatrix S11 , they are not at infinity. As (a, b) reaches the bifurcation curve a = f5 (b) we obtain the phase portrait 1.6, see Figure 17(b). The separatrix S16 of phase portrait 1.5, which connects the infinite equilibrium B ′′ with p1 , now it connects with the separatrix S12 . When (a, b) in Figure 5 provides the phase portrait 1.7, see Figure 17(c), the separatrix S12 of phase portrait 1.6 goes to (1, 0), and the separatrix S16 comes from A′ . When (a, b) provides the phase portrait 1.14, see Figure 17(d), the stability of the focus p2 of the phase portrait 1.7 reverses, a Hopf bifurcation occurs, and a stable limit cycle L1 bifurcates from p2 . Similarly we obtain the phase portraits 1.15 and 1.16 (see Figure 18(a) and 18(b)) from the phase portraits 1.6 and 1.5 , respectively. When (a, b) is in the bifurcation curve a = f4 (b) with b ≥ b4 , we obtain the phase portrait 1.13, see Figure 18(c) (i.e. from the phase portrait 1.14 to the phase portrait 1.17). The limit cycle of the phase portrait 1.14 ends in the loop H1 formed by the connection of the separatrices S15 and S16 . When (a, b) provides the phase portrait 1.17, see Figure 18(d), the homoclinic loop H1 of phase portrait 1.13 breaks up, and the separatrix S16 connects p1 and p2 , and the separatrix S15 connects p1 and (1, 0). When (a, b) is located in the bifurcation curve a = f1 (b) with b > 2 , see Figure 19(b), we obtain the phase portrait 1.22. The finite equilibria p1 and p2 of the phase portraits 1.17 collide providing a saddle-node, the separatrix S16 of the phase portrait of 1.17 disappears. When (a, b) is the bifurcation point (16, 2) we get the phase portrait 1.21, see Figure 19(a). The separatrix S15 of the phase portrait 1.22 connects with the separatrix S13 . Then the saddle-node p1 becomes the cusp (1/2, 1/4). When (a, b) provides the phase portrait 1.23, see Figure 19(c), the finite singular point p1 of the phase portrait 1.22 collides with (1, 0), the separatrices S13,14,15 dis- appear. Hence systems (2) have only one finite singular point which is an attracting node. PHASE PORTRAITS OF THE GRAY-SCOTT MODEL 27 C' C' C' B'' S12 S15 B'' B'' L2 S16 S14 S12 S15 S13 S5 S12 A'' A' A'' S6 A' A'' A' B' B' B' C'' C'' C'' (a) (b) (c) (d) Figure 19. Some separatrices of the phase portraits. (a) phase portrait 1.21, (b) phase portrait 1.22, (c) phase portrait 1.23, (d) phase portrait 1.10. When (a, b) provides the phase portrait 1.17 of Figure 5, systems (2) have six infinite equilibria A′ , A′′ , B ′ , B ′′ , C ′ and C ′′ . A′ and A′′ are saddle-nodes, B ′ and B ′′ are repelling nodes, C ′ and C ′′ are saddles. A′ has one separatrix S5 , A′′ has one separatrix S6 , and C ′′ has one separatrix S11 , these separatrices are not at infinity. There are three finite equilibria (1, 0) and p1,2 . (1, 0) is an attracting node, p1 is a saddle and p2 is an unstable node or focus. p1 has four separatrices S13,14,15,16 , and the separatrix S16 connects p1 and p2 . When (a, b) is in the bifurcation curve a = f2 (b) and 2 < b ≤ 4, the point p2 becomes p′2 = (1/b, (b − 1)/b2 ), and it is an unstable weak focus. When (a, b) provides the phase portrait 1.10, see Figure 19(d), the stability of the focus p2 of the phase portrait of 1.16 reverses, and an unstable limit cycle L2 bifurcates from p′2 . As (a, b) is in the bifurcation curve a = f4 (b) with 2 < b < b2 , we obtain the phase portrait 1.8, see Figure 20(a), the limit cycle of the phase portrait 1.9 ends in the homoclinic loop H1 formed by the connection of sepeartrices S15 and S16 . From the phase portrait 1.8 to 1.7, see Figure 17(c), the homoclinic loop H1 of the phase portrait 1.8 breaks up, and the separatrix S15 connects the point p1 and p2 , and the separatrix S16 connects the point p1 and A′ . Now we consider the phase portraits 1.10 and 1.11. The stability of the focus p2 of the phase portrait 1.10 reverses again, a Hopf bifurcation occurs, and a stable limit cycle L1 bifurcates from this singular point. Hence there exist exactly two limit cycles in the phase portrait 1.11, one is stable and one is unstable, see Figure 20(c). Conversely the limit cycle L1 of the phase portrait 1.11 becomes smaller and finally disappears . When (a, b) is in the bifurcation curve a = f4 (b) with b2 < b < b4 , see Figure 20(b), we have the phase portrait 1.9. The bigger limit cycle L2 of the phase portrait 1.11 ends in the homoclinic loop H1 formed by the connection of separatrices S15,16 . From the phase portrait 1.9 to 1.13, the limit cycle L1 of the phase portrait 1.9 ends in the homoclinic loop H1 . From the phase portrait 1.8 to 1.9, the stability of the focus p2 of the phase portrait 1.8 reverses, a Hopf bifurcation occurs, a stable 28 TING CHEN, JAUME LLIBRE AND SHIMIN LI C' C' C' H1 B'' L1 H1 B'' S15 B'' L1 S16 S12 L2 S15 L1 S16 A'' A' A'' A' A'' A' B' B' B' C'' C'' C'' (a) (b) (c) (d) C' C' B'' S16 B'' S12 S12 S16 S14 S16=S12 S14 S13 S13 S13 A'' S6 A' A'' A' S14 S5 S11 B' B' C'' C'' (e) (f) (g) Figure 20. Some separatrices of the phase portraits. (a) phase portrait 1.8, (b) phase portrait 1.9, (c) phase portrait 1.11, (d) phase portrait 1.12, (e) phase portrait 1.18, (f) phase portrait 1.19, (g) phase portrait 1.20. limit cycle L1 bifurcates from p2 . Conversely the limit cycle L1 of the phase portrait 1.9 becomes smaller and disappears. When (a, b) is the bifurcation point (f4 (b2 ), b2 ), see Figure 20(a), we obtain the phase portrait 1.8. The limit cycle L1 of the phase portrait 1.10 becomes smaller and disappears in a Hopf bifurcation, and the limit cycle L2 ends in the homoclinic loop H1 . When (a, b) is in the bifurcation curve a = f3 (b) with 4 < b < b4 , the two limit cycles of the phase portrait 1.11 collide producing a semi-stable limit cycle in the phase portrait 1.12 (externally unstable and internally stable), see Figure 20(d). When the parameter (a, b) is the bifurcation point (f4 (b4 ), b4 ) we obtain the phase portrait 1.13, see Figure 18(c). The semi-stable limit cycle of the phase portrait 1.12 ends the homoclinic loop H1 . When the parameter (a, b) produces the phase portrait 1.17, see Figure 18(d), the semi-stable limit cycle of the phase portrait 1.12 disappears. On the other hand, when (a, b) provides the phase portrait 1.5, see Figure 17(a), systems (2) have six infinite equilibria A′ , A′′ , B ′ , B ′′ , C ′ and C ′′ . A′ and A′′ are saddle-nodes, B ′ and B ′′ are repelling nodes, and C ′ and C ′′ are saddles. The saddle-node A′ has one separatrix S5 , A′′ has one separatrix S6 , the saddle C ′ has one separatrix S12 and C ′′ has one separatrix S11 , these separatrices are not at infinity. Systems (2) have three finite equilibria (1, 0) and p1,2 . (1, 0) is an attracting node, p1 is a saddle and p2 is an attracting node or focus. The saddle p1 has four separatrices Si (i = 13, 14, 15, 16). PHASE PORTRAITS OF THE GRAY-SCOTT MODEL 29 S14 S15 S13 S12 S2= S5 S3 S16 S2= S5 S3 S6 S10 S1=S6 S2= S5 S9 S1=S6 L3 S8 S7 S4 S4 S11 S11 (a) (b) (c) (d) S3 S3 S1 S1 S6 S11 S6 S11 S2= S5 S2= S5 (e) (f) Figure 21. Some separatrices of the phase portraits. (a) phase portrait 1.24, (b) phase portrait 1.25, (c) phase portrait 1.26, (d) phase portrait 1.27, (e) phase portrait 1.28, (f) phase portrait 1.29. When (a, b) is in the bifurcation curve a = f1 (b) with 0 < b < b1 , we get the phase portrait 1.18, see Figure 20(e). The finite equilibria p1 and p2 of the phase portraits 1.5 collide, and the separatrix S15 connecting them disappears. When (a, b) = (f1 (b1 ), b1 ), we have the phase portrait 1.19, see Figure 20(f). The separatrix S16 of the phase portrait 1.18 connects with the separatrix S12 . When (a, b) is in the bifurcation curve a = f1 (b) with b1 < b < 2, we obtain the phase portrait 1.20, see Figure 20(g). The separatrix S16 of the phase portrait 1.19 connects with the infinite equilibrium A′ , and the separatrix S12 goes to (1, 0). When (a, b) reaches the bifurcation point (16, 2) we get the phase portrait 1.21, see Figure 19(a). The separatrix S16 of the phase portrait 1.20 connects with the separatrix S14 . The saddle-node p1 becomes now the cusp (1/2, 1/4). From the phase portrait 1.6 to the phase portrait 1.19, the finite singular point p2 of the phase portrait 1.6 collides with p1 , and the separatrix S15 disappears. Then p1 becomes a saddle-node. Similarly we obtain the phase portrait 1.20 from the phase portrait 1.7. From the phase portrait 1.8 to the phase portrait 1.21, the finite singular point p2 of the phase portrait 1.8 collides with p1 , and the homoclinic loop H1 disappears. Then p1 becomes a cusp. Hence, we finish the analysis of the bifurcation diagram of Figure 5. Next we summarize the change of the phase portraits from Figure 6. When the parameter (a, b) produces the phase portrait 1.24 , see Figure 21(a), systems (2) 30 TING CHEN, JAUME LLIBRE AND SHIMIN LI C' C' C' C' B'' B'' B'' B'' S3 S3 S3 S3 S5 S1 S5 S1 S1 S5 A'' S1 S5 A'' A'' S4 A' A'' A' A' S6 A' S2 L3 S2 S6= S11 S2 S2 S11 S6 S11 S6 B' S11 B' B' B' C'' C'' C'' C'' (a) (b) (c) (d) Figure 22. Some separatrices of the phase portraits. (a) phase portrait 1.30, (b) phase portrait 1.31, (c) phase portrait 1.32, (d) phase portrait 1.33. have six infinite equilibria A′ , A′′ , B ′ , B ′′ , and C ′ and C ′′ . A′ and A′′ are saddle- nodes, B ′ and B ′′ are attracting nodes, and C ′ and C ′′ are repelling nodes. The saddle-nodes A′ and A′′ have one separatrix S5 and S6 respectively, which are not at infinity. There are three finite singular points (1, 0) and p1,2 . (1, 0) is an attracting node and p1,2 are saddles. p1 and p2 have four separatrices Si (i = 13, · · · , 16) and four separatrices Sj (j = 7, · · · , 10), respectively. On the bifurcation straight line b = 0 and a < 0, we obtain the phase portraits 1.25, see Figure 21(b). The point p1 of the phase portrait 1.24 moves to the infinite point C ′ in U2 , p2 moves to (1, 0), and the separatrices Si (i = 13, 14, 15, 17, 18, 19) disappear. The finite equilibrium (1, 0), the infinite equilibria C ′ and C ′′ become saddle-nodes. The saddle-node (1, 0) has three separatrices S1,2,4 , where S1 = S6 and S2 = S5 . C ′ has one separatrix S12 which is not at infinity. When the parameter (a, b) produces the phase portrait 1.26, see Figure 21(c). The saddle-node (1, 0) of the phase portrait 1.25 splits into an attracting node or a stable focus p2 and a saddle (1, 0). The saddle (1, 0) has four separatrices S1,2,3,4 . The infinite saddle-node C ′′ splits into the same C ′′ (a saddle) and a repelling node or an unstable focus p1 . The saddle C ′′ has one separatrix S11 which is not at infinity. When (a, b) is in the bifurcation curve a = f2 (b) and −1 ≤ b < 0, p1 becomes an unstable weak focus p′1 = (1/b, (b − 1)/b2 ). When the parameter (a, b) produces the phase portrait 1.27, see Figure 21(d), the stability of the focus p′1 of the phase portrait 1.26 reverses, and a Hopf bifurcation occurs, and an unstable limit cycle L3 bifurcates from p1 . On the bifurcation curve a = f7 (b) and −1 ≤ b < 0, we have the phase portrait 1.28, see Figure 21(e). The limit cycle L3 and the separatrix S6,11 of the phase portrait 1.27 collide, and connect with the infinity A′′ producing a heteroclinic loop. When the parameter (a, b) produces the phase portrait 1.29, see Figure 21(f), the orbit of the phase portrait 1.28 at p1 connects with the infinite singular point A′′ . When the parameter (a, b) produces the phase portrait 1.30, see Figure 22(a). The saddle (1, 0) has four separatrices S1,2,3,4 . The separatrix S5 of the phase portrait 1.26 at the saddle-node A′ moves to connect p2 , the separatrix S6 at A′′ PHASE PORTRAITS OF THE GRAY-SCOTT MODEL 31 moves to connect B ′ . Similarly we obtain the phase portrait 1.31 from the phase portrait 1.27, see Figure 22(b). When the parameter (a, b) is in the bifurcation curve a = f7 (b) and b < −1, we have the phase portrait 1.32, see Figure 22(c). The limit cycle L3 and the separatrices S6,11 of the phase portrait 1.31 collide, and it produces a heteroclinic loop. When the parameter (a, b) produces the phase portrait 1.33, see Figure 22(d), the separatrix S6 of the phase portrait 1.32 at the saddle-node A′′ moves to connect with p1 . This completes the analysis of the bifurcation diagrams. 5. Conclusion In this paper the phase portraits in the Poincaré disk for the cubic polynomial systems corresponding to the Gray-Scott model are studied in the Poincaré com- pactification. The phase portraits and the corresponding bifurcation diagrams show the richness and the complicated dynamics of such systems. Acknowledgments The first author is partially supported by National Natural Science Foundation of China (No. 11771059). The second author is partially supported by the MINECO-FEDER grant MTM2016- 77278-P, and the AGAUR grant 2017SGR-1617. The third author is partially supported by the National Natural Science Founda- tions of China (No. 11401111,11661017), China Scholarship Council (No. 201608440447), Science and Technology Program of Guangzhou (No. 201707010426) and Natural Science Foundation of Guangdong Province (2017A030313010). References [1] M. J. Álvarez, A. Ferragut, and X. Jarque, “A survey on the blow up technique,” Int. J. Bifurcation Chaos 31, 3103–3118 (2011). [2] J. Artés and J. Llibre, “Quadratic Hamiltonian vector fields,” J. Differential Equations 107, 80–95 (1994). [3] I. Berenstein and Y. D. Decker, “Defect-mediated turbulence and transition to spatiotemporal intermittency in the Gray–Scott model,” Chaos 24, 043109 (2014). [4] T. Blows and C. Rousseau, “Bifurcation at infinity in polynomial vector fields,” J. Differential Equations 104, 215–242 (1993). [5] T. Chen, L. Huang, P. Yu, and W. Huang, “Bifurcation of limit cycles at infinity in piecewise polynomial systems,” Nonlinear Anal. 41, 82–106 (2018). [6] I. Colak, J. Llibre, and C. Valls, “Hamiltonian linear type centers of linear plus cubic homo- geneous polynomial vector fields,” J. Differential Equations 257, 1623–1661 (2014). [7] I. Colak, J. Llibre, and C. Valls, “Hamiltonian nilpotent centers of linear plus cubic homoge- neous polynomial vector fields,” Adv. Math. 259, 655–687 (2014). [8] I. Colak, J. Llibre, and C. Valls, “Bifurcation diagrams for Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields,” J. Differential Equations 258, 846–879 (2015). [9] I. Colak, J. Llibre, and C. Valls, “Bifurcation diagrams for Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields,” J. Differential Equations 262, 5518–5533 (2017). 32 TING CHEN, JAUME LLIBRE AND SHIMIN LI [10] J. Delgado, L. I. Hernández-Martı́nez, and J. Pérez-López, “Global bifurcation map of the homogeneous states in the Gray-Scott model,” Int. J. Bifurcation Chaos 27, 1730024 (2017). [11] F. Dumortier, J. Llibre, and J. Artés, Qualitative Theory of Planar Differential Systems (Universitext, Springer-Verlag, NewYork, 2006). [12] A. Gasull, A. Guillamon, and V. Mañosa, “Phase portrait of Hamiltonian systems with homogeneous nonlinearities,” Nonlinear Anal. 42, 679–707 (2000). [13] P. Gray and S. K. Scott, “Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Isolas and other forms of multistability,” Chemical Engineering Science 38, 29–43 (1983). [14] A. Guillamon and C. Pantazi, “Phase portraits of separable Hamiltonian systems,” Nonlinear Anal. 74, 4012–4035 (2011). [15] Y. Martı́nez and C. Vidal, “Classification of global phase portraits and bifurcation diagrams of Hamiltonian systems with rational potential,” J. Differential Equations 261, 5923–5948 (2016). [16] D. S. Morgan and T. J. Kaper, “Axisymmetric ring solutions of the 2D Gray–Scott model and their destabilization into spots,” Physica D 192, 33–62 (2004). [17] D. Schlomiuk and N. Vulpe, “Global topological classification of Lotka-Volterra quadratic differential systems,” Electronic J. Differential Equations 64, 1–69 (2012). [18] S. K. Scott, T. Boddington, and P. Gray, “Analytical expressions for effectiveness factors in non-isothermal spherical catalysts,” Chemical Engineering Science 39, 1087–1097 (1984). [19] P. Yu and M. Han, “Twelve limit cycles in a cubic case of the 16th Hilbert problem,” Int. J. Bifurcation Chaos 15, 2191–2205 (2005). 1 College of Mathematics and Econometrics, Hunan University, Changsha, 410082, PR China E-mail address: chenting0715@126.com 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bel- laterra, Barcelona, Catalonia, Spain E-mail address: jllibre@mat.uab.cat 3 School of Statistics and Mathematics, Guangdong University of Finance and Eco- nomics, Guangzhou, 510320, PR China E-mail address: lism1983@126.com