This is a preprint of: “Dynamics and Darboux integrability of the D2 polynomial vector fields of degree 2 in R3 ”, Jean-Marc Ginoux, Jaume Llibre, Clàudia Valls, Math. Phys. Anal. Geom., vol. 24, 1:1–16, 2021. DOI: [10.1007/s11040-020-09372-0] DYNAMICS AND DARBOUX INTEGRABILITY OF THE D2 POLYNOMIAL VECTOR FIELDS OF DEGREE 2 IN R3 JEAN-MARC GINOUX1 , JAUME LLIBRE2 AND CLÀUDIA VALLS3 Abstract. We characterize the Darboux integrability and the global dynamics of the 3-dimensional polynomial differential systems of degree 2 which are invariant under the D2 symmetric group, which have been partially studied previously by several authors. 1. Introduction and statement of the results Differential systems having some symmetries appear often in many ap- plications, and consequently have been studied by several authors, see for instance [6, 8, 9, 12]. In this paper we shall study the dynamics of the 3-dimensional autonomous polynomial differential systems of degree 2 symmetric with respect to the group of symmetries generated by the following transformations of R3 (x, y, z) 7→ (x, y, z), (x, y, z) 7→ (−x, −y, z), (x, y, z) 7→ (−x, y, −z), (x, y, z) 7→ (x, −y, −z). In [11] it is proved that such 3-dimensional autonomous systems are (1) ẋ = ax + yz, ẏ = by + xz, ż = z + xy, and (2) ẋ = ax + yz, ẏ = by + xz, ż = z − xy, where a, b are non-zero parameters and the dot means derivative in t. Ac- cording to [2] systems (1) and (2) are equivariant under the D2 symmetry group. Note that system (2) can be transformed into system (1) by the change (3) (x, y, z) → (iX, iY, Z) with X, Y, Z ∈ R. Hence it is enough to study the integrability of system (1). Moreover, system (1) is invariant by the change (4) (a, b, x, y, z) → (b, a, y, x, z). 2010 Mathematics Subject Classification. 34C05, 34C07, 34C08. Key words and phrases. Integrability, Darboux invariant, Poincaré sphere, global dynam- ics, D2 symmetric polynomial vector fields. 1 2 J.M. GINOUX, J. LLIBRE AND C. VALLS Katsios and Anastassiou in [11] studied the Darboux polynomials of sys- tems (1) and (2), and also provided some few results on the dynamics of these systems, see also [1]. Here our first objective is to study the Dar- boux integrability of system (1), and consequently of system (2) too. After we describe the global dynamics of these systems taking into account their behavior near infinity. A first integral is a nonconstant function h : R3 → R that is constant on all solution curves (x(t), y(t), z(t)) of the system, that is h(x(t), y(t), z(t)) = const. for all values of t for which the solution (x(t), y(t), z(t)) is defined on R3 . We say that system is completely integrable if it has two independent first integrals. One approach to obtain the first integrals of system (1) on a given class consists in using the Darboux theory of integrability (see [10]). The Darboux theory of integrability have been developed strongly these past years (see for instance [5, 7, 13, 15] and the references therein). In particular using this theory we charactarize all the values of the parameters for which systems (1) and (2) has a polynomial, a rational or a Darboux first integral (see the definitions below). We now give a rigorous formulation of our results. We recall that a Dar- boux polynomial of systems (1) is a polynomial f = f (x, y, z) ∈ C[x, y, z] \ C such that ∂f ∂f ∂f (5) (ax + yz) + (by + xz) + (z ± xy) = Kf, ∂x ∂y ∂z for some polynomial (6) K = K(x, y, z) = a0 x + a1 y + a2 z + a3 with ai ∈ C. The polynomial K is called the cofactor of the Darboux polynomial f . We note that f = 0 is an invariant algebraic surface for the flow of systems (1) and (2): if an orbit of the system has a point on this surface then the whole orbit is contained in it. The polynomial K is called the cofactor of the Darboux polynomial f or of the invariant algebraic surface. It was proved in [11] the following theorem. Theorem 1. The following statements hold. (i) System (1) has invariant algebraic surfaces if and only if a = 1, or b = 1, or b = a. If a = 1 the irreducible invariant algebraic surface is x2 − z 2 = 0 with cofactor 2. If b = 1 the irreducible invariant algebraic surface is y 2 − z 2 = 0 with cofactor 2, and if b = a the irreducible invariant algebraic surface is x2 − y 2 = 0 with cofactor 2a. (ii) System (2) has invariant algebraic surfaces or invariant straight lines if and only if a = 1, or b = 1, or b = a. If a = 1 the invariant straight line is x2 +z 2 = 0 with cofactor 2. If b = 1 the invariant straight line DYNAMICS AND DARBOUX INTEGRABILITY OF THE D2 VECTOR FIELD IN R2 3 is y 2 + z 2 = 0 with cofactor 2, and if b = a the invariant algebraic surface is x2 − y 2 = 0 with cofactor 2a. Note that as stated above, statement (ii) can be obtained from statement (i) by simply using (3). Moreover, using (4) it is clear that the case b = 1 is obtained from the case a = 1. Since the cofactors of the Darboux polynomials x2 −z 2 , y 2 −z 2 and x2 −y 2 for a = b = 1 are all equal to 2, from statement (i) of Theorem 8.7 of [7] it follows easily the next corollary. Corollary 2. If b = a = 1 then system (1) is completely integrable with the two first integrals (x2 − z 2 )/(y 2 − z 2 ) and (x2 − z 2 )/(x2 − y 2 ). Since a polynomial first integral is a Darboux polynomial with zero co- factor, from Theorem 1 it follows the next result. Corollary 3. System (1) with (a, b) ∈ R2 \{(0, 0)∪(1, 1)} has no polynomial first integrals. An exponential factor of system (1) is a nonconstant function of the form F = exp(h/g) with g, h ∈ C[x, y, z] coprime polynomials, and the function F satisfies (5) for some polynomial K as in (6), which is also called the cofactor of the exponential factor. Theorem 4. System (1) with (a, b) ∈ R2 \{(0, 0)∪(1, 1)} has no exponential factors. The proof of Theorem 4 is given in section 2. λ µ A first integral is called Darboux if it is of the form f1λ1 · · · fp p F1µ1 · · · Fq q with f1 , . . . , fp Darboux polynomials and F1 , . . . , Fq exponential factors and λj , µj ∈ C for all j and k. Theorem 5. System (1) with (a, b) ∈ R2 \ {(0, 0) ∪ (1, 1)} has no Darboux first integrals. The proof of Theorem 5 is given in section 3. Our second objective is to describe the global dynamics in the com- pactification in R3 in function of the parameters a, b. Roughly speaking the Poincaré ball B is obtained identifying R3 with the interior of the 3- dimensional ball of radius one centered at the origin of coordinates and extending analytically the flow of system (1) to the boundary S2 of B, i.e. the infinity of R3 . In R3 we can go to infinity in as many directions as points of S2 . Doing that we are able to study the behavior of systems (1) and (2) in a neighborhood of infinity and to describe completely the globally dynamics of such differential systems. We recall that the global description of the flow 4 J.M. GINOUX, J. LLIBRE AND C. VALLS of a differential system in R3 in general is very difficult if not impossible. Here we can do it using the Poincaré compactification due to the existence of a Darboux invariant for system (1). An invariant of system (1) on an open subset U of R3 is a nonconstant C1 function I in the variables x, y, z and t such that I(x(t), y(t), z(t), t) is constant on all solution curves (x(t), y(t), z(t)) of system (1) contained in U , i.e. ∂I ∂I ∂I ∂I (ax + yz) + (by + xz) + (z ± xy) + = 0, ∂x ∂y ∂z ∂t for all (x, y, z) ∈ U . An invariant I is called a Darboux invariant if it can be written in the form λ I(x, y, z, t) = f1λ1 · · · fp p es t , where, for i = 1, . . . p, fi = 0 are invariant algebraic surfaces of system (1), λi ∈ C, and s ∈ R \ {0}. Let φp (t) be the solution of system (1) passing through the point p ∈ R3 , defined on its maximal interval (αp , ωp ) and such that φp (0) = p. If ωp = ∞, we define the ω-limit set of p as ω(p) = {q ∈ R3 : ∃{tn } with tn = ∞ and φp (tn ) = q when n = ∞}. In the same way, if αp = −∞, we define the α-limit set of p as α(p) = {q ∈ R3 : ∃{tn } with tn = −∞ and φp (tn ) = q when n = ∞}. For more details on the ω– and α–limit sets see for instance section 1.4 of [7]. The existence of a Darboux invariant of system (1) provides information about the ω– and α–limit sets of all orbits of system (1). More precisely, there is the following result, for a proof see Proposition 5 of [14]. Proposition 6. Let S2 be the infinity of the Poincaré sphere and I(x, y, z, t) = f (x, y, z)est be a Darboux invariant of system (1). Let also p ∈ R3 and φp (t) be the solution of system (1) with maximal interval (αp , ωp ) such that φp (0) = p. Then (1) Assume that s > 0. If ωp = ∞, then the ω(p) is contained in the closure {f (x, y, z) = 0} in the Poincaré ball of {f (x, y, z) = 0}, and if αp = −∞ then α(p) ⊂ {f (x, y, z) = 0} ∩ S2 , being S2 the boundary of the Poincaré ball. (2) Assume that s < 0. If αp = −∞, then the α(p) is contained in {f (x, y, z) = 0}, and if ωp = ∞ then ω(p) ⊂ {f (x, y, z) = 0} ∩ S2 . Now we study the global dynamics of systems (1) and (2). Theorem 7. The following holds for system (1) with (a, b) ∈ R2 \ {(0, 0) ∪ (1, 1)}. (a) The dynamics at infinity are topologically equivalent to the one of Figure 1. DYNAMICS AND DARBOUX INTEGRABILITY OF THE D2 VECTOR FIELD IN R2 5 (b) When b = a 6= 1 the phase portraits in the Poincaré disc defined by the invariant planes x = y, x = −y, are topologically equivalent to the ones given in Figure 2 when a > 0 and Figure 3 when a < 0. Moreover, any orbit starting at a point outside these planes either approach these planes or approach the infinity. In particular such orbits end at infinity if a > 0, and start at infinity if a < 0 (c) When a = 1 with b 6= 1 the phase portraits in the Poincaré disc defined by the invariant planes x = z, x = −z, are topologically equivalent to the ones given in Figures 2 when b > 0 and Figure 3 when b < 0. Moreover, any orbit starting at a point outside these planes either approach these planes or approach the infinity. In par- ticular such orbits end at infinity. The proof of Theorem 7 is given in section 4. Theorem 8. The following holds for system (2) with (a, b) ∈ R2 \ {(0, 0) ∪ (1, 1)}. (a) The dynamics at infinity are topologically equivalent to the one of Figure 4. (b) When b = a 6= 1 the phase portraits in the Poincaré disc defined by planes x = y, x = −y, are topologically equivalent to the ones given in Figure 3 when a > 0 and Figure 5 when a < 0. Moreover, any orbit starting at a point outside these planes either approach these planes or approach the infinity. In particular such orbits end at infinity if a > 0, and start at infinity if a < 0. (c) When a = 1 with b 6= 1 the dynamics reduces to the straight line x = z = 0 containing a unique orbit solution of ẏ = by. In particular all such orbits end at infinity. Moreover if b > 0 they start either at the origin or perhaps at infinity at the endpoins of the y-axis, and if b < 0 they start at infinity at the end points of the y-axis except the ones which are on the 2-dimensional unstable manifold at the origin. The proof of Theorem 8 is given in section 5. 2. Proof of Theorem 4 Let F = exp(h/g) be an exponential factor of system (1). We use the well- known result that ensures that if g = g(x, y, z) is not a constant polynomial, then g = 0 is an invariant algebraic surface (see [4, 5] and Proposition 8.6 of [7] for details). We will also use the following result (see Proposition 8.4 of [7] for a proof): Q α Proposition 9. Let f be a polynomial and f = nj=1 fj j its decomposition in irreducible factors in C[x, y, z]. Then f is a Darboux polynomial if and 6 J.M. GINOUX, J. LLIBRE AND C. VALLS only if all the fj are Darboux polynomials. Moreover, P if K and Kj are the cofactors of f and fj respectively, then K = nj=1 αj Kj . From Theorem 1 and Proposition 9 we get  n1 n2  f1 f2 , f1 = x + y, f2 = x − y, n1 , n2 ≥ 0, if b = a with a 6= 1, g = f3n1 f4n2 , f3 = x + z, f4 = x − z, n1 , n2 ≥ 0, if a = 1 with b 6= 1,   const., if b 6= a and a 6= 1. We separate the proof of Theorem 4 into three different cases. Case 1: b 6= a and a 6= 1 In this case any exponential factor F must be of the form F = exp(h), where h = h(x, y, z) is a nonconstant polynomial satisfying ∂h ∂h ∂h (7) (ax + yz) + (by + xz) + (z + xy) = K, ∂x ∂y ∂z with K as in (6). The singular points of system (1) are (0, 0, 0) and √ √ √ √ √ √ √ √ √ √ √ √ ( b, a, − ab) (− b, − a, − ab), ( b, − a, ab), (− b, a, ab). By substituting them in (7) we get a0 = a1 = a2 = a3 = 0, i.e. K = 0. Hence h is a polynomial first integral of system (1), which is in contradiction with Corollary 3. So in this case there are no exponential factors. Case 2: b = a and a 6= 1. In this case we have that any exponential factor must be of the form F = exp(h/(f1n1 f2n2 )) with n1 , n2 ≥ 0, and h = h(x, y, z) a polynomial coprime with f1n1 f2n2 satisfying (8) ∂h ∂h ∂h (ax + yz) + (ay + xz) + (z + xy) − ((n1 + n2 )a + (n1 − n2 )z)h ∂x ∂y ∂z = Kf1n1 f2n2 , where K is as in (6). We consider two subcases. Subcase 2.1: n1 = n2 = 0. The singular points of system (1) with b = a are (0, 0, 0) and √ √ √ √ √ √ √ √ ( a, a, −a) (− a, − a, −a), ( a, − a, a), (− a, a, a). By substituting them into (8) we obtain a0 = a1 = a2 = a3 = 0, i.e. K = 0. Hence h is a polynomial first integral of system (1), which is in contradiction with Corollary 3. So in this case there are no exponential factors. Subcase 2.2: n1 > 0 or n2 > 0 We first consider the case n2 > 0. Assume first that h = c ∈ C \ {0}. Then from (8) we have (9) −c((n1 + n2 )a + (n1 − n2 )z) = K(x + y)n1 (x − y)n2 . Since n2 > 0 and the left-hand side of (9) is not divisible by x − y, we get a contradiction. DYNAMICS AND DARBOUX INTEGRABILITY OF THE D2 VECTOR FIELD IN R2 7 Assume now that h is not a constant. Now we set y = x and denote by h̄ = h(x, x, z) (the restriction of h to y = x). Then h̄ is a polynomial different from zero, because h and f2 are coprime. The polynomial h̄ satisfies (8) restricted to y = x (that is to f2 = 0) and so ∂ h̄ ∂ h̄ (10) x(a + z) + (z + x2 ) = ((n1 + n2 )a + (n1 − n2 )z)h̄. ∂x ∂z Equation (10) implies that h̄ is a Darbox polynomial of system ẋ = x(a + z), ż = z + x2 , with cofactor (n1 + n2 )a + (n1 − n2 )z. We will show that this is not possible. P Let n ≥ 1 be the degree of h̄. We expand h̄ = h̄(x, y) = ni=0 h̄i (x, y) where h̄i is a homogeneous polynomial of degree i. Then it follows from (10) that h̄n satisfies ∂ h̄n ∂ h̄n (11) xz + x2 = (n1 − n2 )z h̄n . ∂x ∂z The general solution of (11) is h̄n = xn1 −n2 H̄n (z 2 − x2 ) where H̄n is an arbitrary C 1 function, but since h̄n must be a homogeneous polynomial of degree n, we have that without loss of generality we can take h̄n = xn1 −n2 (z 2 − x2 )(n−n1 +n2 )/2 , with n − n1 + n2 even. Computing the terms of degree n in (10) we get ∂ h̄n ∂ h̄n−1 ∂ h̄n ∂ h̄n−1 ax + xz +z + x2 = (n1 + n2 )ah̄n + (n1 − n2 )z h̄n−1 . ∂x ∂x ∂z ∂z Solving this partial differential equation we obtain  n−n1 +n2 n1 −n2 2 2 n1 −n2 2 2 −1 h̄n−1 = x H̄n−1 (z − x ) ∓ x (z − x ) 2 (a − 1)(n + n2 − n1 )z p  x + (n + n2 − 2an2 − n1 ) z 2 − x2 log √ , z 2 − x2 + z z 2 − x2 where H̄n−1 is an arbitrary C 1 function, but since h̄n−1 must be a homoge- neous polynomial of degree n − 1 and n − n1 + n2 is even, then we get h̄n−1 = ±2a(a − 1)n2 zxn−2an2 (z 2 − x2 )an2 −1 , with n1 = n + n2 − 2an2 being an integer. Computing the terms of degree n − 1 in (10) we obtain ∂ h̄n−1 ∂ h̄n−2 ∂ h̄n−1 ∂ h̄n−2 ax +xz +z +x2 = (n+2n2 −2an2 )ah̄n−1 +(n−2an2 )z h̄n−2 . ∂x ∂x ∂z ∂z Solving this partial differential equation we have h̄n−2 = ∓xn−2an2 2an2 (a − 1)(z 2 − x2 )an2 ((a − 1)(an2 − 1)x2 + (z 2 − x2 ) log x) + xn−2an2 H̄n−2 (z 2 − x2 ), 8 J.M. GINOUX, J. LLIBRE AND C. VALLS where H̄n−2 is an arbitrary C 1 function, but since h̄n−2 must be a homoge- neous polynomial of degree n − 2 and a 6∈ {0, 1} we must have n2 = 0, in contradiction because n2 > 0. This contradiction shows that if n2 > 0 there are no exponential factors. The case n1 > 0 is treated analogously obtaining the theorem in this case. Case 3: a = 1 and b 6= a In this case, proceeding in a similar manner as in Case 2 working with f3 and f4 instead of f1 and f2 we also reach to a contradiction and so there are no exponential factors also in this case. This concludes the proof of Theorem 4. 3. Proof of Theorem 5 Let G be a Darboux first integral of system (1). If b 6= a and a 6= 1, by Theorems 1 and 4 there are no Darboux first integrals. If b = a 6∈ {0, 1} then it follows from Theorems 1 and 4 that G = cf1λ1 f2λ2 with c, λ1 , λ2 ∈ C. Imposing that G is a first integral from statement (i) of Proposition 8.7 of [7] it must hold a(λ1 + λ2 ) + z(λ2 − λ1 ) = 0. This implies λ1 = λ2 = 0, so in this case a Darboux first integral does not exist. Finally, if a = 1 with b 6= 1, it follows from Theorems 1 and 4 that G = cf3λ3 f4λ4 with c, λ3 , λ4 ∈ C. Imposing that G is a first integral it must hold λ3 + λ4 + y(λ4 − λ3 ) = 0. This implies λ3 = λ4 = 0, so in this case a Darboux first integral does not exist. This concludes the proof of Theorem 5. 4. Proof of Theorem 7 Now we present the analysis of the flow of system (1) at infinity using the Poincaré compactification of the system in R3 , for more details on the Poincaré compactification see [6]. The expression of the Poincaré compact- ification p(X) of system (1) in the local chart U1 is given by ż1 = z2 + (b − a)z1 z3 − z12 z2 , (12) ż2 = z1 + (1 − a)z2 z3 − z1 z22 , ż3 = −z3 (az3 + z1 z2 ). System (12) has the five singular points (−1, −1, 0), (1, −1, 0), (0, 0, 0), (−1, 1, 0), (1, 1, 0). DYNAMICS AND DARBOUX INTEGRABILITY OF THE D2 VECTOR FIELD IN R2 9 The eigenvalues of the Jacobian matrix at (−1, −1, 0) and (1, −1, 0) are −2, −2, −1 and 2, 2, 1, respectively. Moreover, the eigenvalues of the Jaco- bian matrix at (0, 0, 0) are −1, 1, 0, and (0, 0, 0) restricted to infinity (that is to z3 = 0) it is a saddle. Finally, the eigenvalues of the Jacobian matrix at (−1, 1, 0) and (1, 1, 0) are −2, −2, −1 and 2, 2, 1, respectively. The expression of the Poincaré compactification p(X) of system (1) in the local chart U2 is given by ż1 = z2 + (a − b)z1 z3 − z12 z2 , (13) ż2 = z1 + (1 − b)z2 z3 − z1 z22 , ż3 = −z3 (bz3 + z1 z2 ). In the local chart U2 we only need to study the singular points satisfying z1 = z3 = 0, because the other infinite singular points in this chart were already studied in the local chart U1 . System (13) coincides with system (12) changing b by a. So its origin is such that the eigenvalues of its associated Jacobian matrix are −1, 1, 0, and restricted to infinity is a saddle. Finally, the expression of the Poincaré compactification p(X) of system (2) in the local chart U3 is ż1 = z2 + (a − 1)z1 z3 − z12 z2 , ż2 = z1 + (b − 1)z2 z3 − z1 z22 , ż3 = −z3 (z3 + z1 z2 ). In the local chart U3 we only need to study the origin. We note that the origin is a singular point whose eigenvalues of its associated Jacobian matrix are −1, 1, 0, and (0, 0, 0) restricted to infinity is a saddle. The flow in the local charts Vi (i = 1, 2, 3) are the same as the flows in the local charts Ui (i = 1, 2, 3) reserving in an appropriate way the direction of the time. Taking into account that the systems in the local charts U1 and U2 re- stricted to z3 = 0 are ż1 = z2 (1 − z1 )(1 + z1 ), ż2 = z1 (1 − z2 )(1 + z2 ), we have that the straight lines z1 = ±1 and z2 = ±1 are invariant, and so the local phase portrait on the Poincaré sphere at infinity is topologi- cally equivalent to the one drawn in Figure 1. This completes the proof of statement (a). Consider the case b = a 6= 1. We first study the dynamics on x = y. We have the system (14) ẋ = x(a + z), ż = z + x2 . 10 J.M. GINOUX, J. LLIBRE AND C. VALLS Figure 1. The phase portrait on the sphere S2 of the infin- ity. There are only drawn the orbits contained in the closure of the local chart U1 . √ The unique singular points are (0, 0) and (± a, −a) that exist only when a > 0. The origin is a saddle √ if a < 0 and an unstable node if a > 0. When a > 0, the singular points (± a, −a) are both saddles. The expression of the Poincaré compactification of system (14) in the local chart U1 is (15) u̇ = 1 − u2 + (1 − a)uv, ż2 = −v(u + av). System (23) restricted to v = 0 has the two infinite singular points (±1, 0). Moreover, (1, 0) is a stable node and (−1, 0) is an unstable node. On the local chart U2 system (14) becomes u̇ = u + (a − 1)uv + u3 , v̇ = −v(v + u2 ). The origin is a singular point which is semihyperbolic. Using Theorem 2.19 in [7] we conclude that it is a saddle-node. Using that the straight line x = 0 is invariant it follows easily that the phase portraits in the Poincaré disc of system (14) are given in Figure 2 for a > 0 and in Figure 3 for a < 0. DYNAMICS AND DARBOUX INTEGRABILITY OF THE D2 VECTOR FIELD IN R2 11 Figure 2. Phase portrait of system (14) for a > 0. Now we study the dynamics on x = −y. We have the system (16) ẋ = x(a − z), ż = z − x2 . Introducing the change of variables (X, Z) = (x, −z) we get that system (16) becomes (14), and so the phase portraits in the Poincaré disc of system (16) can be obtained from Figures 2 and 3 doing a symmetry with respect to the x-axis. We recall that if an invariant algebraic surface f (x, y, z) = 0 has a con- stant cofactor κ ∈ R, then f e−κt is a Darboux invariant, see for a proof statement (vi) of Theorem 8.7 of [7]. Therefore, since system (1) with b = a 6= 1 has the invariant algebraic surface x2 − y 2 = 0 with cofactor 2a it has the Darboux invariant (x2 − y 2 )e−2at , and by Proposition 6 all the orbits outside the surface x2 − y 2 = 0 has ω-limit at infinity if a > 0, and α-limit at infinity if a < 0. This completes the proof of the second part of statement (b). Consider the case a = 1 with b 6= 1. We first study the dynamics on z = x. We have the system (17) ẋ = x(1 + y), ẏ = by + x2 . 12 J.M. GINOUX, J. LLIBRE AND C. VALLS Figure 3. Phase portrait of system (14) for a < 0. Introducing the reparameterization of time (X, Z, T ) → (x/b, y/b, bt) we get that (17) becomes (14) with a = 1/b ∈ R\{1}. Therefore the phase portraits in the Poincaré disc of system (17) are topologically equivalent to the ones in Figures 2 for b > 0 and 3 for b < 0. Now we study the dynamics on z = −x. We have the system (18) ẋ = x(1 − y), ẏ = by − x2 . Introducing the reparameterization of time (X, Z, T ) → (x/b, −y/b, bt) we get that (18) becomes (14) with a = 1/b ∈ R \ {1}. Therefore the phase portraits in the Poincaré disc of system (18) are topologically equivalent to the ones in Figures 2 for b > 0 and 3 for b < 0. The second part of statement (c) is proved in a similar way to the second part of the proof of statement (b). This proof of statement (c) is completed. 5. Proof of Theorem 8 Now we present the analysis of the flow of system (2) at infinity using the Poincaré compactification of the system in R3 . The expression of the DYNAMICS AND DARBOUX INTEGRABILITY OF THE D2 VECTOR FIELD IN R2 13 Poincaré compactification p(X) of system (2) in the local chart U1 is given by ż1 = z2 + (b − a)z1 z3 − z12 z2 , (19) ż2 = −z1 + (1 − a)z2 z3 − z1 z22 , ż3 = −z3 (az3 + z1 z2 ). System (19) has the origin as its unique singular point. The eigenvalues of the Jacobian matrix at (0, 0, 0) are i, −i, 0. Moreover, restricted to the infinity (that is to z3 = 0) it is a center because the Jacobian matrix at (0, 0) has eigenvalues ±i and system (19) restricted to z3 = 0 has the rational first integral (1 + z22 )/(1 − z12 ) defined in a neighborhood of the origin. The expression of the Poincaré compactification p(X) of system (2) in the local chart U2 is ż1 = z2 + (a − b)z1 z3 − z12 z2 , (20) ż2 = −z1 + (1 − b)z2 z3 − z1 z22 , ż3 = −z3 (bz3 + z1 z2 ). System (20) coincides with system (19) interchanging b by a. So its unique infinite singular point is the origin that restricted to z3 = 0 is a center. Finally, the expression of the Poincaré compactification p(X) of system (2) in the local chart U3 is ż1 = z2 + (a − 1)z1 z3 + z12 z2 , (21) ż2 = z1 + (b − 1)z2 z3 + z1 z22 , ż3 = z3 (−z3 + z1 z2 ). The origin of system (21) restricted to z3 = 0 is a singular point whose eigenvalues of its associated Jacobian matrix are −1, 1, 0 and restricted to infinity is a saddle. The flow in the local charts Vi (i = 1, 2, 3) are the same as the flows in the local charts Ui (i = 1, 2, 3) reserving in an appropriate way the direction of the time. Taking into account the eigenvectors and eigenvalues of the saddle at the origin of U3 , the local phase portrait in the Poincaré sphere at infinity is topologically equivalent to the one drawn in Figure 4. This completes the proof of statement (a). We consider the case in which b = a 6= 1. In this case system (2) has two invariant algebraic surfaces x − y = 0 and x + y = 0. We will study the dynamics in both of them. We first study the dynamics on x = y. We have the system (22) ẋ = x(a + z), ż = z − x2 . 14 J.M. GINOUX, J. LLIBRE AND C. VALLS Figure 4. The phase portrait on the sphere S2 of the infin- ity. There are only drawn the orbits contained in the closure of the local chart U1 . √ The unique singular points are (0, 0) and (± −a, −a) that exist only when a < 0. The origin is a saddle if a√< 0 and an unstable node if a > 0. When a < 0, the singular points (± −a, −a) are both unstable nodi when 1 + 8a ≥ 0 and unstable foci when 1 + 8a < 0. The expression of the Poincaré compactification of system (22) in the local chart U1 is (23) u̇ = −1 − u2 + (1 − a)uv, ż2 = −v(u + av). System (23) restricted to v = 0 has no singular points. On the local chart U2 system (22) becomes u̇ = u + u3 + (a − 1)uv, v̇ = v(−v + u2 ). The origin is a singular point which is semihyperbolic. Using Theorem 2.19 in [7] we conclude that it is a saddle-node. Combining the above analysis in the finite part and at each local chart at infinity and taking into account that the straight line x = 0 is invariant DYNAMICS AND DARBOUX INTEGRABILITY OF THE D2 VECTOR FIELD IN R2 15 we obtain for a > 0 (resp. a < 0) the phase portrait of Figure 3 doing a symmetry with respect to the x-axis (resp. the phase portrait of Figure 5). Figure 5. Phase portrait of system (24) for a < 0. Now we study the dynamics on x = −y. We have the system (24) ẋ = x(a − z), ż = z + x2 . Introducing the change of variables (X, Z) = (x, −z) we get that system (24) becomes system (14), and so the phase portraits in the Poincaré disc of system (24) are the ones of Figure 3 for a > 0 and the ones of Figure 5 doing a symmetry with respect to the x-axis for a < 0. The second part of the proof of statement (b) of the theorem follows exactly as the second part of the proof of statement (b) of Theorem 7. The proof of statement (b) is completed. Now we note that when a = 1 with b 6= 1 the dynamics of system (2) on x2 + z 2 = 0 reduces to the straight line x = z = 0 containing a unique orbit solution of ẏ = by, b 6= 1. The eigenvalues at the origin are 1, b, 1. Assume first that b > 0. Then the origin is a repeller. By Proposition 6, and since system (2) has the Darboux invariant (x2 + z 2 )e−2t all the orbits outside the y-axis end at infinity, and start either at the origin or 16 J.M. GINOUX, J. LLIBRE AND C. VALLS eventually at the endpoints of the y-axis. Assume now that b < 0. Then the origin has a one-dimensional stable manifold and a 2-dimensional unstable manifold. As before all the orbits outside the y-axis end at infinity. But by Proposition 6 all these orbits start at infinity at the endpoints of the y-axis except the ones which are on the two-dimensional unstable manifold at the origin. This completes the proof of statement (c). Acknowledgements The first author is supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA- RISE-2017-777911. The third author is partially supported by FCT/Portugal through UID/MAT/04459/2013. References [1] S. Anastassiou, S. Pnevmatikos and T. Bountis, Quadratic vector fields equi- variant under the D2 symmetry group, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 23 (2013), 1350017, 14 pp. [2] G.M. Chechin and D.S. Ryabov, Three-dimensional chaotic flows with discrete symmetries, Phys. Rev. E. 69 (2004), 036202, 9 pp. [3] P. Chossat and R. Lauterbach, Methods in equivariant bifurcations and dynamical systems, World Scientific, 2000. [4] C. Christopher, Invariant algebraic curves and conditions for a center, Proc. R. Soc. Edinburgh, Sect. A: Math. 124 A (1994), 1209–1229. [5] C. 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[13] J. Llibre, Handbook of Differential Equations, Ordinary Differential Equations, edited by Cañada, A., Drabek, P., and Fonda, (Elsevier 2004), Vol. 1, pp. 437–533. [14] J. Llibre and R. Oliveira, Quadratic systems with an invariant conic having Dar- boux invariants, Comm. Contem. Math. 20 (2018), 1750033, 15 pp. [15] J. Llibre and X. Zhang, Darboux integrability in Cn taking into account the mul- tiplicity, J. Differential Equations 246 (2009), 541–551. DYNAMICS AND DARBOUX INTEGRABILITY OF THE D2 VECTOR FIELD IN R2 17 1 Centre de Physique Théorique, UMR 7332, CNRS, Université de Toulon, France Email address: ginoux@univ-tln.fr 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat 3 Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1049–001, Lisboa, Portugal Email address: cvalls@math.ist.utl.pt