This is a preprint of: “Periodic orbits bifurcating from a Hopf equilibrium of 2-dimensional polyno- mial Kolmogorov systems of arbitrary degree”, Djamila Djedid, Jaume Llibre, Ammar Makhlouf, Chaos Solitons Fractals, vol. 142, 110489, 2021. DOI: [10.1016/j.chaos.2020.110489] PERIODIC ORBITS BIFURCATING FROM A HOPF EQUILIBRIUM OF 2–DIMENSIONAL POLYNOMIAL KOLMOGOROV SYSTEMS OF ARBITRARY DEGREE DJAMILA DJEDID1 , JAUME LLIBRE2 AND AMAR MAKHLOUF3 Abstract. A Hopf equilibrium of a differential system in R2 is an equilibrium point whose linear part has eigenvalues ±ωi with ω 6= 0. We provide necessary and sufficient conditions for the existence of a limit cycle bifurcating from a Hopf equilibrium of 2–dimensional polynomial Kolmogorov systems of arbitrary degree. We provide an estimation of the bifurcating small limit cycle and also characterize the stability of this limit cycle. 1. Introduction and statements of the main results A polynomial differential system dx dy ẋ = = P (x, y), ẏ = = Q(x, y), dt dt in R2 has degree n if the maximum of the degrees of the polynomials P and Q is n. A quadratic polynomial vector field X = (P, Q) with x a factor of P and y a factor of Q is a Lotka–Volterra system. While an n–degree polynomial vector field X = (P, Q) with x a factor of P and y a factor of Q is a Kolmogorov system. Lotka–Volterra systems were initially considered independently by Alfred J. Lotka in 1925 [15] and by Vito Volterra in 1926 [20], as a model for studying the interactions between two species. Later on Kolmogorov [11] in 1936 extended these systems to arbitrary dimension and arbitrary degree, these kinds of systems are now called Kolmogorov systems. Many natural phenomena can be modeled by the Kolmogorov systems such as the time evolution of conflicting species in biology [16], chemical reactions [10], hydrodynamics [5], economics [18], the coupling of waves in laser physics [12], the evolution of electrons, ions and neutral species in plasma physics [13], etc. Here we study the polynomial Kolmogorov systems in the plane, i.e. differential systems of the form (1) ẋ = xf (x, y), ẏ = yg(x, y), where f and g are polynomials of degree larger than 1. In fact we are interested in the existence of limit cycles of Kolmogorov systems living in the positive quadrant of the plane, and consequently surrounding some equilibrium points (see for instance Theorem 1.31 of [8]) which are in the positive quadrant. We recall that a limit cycle of the Kolmogorov system (1) is a periodic solution of system (1) isolated in the set of all periodic solutions of (1). In general to detect the existence of limit cycles is a difficult problem. A Hopf equilibrium of a differential system in R2 is an equilibrium point whose linear part has eigenvalues ±ωi with ω 6= 0. Here a Hopf bifurcation means that some limit cycles bifurcate from a Hopf equilibrium when some parameter of the differential system varies, but in the literature not always a Hopf bifurcation has this meaning, see for instance [1]. Our objective is to provide necessary and sufficient conditions for the existence of a limit cycle bifurcating from a Hopf equilibrium of 2–dimensional polynomial Kolmogorov systems of arbitrary degree, using the averaging theory of second order. We also provide the stability of the small limit cycle which is born in the Hopf bifurcation, and an estimation of its size in function of the bifurcation parameter. 2010 Mathematics Subject Classification. 37G15. Key words and phrases. Lotka-Volterra system, periodic orbit, averaging theory, zero Hopf bifurcation, zero–Hopf equilibria. 1 2 D. DJEDID ,J. LLIBRE AND A. MAKHLOUF We translate an equilibrium point (a, b) of the positive quadrant to the point (1, 1) doing the rescaling (x, y) → (x/a, y/b), and we obtain that any Kolmogorov system in the plane of degree 2, i.e. any Lotka– Volterra system can be written as (2) ẋ = x(a1 (x − 1) + a2 (y − 1)), ẏ = y(b1 (x − 1) + b2 (y − 1)). This is a quadratic polynomial differential system with two invariant straight lines and by a Bautin’s result [2] we know that such differential systems have no limit cycles (see also [7]). A Kolmogorov system having an equilibrium in the positive quadrant can be written as  ẋ = x a1 (x − 1) + a2 (y − 1) + a3 (x − 1)2 + a4 (x − 1)(y − 1) + a5 (y − 1)2 , (3) ẏ = y b1 (x − 1) + b2 (y − 1) + b3 (x − 1)2 + b4 (x − 1)(y − 1) + b5 (y − 1)2 , if its degree is 3, and it can be written as ẋ = x a1 (x − 1) + a2 (y − 1) + a3 (x − 1)2 + a4 (x − 1)(y − 1) + a5 (y − 1)2 +  a6 (x − 1)3 + a7 (x − 1)2 (y − 1) + a8 (x − 1)(y − 1)2 + a9 (y − 1)3 + h.o.t. , (4) ẏ = y b1 (x − 1) + b2 (y − 1) + b3 (x − 1)2 + b4 (x − 1)(y − 1) + b5 (y − 1)2 + b6 (x − 1)3 + b7 (x − 1)2 (y − 1) + b8 (x − 1)(y − 1)2 + b9 (y − 1)3 + h.o.t. , if its degree is larger than 3, where h.o.t. denotes higher order terms in x − 1 and y − 1, i.e. terms of order at least four in the variables x − 1 and y − 1. Proposition 1. Let ω be a positive real number and b1 6= 0. A Kolmogorov system (3) or (4) has a Hopf equilibrium at (1, 1) with eigenvalues ±ωi if and only if b22 + ω 2 a1 = −b2 , a2 = − . b1 Proposition 1 is proved in section 3. We define the quantities A = −2a23 b1 b32 −2a23 b1 b2 ω 2 +3a3 a4 b21 b22 +a3 a4 b21 ω 2 −2a3 a5 b31 b2 +a3 b21 b32 +a3 b21 b2 ω 2 + a3 b1 b42 + a3 b1 b32 b4 + 3a3 b1 b22 ω 2 + a3 b1 b2 b4 ω 2 + 2a3 b1 ω 4 − 2a3 b42 b3 − 4a3 b22 b3 ω 2 − 2a3 b3 ω 4 − a24 b31 b2 + a4 a5 b41 − a4 b31 b22 −a4 b31 b2 b5 −a4 b21 b32 −a4 b21 b2 ω 2 +a4 b1 b32 b3 +a4 b1 b2 b3 ω 2 +a5 b41 b2 + 2a5 b41 b5 +a5 b31 b22 −a5 b31 b2 b4 +b31 b22 b5 + 2b31 b2 b25 +2b31 b5 ω 2 −b21 b32 b4 +b21 b32 b5 −3b21 b22 b4 b5 −b21 b2 b4 ω 2 +b21 b2 b5 ω 2 −b21 b4 b5 ω 2 +b1 b42 b3 −b1 b42 b4 +2b1 b32 b3 b5 + b1 b32 b24 + b1 b22 b3 ω 2 − b1 b22 b4 ω 2 + 2b1 b2 b3 b5 ω 2 + b1 b2 b24 ω 2 + b52 b3 − b42 b3 b4 + 2b32 b3 ω 2 − 2b22 b3 b4 ω 2 + b2 b3 ω 4 − b3 b4 ω 4 , and  C = A + b1 ω 2 3a6 b22 + 3a6 ω 2 − 2a7b1 b2 + a8 b21 + 3b21 b9 − 2b1 b2 b8 + b22 b7 + b7 ω 2 . Theorem 2. Consider the Kolmogorov systems (3) or (4) with b22 + ω 2 (5) a1 = −b2 + ε2 B, a2 = − , b1 6 0 and ε a small parameter (the bifurcation parameter). with ω > 0, b1 = (a) System (3) has one limit cycle (x(t, ε), y(t, ε)) bifurcating from the Hopf equilibrium (1, 1) when ε = 0 if and only if b1 AB < 0. This limit cycle is stable if B > 0 and unstable if B < 0. Moreover   b1 ωr∗ (6) (x(0, ε), y(0, ε)) = 1 + O(ε2 ), 1 − ε 2 + O(ε 2 ) , b2 + ω 2 where r b1 B(b22 + ω 2 ) ∗ r = 2ω − . A (b) System (4) has one limit cycle (x(t, ε), y(t, ε)) bifurcating from the Hopf equilibrium (1, 1) when ε = 0 if and only if b1 CB < 0. This limit cycle is stable if B > 0 and unstable if B < 0. Moreover the equality (6) holds with r ∗ b1 B(b22 + ω 2 ) r = 2ω − . C Theorem 2 is proved in section 3 using the averaging theory of second order. A brief summary of the averaging theory that we need for proving Theorem 2 is stated in section 2. Note that Theorem 2 provides sufficient conditions in order that one limit cycle bifurcates from a Hopf equilibrium point localized in the positive quadrant of a planar Kolmogorov system of arbitrary degree, and HOPF BIFURCATIONS OF 2–DIMENSIONAL KOLMOGOROV SYSTEMS 3 that (6) gives an estimation of the size of the small limit cycle which bifurcates from the equilibrium (1, 1) in function of the bifurcation parameter ε. 2. The averaging theory for periodic orbits The method of averaging started with the classical works of Lagrange and Laplace who provided an intuitive justification of this theory. The first formalization of the averaging theory is due to Fatou [9] in 1928. Important contributions to this theory were made by Krylov and Bogoliubov [4] in the 1930s and Bogoliubov [3] in 1945. The averaging theory of first order for studying periodic orbits can be found in [19]. Now we shall present the basic results from averaging theory that we need for proving the results of this paper. The next theorem provides a first and the second order approximation for the periodic solutions of a periodic differential system, for the proof see Theorems 11.5 and 11.6 of Verhulst [19], and Buica and Llibre [6]. Consider the differential equation (7) ẋ(t) = εF1 (t, x) + ε2 F2 (t, x) + ε3 R(t, x, ε), where F1 , F2 : R × D → Rn , R : R × D × (−εf , εf ) → Rn are continuous functions, T -periodic in the first variable, and D is an open subset of Rn . Assume that the following hypotheses (i) and (ii) hold. Assume: (i) F1 , F2 , R are locally Lipschitz with respect to x, F1 (t, .) ∈ C 1 (D) for all t ∈ R, and R is differentiable with respect to ε. We define the averaging functions of first and second order f1 , f2 : D −→ Rn as Z 1 T f1 (z) = F1 (s, z)ds, T 0 (8) Z  Z s  1 T f2 (z) = Dz F1 (s, z) F1 (t, z)dt + F2 (s, z) ds. T 0 0 (ii) For V ⊂ D an open and bounded set and for each ε ∈ (−εf , εf ) \ {0}, there exists a ∈ V such that f1 (a) + εf2 (a) = 0 and dB (f1 + εf2 , V, a) 6= 0. Then for |ε| > 0 sufficiently small there exists a T -periodic solution x(t, ε) of the system (7) such that x(0, ε) → a when ε → 0. Here dB (f1 + εf2 , V, 0) denotes the Brouwer degree of the function f1 + εf2 in the neighborhood V of zero. It is known that if the function f1 + εf2 is C 1 then it is sufficient to check that the determinant of the Jacobian matrix D(f1 + εf2 (aε )) is non-zero in order to have that dB (f1 + εf2 , V, 0) 6= 0, for more details see [14]. On the other hand if one of the real parts of the eigenvalues of the Jacobian matrix D(f1 + εf2 )(aε ) is positive the periodic solution x(t; ε) is unstable. If all the real parts of the eigenvalues of this matrix are negative the periodic solution is locally stable. For a proof see Theorem 11.6 of [19]. For a general information on the averaging theory see for instance the books [17, 19]. 3. Proofs Proof of Proposition 1. The characteristic polynomial of the linear part of the Lotka-Volterra system (3) at the equilibrium point (1, 1) is p(λ) = λ2 − (a1 + b2 )λ + a1 b2 − a2 b1 . Imposing that p(λ) = λ2 + ω 2 in order to have a Hopf equilibrium, we obtain the system a1 + b2 = 0, a1 b2 − a2 b1 = ω 2 . Solving this system we get the family of zero–Hopf equilibria described in Proposition 1. This completes the proof.  4 D. DJEDID ,J. LLIBRE AND A. MAKHLOUF Proof of Theorem 2. We shall prove that a periodic orbit bifurcates from the zero–Hopf equilibrium point (1, 1) of the Kolmogorov system (3) for the parameters of system (3) given in the statement of Proposition 1. We perturb system (3) with the parameters given in (5). We translate the equilibrium point (1, 1) to the origin of coordinates doing the change of variables x = X + 1, y = Y + 1. Then system (3) becomes 1  Ẋ = (1 + X) (Bε2 − b2 )b1 X − (b22 + ω 2 )Y + a3 b1 X 2 + a4 b1 XY + a5 b1 Y 2 , (9) b1  Ẏ = (1 + Y ) b1 X + b2 Y + b3 X 2 + b4 XY + b5 Y 2 . In order to simplify the application of the averaging theory, for computing the Hopf bifurcation we write the linear part of system (9) with ε = 0 at the equilibrium point (0, 0) into its real Jordan normal form, i.e. into the form   0 −ω . ω 0 Then doing the change of variables     b2 b22 + ω 2   u − − X = ω b1 ω  Y , v 1 0 whose inverse is     0 1   X b1 b2  u , = b1 ω Y − 2 − 2 v b2 + ω 2 b2 + ω 2 The differential system (9) in the variables (u, v) writes 1  u̇ = − (a5 b31 b32 − a4 b21 b42 + a3 b1 b52 + b62 b3 − b1 b52 b4 + b21 b42 b5 )v 2 + (a5 b31 b32 b1 ω(b22 + ω 2 )2 −a4 b21 b42 + a3 b1 b52 − b1 b52 b3 + b21 b42 b4 − b31 b32 b5 )v 3 + (2a5 b31 b22 − a4 b21 b32 + b21 b42 + b1 b52 −b1 b42 b4 + 2b21 b32 b5 )uvω + (2a5 b31 b22 − a4 b21 b32 − b1 b42 b3 + 2b21 b32 b4 − 3b31 b22 b5 )uv 2 ω+ (a5 b31 b2 + b21 b32 + b21 b22 b5 )u2 ω 2 + b1 b42 vω 2 + (a5 b31 b2 + b21 b22 b4 − 3b31 b2 b5 )u2 vω 2 + (−a4 b21 b22 + 2a3 b1 b32 − b21 b32 + 3b42 b3 − 2b1 b32 b4 + b21 b22 b5 )v 2 ω 2 + (−a4 b21 b22 + 2a3 b1 b32 −2b1 b32 b3 + b21 b22 b4 )v 3 ω 2 − b31 b5 u3 ω 3 + (−a4 b21 b2 + 2b1 b32 − 2b1 b22 b4 + 2b21 b2 b5 )uvω 3 (10) +(2b21 b2 b4 − a4 b21 b2 − 2b1 b22 b3 )uv 2 ω 3 + (b21 b2 + b21 b5 )u2 ω 4 + 2b1 b22 vω 4 + b21 b4 u2 vω 4 +(a3 b1 b2 − b21 b2 + 3b22 b3 − b1 b2 b4 )v 2 ω 4 + (a3 b1 b2 − b1 b2 b3 )v 3 ω 4 + (b1 b2 − b21 −  1 b1 b4 )uvω 5 − b1 b3 uv 2 ω 5 + b1 vω 6 + b3 v 2 ω 6 − ε2 Bb2 v(1 + v), ω (1 + v)  v̇ = 2 (a5 b21 b22 − a4 b1 b32 + a3 b42 )v 2 + b42 uω + (2a5 b21 b2 − a4 b1 b22 )uvω + ω 2 (b2 + ω 2 )2  a5 b21 u2 + (−a4 b1 b2 + 2a3 b22 )v 2 ω 2 + 2b22 uω 3 − a4 b1 uvω 3 + a3 v 2 ω 4 + uω 5 +ε2 Bv(1 + v). HOPF BIFURCATIONS OF 2–DIMENSIONAL KOLMOGOROV SYSTEMS 5 Doing the rescaling of the variables (u, v) = (εU, εV ) system (10) in the new variables (U, V ) writes 1  U̇ = −V ω − ε (a5 b31 b32 − a4 b21 b42 + a3 b1 b52 + b62 b3 − b1 b52 b4 + b21 b42 b5 )V 2 b1 ω(b22 + ω 2 )2 +(2a5 b31 b22 − a4 b21 b32 + b21 b42 + b1 b52 − b1 b42 b4 + 2b21 b32 b5 )U V ω + (a5 b31 b2 + b21 b32 + b21 b22 b5 )U 2 ω 2 + (−a4 b21 b22 + 2a3 b1 b32 − b21 b32 + 3b42 b3 − 2b1 b32 b4 + b21 b22 b5 )V 2 ω 2 + (−a4 b21 b2 + 2b1 b32 − 2b1 b22 b4 + 2b21 b2 b5 )U V ω 3 + (b21 b2 + b21 b5 )U 2 ω 4 + (a3 b1 b2 −  1 b21 b2 + 3b22 b3 − b1 b2 b4 )V 2 ω 4 + (−b21 + b1 b2 − b1 b4 )U V ω 5 + b3 V 2 ω 6 + ε2 2 + ω 2 )2  ω(b 2 − Bb52 V + (−a5 b21 b32 + a4 b1 b42 − a3 b52 + b52 b3 − b1 b42 b4 + b21 b32 b5 )V 3 − (2a5 b21 b22 −a4 b1 b32 − b42 b3 + 2b1 b32 b4 − 3b21 b22 b5 )U V 2 ω − 2Bb32 V ω 2 − (a5 b21 b2 + b1 b22 b4 − (11) 3b21 b2 b5 )U 2 V ω 2 + (a4 b1 b22 − 2a3 b32 + 2b32 b3 − b1 b22 b4 )V 3 ω 2 + b21 b5 U 3 ω 3 + (a4 b1 b2  +2b22 b3 − 2b1 b2 b4 )U V 2 ω 3 − Bb2 V ω 4 − b1 b4 U 2 V ω 4 + (b2 b3 − a3 b2 )V 3 ω 4 + b3 U V 2 ω 5 +O(ε3 ), 1  V̇ = U ω + ε 2 (a5 b21 b22 − a4 b1 b32 + a3 b42 )V 2 + (2a5 b21 b2 − a4 b1 b22 + b42 )U V ω (b2 + ω 2 )2  +a5 b21 U 2 ω 2 + (−a4 b1 b2 + 2a3 b22 )V 2 ω 2 + (−a4 b1 + 2b22 )U V ω 3 + a3 V 2 ω 4 + U V ω 5 1  +ε2 2 V Bb42 + (a5 b21 b22 − a4 b1 b32 + a3 b42 )V 2 + (2a5 b21 b2 − a4 b1 b22 )U V ω (b2 + ω 2 )2 +2Bb22 ω 2 + a5 b21 U 2 ω 2 + (−a4 b1 b2 + 2a3 b22 )V 2 ω 2 − a4 b1 U V ω 3 + Bω 4 + a3 V 2 ω 4 ) + O(ε3 ). We must pass to polar coordinates for introducing the angle θ which later on we shall take as the new independent variable in order that the differential system (11) becomes periodic in θ and we can apply the averaging theory, where (r, θ) are defined by U = r cos θ and V = r sin θ, and we obtain 1  ṙ = −ε r2 b21 ω 2 (a5 b1 b2 + (b2 + b5 )(b22 + ω 2 )) cos3 θ − b1 ω(a5 b21 (−2b22 + ω 2 ) b1 ω(b22 + ω 2 )2 +(b22 + ω 2 )(a4 b1 b2 − (b2 − b4 )(b22 + ω 2 ) + b1 (−b22 − 2b2 b5 + ω 2 ))) cos2 θ sin(θ) + (a5 b31 b2 (b22 − 2ω 2 ) + (b22 + ω 2 )(b42 b3 − b1 b32 b4 + b21 b22 b5 − b21 b2 ω 2 − b1 b22 ω 2 + 2b22 b3 ω 2 − b1 b2 b4 ω 2 −b1 ω 4 + b3 ω 4 + a4 b21 (−b22 + ω 2 ) + a3 b1 b2 (b22 + ω 2 ))) cos θ sin θ2 − b1 ω(a5 b21 b22 +  1  (b22 + ω 2 )(−a4 b1 b2 + a3 (b22 + ω 2 ))) sin3 θ + ε2 b2 b5 r2 ω 3 cos(θ)4 − b1 r2 ω(b22 + ω 2 )2 1 ω 2 (a5 b1 b2 + b4 b22 − 3b1 b2 b5 + b4 ω 2 ) cos3 θ sin θ + r2 ω(b42 b3 − 2b1 b32 b4 + 3b21 b22 b5 + 2b22 b3 ω 2 − 2b1 b2 b4 ω 2 + b3 ω 4 + a5 b21 (−2b22 + ω 2 ) + a4 b1 b2 (b22 + ω 2 )) cos2 θ sin2 θ + ω sin2 θ (B(b22 + ω 2 )2 + r2 (a5 b21 b22 + (b22 + ω 2 )(−a4 b1 b2 + a3 (b22 + ω 2 ))) sin2 θ) − cos θ sin θ(Bb2 (b22 + ω 2 )2 + r2 (a5 b21 b2 (b22 − 2ω 2 ) + a4 b1 (−b 4 4 4 3 2  2 + ω ) + b2 (−b2 b3 + b1 b2 b4 + b1 b2 b4 ω − b3 ω 4 (12) +a3 (b22 + ω 2 )2 − b22 (b21 b5 + 2b3 ω 2 ))) sin2 θ) + O(ε3 ), 1  θ̇ = ω + ε r a5 b31 ω 3 cos3 θ + b1 ω 2 (3a5 b21 b2 − (a4 b1 − b1 b2 − b22 − b1 b5 − ω 2 )(b22 + b1 ω(b22 + ω 2 )2 ω 2 )) cos2 θ sin θ + b1 ω(3a5 b21 b22 + (b22 + ω 2 )(−2a4 b1 b2 + b1 b22 + b32 − b22 b4 + 2b1 b2 b5 − b1 ω 2 +b2 ω 2 − b4 ω 2 + a3 (b22 + ω 2 ))) cos θ sin2 θ + (a5 b31 b32 + (b22 + ω 2 )(−a4 b21 b22 + b42 b3 − b1 b32 b4  1 +b21 b22 b5 − b21 b2 ω 2 + 2b22 b3 ω 2 − b1 b2 b4 ω 2 + b3 ω 4 + a3 b1 b2 (b22 + ω 2 ))) sin3 θ + ε2 2 + ω 2 )2  2ω(b 2 sin θ ω cos θ + b2 sin θ)((b22 + ω 2 )(2B(b22 + ω 2 ) + r2 (a5 b21 − a4 b1 b2 + a3 b22 − b22 b3 + b1 b2 b4 −b21 b5 + a3 ω 2 − b3 ω 2 )) − r2 (a3 b42 − b42 b3 + b1 b32 b4 − b21 b22 b5 + 2a3 b22 ω 2 − 2b22 b3 ω 2 + b1 b2 b4 ω 2 + b21 b5 ω 2 + a3 ω 4 − b3 ω 4 + a5 b21 (b22 − ω 2 )− a4 b1 b2 (b22 + ω 2 )) cos(2θ) + b1 r2 ω(2a5 b1 b2 + b22 b4 − 2b1 b2 b5 + b4 ω 2 − a4 (b22 + ω 2 )) sin(2θ) + O(ε3 ). 6 D. DJEDID ,J. LLIBRE AND A. MAKHLOUF We take θ as the new independent variable and system (12) becomes into the normal form for applying the averaging theory 1  ṙ = −ε 2 2 2 2 r2 b21 ω 2 (a5 b1 b2 + (b2 + b5 )(b22 + ω 2 )) cos3 θ − b1 ω(a5 b21 (−2b22 + ω 2 ) b1 ω (b2 + ω ) +(b22 + ω 2 )(a4 b1 b2 − (b2 − b4 )(b22 + ω 2 ) + b1 (−b22 − 2b2 b5 + ω 2 ))) cos2 θ sin θ + (a5 b31 b2 (b22 − 2ω 2 ) + (b22 + ω 2 )(b42 b3 − b1 b32 b4 + b21 b22 b5 − b21 b2 ω 2 − b1 b22 ω 2 + 2b22 b3 ω 2 − b1 b2 b4 ω 2 −b1 ω 4 + b3 ω 4 + a4 b21 (−b22 + ω 2 ) + a3 b1 b2 (b22 + ω 2 ))) cos θ sin2 θ − b1 ω(a5 b21 b22 + (b22 +   1 ω 2 )(−a4 b1 b2 + a3 (b22 + ω 2 ))) sin3 θ + ε2 2 4 2 r3 b21 ω 2 (a5 b1 b2 + (b2 + b5 ) b1 ω (b2 + ω 2 )4 ) (b22 + ω 2 )) cos3 θ − b1 ω(a5 b21 (−2b22 + ω 2 ) + (b22 + ω 2 )(a4 b1 b2 − (b2 − b4 )(b22 + ω 2 ) + b1 (−b22 − 2b2 b5 + ω 2 ))) cos2 θ sin θ + (a5 b31 b2 (b22 − 2ω 2 ) + (b22 + ω 2 )(b42 b3 − b1 b32 b4 + b21 b22 b5 −b21 b2 ω 2 − b1 b22 ω 2 + 2b22 b3 ω 2 − b1 b2 b4 ω 2 − b1 ω 4 + b3 ω 4 + a4 b21 (−b22 + ω 2 ) + a3 b1 b2 (b22 + (13) ω 2 ))) cos θ sin2 θ − b1 ω(a5 b21 b22 + (b22 + ω 2 )(−a4 b1 b2 + a3 (b22 + ω 2 ))) sin3 θ)(a5 b31 ω 3 cos3 θ + b1 ω 2 (3a5 b21 b2 − (a4 b1 − b1 b2 − b22 − b1 b5 − ω 2 )(b22 + ω 2 )) cos2 θ sin θ + b1 ω (3a5 b21 b22 + (b22 + ω 2 )(−2a4 b1 b2 + b1 b22 + b32 − b22 b4 + 2b1 b2 b5 − b1 ω 2 + b2 ω 2 − b4 ω 2 + a3 (b22 + ω 2 ))) cos θ sin2 θ + (a5 b31 b32 + (b22 + ω 2 )(−a4 b21 b22 + b42 b3 − b1 b32 b4 + b21 b22 b5 − b21 b2  1 ω 2 + 2b22 b3 ω 2 − b1 b2 b4 ω 2 + b3 ω 4 + a3 b1 b2 (b22 + ω 2 ))) sin3 θ + 2 2 r b21 b5 r2 ω 3 ω (b2 + ω 2 )2 cos4 θ − b1 r2 ω 2 (a5 b1 b2 + b22 b4 − 3b1 b2 b5 + b4 ω 2 ) cos3 θ sin θ + r2 ω(b42 b3 − 2b1 b32 b4 + 3b21 b22 b5 + 2b22 b3 ω 2 − 2b1 b2 b4 ω 2 + b3 ω 4 + a5 b21 (−2b22 + ω 2 ) + a4 b1 b2 (b22 + ω 2 )) cos2 θ sin2 θ +ω sin2 θ(B(b22 + ω 2 )2 + r2 (a5 b21 b22 + (b22 + ω 2 )(−a4 b1 b2 + a3 (b22 + ω 2 ))) sin2 θ) − cos θ sin θ(Bb2 (b22 + ω 2 )2 + r2 (a5 b21 b2 (b22 − 2ω 2 ) + a4 b1 (−b42 + ω 4 ) + b2 (−b42 b3 + b1 b32 b4 + b1 b2 b4  ω 2 − b3 ω 4 + a3 (b22 + ω 2 )2 − b22 (b21 b5 + 2b3 ω 2 ))) sin2 θ)2 + O(ε3 ). Here the prime denotes derivative with respect to θ. We shall apply the averaging theory described in section 2 to the differential system (13). Using the notation of section 2 we have t = θ, T = 2π, x = r. It is immediate to check that system (13) satisfies all the assumptions for applying the averaging theory. Now we compute the integrals (8), and we obtain f1 (r) = 0, 1 f2 (r) = r(4b1 Bω 2 (b22 + ω 2 ) + Ar2 ). 8b1 ω 3 (b22 + ω 2 ) The system f2 (r) = 0 has a unique positive solution r b1 B(b22 + ω 2 ) r∗ = 2ω − . A The derivative of f2 (r) at r∗ is Ar∗ /(4b1 ω 3 (b22 + ω 2 ). Therefore from section 2 we obtain that the small limit cycle is stable if B > 0, and unstable if B < 0. Going back through the changes of variables we obtain the expression (6). This completes the proof of statement (a) of the theorem. By following for system (4) the same steps done for the system (3) we find that the averaged functions f1 and f2 are f1 (r) = 0, 1 f2 (r) = r(4b1 Bω 2 (b22 + ω 2 ) + Cr2 ). 8b1 ω 3 (b22 + ω 2 ) The equation f2 (r) = 0 has a unique positive solution r ∗ b1 B(b22 + ω 2 ) r = 2ω − . C Since the derivative of f2 (r) at r∗ is Cr∗ /(4b1 ω 3 (b22 + ω 2 ), the statement (b) follows in a similar way to statement (a).  Acknowledgments This work 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. HOPF BIFURCATIONS OF 2–DIMENSIONAL KOLMOGOROV SYSTEMS 7 References [1] B. Alarcón and R. Rabanal, Hopf bifurcation at infinity and dissipative vector fields of the plane, Proc. Amer. Math. Soc. 145 (2017), 3033–3046. [2] N.N. 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Accademia Nazionale dei Lincei, CCCCXXIII(II) (1926), 558–560. 1 Department of Mathematics, University of Annaba, Laboratory LMA PO.Box.12;23 Annaba, Algeria Email address: roubachedjedid@gmail.com 2 Departament de Matematiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalo- nia, Spain Email address: jllibre@mat.uab.cat 3 Department of Mathematics, University of Annaba, Laboratory LMA PO.Box.12;23 Annaba, Algeria Email address: makhloufamar@yahoo.fr