This is a preprint of: “The zero-Hopf bifurcations in the Kolmogorov systems of degree 3 in R3 ”, Erika Diz Pita, Jaume Llibre, M. Victoria Otero Espinar, Clàudia Valls, Commun. Nonlinear Sci. Numer. Simul., vol. 95, 105621, 2021. DOI: [10.1016/j.cnsns.2020.105621] The zero-Hopf bifurcations in the Kolmogorov systems of degree 3 in R3 Érika Diz-Pitaa , Jaume Llibreb , M. Victoria Otero-Espinara , Claudia Vallsc a Departamento de Estatística, Análise Matemática e Optimización, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain b Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain c Center for Mathematical Analysis, Geometry and Dynamical Systems, Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001, Lisboa, Portugal Abstract In this work we study the periodic orbits which bifurcate from all zero-Hopf bifurcations that an arbitrary Kolmogorov system of degree 3 in R3 can exhibit. The main tool used is the averaging theory. Keywords: Lotka–Volterra system, Kolmogorov systems, phase portraits, Hopf bifurcation, zero-Hopf bifurcation, limit cycle 1. Introduction and statement of the main results Lotka-Volterra systems were initially proposed, independently, by Alfred J. Lotka in 1925 [1] and Vito Volterra in 1926 [2], both in the context of competing species. These Lotka-Volterra systems are polynomial differential systems of the form ẋ = xP (x, y), ẏ = yQ(x, y), where P and Q are polynomials of degree 1. Later on the Lotka-Volterra systems were generalized and considered on arbitrary dimension n ≥ 2, i.e. ẋi = xi Pi (x1 , . . . , xn ), where Pi are polynomials of degree 1. Finally in 1936 Andrei Kolmogorov [3] extended those systems to arbitrary degree, i.e. the polynomials Pi can have any degree. These last systems are now called Kolmogorov systems. The Lotka-Volterra and Kolmogorov systems have been used for modelling many natural phenomena, such as the time evolution of conflicting species in biology [4], chemical reactions [5], plasma physics [6], hydrodynamics [7], and many other phenomena as social science and economics [8]. Recently limit cycles for differential systems in R3 also are studied for discontinuous differential systems see for instance [9] and the references quoted therein. We want to study the limit cycles of the Kolmogorov systems of degree 3 in R3 which bifurcate in the zero-Hopf bifurcations of the singular points (a, b, c) which are not on the invariant planes x = 0, y = 0 and z = 0 of the Kolmogorov system ẋ = xP (x, y, z), ẏ = yQ(x, y, z), ż = zR(x, y, z), with P , Q and R polynomials of degree 2. Doing the scaling (x, y, z) → (x/a, y/b, z/c) we can assume without loss of generality that (a, b, c) = (1, 1, 1). Therefore it is sufficient to study the limit cycles which can bifurcate Email addresses: erikadiz.pita@usc.es (Érika Diz-Pita), jllibre@mat.uab.cat (Jaume Llibre), mvictoria.otero@usc.es (M. Victoria Otero-Espinar), cvalls@mat.tecnico.ulisboa.pt (Claudia Valls) Preprint submitted to Journal of Computational and Applied Mathematics June 11, 2020 from the singular point (1, 1, 1) of the system ẋ = x a1 (x − 1) + a2 (y − 1) + a3 (z − 1) + a4 (x − 1)2 + a5 (x − 1)(y − 1)  + a6 (x − 1)(z − 1) + a7 (y − 1)2 + a8 (y − 1)(z − 1) + a9 (z − 1)2 , ẏ = y b1 (x − 1) + b2 (y − 1) + b3 (z − 1) + b4 (x − 1)2 + b5 (x − 1)(y − 1)  (1.1) + b6 (x − 1)(z − 1) + b7 (y − 1)2 + b8 (y − 1)(z − 1) + b9 (z − 1)2 , ż = z c1 (x − 1) + c2 (y − 1) + c3 (z − 1) + c4 (x − 1)2 + c5 (x − 1)(y − 1)  + c6 (x − 1)(z − 1) + c7 (y − 1)2 + c8 (y − 1)(z − 1) + c9 (z − 1)2 , when this singular point is a zero-Hopf equilibrium, i.e. when the eigenvalues of the linear part of the system at (1, 1, 1) are of the form 0 and ±βi with β > 0. Here the dot denotes derivative with respect to the time t. In the next result we characterize when the singular point (1, 1, 1) is zero-Hopf. Proposition 1.1. The singular point (1, 1, 1) of system (1.1) is zero-Hopf if and only if one of the following sets of conditions hold. (i) γ = a3 b3 (b2 − a1 ) − a2 b23 + a23 b1 6= 0, c3 = −a1 − b2 , 1 3   c1 = a b3 − a21 a3 b1 − a1 a3 b1 b2 − b3 2a2 b1 + β 2 − b1 (a2 (a3 b1 − b2 b3 ) + a3 β 2 + b22 ) and γ 1 1 2    c2 = a1 a2 b3 + a1 a2 (b2 b3 − a3 b1 ) + a22 b1 b3 − a3 b2 β 2 + b22 + a2 b3 β 2 + b22 − 2a3 b1 b2 . γ a3 b2 a1 b3 (a1 + b2 )2 + a3 c1 + β 2 (ii) a3 b3 6= 0, a2 = , b1 = , c3 = −a1 − b2 and c2 = − . b3 a3 b3 b22 + β 2 (iii) b3 6= 0, a1 = a2 = a3 = 0, c2 = − and c3 = −b2 . b3 a21 + β 2 (iv) a3 6= 0, b1 = b2 = b3 = 0, c1 = − and c3 = −a1 . a3 a21 + β 2 (v) b1 6= 0, a2 = − , a3 = b3 = c3 = 0 and b2 = −a1 . b1 Proposition 1.1 is proved in section 2. Using the averaging theory of first order in Theorem 3 of [10] are provided sufficient conditions in order that the Kolmogorov systems (1.1) under conditions (i) exhibit a zero-Hopf bifurcation from which two limit cycles bifurcate, the kind of stability or inestability of these limit cycles is also provided. We include the result here for completeness. The expressions of c, d, e, and f , are the ones given in [10], and we denote k = (2cf + dS2 )2 − 8cdf S1 , S2 = (c31 + 2cβ)/β and S1 = (dS2 + cf )/f .   Theorem 1.2. If γ 6= 0, c1 = (a31 b3 −a21 a3 b1 −a1 a3 b1 b2 − b3 2a2 b1 + β2 −b1 (a2 (a3 b1 −b2 b3 )+a3 β 2 + b22 ))/γ, c2 = (a21 a2 b3 + a1 a2 (b2 b3 − a3 b1 ) + a22 b1 b3 + a2 b3 β 2 + b22 − 2a3 b1 b2 − a3 b2 (β 2 + b22 ))/γ, c3 = −a1 − b2 , a2 b1 − a1 b2 6= 0, c 6= 0, d 6= 0, e 6= 0, f 6= 0, c31 + 2cβ 6= 0 and ec(c31 d + cβ(2d + f )) < 0, then the Kolmogorov system has two limit cycles bifurcating from the zero-Hopf equilibrium point (1, 1, 1). Moreover the following statements hold. √ (a) If S2 S1 < 0, k > 0 and |−2cf − dS2 | < k, then the two limit cycles are unstable and have a stable manifold formed by two cylinders and an unstable manifold formed by two cylinders. (b) If S2 < 0, S1 < 0 and √ √ • either k > 0, d(−2cf − dS2 + k) > 0 and d(−2cf − dS2 − k) > 0, • or k ≤ 0 and d(−2cf − dS2 ) > 0; or if S2 > 0, S1 > 0 and 2 √ √ • either k > 0, d(−2cf − dS2 + k) < 0 and d(−2cf − dS2 − k) < 0, • or k ≤ 0 and d(−2cf − dS2 ) < 0; then one limit cycle is a local repeller, and the other is a local attractor. √ (c) If S2 > 0, S1 > 0, k > 0 and |−2cf − dS2 | < k; or if S2 S1 < 0 and √ √ • either k > 0, d(−2cf − dS2 + k) > 0 and d(−2cf − dS2 − k) > 0, • or k ≤ 0 and d(−2cf − dS2 ) > 0; then both limit cycles are unstable. One limit cycle is a local repeller, and the other has a stable manifold formed by two cylinders and an unstable manifold formed by two cylinders. √ (d) If S2 < 0, S1 < 0, k > 0 and |−2cf − dS2 | < k; or if S2 S1 < 0 and √ √ • either k > 0, d(−2cf − dS2 + k) < 0 and d(−2cf − dS2 − k) < 0, • or k ≤ 0 and d(−2cf − dS2 ) < 0; then one limit cycle is a local attractor, and the other is unstable and has a stable manifold formed by two cylinders and an unstable manifold formed by two cylinders. (e) If S2 S1 < 0 and k < 0 and −2cf = dS2 , then one limit cycle is unstable and has a stable manifold formed by two cylinders and an unstable manifold formed by two cylinders and we cannot decide about the stability of the other. Here we provide examples showing that all the sets of conditions given in this theorem are non-empty, see section 3. Such examples were not given in [10]. In this paper we use the averaging theory of first order for studying the limit cycles bifurcating from the zero-Hopf bifurcations of the Kolmogorov systems (1.1) under conditions (ii) to (v). Our main result concerning the Kolmogorov systems (1.1) under the conditions (ii) is the following. The expressions of Ai , with i = 0, ..., 4, K1 and N are defined in Appendix B. Theorem 1.3. If a3 b3 6= 0, N 6= 0, a2 = a3 b2 /b3 , b1 = a1 b3 /a3 , c3 = −a1 − b2 , c2 = −((a1 + b2 )2 + a3 c1 + β 2 )/b3 , A1 6= 0, A2 6= 0, A3 6= 0,and A0 A4 (A1 A2 − A0 A3 ) > 0, then the Kolmogorov system (1.1) has two limit cycles bifurcating from the zero-Hopf equilibrium point (1, 1, 1). Moreover the following statements hold. √ (a) If K1 > 0, A2 A3 (A0 A3 − A1 A2 )N < 0 and |2A0 A3 − A1 A2 | < K1 , then the two limit cycles have a stable manifold formed by two cylinders and an unstable manifold formed by two cylinders. (b) If b3 A2 N > 0, b3 A3 (A0 A3 − A1 A2 ) > 0 and √ √ • either K1 > 0, b3 A1 N (2A0 A3 − A1 A2 − K1 ) < 0 and b3 A1 N (2A0 A3 − A1 A2 + K1 ) < 0, • or K1 ≤ 0 and b3 A1 N (2A0 A3 − A1 A2 ) < 0; or if b3 A2 N < 0, b3 A3 (A0 A3 − A1 A2 ) < 0 and √ √ • either K1 > 0, b3 A1 N (2A0 A3 − A1 A2 − K1 ) > 0 and b3 A1 N (2A0 A3 − A1 A2 + K1 ) > 0, • or K1 ≤ 0 and b3 A1 N (2A0 A3 − A1 A2 ) > 0; then one limit cycle is local repeller, and the other is a local attractor. √ (c) If b3 A2 N > 0, b3 A3 (A0 A3 − A1 A2 ) > 0, K1 > 0 and |2A0 A3 − A1 A2 | < K1 ; or if A2 A3 (A0 A3 − A1 A2 )N < 0 and √ √ • either K1 > 0, b3 A1 N (2A0 A3 − A1 A2 − K1 ) > 0 and b3 A1 N (2A0 A3 − A1 A2 + K1 ) > 0, • or K1 ≤ 0 and b3 A1 (2A0 A3 − A1 A2 )N > 0; then both limit cycles are unstable. One limit cycle is a local repeller, and the other has a stable manifold formed by two cylinders and an unstable manifold formed by two cylinders. 3 √ (d) If b3 A2 N < 0, b3 A3 (A0 A3 − A1 A2 ) < 0, K1 > 0 and |2A0 A3 − A1 A2 | < K1 ; or if A2 A3 (A0 A3 − A1 A2 )N < 0 and √ √ • either K1 > 0, b3 A1 N (2A0 A3 − A1 A2 − K1 ) < 0 and b3 A1 N (2A0 A3 − A1 A2 + K1 ) < 0, • or K1 ≤ 0 and b3 A1 (2A0 A3 − A1 A2 )N < 0; then one limit cycle is a local attractor, and the other is unstable and has a stable manifold formed by two cylinders and an unstable manifold formed by two cylinders. (e) If K1 < 0, A2 A3 (A0 A3 − A1 A2 )N < 0 and 2A0 A3 = A1 A2 ; then one limit cycle is unstable and has a stable manifold formed by two cylinders and an unstable manifold formed by two cylinders and we cannot decide about the stability of the other. The main result concerning the Kolmogorov systems (1.1) under the conditions (iii) is the following. The expressions of Bi with i = 0, ..., 4, and K2 are given in Appendix B. Theorem 1.4. If b3 6= 0, a1 = a2 = a3 = 0, c2 = −(b22 + β 2 )/b3 , c3 = −b2 , B1 6= 0, B2 6= 0, B3 6= 0 and B0 B4 (B1 B2 − B0 B3 ) > 0, then the Kolmogorov system (1.1) has two limit cycles bifurcating from the zero-Hopf equilibrium point (1, 1, 1). Moreover the following statements hold. √ (a) If K2 > 0, B2 B3 (B0 B3 −B1 B2 ) > 0 and |B1 B2 − 2B0 B3 | < K2 ; then the two limit cycles are unstable and have a stable manifold formed by two cylinders and an unstable manifold formed by two cylinders. (b) If B2 > 0, B3 (B0 B3 − B1 B2 ) < 0 and √ • either K2 > 0, B1 > 0 and B1 B2 − 2B0 B3 < − K2 , • or K2 ≤ 0 and B1 (B1 B2 − 2B0 B3 ) < 0; then the two limit cycles are local attractors. √ (c) If B2 < 0, B3 (B0 B3 − B1 B2 ) > 0, K2 > 0 √and |B1 B2 − 2B0 B3 | < K2 ; or if B2 B3 (B0 B3 − B1 B2 ) > 0, K2 > 0, B1 < 0 and B1 B2 − 2B0 B3 < − K2 ; then both limit cycles are unstable. One limit cycle is a local repeller, and the other has a stable manifold formed by two cylinders and an unstable manifold formed by two cylinders. (d) If B2 < 0, B3 (B0 B3 − B1 B2 ) > 0 and √ √ • either K2 > 0, B1 (B1 B2 − 2B0 B3 − K2 ) < 0 and B1 (B1 B2 − 2B0 B3 + K2 ) < 0, • or K2 ≤ 0 and B1 (B1 B2 − 2B0 B3 ) < 0; √ or if B2 > 0, B3 (B0 B3 − B1 B2 ) < 0, K2 > 0, B1 < 0 and B1 B2 − 2B0 B3 < − K2 ; then one limit cycle is a local attractor and the other limit cycle is a local repeller. √ (e) If B2 > 0, B3 (B0 B3 − B1 B2 ) < 0, K2 > 0 and |B1 B2 − 2B0 B3 | < K2 ; or if B2 B3 (B0 B3 − B1 B2 ) > 0 and √ √ • either K2 > 0, B1 (B1 B2 − 2B0 B3 − K2 ) < 0 and B1 (B1 B2 − 2B0 B3 + K2 ) < 0, • or K2 ≤ 0 and B1 (B1 B2 − 2B0 B3 ) < 0, then one limit cycle is a local attractor and the other limit cycle is unstable and has a stable manifold formed by two cylinders and an unstable manifold formed by two cylinders. (f ) B2 B0 < 0, K2 < 0 and B1 B2 = 2B0 B3 ; then one limit cycle is unstable and has a stable manifold formed by two cylinders and an unstable manifold formed by two cylinders and we cannot decide about the stability of the other. 4 Theorems 1.3 and 1.4 are proved in section 2. Examples showing that the conditions provided by both theorems are non-empty are given in section 3. Kolmogorov systems (1.1) under conditions (iv) are the same as under conditions (iii) but interchanging the variables x and y, so if we change the conditions b3 6= 0, a1 = a2 = a3 = 0, c2 = −(b22 + β 2 )/b3 , c3 = −b2 into a3 6= 0, b1 = b2 = b3 = 0, c1 = −(a21 + β 2 )/a3 , c3 = −a1 , and redefine the constants Bi for i = 0, ..., 4 as it is indicated in Appendix B the same Theorem 1.4 holds. At last our main result concerning the Kolmogorov systems (1.1) under the conditions (v) is the following, with the expressions of Di , for i = 0, ..., 4, and K4 given in Appendix B. Theorem 1.5. If b1 6= 0, a3 = b3 = c3 = 0, a2 = −(a21 + β 2 )/b1 , b2 = −a1 , D1 6= 0, D2 6= 0, D3 6= 0 and D0 D4 (D1 D2 − D0 DB3 ) > 0, then the Kolmogorov system (1.1) has two limit cycles bifurcating from the zero-Hopf equilibrium point (1, 1, 1). Moreover the following statements hold. √ (a) If K4 > 0, D2 D3 (a1 c1 + b1 c2 )(D0 D3 − D1 D2 ) > 0 and |D1 D2 − 2D0 D3 | < K4 ; then the two limit cycles are unstable and have a stable manifold formed by two cylinders and an unstable manifold formed by two cylinders. (b) If b1 D2 (a1 c1 + b1 c2 ) < 0, b1 D3 (D0 D3 − D1 D2 ) > 0 and √ • either K4 √> 0, b1 D1 (a1 c1 + b1 c2 )(D1 D2 − 2D0 D2 − K4 ) < 0 and b1 D1 (a1 c1 + b1 c2 )(D1 D2 − 2D0 D2 + K4 ) < 0, • or K4 ≤ 0 and b1 D1 (a1 c1 + b1 c2 )(D1 D2 − 2D0 D3 ) < 0; or if b1 D2 (a1 c1 + b1 c2 ) > 0, b1 D3 (D0 D3 − D1 D2 ) < 0 and √ • either K4 √> 0, b1 D1 (a1 c1 + b1 c2 )(D1 D2 − 2D0 D2 − K4 ) > 0 and b1 D1 (a1 c1 + b1 c2 )(D1 D2 − 2D0 D2 + K4 ) > 0, • or K4 ≤ 0 and b1 D1 (a1 c1 + b1 c2 )(D1 D2 − 2D0 D3 ) > 0; then one limit cycle is a local repeller, and the other is a local attractor. √ (c) If b1 D2 (a1 c1 + b1 c2 ) < 0, b1 D3 (D0 D3 − D1 D2 ) > 0, K4 > 0 and |D1 D2 − 2D0 D3 | < K4 ; or if D2 D3 (a1 c1 + b1 c2 )(D0 D3 − D1 D2 ) > 0 and √ • either K4 √> 0, b1 D1 (a1 c1 + b1 c2 )(D1 D2 − 2D0 D2 − K4 ) > 0 and b1 D1 (a1 c1 + b1 c2 )(D1 D2 − 2D0 D2 + K4 ) > 0, • or K4 ≤ 0 and b1 D1 (a1 c1 + b1 c2 )(D1 D2 − 2D0 D3 ) > 0; then both limit cycles are unstable. One limit cycle is a local repeller, and the other has a stable manifold formed by two cylinders and an unstable manifold formed by two cylinders. √ (d) If b1 D2 (a1 c1 + b1 c2 ) > 0, b1 D3 (D0 D3 − D1 D2 ) < 0 K4 > 0 and |D1 D2 − 2D0 D3 | < K4 ; or if D2 D3 (a1 c1 + b1 c2 )(D0 D3 − D1 D2 ) > 0 and √ • either K4 √> 0, b1 D1 (a1 c1 + b1 c2 )(D1 D2 − 2D0 D2 − K4 ) < 0 and b1 D1 (a1 c1 + b1 c2 )(D1 D2 − 2D0 D2 + K4 ) < 0, • or K4 ≤ 0 and b1 D1 (a1 c1 + b1 c2 )(D1 D2 − 2D0 D3 ) < 0; then one limit cycle is a local attractor, and the other is unstable and has a stable manifold formed by two cylinders and an unstable manifold formed by two cylinders. (e) If D2 D3 (a1 c1 + b1 c2 )(D0 D2 − D1 D2 ) > 0, K4 < 0 and D1 D2 = 2D0 D3 ; then one limit cycle is unstable and has a stable manifold formed by two cylinders and an unstable manifold formed by two cylinders and we cannot decide about the stability of the other. Theorem 1.5 is proved in section 2. In section 3 can be found examples showing that the conditions provided by this theorem are non-empty. 5 2. Proof of results Proof of Proposition 1.1. We want to characterize when the singular point (1, 1, 1) of system (1.1) is a zero- Hopf equilibrium. At first, through the change of variables (x, y, z) → (x + 1, y + 1, z + 1), we translate the point (1, 1, 1) to the origin of coordinates, obtaining the system: ẋ = (1 + x)(a1 z + a2 y + a3 z + a4 x2 + a5 xy + a6 xz + a7 y 2 + a8 yz + a9 z 2 ), ẏ = (1 + y)(b1 x + b2 y + b3 z + b4 x2 + b5 xy + b6 xz + b7 y 2 + b8 yz + b9 z 2 ), (2.1) ż = (1 + z)(c1 x + c2 y + c3 z + c4 x2 + c5 xy + c6 xz + c7 y 2 + c8 yz + c9 z 2 ). In order that the origin of system (2.1) can exhibit a zero-Hopf bifurcation we must require that the eigenvalues of the linear part of the system at the origin be of the form 0 and ±βi with β > 0. We compute the characteristic polynomial and require that it has the form λ(λ2 + β 2 ). Solving the resultant equation we get the five solutions given in (i)–(v). Proof of Theorem 1.3. We consider system (1.1) under conditions (ii) of Proposition 1.1, and we proceed to study the limit cycles bifurcating from the zero-Hopf equilibrium point, applying the averaging theory of first order, summarized in Theorem A.1 of Appendix A. To do so we perturb the parameters a2 , b1 , c2 and c3 which define the zero-Hopf equilibrium under the assumption (ii) as follows a3 b2 a1 b3 (a1 + b2 )2 + a3 c1 + β 2 a2 = + εa21 , b1 = + εb11 , c2 = − + εc21 , c3 = −a1 − b2 + εc31 , b3 a3 b3 where ε is a small parameter. We write the lineal part of system (2.1) at the origin in its real Jordan normal form, and the associated system becames system (1) in file ss[[2]]. Now we want to write the system in such a way that conditions of Theorem A.1 are satisfied. For this we write the system in cylindrical coordinates by means of the change of variables (X, Y, Z) → (r cos θ, r sin θ, Z) obtaining system (2) of file ss[[2]]. In order to study the periodic solutions in a neigborhood of the origin, i.e. in a neigborhood of the zero-Hopf equilibrium, we do the scaling (r, Z) → (εR, εZ), where ε > 0 is the same parameter used before. We obtain system (3) of file ss[[2]]. We take the variable θ as the new independent variable and so we obtain the system R0 = εF11 + O(ε2 ), Z 0 = εF12 + O(ε2 ), (2.2) with coefficients F11 and F12 given in the file ss[[2]]. Note that system (2.2) is in the normal form (A.1), so we can apply the averaging theory with T = 2π, x = (R, Z), t = θ and εR(θ, x, ε) = O(ε2 ). The functions F11 , F12 and R are C 2 in x and 2π-periodic in θ. Applying Theorem A.1 we compute the averaging function of first order f1 = (f11 (R, Z), f12 (R, Z)), and we obtain πR(A0 + A1 Z) π(A2 Z + A3 Z 2 + A4 R2 ) f11 = 2 , f12 = − , a3 b3 β 5 a23 b3 β 5 N and Ai , for i = 0, ..., 4, K1 and N are given in Appendix B. We look for the isolated solutions of the equation (f11 (R, Z), f12 (R, Z)) = (0, 0), and we obtain,  appart p √ from the origin, (R1 , Z1 ) = (0, −A2 /A3 ) and (R2 , Z2 ) = ± A0 (A1 A2 − A0 A3 )/(A1 A4 ), −A0 /A1 . We consider always the positive expression of R2 , i.e. we consider the positive sign if A1 > 0 and the negative sign if A1 < 0. We compute the Jacobian matrix of f1 , which is   π(A0 + A1 Z) πRA1  a23 b2 β 5 a23 b3 β 5     ,  2πRA4 π(A2 + 2A3 Z)  − 2 − a3 b3 β 5 N a23 b3 β 5 N 6 and its determinant is π 2 (−2A1 A4 R2 + (A0 + A1 Z)(A2 + 2A3 Z))/(a43 b23 β 10 N ). Evaluating the determinant at the solution (R1 , Z1 ) we get that it is equal to π 2 A2 (A0 A3 − A1 A2 )/(a43 b23 A3 β 10 N ), and at the solutions (R2 , Z2 ) we get that it is equal to 2π 2 A0(A1 A2 − A0 A3 )/(a43 b23 A1 β 10 N ). From the hypothesis considered these determinants are nonzero, therefore it follows from Theorem A.1 that for ε sufficiently small system (2.2) has two 2π-periodic solutions (R1 (θ, ε), Z1 (θ, ε)) and (R2 (θ, ε), Z2 (θ, ε)) such that (Rj (θ, ε), Zj (θ, ε)) → (Rj , Zj ) for j = 1, 2 when ε → 0. Moreover the Jacobian matrix evaluated at the solution (R1 , W1 ) has eigenvalues equal to πA2 /(a23 b3 β 5 N ) and π(A0 A3 − A1 A2 )/(a23 b3 A3 β 5 ). Since the eigenvalues of the Jacobian matrix evaluated at the solutions provide the stability of the fixed point corresponding to the Poincaré map defined in a neighborhood of the solution, if A2 b3 N > 0 and A3 b3 (A0 A3 − A1 A2 ) > 0, then the fixed point of the Poincaré map has an unstable manifold of dimension two, and the corresponding periodic solution is unstable and has an unstable manifold of dimension three, which is equivalent to say that is a repelling periodic orbit. If A2 b3 N < 0 and A3 b3 (A0 A3 − A1 A2 ) < 0, then the fixed point of the Poincaré map has a stable manifold of dimension two, and the associated periodic solution is stable and has a stable manifold of dimension three, which is equivalent to say that is a attracting periodic orbit. Finally, if A2 A3 N (A0 A3 − A1 A2 ) < 0, then the fixed point of the Poincaré application is a saddle point with a stable manifold of degree one and an unstable manifold of degree one, and the corresponding periodic solution is unstable and has a stable manifold formed by two cylinders and an unstable manifold formed by two cylinders. √ On the2other hand, the Jacobian matrix evaluated at (R2 , Z2 ) has eigenvalues equal to π(2A √0 A3 − A1 A2 ± K1 )/(2a3 b3 A1 β 5 N ), and√so its stability is as follows. If K1 > 0, b3 A1 N (2A0 A3 − A1 A2 + K1 ) > 0 and b3 A1 N (2A0 A3 − A1 A2 − K1 ) > 0 or if K1 < 0 and b3 A1 N (2A0 A3 − A1 A2 ) > 0, then the fixed point of the Poincaré map has an unstable manifold of dimension two, and the periodic solution √ is unstable and has an unstable manifold √ of dimension three. If K1 > 0, b 3 A 1 N (2A 0 A 3 − A 1 A 2 + K1 ) < 0 and b3 A1 N (2A0 A3 − A1 A2 K1 ) < 0 or if K1 < 0 and b3 A1 N (2A0 A3 − A1 A2 ) < 0, then the fixed point of the Poincaré map has an unstable manifold of dimension √ two, and the periodic √ solution is stable and has a stable manifold of dimension three. If K1 > 0 and − K1 < 2A0 A3 − A1 A2 < K1 , then the fixed point of the Poincaré map is a saddle point with a stable manifold of degree one and an unstable manifold of degree one, and the associated periodic solution is unstable and has a stable manifold formed by two cylinders and an unstable manifold formed by two cylinders. If K1 < 0 and A1 A2 = 2A0 A3 , the fixed point of the Poincaré map asociated with the periodic orbit is linearly stable, and we cannot decide about the stability of the periodic orbit. Combining the above information of the eigenvalues of the Jacobian matrix for both (R1 , Z1 ) and (R2 , Z2 ) we get statements (a)–(e) in the theorem. Now we shall go back through the changes of variables and we obtain two periodic solutions, for j = 1, 2, (xj (t, ε), yj (t, ε), zj (t, ε)) bifurcating from (1, 1, 1) with a period tending to 2π when ε → 0. Moreover, (xj (t, ε), yj (t, ε), zj (t, ε)) = (1, 1, 1) + O(ε) for j = 1, 2. This completes the proof of the theorem. Proof of Theorem 1.4. We consider system (1.1) under conditions (iii) of Proposition 1.1. In order to study the zero-Hopf bifurcation we perturb the parameters a1 , a2 , a3 , c2 and c3 which define the zero-Hopf equi- librium under conditions (iii) as follows b22 + β 2 a1 = εa11 , a2 = εa21 , a3 = εa31 , c2 = − + εc21 , c3 = −b2 + εc31 , b3 where ε is a parameter to be taken sufficiently small. We write the lineal part of system (2.1) at the origin in its real Jordan normal form, and the associated system becomes system (1) of file ss[[3]]. Then we write the system in cylindrical coordinates obtaining system (2) of file ss[[3]], and we do the scaling (r, Z) → (εR, εZ) obtaining system (3) in file ss[[3]]. As in the proof of Theorem 1.3 in order to apply Theorem A.1, we take the variable θ as the new independent variable obtaining a system R0 = εF11 + O(ε2 ), Z 0 = εF12 + O(ε2 ) (2.3) which coefficients F11 and F12 are given in the file ss[[3]]. The averaged function of first order f1 = (f11 (R, Z), f12 (R, Z)) is πR(B0 + B1 Z) π(B2 Z + B3 Z 2 + B4 R2 ) f11 = , f12 = , b23 β 5 b23 β 5 7 with Bi , for i = 0, ..., 4, and K2 given in Appendix B.Solving the  pequation (f11 (R, Z), f12 (R, Z)) = (0, 0) we  √ obtain two solutions (R1 , Z1 ) = (0, −B2 /B3 ) and (R2 , Z2 ) = ± B0 (B1 B2 − B0 B3 )/(B1 B4 ), −B0 /B1 . Again we consider always the positive expression of R2 . We compute the Jacobian matrix of f1 and we get   π(B0 + B1 Z) πRB1  b23 β 5 b23 β 5     ,  2πRB4 π(B2 + 2B3 Z)  b23 β 5 b23 β 5 whose determinant is π 2 (−2B1 B4 R2 + (B0 + B1 Z)(B2 + 2B3 Z))/(b43 β 10 ). The determinant at the solution (R1 , Z1 ) is π 2 B2 (B1 B2 − B0 B3 )/(b43 B3 β 10 ), and at the solutions (R2 , Z2 ) is 2π 2 B0 (B0 B3 − B1 B2 )/(b43 B1 β 10 ). From the hypothesis considered these determinants are nonzero, so it follows from Theorem A.1 that for ε sufficiently small, system (2.2) has two solutions (R1 (θ, ε), Z1 (θ, ε)) and (R2 (θ, ε), Z2 (θ, ε)) such that (Rj (θ, ε), Zj (θ, ε)) → (Rj , Zj ) for j = 1, 2 when ε → 0. The Jacobian matrix evaluated at the solution (R1 , W1 ) has eigenvalues equal to −πB2 /(b23 β 5 ) and π(B0 B3 − B1 B2 )/(b23 B3 β 5 ). We study the stability of the associated periodic orbit which is provided by these eigenvalues. If B2 < 0 and B3 (B0 B3 − B1 B2 ) > 0, the associated periodic solution is unstable and has an unstable manifold of dimension three. If B2 > 0 and B3 (B0 B3 − B1 B2 ) < 0, then the associated periodic solution is stable and has a stable manifold of dimension three. Finally if B2 B3 (B0 B3 − B1 B2 ) > 0, the periodic solution is unstable and has a stable manifold formed by two cylinders and an unstable manifold formed by two cylinders. √ On the2 other5 hand, the Jacobian matrix evaluated at (R√2 , Z2 ) has eigenvalues equal to π(B1 B √2 − 2B0 B3 ± K2 )/(2b3 B1 β ), and so if K2 > 0, B1 (B1 B2 − 2B0 B3 + K2 ) > 0 and B1 (B1 B2 − 2B0 B3 − K2 ) > 0, or if K2 < 0 and B1 (B1 B2 − 2B0 B3 ) > 0, then the associated periodic √ solution is unstable and has an √ unstable manifold of dimension three. If K2 > 0, B1 (B1 B2 − 2B0 B3 + K2 ) < 0 and B1 (B1 B2 − 2B0 B3 − K2 ) < 0, or if K2 < 0 and B1 (B1 B2 − 2B0 B3 ) < 0, then √ the associated periodic solution √ is stable and has a stable manifold of dimension three. If K2 > 0 and − K2 < B1 B2 − 2B0 B3 < K2 , then the associated periodic solution is unstable and has a stable manifold formed by two cylinders and an unstable manifold formed by two cylinders. If K2 < 0 and B1 B2 − 2B0 B3 = 0, the fixed point of the Poincaré map associated with the periodic orbit is linearly stable, and we cannot decide about the stability of the periodic orbit. Combining the above information of the eigenvalues of the Jacobian matrix for both (R1 , Z1 ) and (R2 , Z2 ) we get statements (a)–(f) of the theorem. Now we shall go back through the changes of variables and we obtain two periodic solutions,for j = 1, 2, (xj (t, ε), yj (t, ε), zj (t, ε)) bifurcating from (1, 1, 1) with a period tending to 2π when ε → 0. Moreover, (xj (t, ε), yj (t, ε), zj (t, ε)) = (1, 1, 1) + O(ε) for j = 1, 2. This completes the proof of the theorem. Proof of Theorem 1.5. We consider system (1.1) under conditions (v) of Proposition 1.1. In order to study the zero-Hopf bifurcation we perturb the parameters a2 , a3 , b2 , b3 and c3 which define the zero-Hopf equilibrium point into the form a21 + β 2 a2 = − + εa21 , a3 = εa31 , b2 = −a1 + εb21 , b3 = εb31 c3 = εc31 b1 where ε is a sufficiently small parameter. We write the system with the linear part at the origin in its real Jordan normal form, then we write it in cylindrical coordinates, and finally we do the scaling (r, Z) → (εR, εZ). Thus we obtain respectively systems (1), (2) and (3) of file ss[[5]]. Taking θ as the new independent variable we obtain a system in the form R0 = εF11 + O(ε2 ), Z 0 = εF12 + O(ε2 ) (2.4) with coefficients F11 and F12 given in the file ss[[5]]. As in previous proofs we are in conditions to apply Theorem A.1. Now the averaged function of first order f1 = (f11 (R, Z), f12 (R, Z)) is πR(D0 + D1 Z) π(D2 Z + D3 Z 2 + D4 R2 ) f11 = , f12 = , b1 β 5 b1 (a1 c1 + b1 c2 )β 5 8 with Di , for i = 0, ..., 4, and K4 given in Appendix B. We look  p for the solutions of (f11 (R, Z), f12 (R, Z)) = √ (0, 0), and we obtain (R1 , Z1 ) = (0, −D2 /D3 ) and (R2 , Z2 ) = ± D0 (D1 D2 − D0 D3 )/(D1 D4 ), −D0 /D1 , considering the positive expression of R2 . We compute the Jacobian matrix of f1 and we get   π(D0 + D1 Z) πRD1  b1 β 5 b1 β 5     ,  2πRD4 π(D2 + 2D3 Z)  b1 β 5 (a1 c1 + b1 c2 ) b1 β 5 (a1 c1 + b1 c2 ) whose determinant is π 2 (−2D1 D4 R2 + (D0 + D1 Z)(D2 + 2D3 Z))/(b21 β 10 (a1 c1 + b1 c2 )). The determinant at the solution (R1 , Z1 ) is π 2 D2 (D1 D2 − D0 D3 )/(b21 D3 β 10 (a1 c1 + b1 c2 )), and at the solution (R2 , Z2 ) is 2π 2 D0 (D0 D3 − D1 D2 )/(b21 D1 β 10 (a1 c1 + b1 c2 )), and both are nonzero by the hypotheses. By Theorem A.1, for ε sufficiently small system (2.4) has two solutions (R1 (θ, ε), Z1 (θ, ε)) and (R2 (θ, ε), Z2 (θ, ε)) such that (Rj (θ, ε), Zj (θ, ε)) → (Rj , Zj ) for j = 1, 2 when ε → 0. The Jacobian matrix evaluated at the solution (R1 , W1 ) has eigenvalues equal to −πD2 /(b1 β 5 (a1 c1 +b1 c2 )) and π(D0 D3 − D1 D2 )/(b1 D3 β 5 ). We study the stability of the associated periodic orbit which is provided by these eigenvalues. If b1 D2 (a1 c1 + b1 c2 ) < 0 and b1 D3 (D0 D3 − D1 D2 ) > 0, the associated periodic solution is unstable and has an unstable manifold of dimension three. If b1 D2 (a1 c1 +b1 c2 ) > 0 and b1 D3 (D0 D3 −D1 D2 ) < 0, then the associated periodic solution is stable and has a stable manifold of dimension three. Finally if D2 D3 (a1 c1 + b1 c2 )(D0 D3 − D1 D2 ) > 0, the associated periodic solution is unstable and has a stable manifold formed by two cylinders and an unstable manifold formed by two cylinders. √ On the other5 hand, the Jacobian matrix evaluated at (R2 , Z2 ) has eigenvalues equal √ to π(D1 D2 −2D0 D3 ± K4 )/(2b1 D1 β (a1 c1 +b√ 1 c2 )). Then if K 4 > 0, b 1 D 1 (a 1 c1 +b1 c2 )(D1 D2 −2D 0 D 3 + K4 ) > 0 and b1 D1 (a1 c1 + b1 c2 )(D1 D2 −2D0 D3 − K4 ) > 0, or if K4 < 0 and b1 D1 (a1 c1 +b1 c2 )(D1 D2 −2D0 D3 ) > 0, then the associated periodic solution is unstable √ and has an unstable manifold of dimension three. √ If K4 > 0, b1 D1 (a1 c1 + b1 c2 )(D1 D2 − 2D0 D3 + K4 ) < 0 and b1 D1 (a1 c1 + b1 c2 )(D1 D2 − 2D0 D3 − K4 ) < 0, or if K4 < 0 and b1 D1 (a1 c1 + b1 c2 )(D1 D2 − 2D0 D3 ) < 0, then√the associated periodic solution √ is stable and has a stable manifold of dimension three. If K4 > 0 and − K4 < D1 D2 − 2D0 D3 < K4 , then the associated periodic solution is unstable and has a stable manifold formed by two cylinders and an unstable manifold formed by two cylinders. If K4 < 0 and D1 D2 = 2D0 D3 , the fixed point of the Poincaré map associated with the periodic orbit is linearly stable, and we cannot decide about the stability of the periodic orbit. Combining the above information we get statements (a)–(e) in the theorem, and going back through the changes of variables we obtain two periodic solutions (xj (t, ε), yj (t, ε), zj (t, ε)) for j = 1, 2 bifurcating from (1, 1, 1) with a period tending to 2π when ε → 0. Moreover, (xj (t, ε), yj (t, ε), zj (t, ε)) = (1, 1, 1) + O(ε) for j = 1, 2. This completes the proof of the theorem. 3. Examples 3.1. Examples of Theorem 1.2 We give examples showing that the conditions provided by Theorem 1.2 are non-empty. The system   11032749 2 ẋ = x 1 − 2(x − 1) + (x − 1) − 6(y − 1) + (x − 1)(y − 1) − z , 65536   193 ẏ = y 1 + 5(x − 1) − (y − 1) − z , 16      1048639 94765 225 12865 279 ż = z − (x − 1)2 − (y − 1) + (z − 1) + ε + (x − 1) + ε , 128 768 16 48 32 has two limit cycles with the stability given in (a) of Theorem 1.2. The following systems verify, respec- tively, the four sets of conditions of statement (b) of Theorem 1.2, so all of them have two limit cycles with 9 the stability given in (b). ẋ = x 1 + 2(x − 1)2 − x − 2(y − 1) + 2(x − 1)(y − 1) + 2(y − 1)2 − 2(z − 1) + 2(x − 1)(z − 1) +(y − 1)(z − 1)) , ẏ = y (−2 + 2(x − 1) + y + z) ,   11 ż = z − (x − 1) + (y − 1)2 − (z − 1)2 − (y − 1)ε + (z − 1)ε , 8  ẋ = x −1 + 2(x − 1)2 + x − 2(y − 1) + 2(y − 1)2 + 2(z − 1) + 2(x − 1)(z − 1) , ẏ = y (−2 + x + z) ,     1 ż = z −1 + (y − 1)2 + y − (z − 1)2 + (x − 1) − − 2ε + (z − 1)(−1 − ε) , 2   1 ẋ = x 1 + 2(x − 1) + (y − 1) + (y − 1)2 − 100(x − 1)(z − 1) − z , 2   10 ẏ = y −4(x − 1) − (y − 1) − 2(z − 1) , 9      668 2953 8 ż = z (x − 1) + (y − 1) + ε + (z − 1) − + 5ε , 495 8910 9   1 ẋ = x 1 + 2(x − 1) + (y − 1) + 2(y − 1)2 − 100(x − 1)(z − 1) − z , 2   10 ẏ = y −4(x − 1) − (y − 1) − 2(z − 1) , 9      2953 668 8 ż = z (y − 1) + (x − 1) − 2ε + (z − 1) − + 2ε , 8910 495 9 The following systems verify, respectively, the three sets of conditions in statement (c) of Theorem 1.2, so all of them have two limit cycles with the stability given in (c).  ẋ = x −2 + (x − 1)2 + x + z ,  ẏ = y −3 + (x − 1)2 + x + (x − 1)(y − 1) + y + z , ż = z ((x − 1)(y − 1) + (x − 1)(−2 + ε) + (y − 1)(−2 + ε) + (z − 1)(−2 + ε)) , ẋ = x 2(x − 1)2 + x + 2(x − 1)(y − 1) + 2(y − 1)2 − y +2(z − 1) + 2(x − 1)(z − 1) + 2(y − 1)(z − 1)) , ẏ = y (−3 + x + y + z) ,     1 7 ż = z (y − 1) + (y − 1)2 − (z − 1)2 + (x − 1) − − 2ε + (z − 1)(−2 − ε) , 5 5  ẋ = x −1 + x + (y − 1)2 − y + (z − 1)2 + z , ẏ = y (−2 + x + y) , ż = z (−1 + y + (z − 1)(−2 − ε) + (x − 1)(−3 + ε)) . 10 The following systems verify, respectively, the three sets of conditions in statement (d) of Theorem 1.2, so all of them have two limit cycles with the stability given in (d). ẋ = x (2 − y − z) ,  ẏ = y −2 + (x − 1)2 + x + (x − 1)(y − 1) + y ,    1 3 ż = z 1 + (x − 1) + (x − 1)(y − 1) − z + (y − 1) − − ε , 4 4   1 ẋ = x 1 − x + (y − 1) , 2 ẏ = y (−3 + x + y + z) , 5  ż = z − (x − 1)2 − (y − 1) − (x − 1)(y − 1) + (z − 1)2 + (z − 1)ε + (x − 1)(5 + ε) , 2  ẋ = x −2 + (x − 1)2 + x + z ,  ẏ = y −3 + (x − 1)2 + x + (x − 1)(y − 1) + y + z , ż = z (−2(y − 1) − 2(z − 1) + (x − 1)(−2 + ε)) . Finally, the system ẋ = x (2 − x − z) ,   69 2 231 2 ẏ = y 2 − (x − 1) − x − y − (z − 1) , 5 10 8 58 109  ż = z − (x − 1)2 + 2(y − 1) − (x − 1)(y − 1) − (y − 1)2 + 2(z − 1) − (z − 1)2 + (x − 1)(4 − ε) , 5 5 5 has two limit cycles with the stability given in (e) of Theorem 1.2. 3.2. Examples of Theorem 1.3 We give examples showing that the conditions provided by Theorem 1.3 are non-empty. The system   1 2 ẋ = x 2 + (x − 1) − x − (x − 1)(y − 1) − z − (1 + ε)(y − 1) , 2 ẏ = y (3 − x − y − z) ,   3 7 ż = z (x − 1) + (y − 1) + 2(z − 1) + 4(x − 1)(z − 1) , 2 2 has two limit cycles with the stability given in (a) of Theorem 1.3. The following systems verify, respectively, the four sets of conditions in statement (b) of Theorem 1.3, so all of them have two limit cycles with the stability given in (b).   7 ẋ = x 2 − (x − 1)2 − x + (x − 1)(y − 1) − z + (−1 + ε)(y − 1) , 8 ẏ = y (3 − x − y − z) ,   7 37 ż = z (x − 1) + (y − 1) + 6(x − 1)(z − 1) + (2 + 5ε)(z − 1) , 4 16 11   1 ẋ = x 2 − (x − 1)2 − x + (x − 1)(y − 1) − z + (−1 + ε)(y − 1) , 2 ẏ = y (3 − x − y − z) ,   3 7 ż = z (x − 1) + (y − 1) + 5(x − 1)(z − 1) + (2 + ε)(z − 1) , 2 2   7 ẋ = x 2 − (x − 1)2 − x − (x − 1)(y − 1) − z + (1 + ε)(y − 1) , 8 ẏ = y (−1 + x − y + z) ,   7 37 ż = z (x − 1) − (y − 1) + 5(x − 1)(z − 1) + (2 − 5ε)(z − 1) , 4 16   3 2 ẋ = x 2 + (x − 1) − x + (x − 1)(y − 1) − z + (1 + ε)(y − 1) , 4 ẏ = y (−1 + x − y + z) , ż = z (−5(y − 1) + (2 − 3ε)(z − 1)) , The following systems verify, respectively, the three sets of conditions in statement (c) of Theorem 1.3, so all of them have two limit cycles with the stability given in (c).   1 ẋ = x 2 + (x − 1)2 − x − (x − 1)(y − 1) − z + (ε − 1)(y − 1) , 2 ẏ = y (3 − x − y − z) ,   3 7 ż = z (x − 1) + (y − 1) + 4(x − 1)(z − 1) + (2 + ε)(z − 1) , 2 2     3 ẋ = x −x − (x − 1)(y − 1) + z − + ε (y − 1) , 4   3 ẏ = y x + (y − 1) − z , 4     17 1 ż = z (y − 1) + 2(x − 1)(z − 1) + + ε (z − 1) , 16 4   1 2 ẋ = x 1 + (x − 1) − x − (x − 1)(y − 1) − 3(z − 1) − (3 + ε)(y − 1) , 2   1 ẏ = y 2 − (x − 1) − y − z , 3 ż = z (1 + 2(x − 1) − y + 2(x − 1)(z − 1) + (2 + 2ε)(z − 1)) , The following systems verify, respectively, the three sets of conditions in statement (d) of Theorem 1.3, so all of them have two limit cycles with the stability given in (d).   1 ẋ = x 2 − (x − 1)2 − x − (x − 1)(y − 1) − z + (1 + ε)(y − 1) , 2 ẏ = y (−1 + x − y + z) ,   3 7 27 ż = z (x − 1) − (y − 1) + (x − 1)(z − 1) + (2 − ε)(z − 1) , 2 2 8 12     1 1 ẋ = x 3(x − 1)2 − (x − 1)(y − 1) − (z − 1) + − + ε (y − 1) , 2 4   1 ẏ = y 1 − (y − 1) − z , 2     1 393 1 ż = z − (x − 1) + (y − 1) + 33(x − 1)(z − 1) + + 29ε (z − 1) , 4 1024 2  ẋ = x 2 − 2(x − 1)2 − x − (x − 1)(y − 1) − z + (1 + ε)(y − 1) , ẏ = y (−1 + x − y + z) ,   3 7 ż = z (x − 1) − (y − 1) − (x − 1)(z − 1) + (2 − ε)(z − 1) , 2 2 Finally, the system ẋ = x (2 − x − z + (−1 + ε)(y − 1)) ,  ẏ = y 3 − x − y − (z − 1)2 − z ,   3 7 44 ż = z (x − 1) + (y − 1) + (x − 1)(z − 1) + (2 + ε)(z − 1) , 2 2 3 has two limit cycles with the stability given in (e) of Theorem 1.3. 3.3. Examples of Theorem 1.4 We give examples showing that the conditions provided by Theorem 1.4 are non-empty. The system  ẋ = x 2(x − 1)2 − (y − 1)2 − 2ε(x − 1) + ε(y − 1) , ẏ = y (3 − x − y − z) , ż = z (−1 + 2(y − 1) + z) , has two limit cycles with the stability given in (a) of Theorem 1.4. The system  ẋ = x −2(x − 1)2 + (y − 1)2 − ε(y − 1) , ẏ = y (3 − x − y − z) , ż = z (−1 + 2(y − 1) + z) , verfies the first set of conditions in statement (b) of Theorem 1.4, and the system  ẋ = x 4(x − 1)2 − 2(x − 1)(y − 1) − (y − 1)2 − ε(y − 1) , ẏ = y (3 − x − y − z) , ż = z (−1 + 2(y − 1) + z) , verifies the second set of conditions. In both of them there exist two limit cycles with the stability given in (b). The system   7 ẋ = x − (x − 1)2 + 4(x − 1)(y − 1) − (y − 1)2 − 2ε(x − 1) + ε(y − 1) , 2 ẏ = y (1 − x − y + z) , ż = z (−1 − 2(y − 1) + z) , 13 verfies the first set of conditions in statement (c) of Theorem 1.4, and the system  ẋ = x −3(x − 1)2 + (y − 1)2 − 2ε(x − 1) + ε(y − 1) , ẏ = y (1 − x − y + z) , ż = z (−1 − 2(y − 1) + z) , verifies the second set of conditions. In both of them there exist two limit cycles with the stability given in (c). The following systems verify, respectively, the three sets of conditions in statement (d) of Theorem 1.4, so all of them have two limit cycles with the stability given in (d).  ẋ = x −2(x − 1)2 + (y − 1)2 − (x − 1)(z − 1) − 4ε(x − 1) + ε(z − 1) , ẏ = y (2 − x − z) , ż = z (y − 1) ,  ẋ = x −2(x − 1)2 + (y − 1)2 − ε(y − 1) , ẏ = y (3 − x − y − z) , ż = z (−1 + 2(y − 1) + z) ,  ẋ = x −2(x − 1)2 + (y − 1)2 − (x − 1)(z − 1) + 4ε(x − 1) − ε(z − 1) , ẏ = y (2 − x − z) , ż = z (y − 1) . The following systems verify, respectively, the three sets of conditions in statement (e) of Theorem 1.4, so all of them have two limit cycles with the stability given in (e).   1 ẋ = x (x − 1)2 − (y − 1)2 + 2ε(x − 1) − ε(y − 1) , 2 ẏ = y (3 − x − y − z) , ż = z (−1 + 2(y − 1) + z) ,   1 1 ẋ = x − (x − 1)2 − (x − 1)(y − 1) + (y − 1)2 + ε(y − 1) , 16 2 ẏ = y (1 − x − y − z) , ż = z (−1 − 2(y − 1) + z) ,  ẋ = x (y − 1)2 + (x − 1)(z − 1) − 2ε(x − 1) − ε(z − 1) , ẏ = y (−x + z) , ż = z (y − 1) . Finally, the system  ẋ = x −(x − 1)2 − (y − 1)2 − ε(z − 1) , ẏ = y (2 − x − z) ,  ż = z −1 + y − (z − 1)2 , has two limit cycles with the stability given in (f) of Theorem 1.4. 14 3.4. Examples of Theorem 1.5 We give examples showing that the conditions provided by Theorem 1.5 are non-empty. The system  ẋ = x 1 + 2(x − 1)2 − x + 2(y − 1) − (z − 1)2 + ε(z − 1) , ẏ = y (1 − x + (1 + 2ε)(y − 1)) , ż = z (1 − x) , has two limit cycles with the stability given in (a) of Theorem 1.5. The following systems verify, respectively, the four sets of conditions in statement (b) of Theorem 1.5, so all of them have two limit cycles with the stability given in (b).  ẋ = x 1 − 2(x − 1)2 − x + 2(y − 1) + 3(x − 1)(z − 1) + (z − 1)2 − ε(z − 1) ,     7ε ẏ = y 1 − x + 1 − (y − 1) , 8 ż = z (1 − x) ,  ẋ = x 1 − 2(x − 1)2 − x + 2(y − 1) − 5(x − 1)(z − 1) + (z − 1)2 + ε(z − 1) , ẏ = y (−x + y) , ż = z (x − 1) ,  ẋ = x 1 + 2(x − 1)2 − x − 2(y − 1) + 3(x − 1)(z − 1) − (z − 1)2 − ε(z − 1) , ẏ = y (−2 + x + y) , ż = z (x − 1) ,  ẋ = x 1 + 2(x − 1)2 − x − 2(y − 1) + 5(x − 1)(z − 1) − (z − 1)2 − ε(z − 1) , ẏ = y (−2 + x + y) , ż = z (x − 1) , The following systems verify, respectively, the three sets of conditions in statement (c) of Theorem 1.5, so all of them have two limit cycles with the stability given in (c).  ẋ = x 1 + 3(x − 1)2 − x + 2(y − 1) − (x − 1)(z − 1) + ε(z − 1) , ẏ = y (−x + y) ,  ż = z −1 + x + 2(z − 1)2 ,  ẋ = x 1 − x − 2(y − 1) − (x − 1)(z − 1) + (z − 1)2 − ε(z − 1) ,     7ε ẏ = y −1 + x + 1 + (y − 1) , 8 ż = z (x − 1) ,  ẋ = x 1 + (x − 1)2 − x − 2(y − 1) − (x − 1)(z − 1) − ε(z − 1) , ẏ = y (−1 + x + (1 − 4ε)(y − 1)) ,   1 ż = z 1 − x − (z − 1)2 . 2 The following systems verify, respectively, the three sets of conditions in statement (d) of Theorem 1.5, 15 so all of them have two limit cycles with the stability given in (d).  ẋ = x 1 − 3(x − 1)2 − x − 2(y − 1) + (x − 1)(z − 1) − ε(z − 1) , ẏ = y (−2 + x + y) ,  ż = z −1 + x − 2(z − 1)2 ,  ẋ = x 1 − 3(x − 1)2 − x + 2(y − 1) + (x − 1)(z − 1) + ε(z − 1) , ẏ = y (1 − x + (1 + 2ε)(y − 1)) ,  ż = z 1 − x + 2(z − 1)2 ,  ẋ = x 1 − (x − 1)2 − x + 2(y − 1) + (x − 1)(z − 1) + ε(z − 1) , ẏ = y (1 − x + (1 + 4ε)(y − 1)) ,   1 2 ż = z 1 − x + (z − 1) . 2 Finally, the system   7 ẋ = x 1 − x + 2(y − 1) − (x − 1)(z − 1) + ε(z − 1) , 8 ẏ = y (−x + y) ,   1 ż = z −1 + x + (z − 1)2 , 2 has two limit cycles with the stability given in (e) of Theorem 1.5. A. Averaging theory We summarize the averaging theory of first order, which provides sufficient conditions for the existence of periodic orbits for a periodic differential system depending on small parameters. For additional details and the proof of the result stated in this appendix, see [11, 12, 13] and [14, Theorems 11.5, 11.6]. Theorem A.1. We consider the following differential system x0 (t) = εF1 (t, x) + ε2 R(t, x, ε), (A.1) where F1 : 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 . We define f1 : D → Rn as Z T f1 (z) = F1 (s, z)ds, (A.2) 0 and assume that: 1. F1 and R are locally Lipschitz with respect to x; 2. for a ∈ D with f1 (a) = 0, there exists a neighboirhood V of a such that f1 (z) 6= 0 for all z ∈ V \(a) and dB (f1 , V, 0) 6= 0, where dB (f1 , V, 0) is the Brouwer degree. Then for |ε| > 0 sufficiently small, there exists a T -periodic solution ϕ(·, ε) of system (A.1) such that ϕ(·, ε) → a as ε → 0. The kind of stability of the limit cycle is given by the eigenvalues of the Jacobian matrix at the point a. Note that a sufficient condition for showing that the Brouwer degree of a function f at a point a is nonzero, is that the Jacobian of the function f at a, when it is defined, is nonzero, see [15]. 16 B. Notation  A0 = a3 β 2 (a23 b11 − b23 a21 )(a1 (a1 + b2 ) + a3 c1 ) + a3 β 2 (a3 b11 + b3 c31 ) ,  A1 = a61 (−2a9 b3 + 2a3 b9 ) + 2a51 b3 (a8 b3 − 4a9 b2 ) − a23 b6 + a3 (a6 b3 − b3 b8 + 4b2 b9 )  + a23 2 −b3 a9 b22 + b3 (a7 b3 − a8 b2 ) + a33 b4 − a23 (a4 b3 − b3 b5 + b2 b6 )  + a3 b3 (a6 b2 − a5 b3 + b3 b7 − b2 b8 ) + a3 b22 b9 c21 + c1 4a23 b4 − 2a3 b2 b6 + a3 b3  (3b5 − 2a4 + c6 ) + b3 a6 b2 + b22 + b3 (2b7 − a5 + c8 ) + b2 (b3 − b8 − 2c9 ) β 2  + (2a3 b4 + b3 (b5 + c6 ))β 4 + a1 a3 −2a33 c1 (b6 c1 − 2b2 b4 ) + a23 (2c1 (−2b2  (a4 b3 − b3 b5 + b2 b6 ) + (a6 b3 − b3 b8 + 2b2 b9 )c1 ) + (4b2 b4 + (b3 − 4b6 )c1 )β 2  a3 2c1 2b2 b3 (a6 b2 − a5 b3 + b3 b7 − b2 b8 ) + 2b32 b9 + b3 (a8 b3 − 2a9 b2 )c1  + −2a4 b2 b3 − 2b22 b6 + 4b2 b9 c1 + b2 b3 (3b5 + 2c1 + c6 ) + b3 c1 (3a6 − 3b8 − 2c9 )  β 2 + (b3 − 2b6 )β 4 + b3 4b2 b3 (a8 b2 − a7 b3 )c1 + a6 b22 + b32 + 2a8 b3 c1 + b2 b3  (2b7 − a5 + c8 ) + b22 (b3 − b8 − 2c9 ) β 2 + (a6 + b2 − b8 − 2c9 )β 4 − 4a9 b2 c1 b22   + β2 + a41 2a33 b4 + 2a3 3a6 b2 b3 − a5 b23 + b23 b7 − 3b2 b3 b8 + 6b22 b9 − 2a9 b3 c1 + a23 (2b3 b5 − 2a4 b3 − 6b2 b6 + 4b9 c1 ) + a3 (b3 + 4b9 )β 2 − 2b3 (b3 (a7 b3 − 3a8 b2 )  + 2a9 (3b22 + β 2 ) + a21 4a43 b4 c1 + 2a33 b22 b4 − 4b2 b6 c1 + c1 (2b3 b5 − 2a4 b3 + b9 c1 )  + 2b4 β 2 + a3 2b22 b3 (a6 b2 − a5 b3 + b3 b7 − b2 b8 ) + 2b42 b9 − 4b3 3a9 b22 + b3 (a7 b3 − 2a8 b2 )) c1 + 4a6 b2 b3 + b22 (3b3 + 4b9 ) + b3 (−4a9 c1 + b3 (2b7 − a5 + c8 ))  + b2 b3 (b3 − 4(b8 + c9 ))) β 2 + (b3 + 2b9 )β 4 + a23 −2b32 b6 + 8b2 b3 (a6 − b8 )c1 − 2b3 c1 (2a5 b3 − 2b3 b7 + a9 c1 ) + 2b22 (b3 b5 + 6b9 c1 ) + (b2 (b3 − 6b6 ) + 4b9 c1  + b3 (3b5 + c1 + c6 )) β 2 − 2a4 b3 (b22 + β 2 ) − 2b3 a9 (b22 + β 2 )2 − b2 b3 (−a7 b2 b3  + a8 (b22 + β 2 ) + a31 2a3 b2 b3 (3a6 b2 − 2a5 b3 + 2b3 b7 − 3b2 b8 ) + 4b32 b9 + 2b3 c1 (a8 b3 − 3a9 b2 )) + 4a33 (b2 b4 − b6 c1 ) + a3 (3a6 b3 + 3b2 b3 − 3b3 b8 + 8b2 b9 − 2b3 c9 )  β 2 + a23 −4a4 b2 b3 − 6b22 b6 + 4b3 (a6 − b8 )c1 + 4b2 (b3 b5 + 3b9 c1 ) + (b3 − 4b6 )β 2  + 2b3 −4a9 b2 (b22 + β 2 ) + b3 −2a7 b2 b3 + a8 (3b22 + β 2 ) ,    2 A2 = − 2a3 β 2 a23 b11 − b23 a21 (a1 (a1 + b2 ) + a3 c1 ) + a23 b11 β 2 (a1 (a1 + b2 ) + a3 c1 )   + 2a21 + 2a1 b2 + b22 + 2a3 c1 β 2 + β 4 ,    2 A3 = − 2 (a1 (a1 + b2 ) + a3 c1 ) + 2a21 + 2a1 b2 + b22 + 2a3 c1 β 2 + β 4 a61 (a3 b9  − a9 b3 ) + a51 b3 (a8 b3 − 4a9 b2 ) − a23 b6 + a3 (a6 b3 − b3 b8 + 4b2 b9 )  + a23 −b3 a9 b22 + b3 (a7 b3 − a8 b2 ) + a33 b4 − a23 (a4 b3 − b3 b5 + b2 b6 ) + a3 b3  (a6 b2 − a5 b3 + b3 b7 − b2 b8 ) + a3 b22 b9 c21 + b3 (a6 b2 − a5 b3 ) + 2a23 b4  + a3 (b3 b5 − 2a4 b3 − b2 b6 )) c1 β 2 + (a3 b4 − a4 b3 )β 4 + a1 a3 a33 c1 (2b2 b4 − b6 c1 ) + a23 c1 (−2b2 (a4 b3 − b3 b5 + b2 b6 ) + (a6 b3 − b3 b8 + 2b2 b9 ) c1 ) + 2β 2 (b2 b4 − b6 c1 )) + a3 c1 2b2 b3 (a6 b2 − a5 b3 + b3 b7 − b2 b8 ) + 2b32 b9 + b3 c1 (a8 b3   − 2a9 b2 )) + b2 b3 b5 − 2a4 b2 b3 − b22 b6 + 2a6 b3 c1 − b3 b8 c1 + 2b2 b9 c1 β 2 − b6 β 4  + b3 2b2 b3 c1 (a8 b2 − a7 b3 ) + a6 b22 − a5 b2 b3 + a8 b3 c1 β 2 + a6 β 4 − 2a9 b2 c1 b22  + β2 + a41 a33 b4 + a23 (b3 b5 − a4 b3 − 3b2 b6 + 2b9 c1 ) + a3 3a6 b2 b3 − a5 b23  + b23 b7 − 3b2 b3 b8 + 6b22 b9 − 2a9 b3 c1 + 2b9 β 2 − b3 b3 (a7 b3 − 3a8 b2 ) + 2a9 3b22 17  + β2 + a31 2a33 (b2 b4 − b6 c1 ) + a23 2b2 b3 (b5 − a4 ) − 3b22 b6 + 2a6b3 c1  − 2b3 b8 c1 + 6b2 b9 c1 − 2b6 β 2 + a3 3b22 b3 (a6 − b8 ) − 2b2 b23 (b7 − a5 ) + 4b32 b9   − 6a9 b2 b3 c1 + 2a8 b23 c1 + (2a6 b3 − b3 b8 + 4b2 b9 ) β 2 + b3 −4a9 b2 b22 + β 2  + b3 3a8 b22 − 2a7 b2 b3 + a8 β 2 + a21 2a43 b4 c1 + a33 b22 b4 − 4b2 b6 c1 + c1 (2b3 b5  − 2a4 b3 + b9 c1 ) + 2b4 β 2 − a23 b32 b6 + 4b2 b3 c1 (b8 − a6 ) + b3 c1 (2a5 b3 − 2b3 b7  + a9 c1 ) − b22 (b3 b5 + 6b9 c1 ) + (3b2 b6 − b3 b5 − 2b9 c1 )β 2 + a4 b3 (b22 + 2β 2 ) + a3 b22 b23 b7 − b32 b3 b8 + b42 b9 − 6a9 b22 b3 c1 + 4a8 b2 b23 c1 − 2a7 b33 c1 + β 2 (−b2 b3 b8   + 2b22 b9 − 2a9 b3 c1 + b9 β 4 − a5 b23 (b22 + β 2 ) + a6 b2 b3 (b22 + 3β 2 ) + b3 −a9 b22 2  + β 2 + b2 b3 −a7 b2 b3 + a8 (b22 + β 2 ) ,  2 A4 = − (a1 + b2 ) (a1 (a1 + b2 ) + a3 c1 ) β + a1 β 2 a33 b4 − a23 (a4 b3 − b3 b5 + b6 (a1 + b2 ))  − b3 a21 a9 + 2a1 a9 b2 + a9 b22 − a1 a8 b3 − a8 b2 b3 + a7 b23 + a9 β 2 + a3 (a6 b2 b3  − a5 b23 + b23 b7 − b2 b3 b8 + a21 b9 + b22 b9 + a1 (a6 b3 − b3 b8 + 2b2 b9 ) + b9 β 2 ,  K1 = A21 A22 − 4A0 A1 A2 A3 + 2A1 (a1 (a1 + b2 ) + a3 c1 )2 + β 2 2a21 + 2a1 b2 + b22 + 2a3 c1   + β 4 + 4A20 A3 A3 + 2A1 β 2 2a21 + 2a1 b2 + 2a3 c1 + b22 + (a1 (a1 + b2 )  2 + a3 c1 ) + β 4 , N = (a1 (a1 + b2 ) + a3 c1 )2 + (2a21 + 2a1 b2 + b22 + 2a3 c1 )β 2 + β 4 6= 0. Under conditions (iii) of Proposition 1.1, the expressions of Bi with i = 0, ..., 4 and K2 are the following.  B0 = β 2 b3 (b1 b2 + b3 c1 )(a31 b2 − a21 b3 ) + β 2 (a31 b1 + b3 c31 ) , B1 = b3 (b1 b2 + b3 c1 )(2(a8 b2 − a7 b3 )(b1 b2 + b3 c1 ) + β 2 (2a8 b1 + a6 b2 + b22 + b3 (−a5 + 2b7 + c8 ) +b2 (b3 − b8 − 2c9 ))) + β 4 b3 (b3 (b5 + c6 ) + b1 (a6 + b2 − b8 − 2c9 )) − 2a9 (b2 b3 c1 + b1 (b22 + β 2 ))2 , B2 = 2b3 β 2 ((b1 b2 + b3 c1 )(a21 b3 − a31 b2 ) + β 2 (a11 b3 − a31 b1 )), B3 = 2(b3 ((a7 b3 − a8 b2 )(b1 b2 + b3 c1 )2 ) − (a8 b1 + a6 b2 − a5 b3 )(b1 b2 + b3 c1 )β 2 + (a4 b3 − a6 b1 )β 4 ) + a9 (b2 b3 c1 + b1 (b22 + β 2 ))2 ), B4 = β 4 (b3 (a7 b3 − a8 b2 ) + a9 (b22 + β 2 )), K2 = B12 B2 (8B0 + B2 ) − 4B0 B1 (2B0 + B2 )B3 + 4B02 B32 . On the other hand, under conditions (iv) of Proposition 1.1, we redefine the expressions of Bi with i = 0, ..., 4 as follows: B0 = β 2 a3 (a1 a2 + a3 c2 )(b31 a1 − b11 a3 ) + β 4 a3 (b31 a2 + a3 c31 ), B1 = a3 (a1 a2 + a3 c2 ) 2(b6 a1 − b4 a3 )(a1 a2 + a3 c2 ) + β 2 (2a2 b6 + a1 b8 + a21 + a3 (−b5 + 2a4 + c6 ) +a1 (a3 − a6 − 2c9 ))) + β 4 a3 (a3 (a5 + c8 ) + a2 (b8 + a1 − a6 −2c9 )) − 2b9 (a1 a3 c2 + a2 (a21 + β 2 ))2 , B2 = 2a3 β 2 ((a1 a2 + a3 c2 )(b11 a3 − b31 a1 ) + β 2 (b21 a3 − b31 a2 )), B3 = 2(a3 ((a3 b4 − a1 b6 )(a1 a2 + a3 c2 )2 ) − (b6 a2 + a1 b8 − a3 b5 )(a1 a2 + a3 c2 )β 2 + (a3 b7 − a2 b8 )β 4 ) + b9 (a1 a3 c2 + a2 (a21 + β 2 ))2 ), B4 = β 4 (a3 (a3 b4 − a1 b6 ) + b9 (a21 + β 2 )), The expression of K2 is the same as under conditions (iii). 18 D0 = (a1 b31 − b1 a31 )(a1 c1 + b1 c2 )β 2 + (b1 b21 + c1 b31 )β 4 , D1 = (a1 c1 + b1 c2 )(2(a1 b9 − a9 b1 )(a1 c1 + b1 c2 ) + (b1 (a6 + b8) + 2b9 c1 )β 2 ), D2 = 2(a1 c1 + b1 c2 )β 2 ((a31 b1 − a1 b31 )(a1 c1 + b1 c2 ) + (b1 c31 − b31 c1 )β 2 ) D3 = 2(a1 c1 + b1 c2 )2 ((a9 b1 − a1 b9 )(a1 c1 + b1 c2 ) + (b1 c9 − b9 c1 )β 2 ), D4 = (a9 b1 − a1 b9 )(a1 c1 + b1 c2 )3 − (a1 c1 + b1 c2 ) a31 b4 + a21 b1 (b5 − a4 ) + a1 −a5 b21  +b21 b7 + b1 b8 c1 + 2b9 c21 − b1 b6 c2 − b1 c1 c9 − b1 a7 b21 + a8 b1 c1 + a9 c21 −a6 b1 c2 − b9 c1 c2 + b1 c2 c9 )) β 2 + −b9 c31 + a21 (b1 c4 − 2b4 c1 ) + a1 b1 (a4 c1 −b5 c1 − b4 c2 + b1 c5 ) + b31 c7 + b21 (−c1 b7 + a4 c2 − c2 c6 + c1 c8 ) + b1 c1 (−b8 c1 +b6 c2 + c1 c9 )) β 4 + (b1 c4 − b4 c1 )β 6 , K4 = (D1 D2 − 2D0 D3 )2 + 8a1 c1 D0 D1 (D1 D2 − D0 D3 ) + 8b1 c2 D0 D1 (D1 D2 − D0 D3 ). Acknowledgements The first and third authors are partially supported by the Ministerio de Economía, Industria y Com- petitividad, Agencia Estatal de Investigación (Spain), grant MTM2016-79661-P (European FEDER support included, UE) and the Consellería de Educación, Universidade e Formación Profesional (Xunta de Gali- cia), grant ED431C 2019/10 with FEDER funds. The first author is also supported by the Ministerio de Educacion, Cultura y Deporte de España, contract FPU17/02125. The second author is partially supported by the Ministerio de Economía, Industria y Competitividad, Agencia Estatal de Investigación grants MTM2016 - 77278 - P (FEDER) and MDM - 2014 - 0445, the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. The fourth author is partially supported by FCT/Portugal through UID/ MAT/ 04459/2013. Bibliography [1] A.J. 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