The zero-Hopf bifurcations in the Kolmogorov systems of degree 3 in R3

The zero-Hopf bifurcations in the Kolmogorov systems of degree 3in R3 Érika Diz-Pitaa, Jaume Llibreb, M. Victoria Otero-Espinara, Claudia Vallsc aDepartamento de Estatística, Análise Matemática e Optimización, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain bDepartament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain cCenter 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 3in R3can exhibit. The main tool used is the averaging theory. Keywords: Lotka–Volterra system, Kolmogorov systems, 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 differential systems of the form ˙x=xP(x, y),˙y=yQ(x, y), where Pand Qare polynomials of degree 1. Later on the Lotka-Volterra systems were generalized and considered on arbitrary dimension n≥2, i.e. ˙xi=xiPi(x1, . . . , xn), where Piare polynomials of degree 1. Finally in 1936 Andrei Kolmogorov [3] extended those systems to arbitrary degree, i.e. the polynomials Pican 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]. We want to study the limit cycles of the Kolmogorov systems of degree 3in R3which 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 ˙x=xP(x, y, z),˙y=yQ(x, y, z),˙z=zR(x, y, z), with P,Qand Rpolynomials 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: [email protected] (Érika Diz-Pita), [email protected] (Jaume Llibre), [email protected] (M. Victoria Otero-Espinar), [email protected] (Claudia Valls) Preprint submitted to Journal of Computational and Applied Mathematics January 22, 2025 from the singular point (1,1,1) of the system ˙x=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, ˙y=yb1(x−1) + b2(y−1) + b3(z−1) + b4(x−1)2+b5(x−1)(y−1) +b6(x−1)(z−1) + b7(y−1)2+b8(y−1)(z−1) + b9(z−1)2, ˙z=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, (1.1) 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 0and ±βi with β > 0. Here the dot denotes derivative with respect to the time t. Limit cycles, i.e. isolated periodic orbits in the set of all periodic orbits of a differential system, play an important role in the qualitative theory of differential equations. The behavior of many real-world oscillatory systems have been modelized by limit cycles, see for instance the famous limit cycle of van der Pol [9]. The study of limit cycles was initiated by Poincaré [10]. A big interest in their study was motivated by the famous 16th Hilbert problem [11, 12, 13]. Limit cycles are also studied in dimension higher than two, see for instance [14]. 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, with γ=a3b3(b2−a1)−a2b2 3+a2 3b1and β > 0: (i) γ= 0,c3=−a1−b2, c1=1 γa3 1b3−a2 1a3b1−a1a3b1b2−b32a2b1+β2−b1(a2(a3b1−b2b3) + a3β2+b2 2)and c2=1 γa2 1a2b3+a1a2(b2b3−a3b1) + a2 2b1b3−a3b2β2+b2 2+a2b3β2+b2 2−2a3b1b2. (ii) γ= 0,a3b3= 0, a2=a3b2 b3 , b1=a1b3 a3 , c3=−a1−b2and c2=−(a1+b2)2+a3c1+β2 b3 . (iii) γ= 0,b3= 0, a1=a2=a3= 0, c2=−b2 2+β2 b3 and c3=−b2. (iv) γ= 0,a3= 0, b1=b2=b3= 0, c1=−a2 1+β2 a3 and c3=−a1. (v) γ= 0,b1= 0, a2=−a2 1+β2 b1 , a3=b3=c3= 0 and b2=−a1. Proposition 1.1 is proved in section 2. In Theorem 3 of [15] 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. 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)–(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,K1and Nare defined in Appendix B. Theorem 1.2. If a3b3= 0,N= 0,a2=a3b2/b3,b1=a1b3/a3,c3=−a1−b2,c2=−((a1+b2)2+a3c1+ β2)/b3,A1= 0,A2= 0,A3= 0,and A0A4(A1A2−A0A3)>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. 2 (a) If K1>0,A2A3(A0A3−A1A2)N < 0and |2A0A3−A1A2|<√K1, then the two limit cycles have a stable manifold formed by two cylinders and an unstable manifold formed by two cylinders. (b) If b3A2N > 0,b3A3(A0A3−A1A2)>0and •either K1>0,b3A1N(2A0A3−A1A2−√K1)<0and b3A1N(2A0A3−A1A2+√K1)<0, •or K1≤0and b3A1N(2A0A3−A1A2)<0; or if b3A2N < 0,b3A3(A0A3−A1A2)<0and •either K1>0,b3A1N(2A0A3−A1A2−√K1)>0and b3A1N(2A0A3−A1A2+√K1)>0, •or K1≤0and b3A1N(2A0A3−A1A2)>0; then one limit cycle is local repeller, and the other is a local attractor. (c) If b3A2N > 0,b3A3(A0A3−A1A2)>0,K1>0and |2A0A3−A1A2|<√K1; or if A2A3(A0A3− A1A2)N < 0and •either K1>0,b3A1N(2A0A3−A1A2−√K1)>0and b3A1N(2A0A3−A1A2+√K1)>0, •or K1≤0and b3A1(2A0A3−A1A2)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. (d) If b3A2N < 0,b3A3(A0A3−A1A2)<0,K1>0and |2A0A3−A1A2|<√K1; or if A2A3(A0A3− A1A2)N < 0and •either K1>0,b3A1N(2A0A3−A1A2−√K1)<0and b3A1N(2A0A3−A1A2+√K1)<0, •or K1≤0and b3A1(2A0A3−A1A2)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,A2A3(A0A3−A1A2)N < 0and 2A0A3=A1A2; 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 Biwith i= 0, ..., 4, and K2are given in Appendix B. Theorem 1.3. If b3= 0,a1=a2=a3= 0,c2=−(b2 2+β2)/b3,c3=−b2,B1= 0,B2= 0,B3= 0 and B0B4(B1B2−B0B3)>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,B2B3(B0B3−B1B2)>0and |B1B2−2B0B3|<√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(B0B3−B1B2)>0,K2>0and |B1B2−2B0B3|<√K2; or if B2B3(B0B3−B1B2)>0, K2>0,B1<0and B1B2−2B0B3<−√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. (c) If B2<0,B3(B0B3−B1B2)>0and •either K2>0,B1(B1B2−2B0B3−√K2)<0and B1(B1B2−2B0B3+√K2)<0, •or K2≤0and B1(B1B2−2B0B3)<0; or if B2>0,B3(B0B3−B1B2)<0,K2>0,B1<0and B1B2−2B0B3<−√K2; then one limit cycle is a local attractor and the other limit cycle is a local repeller. 3 (d) If B2>0,B3(B0B3−B1B2)<0,K2>0and |B1B2−2B0B3|<√K2; or if B2B3(B0B3−B1B2)>0 and •either K2>0,B1(B1B2−2B0B3−√K2)<0and B1(B1B2−2B0B3+√K2)<0, •or K2≤0and B1(B1B2−2B0B3)<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. (e) B2B0<0,K2<0and B1B2= 2B0B3; 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. Theorems 1.2 and 1.3 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 xand y, so if we change the conditions b3= 0,a1=a2=a3= 0,c2=−(b2 2+β2)/b3,c3=−b2 into a3= 0,b1=b2=b3= 0,c1=−(a2 1+β2)/a3,c3=−a1, and redefine the constants Bifor i= 0, ..., 4as it is indicated in Appendix B the same Theorem 1.3 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 K4given in Appendix B. Theorem 1.4. If b1= 0,a3=b3=c3= 0,a2=−(a2 1+β2)/b1,b2=−a1,D1= 0,D2= 0,D3= 0 and D0D4(D1D2−D0DB3)>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,D2D3(a1c1+b1c2)(D0D3−D1D2)>0and |D1D2−2D0D3|<√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 b1D2(a1c1+b1c2)<0,b1D3(D0D3−D1D2)>0and •either K4>0,b1D1(a1c1+b1c2)(D1D2−2D0D2−√K4)<0and b1D1(a1c1+b1c2)(D1D2− 2D0D2+√K4)<0, •or K4≤0and b1D1(a1c1+b1c2)(D1D2−2D0D3)<0; or if b1D2(a1c1+b1c2)>0,b1D3(D0D3−D1D2)<0and •either K4>0,b1D1(a1c1+b1c2)(D1D2−2D0D2−√K4)>0and b1D1(a1c1+b1c2)(D1D2− 2D0D2+√K4)>0, •or K4≤0and b1D1(a1c1+b1c2)(D1D2−2D0D3)>0; then one limit cycle is a local repeller, and the other is a local attractor. (c) If b1D2(a1c1+b1c2)<0,b1D3(D0D3−D1D2)>0,K4>0and |D1D2−2D0D3|<√K4; or if D2D3(a1c1+b1c2)(D0D3−D1D2)>0and •either K4>0,b1D1(a1c1+b1c2)(D1D2−2D0D2−√K4)>0and b1D1(a1c1+b1c2)(D1D2− 2D0D2+√K4)>0, •or K4≤0and b1D1(a1c1+b1c2)(D1D2−2D0D3)>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 b1D2(a1c1+b1c2)>0,b1D3(D0D3−D1D2)<0K4>0and |D1D2−2D0D3|<√K4; or if D2D3(a1c1+b1c2)(D0D3−D1D2)>0and 4 •either K4>0,b1D1(a1c1+b1c2)(D1D2−2D0D2−√K4)<0and b1D1(a1c1+b1c2)(D1D2− 2D0D2+√K4)<0, •or K4≤0and b1D1(a1c1+b1c2)(D1D2−2D0D3)<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 D2D3(a1c1+b1c2)(D0D2−D1D2)>0,K4<0and D1D2= 2D0D3; 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 1.4 is proved in section 2. In section 3 can be found examples showing that the conditions provided by this theorem are non-empty. All the necessary computations for proving our results have been made with the algebraic manipulator Mathematica 12.0.0.0 (for a Mac OS X x86) in a computer MacBook Air of 2019. The computations done with Mathematica were verified for families (i) and (ii) also with the software Maple. 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 zeroHopf 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: ˙x= (1 + x)(a1x+a2y+a3z+a4x2+a5xy +a6xz +a7y2+a8yz +a9z2), ˙y= (1 + y)(b1x+b2y+b3z+b4x2+b5xy +b6xz +b7y2+b8yz +b9z2), ˙z= (1 + z)(c1x+c2y+c3z+c4x2+c5xy +c6xz +c7y2+c8yz +c9z2).(2.1) 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 0and ±βi with β > 0. We compute the characteristic polynomial and require that it has the form λ(λ2+β2). Solving the resultant equation we get the five solutions given in (i)–(v). Proof of Theorem 1.2.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,c2and c3 which define the zero-Hopf equilibrium under the assumption (ii) as follows a2=a3b2 b3 +εa21, b1=a1b3 a3 +εb11, c2=−(a1+b2)2+a3c1+β2 b3 +εc21, c3=−a1−b2+εc31, where εis a small parameter and β > 0. We write the linear part of system (2.1) at the origin in its real Jordan normal form J=  0−β0 β0 0 0 0 0  .(2.2) The variables of the system having its linear part in the real Jordan normal form are (X, Y, Z). Then system (1.1) under conditions (ii) of Proposition 1.1 becomes of the form ˙ X=−βY +O(ε), ˙ Y=βX +O(ε), ˙ Z=O(ε). 5 The complete explicit expression of this system is given in system (˙ X, ˙ Y , ˙ Z)in file ss[[2]]. We note that there are infinitely many linear changes of variables for writing the linear part of system (2.1) at the origin in its real Jordan normal form. This forces to choose some of the entries (denoted by yi for i= 1,...,9in file ss[[2]]) of the changing matrix. Thus in the file ss[[2]] we choose y1=y7= 1 and y2= 0 in order to fix a unique changing matrix. We note that the choice of the matrix Jgiven in (2.2) instead of the matrix J=  0 0 0 0−β0 β0 0   is irrelevant. If we choose this last expression for the matrix in its real Jordan form, then instead of doing the change to cylindrical coordinates (X, Y, Z)→(rcos θ, r sin θ, Z), we must do the change (X, Y, Z)→ (X, r cos θ, r sin θ). This change to cylindrical coordinates is necessary in order to arrive to write the differential system in the normal form for applying the averaging theory described in Theorem A.1. 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)→(rcos θ, r sin θ, Z) obtaining system ( ˙r, ˙ θ, ˙ Z)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 > 0is the same parameter used before. We obtain system (˙ R, ˙ Z)of file ss[[2]]. We take the variable θas the new independent variable and so we obtain the system R′=εF11 +O(ε2), Z′=εF12 +O(ε2),(2.3) with coefficients F11 and F12 given in the file ss[[2]]. Note that system (2.3) is in the normal form (A.1), so we can apply the averaging theory with T= 2π, x= (R, Z),t=θand εR(θ, x, ε) = O(ε2). The functions F11,F12 and Rare C2in xand 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 f11 =πR(A0+A1Z) a2 3b3β5, f12 =−π(A2Z+A3Z2+A4R2) a2 3b3β5N,(2.4) and Ai, for i= 0, ..., 4,K1and Nare 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 from the origin, (R1, Z1) = (0,−A2/A3)and (R2, Z2) = ±pA0(A1A2−A0A3)/(A1√A4),−A0/A1. We consider always the positive expression of R2, i.e. we consider the positive sign if A1>0and the negative sign if A1<0. We compute the Jacobian matrix of f1, which is      π(A0+A1Z) a2 3b2β5 πRA1 a2 3b3β5 −2πRA4 a2 3b3β5N−π(A2+ 2A3Z) a2 3b3β5N      , and its determinant is π2(−2A1A4R2+ (A0+A1Z)(A2+ 2A3Z))/(a4 3b2 3β10N). Evaluating the determinant at the solution (R1, Z1)we get that it is equal to π2A2(A0A3−A1A2)/(a4 3b2 3A3β10N), and at the solutions (R2, Z2)we get that it is equal to 2π2A0(A1A2−A0A3)/(a4 3b2 3A1β10N). From the hypothesis considered these determinants are nonzero, therefore it follows from Theorem A.1 that for εsufficiently small system (2.3) 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= 1,2when ε→0. 6 Moreover the Jacobian matrix evaluated at the solution (R1, Z1)has eigenvalues equal to πA2/(a2 3b3β5N) and π(A0A3−A1A2)/(a2 3b3A3β5). 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 A2b3N > 0and A3b3(A0A3−A1A2)>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 A2b3N < 0 and A3b3(A0A3−A1A2)<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 A2A3N(A0A3−A1A2)<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 the other hand, the Jacobian matrix evaluated at (R2, Z2)has eigenvalues equal to π(2A0A3−A1A2± √K1)/(2a2 3b3A1β5N), and so its stability is as follows. If K1>0,b3A1N(2A0A3−A1A2+√K1)>0and b3A1N(2A0A3−A1A2−√K1)>0or if K1<0and b3A1N(2A0A3−A1A2)>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,b3A1N(2A0A3−A1A2+√K1)<0and b3A1N(2A0A3−A1A2√K1)<0or if K1<0and b3A1N(2A0A3−A1A2)<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>0and −√K1<2A0A3−A1A2<√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<0and A1A2= 2A0A3, 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 2πwhen 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 1.3.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,c2and c3which define the zero-Hopf equilibrium under conditions (iii) as follows a1=εa11, a2=εa21, a3=εa31, c2=−b2 2+β2 b3 +εc21, c3=−b2+εc31, where εis 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 (˙ X, ˙ Y , ˙ Z)of file ss[[3]]. Then we write the system in cylindrical coordinates obtaining system ( ˙r, ˙ θ, ˙ Z)of file ss[[3]], and we do the scaling (r, Z)→(εR, εZ)obtaining system (˙ R, ˙ Z)in file ss[[3]]. As in the proof of Theorem 1.2 in order to apply Theorem A.1, we take the variable θas the new independent variable obtaining a system R′=εF11 +O(ε2), Z′=εF12 +O(ε2)(2.5) 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 f11 =πR(B0+B1Z) b2 3β5, f12 =π(B2Z+B3Z2+B4R2) b2 3β5, 7 with Bi, for i= 0, ..., 4, and K2given in Appendix B.Solving the equation (f11(R, Z), f12(R, Z)) = (0,0) we obtain two solutions (R1, Z1) = (0,−B2/B3)and (R2, Z2) = ±pB0(B1B2−B0B3)/(B1√B4),−B0/B1. Again we consider always the positive expression of R2. We compute the Jacobian matrix of f1and we get      π(B0+B1Z) b2 3β5 πRB1 b2 3β5 2πRB4 b2 3β5 π(B2+ 2B3Z) b2 3β5      , whose determinant is π2(−2B1B4R2+ (B0+B1Z)(B2+ 2B3Z))/(b4 3β10). The determinant at the solution (R1, Z1)is π2B2(B1B2−B0B3)/(b4 3B3β10), and at the solutions (R2, Z2)is 2π2B0(B0B3−B1B2)/(b4 3B1β10). From the hypothesis considered these determinants are nonzero, so it follows from Theorem A.1 that for εsufficiently small, system (2.3) 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= 1,2when ε→0. The Jacobian matrix evaluated at the solution (R1, Z1)has eigenvalues equal to −πB2/(b2 3β5)and π(B0B3−B1B2)/(b2 3B3β5). We study the stability of the associated periodic orbit which is provided by these eigenvalues. If B2<0and B3(B0B3−B1B2)>0, the associated periodic solution is unstable and has an unstable manifold of dimension three. If B2>0and B3(B0B3−B1B2)<0, then the associated periodic solution is stable and has a stable manifold of dimension three. Finally if B2B3(B0B3−B1B2)>0, the periodic solution is unstable and has a stable manifold formed by two cylinders and an unstable manifold formed by two cylinders. On the other hand, the Jacobian matrix evaluated at (R2, Z2)has eigenvalues equal to π(B1B2−2B0B3± √K2)/(2b2 3B1β5), and so if K2>0,B1(B1B2−2B0B3+√K2)>0and B1(B1B2−2B0B3−√K2)>0, or if K2<0and B1(B1B2−2B0B3)>0, then the associated periodic solution is unstable and has an unstable manifold of dimension three. If K2>0,B1(B1B2−2B0B3+√K2)<0and B1(B1B2−2B0B3−√K2)<0, or if K2<0and B1(B1B2−2B0B3)<0, then the associated periodic solution is stable and has a stable manifold of dimension three. If K2>0and −√K2< B1B2−2B0B3<√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<0and B1B2−2B0B3= 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)–(e) 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 2πwhen 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 (v) of Proposition 1.1. In order to study the zero-Hopf bifurcation we perturb the parameters a2,a3,b2,b3and c3which define the zero-Hopf equilibrium point into the form a2=−a2 1+β2 b1 +εa21, a3=εa31, b2=−a1+εb21, b3=εb31, c3=εc31, where εis 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 (˙ X, ˙ Y , ˙ Z),( ˙r, ˙ θ, ˙ Z)and (˙ R, ˙ Z)of file ss[[5]]. Taking θas the new independent variable we obtain a system in the form R′=εF11 +O(ε2), Z′=εF12 +O(ε2)(2.6) 8 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 f11 =πR(D0+D1Z) b1β5, f12 =π(D2Z+D3Z2+D4R2) b1(a1c1+b1c2)β5, with Di, for i= 0, ..., 4, and K4given in Appendix B. We look for the solutions of (f11(R, Z), f12(R, Z)) = (0,0), and we obtain (R1, Z1) = (0,−D2/D3)and (R2, Z2) = ±pD0(D1D2−D0D3)/(D1√D4),−D0/D1, considering the positive expression of R2. We compute the Jacobian matrix of f1and we get      π(D0+D1Z) b1β5 πRD1 b1β5 2πRD4 b1β5(a1c1+b1c2) π(D2+ 2D3Z) b1β5(a1c1+b1c2)      , whose determinant is π2(−2D1D4R2+ (D0+D1Z)(D2+ 2D3Z))/(b2 1β10(a1c1+b1c2)). The determinant at the solution (R1, Z1)is π2D2(D1D2−D0D3)/(b2 1D3β10 (a1c1+b1c2)), and at the solution (R2, Z2)is 2π2D0(D0D3−D1D2)/(b2 1D1β10(a1c1+b1c2)), and both are nonzero by the hypotheses. By Theorem A.1, for εsufficiently small system (2.6) 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= 1,2when ε→0. The Jacobian matrix evaluated at the solution (R1, Z1)has eigenvalues equal to −πD2/(b1β5(a1c1+b1c2)) and π(D0D3−D1D2)/(b1D3β5). We study the stability of the associated periodic orbit which is provided by these eigenvalues. If b1D2(a1c1+b1c2)<0and b1D3(D0D3−D1D2)>0, the associated periodic solution is unstable and has an unstable manifold of dimension three. If b1D2(a1c1+b1c2)>0and b1D3(D0D3−D1D2)< 0, then the associated periodic solution is stable and has a stable manifold of dimension three. Finally if D2D3(a1c1+b1c2)(D0D3−D1D2)>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 other hand, the Jacobian matrix evaluated at (R2, Z2)has eigenvalues equal to π(D1D2−2D0D3± √K4)/(2b1D1β5(a1c1+b1c2)). Then if K4>0,b1D1(a1c1+b1c2)(D1D2−2D0D3+√K4)>0and b1D1(a1c1+ b1c2)(D1D2−2D0D3−√K4)>0, or if K4<0and b1D1(a1c1+b1c2)(D1D2−2D0D3)>0, then the associated periodic solution is unstable and has an unstable manifold of dimension three. If K4>0,b1D1(a1c1+ b1c2)(D1D2−2D0D3+√K4)<0and b1D1(a1c1+b1c2)(D1D2−2D0D3−√K4)<0, or if K4<0and b1D1(a1c1+b1c2)(D1D2−2D0D3)<0, then the associated periodic solution is stable and has a stable manifold of dimension three. If K4>0and −√K4< D1D2−2D0D3<√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<0and D1D2= 2D0D3, 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= 1,2bifurcating from (1,1,1) with a period tending to 2πwhen 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. We give only the values of the parameters which are nonzero. System (1.1) with the set of parameters {a1=a3=a5=b1=b2=b3=−1, a2=−1−ε, a4= 1/2, c1= 3/2, c2= 7/2, c3= 2, c6= 4} 9 D0= (a1b31 −b1a31)(a1c1+b1c2)β2+ (b1b21 +c1b31)β4, D1= (a1c1+b1c2)(2(a1b9−a9b1)(a1c1+b1c2)+(b1(a6+b8) + 2b9c1)β2), D2= 2(a1c1+b1c2)β2((a31b1−a1b31)(a1c1+b1c2)+(b1c31 −b31c1)β2) D3= 2(a1c1+b1c2)2((a9b1−a1b9)(a1c1+b1c2)+(b1c9−b9c1)β2), D4= (a9b1−a1b9)(a1c1+b1c2)3−(a1c1+b1c2)a3 1b4+a2 1b1(b5−a4) + a1−a5b2 1 +b2 1b7+b1b8c1+ 2b9c2 1−b1b6c2−b1c1c9−b1a7b2 1+a8b1c1+a9c2 1 −a6b1c2−b9c1c2+b1c2c9)) β2+−b9c3 1+a2 1(b1c4−2b4c1) + a1b1(a4c1 −b5c1−b4c2+b1c5) + b3 1c7+b2 1(−c1b7+a4c2−c2c6+c1c8) + b1c1(−b8c1 +b6c2+c1c9)) β4+ (b1c4−b4c1)β6, K4= (D1D2−2D0D3)2+ 8a1c1D0D1(D1D2−D0D3)+8b1c2D0D1(D1D2−D0D3). Acknowledgements The first and third authors are partially supported by the Ministerio de Economía, Industria y Competitividad, 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 Galicia), 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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