Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 90, pp. 1–15. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu FINITE CYCLICITY OF THE CONTACT POINT IN SLOW-FAST INTEGRABLE SYSTEMS OF DARBOUX TYPE RENATO HUZAK Abstract. Using singular perturbation theory and family blow-up we prove that nilpotent contact points in deformations of slow-fast Darboux integrable systems have finite cyclicity. The deformations are smooth or analytic depend- ing on the region in the parameter space. This article is a natural continuation of [1, 3], where one studies limit cycles in polynomial deformations of slow-fast Darboux integrable systems, around the “integrable” direction in the parame- ter space. We extend the existing finite cyclicity result of the contact point to analytic deformations, and under some assumptions we prove that the contact point has finite cyclicity around the “slow-fast” direction in the parameter space. 1. Introduction A typical example of slow-fast Darboux integrable systems studied in [1, 3] is given by Xε : { ẋ = y − x2 − ε+ εy ẏ = −2εx+ 2εxy, (1.1) where ε ≥ 0, ε ∼ 0, is the singular perturbation parameter. System Xε has the first integral (of Darboux type) H(x, y) = (y − x2)ε(1− y). When ε > 0, Xε has a family of periodic orbits Oε bounded by {y − x2 = 0} and {1 − y = 0} (see Figure 1). The fast subsystem X0 of Xε, has the curve of singularities {y−x2 = 0}, often called the critical or slow curve, and (fast) regular horizontal orbits (see Figure 2). The goal in [1, 3] was to prove ε-uniform finiteness of the number of limit cycles bifurcating from the compact set Ō := Oε in a polynomial deformation Xε,δ of Xε, with δ = (δ1, . . . , δm) ∈ Rm, m ∈ N \ {0}, and δ ∼ 0. (Note that Ō is the ε-independent region bounded by {y − x2 = 0} and {1 − y = 0}, including the boundary.) More precisely, we consider a vector field Xε,δ := Xε +Q1(x, y, δ) ∂ ∂x + 2010 Mathematics Subject Classification. 34C26, 34E15, 34E17. Key words and phrases. Blow-up; cyclicity; Darboux systems; singular perturbation theory; slow-fast systems. c©2020 Texas State University. Submitted December 1, 2019. Published September 6, 2020. 1 2 R. HUZAK EJDE-2020/90 Oε Figure 1. Dynamics of Xε, with ε > 0. Figure 2. Dynamics of X0. Q2(x, y, δ) ∂∂y where Qi is an analytic δ-family of real polynomials in (x, y) and Qi(x, y, 0) ≡ 0, i = 1, 2. Let ε > 0 be small and fixed. Let C(ε) := Cycl(Xε,δ, Ō) be the maximal number of limit cycles of the system Xε,δ, bifurcating from Ō, for δ ∼ 0. Following [1, Theorem 2.1], the cyclicity C(ε) is finite and uniformly bounded in ε > 0, under the assumption that the parameter (ε, δ) is kept in a narrow region around the ε-axis, i.e. around the integrable direction (see Figure 3). ε δ Figure 3. Integrable direction in the parameter space (ε, δ) stud- ied in [1, 3]. Let us recall that article [1] is a natural continuation of [2, 3] where the ε-uniform finiteness property has been obtained in the integrable direction (see Figure 3) for any compact set K contained in IntŌ by studying zeros of pseudo-Abelian integrals. Thus, the cyclicity of the polycycle, i.e. the boundary of Ō, has not been studied in [3]. In [1], Dulac maps and a technique based on the so-called “Petrov trick” (see also [14]) have been used to obtain the ε-uniform finiteness property for Ō. One of the reasons why the polynomial deformations of Xε have been studied in [1, 3] is Hilbert’s 16th problem (see [13, 21]). The main purpose of this article is to initiate the study of limit cycles of Xε,δ, bifurcating from the compact set Ō (or any other compact set in the phase space), EJDE-2020/90 FINITE CYCLICITY IN SLOW-FAST DARBOUX SYSTEMS 3 around the δ-axis in the (ε, δ)-space. Our focus is on the cyclicity of the (generic) nilpotent contact point (0, 0) ∈ Ō, i.e. the nilpotent singularity of X0 at which the critical curve {y − x2 = 0} of X0 has a (quadratic) contact with fast orbits (see Figure 2). We use geometric singular perturbation theory and the family blow-up. All singularities of the critical curve, located away from the origin, are normally hyperbolic (attracting when x > 0 and repelling when x < 0). The contact point is called turning point if it allows the passage from the attracting part of the critical curve to the repelling part of the critical curve in Xε,δ, with (ε, δ) 6= (0, 0). We can have diverse (nilpotent) contact points, depending on the region in the (ε, δ)-space: jump points (see [11, 20]), slow-fast saddle points, slow-fast Hopf points (sometimes called generic turning points)(see [11, 12, 16, 19]), slow-fast Bogdanov-Takens points (see [8]), slow-fast codimension 3 saddle or elliptic points (see [9, 10, 15, 18]), slow- fast codimension 4 saddle-node points (see [17]), etc. To determine the type of our nilpotent contact point, we first blow up the origin in the parameter space (ε, δ) by using the following homogeneous rescaling: (ε, δ) = (ε̄E, ε̄D), ε̄ ≥ 0, ε̄ ∼ 0, (E,D) ∈ Sm, (1.2) where D = (D1, . . . , Dm) ∈ Rm. The advantage of using the blow-up (1.2) is that we now have only one small parameter ε̄ which is a singular perturbation parameter ((E,D) 6= (0, 0) because (E,D) ∈ Sm). Note that if we vary ε̄ ≥ 0 and (E,D), kept on the unit sphere, then we cover a complete (small) neighborhood of the origin in the (ε, δ)-space. As usual, we use different charts of the sphere: (1) (“Integrable” direction E = ±1) We have (ε, δ) = (±ε̄, ε̄D) where ε̄ ≥ 0, ε̄ ∼ 0 and D ∼ 0 ∈ Rm. See Figure 4(a). This region is covered by [1, 3] working with polynomial deformations Xε,δ of Xε as explained above. When E = 1 (resp. E = −1) we deal with a slow-fast Hopf point (resp. a slow-fast saddle point). In the slow-fast Hopf region we prove ε-uniform finiteness of the number of limit cycles Hausdorff close to the contact point (x, y) = (0, 0) under analytic deformations of Xε (Q1 and Q2 are analytic). In the slow-fast saddle region the compact set Ō produces no limit cycles under smooth deformations (Q1 and Q2 are smooth). For more details see Theorem 2.1. (2) (“Slow-fast” direction D ∈ Sm−1) We have (ε, δ) = (ε̄E, ε̄D) where ε̄ ≥ 0, ε̄ ∼ 0, D ∈ Sm−1 and E is kept in a large compact set in R. See Figure 4(b). This region is covered by our paper. As we will see in Sections 2 and 3, the type of the nilpotent contact point is closely related to the order of vanishing of the slow dynamics of Xε,δ (defined along the slow curve {y − x2 = 0}) at the contact point. The slow dynamics is given by x′ = −E + Ex2 + 〈D, Q̃(x, x2, 0)〉 2x , x 6= 0, where we write Q2(x, y, δ) = 〈δ, Q̃(x, y, δ)〉 (〈·, ·〉 denotes the inner product in Rm). For more details about the definition of the slow dynamics see Section 2.1. Using 〈D, Q̃(x, x2, 0)〉 = α0+α1x+α2x 2+α3x 3+α4x 4+O(x5), the slow dynamics can be written as x′ = α0 2x + ( α1 2 − E) + α2 2 x+ ( α3 2 + E)x2 + α4 2 x3 +O(x4), x 6= 0. 4 R. HUZAK EJDE-2020/90 When α0 6= 0 for some D = D0 ∈ Sm−1, we deal with a jump contact point and any compact set in the phase space (x, y) produces no limit cycles under smooth deformations, for each D ∼ D0 and E kept in a large compact set. See Theorem 2.2. When α0 = 0 for some D = D0 ∈ Sm−1, then the contact point (resp. any compact set in the phase space (x, y)) produces a finite number of limit cycles under analytic deformations (resp. no limit cycles under smooth deformations) uniformly in (E,D), with D ∼ D0 and α1 2 − E < 0 (resp. α1 2 −E > 0). Like in the integrable direction, we deal with a slow-fast Hopf point or a slow-fast saddle point. For more details see Theorem 2.4. When α0 = 0, α1 2 − E = 0 and α2 6= 0, for some (E,D) = (E0, D0) ∈ R × Sm−1, then we deal with a slow-fast Bogdanov-Takens point and any compact set in the phase space (x, y) produces at most 1 limit cycle under smooth perturbations, with (E,D) ∼ (E0, D0). See Theorem 2.6. When α0 = α1 2 − E = α2 = 0, α3 2 + E > 0 (resp. α3 2 + E < 0) and α4 6= 0 for some (E,D) = (E0, D0) ∈ R × Sm−1, then any compact set in the phase space (x, y) (resp. the contact point) produces at most 2 limit cycles under smooth perturbations, for (E,D) ∼ (E0, D0). We deal with a slow-fast codimension 3 saddle (resp. elliptic) point. See Theorem 2.8. When α0 = α1 2 − E = α2 = α3 2 + E = 0 and α4 6= 0 for some (E,D) = (E0, D0) ∈ R× Sm−1, then we deal with a slow-fast codimension 4 saddle- node point and any compact set in the phase space (x, y) produces at most 2 limit cycle under smooth perturbations, with (E,D) ∼ (E0, D0). See Theorem 2.9. The cases “α0 = α1 2 − E = α2 = α4 = 0, α3 2 + E 6= 0” and “α0 = α1 2 − E = α2 = α3 2 + E = α4 = 0” are topics of further study. ε δ ε δ E = −1 E = 1 (a) (b) Figure 4. (a) The “integrable” direction in the parameter space (ε, δ). (b) The “slow-fast” direction in the (ε, δ)-space. Besides the small-amplitude limit cycles of Xε,δ studied in this paper, we can also have limit cycles in Xε,δ bifurcating from so-called detectable canard limit periodic sets consisting of a fast orbit of X0,0 and the part of the critical curve of X0,0 between the α-limit set and the ω-limit set of the fast orbit (see Figure 2). In the “slow-fast” direction, these canard limit cycles are possible only if the slow dynamics points from the attracting part of the critical curve to the repelling part of the critical curve. This happens, for instance, in the slow-fast Hopf case and the slow-fast codimension 3 elliptic case. More precisely, we have to deal with EJDE-2020/90 FINITE CYCLICITY IN SLOW-FAST DARBOUX SYSTEMS 5 detectable canard limit cycles when the order of the slow dynamics at the contact point is 0, 2, 4, 6, . . . , with a negative coefficient. In the slow-fast Hopf case, i.e. when the order of the slow dynamics is 0, we can study zeros of the so-called slow divergence integral if the slow dynamics is regular (see [4]). If the slow dynamics has isolated singularities (away from the contact point), we can use the results of [5]. When the order of the slow dynamics is ≥ 2, we study zeros of the derivative of the slow divergence integral if the slow dynamics is regular (see [6, 7]). If the slow dynamics has isolated zeros, away from the contact point, we can combine [6, 7] with [5]. The same can be done in the “integrable” direction (the slow-fast Hopf case). Since there are a lot of different possibilities, we prefer to deal with the detectable canard limit cycles in a separate paper. This is a topic of further study. In the “slow-fast” direction, compact regions in which we have to study de- tectable canard cycles can be larger than the compact region Ō in the “integrable” direction. This happens, for example, in the slow-fast Hopf case with regular slow dynamics. In the “integrable” direction, the slow dynamics has two hyperbolic saddles x = ±1 for (E,D) = (1, 0). See also Figure 1. In Section 2 we state our main results. We prove the results in Section 3. Al- though in this paper we are mostly interested in the slow-fast Darboux system (1.1), our methods can be used in a more general framework of slow-fast integrable systems of Darboux type introduced in [3], under assumption that our contact point is of generic nilpotent type. See Section 4. 2. Slow dynamics and statement of results In Section 2.1 we define the notion of slow dynamics of (2.1) along the critical curve {y − x2 = 0}. We state our main results in Section 2.2. 2.1. The fast subsystem and slow dynamics. We consider an (ε, δ, λ)-family of 2-dimensional vector fields Xε,δ,λ : { ẋ = y − x2 − ε+ εy +Q1(x, y, δ, λ) ẏ = −2εx+ 2εxy +Q2(x, y, δ, λ) (2.1) where ε ∈ (R, 0), δ ∈ (Rm, 0), λ ∈ (Rn, λ0) (m,n ∈ N \ {0}) and Q1 and Q2 are smooth functions withQ1(x, y, 0, λ) = Q2(x, y, 0, λ) ≡ 0. (For the sake of generality, Q1 and Q2 may depend on extra parameter λ.) We focus on system Xε̄E,ε̄D,λ where ε̄ ≥ 0, ε̄ ∼ 0 and (E,D) ∈ Sm (see (1.2)). The dynamics of the fast subsystem X0,0,λ is given in Figure 2, and the dynamics of Xε̄E,ε̄D,λ, with ε̄ ∼ 0 and ε̄ > 0, away from the critical curve, is governed by the dynamics of X0,0,λ. The dynamics of Xε̄E,ε̄D,λ near the critical curve, away from the contact point (x, y) = (0, 0), can be studied using the slow dynamics. Let us define the slow dynamics. Center manifolds at the normally hyperbolic (or semi-hyperbolic) singularity (x, y, ε̄) = (x, x2, 0), x 6= 0, of Xε̄E,ε̄D,λ + 0 ∂ ∂ε̄ can be written as y = x2 + ε̄ ( − 〈D, Q̄(x, x2, 0, λ)〉+ 〈D, Q̃(x, x2, 0, λ)〉 2x +O(ε̄) ) , with Q1(x, y, δ, λ) = 〈δ, Q̄(x, y, δ, λ)〉 and Q2(x, y, δ, λ) = 〈δ, Q̃(x, y, δ, λ)〉. The dynamics inside these center manifolds can be obtained from the first equation in 6 R. HUZAK EJDE-2020/90 (2.1): ẋ = ε̄ ( − E + Ex2 + 〈D, Q̃(x, x2, 0, λ)〉 2x +O(ε̄) ) . If we divide this equation by ε̄ and if we let ε̄→ 0, we find the slow dynamics x′ = f(x,E,D, λ) := −E + Ex2 + 〈D, Q̃(x, x2, 0, λ)〉 2x , x 6= 0. (2.2) Using (2.2) we can write f(x,E,D, λ) = 3∑ k=−1 βkx k +O(x4), where β−1(D,λ) = 〈D, Q̃(0, 0, 0, λ)〉 2 ; β0(E,D, λ) = 〈D, ∂Q̃∂x (0, 0, 0, λ)〉 2 − E; β1(D,λ) = 〈D, ( ∂2Q̃ ∂x2 + 2∂Q̃∂y ) (0, 0, 0, λ)〉 4 ; β2(E,D, λ) = 〈D, ( ∂3Q̃ ∂x3 + 6 ∂2Q̃ ∂x∂y ) (0, 0, 0, λ)〉 12 + E; β3(D,λ) = 〈D, ( ∂4Q̃ ∂x4 + 12 ∂3Q̃ ∂x2∂y + 12∂ 2Q̃ ∂y2 ) (0, 0, 0, λ)〉 48 . Clearly, the order of vanishing of the slow dynamics f , with λ = λ0, at x = 0 depends on (E,D) ∈ Sm, the function Q̃ and its partial derivatives at (x, y, δ, λ) = (0, 0, 0, λ0). As explained in Section 1, limit cycles of (2.1) cannot be studied uniformly. We have to use different techniques depending on the order of vanishing of f at x = 0. 2.2. Statement of results. In the parameter space (ε, δ), we distinguish between the integrable direction {E = ±1}, with D ∼ 0 ∈ Rm, and the slow-fast direction {D ∈ Sm−1}, with E kept in a large compact set in R. 2.2.1. Integrable direction. In this section we study limit cycles of system X±ε̄,ε̄D,λ where (ε̄, D, λ) ∼ (0, 0, λ0) ∈ R×Rm×Rn and ε̄ ≥ 0. We have β−1 = 0 and β0 6= 0 for (E,D, λ) = (±1, 0, λ0). We obtain the following result. Theorem 2.1. Let the family X±ε̄,ε̄D,λ be as defined above. The following state- ments are true: (1) (Slow-fast Hopf case E = 1) Suppose that Q1 and Q2 are analytic. Then there exists ε̄0 > 0, a neighborhood W of (0, λ0) in the (D,λ)-space, a neighborhood V of (x, y) = (0, 0) and N ∈ N such that system Xε̄,ε̄D,λ has at most N limit cycles in V, for each value (ε̄, D, λ) ∈ [0, ε̄0]×W. (2) (Slow-fast saddle case E = −1) Let Q1 and Q2 be smooth and let K be an arbitrary compact set in the phase space (x, y). Then there exists ε̄0 > 0 and a neighborhood W of (0, λ0) in the (D,λ)-space such that system X−ε̄,ε̄D,λ has no limit cycles in K, for each value (ε̄, D, λ) ∈ [0, ε̄0]×W. EJDE-2020/90 FINITE CYCLICITY IN SLOW-FAST DARBOUX SYSTEMS 7 To prove Theorem 2.1 in the region E = 1, first we bring the analytic family Xε̄,ε̄D,λ near (x, y) = (0, 0) to a normal form of Liénard type using [16]. Then, we use a finite cyclicity result for analytic slow-fast Hopf points of Liénard type obtained in [12]. See Section 3.1. 2.2.2. Slow-fast direction. In this section we deal with limit cycles of systemXε̄E,ε̄D,λ where ε̄ ∼ 0, ε̄ > 0, E is kept in a large compact set in R, D ∈ Sm−1 and λ ∼ λ0. We start with the simplest case when β−1 6= 0, i.e. with the jump case. Theorem 2.2 (Jump case). Let Q1 and Q2 be smooth and β−1(D0, λ0) 6= 0 for some D0 ∈ Sm−1. For each compact set K in the phase space (x, y) and for each compact set C in the parameter space E there exists ε̄0 > 0 and a neighborhood W of (D0, λ0) in the (D,λ)-space such that Xε̄E,ε̄D,λ has no limit cycles in K for each value (ε̄, E,D, λ) ∈ [0, ε̄0]× C ×W. The above theorem will be proved in Section 3.2.2. When m = 1, we have D = ±1 and the condition β−1 6= 0 is equivalent to Q̃(0, 0, 0, λ0) 6= 0. Now, as a direct consequence of Theorem 2.1 and Theorem 2.2, we obtain the following finite cyclicity result of the contact point in analytic families (2.1), in a full neighborhood of (ε, δ) = (0, 0). Theorem 2.3. Suppose that Q1 and Q2 are analytic, m = 1 and ∂Q2 ∂δ (0, 0, 0, λ0) 6= 0. There exists a neighborhood W of (ε, δ, λ) = (0, 0, λ0), a neighborhood V of (x, y) = (0, 0) and N ∈ N such that system Xε,δ,λ has at most N limit cycles in V, for each (ε, δ, λ) ∈ W. As in Section 2.2.1, in the slow-fast direction we can also encounter slow-fast Hopf and saddle cases (β−1 = 0 and β0 6= 0). Theorem 2.4. Suppose that β−1(D0, λ0) = 0 and β0(E0, D0, λ0) 6= 0 for some (E0, D0) ∈ R× Sm−1. The following statements are true: (1) (Slow-fast Hopf case β0 < 0) Suppose that Q1 and Q2 are analytic. If β0(E0, D0, λ0) < 0, then there exists ε̄0 > 0, a neighborhoodW of (E0, D0, λ0) in the (E,D, λ)-space, a neighborhood V of (x, y) = (0, 0) and N ∈ N such that system Xε̄E,ε̄D,λ has at most N limit cycles in V, for each value (ε̄, E,D, λ) ∈ [0, ε̄0]×W. (2) (Slow-fast saddle case β0 > 0) Let Q1 and Q2 be smooth functions, let β0(E0, D0, λ0) > 0 and let K be an arbitrary compact set in the phase space (x, y). Then there exists ε̄0 > 0 and a neighborhood W of (E0, D0, λ0) in the (E,D, λ)-space such that system Xε̄E,ε̄D,λ has no limit cycles in K, for each value (ε̄, E,D, λ) ∈ [0, ε̄0]×W. The proof of the above theorem is similar to the proof of Theorem 2.1 (see Section 3.1). When m = 1, Theorem 2.1, Theorem 2.4 and the definition of β−1, β0 imply Theorem 2.5. Let Q1 and Q2 be analytic functions, m = 1, ∂Q2 ∂δ (0, 0, 0, λ0) = 0 and let S±(ρ) := { (ε̄E,±ε̄) | ε̄ ∈]0, 1[, E ∈ B ( ± ∂2Q2 ∂x∂δ (0, 0, 0, λ0) 2 , ρ )} , ρ > 0, (2.3) where B is an open ball in R. For each (small) ρ > 0 there exists a neighborhoodW1 of (ε, δ) = (0, 0), a neighborhood W2 of λ = λ0, a neighborhood V of (x, y) = (0, 0) and N ∈ N such that system Xε,δ,λ has at most N limit cycles in V, for each (ε, δ, λ) ∈ (W1 \ (S+(ρ) ∪ S−(ρ)))×W2. 8 R. HUZAK EJDE-2020/90 When β−1 = β0 = 0 and β1 6= 0, we deal with the slow-fast Bogdanov-Takens case. Theorem 2.6 (Slow-fast Bogdanov-Takens case). Let Q1 and Q2 be smooth func- tions, β−1(D0, λ0) = β0(E0, D0, λ0) = 0 and β1(D0, λ0) 6= 0 for some (E0, D0) ∈ R × Sm−1. For each compact set K in the (x, y)-space there exists ε̄0 > 0 and a neighborhood W of (E0, D0, λ0) in the (E,D, λ)-space such that system Xε̄E,ε̄D,λ has at most 1 limit cycle in K, for each value (ε̄, E,D, λ) ∈ [0, ε̄0]×W. We prove Theorem 2.6 in Section 3.2.3. When m = 1, Theorem 2.6 gives us better understanding of limit cycles in the narrow parameter region S+(ρ)∪ S−(ρ) defined in (2.3). Theorem 2.7. Let Q1 and Q2 be smooth functions, m = 1, ∂Q2 ∂δ (0, 0, 0, λ0) = 0 and ( ∂3Q2 ∂x2∂δ + 2∂ 2Q2 ∂y∂δ ) (0, 0, 0, λ0) 6= 0. For each compact set K in the (x, y)-space there exists ρ > 0, a neighborhood W1 of (ε, δ) = (0, 0) and a neighborhood W2 of λ = λ0 such that system Xε,δ,λ has at most 1 limit cycle in K, for each (ε, δ, λ) ∈ (W1 ∩ (S+(ρ) ∪ S−(ρ)))×W2. When β−1 = β0 = β1 = 0 and β2 > 0 (resp. β2 < 0), we deal with the slow-fast codimension 3 saddle (resp. elliptic) case. Moreover, when β3 6= 0, we have the following result. Theorem 2.8. Let Q1 and Q2 be smooth functions, β−1 = β0 = β1 = 0, β2 6= 0 and β3 6= 0 for some (E0, D0) ∈ R× Sm−1. The following statements are true: (1) (Slow-fast codimension 3 saddle case) Let β2(E0, D0, λ0) > 0 and let K be an arbitrary compact set in the phase space (x, y). Then there ex- ists ε̄0 > 0 and a neighborhood W of (E0, D0, λ0) in the (E,D, λ)-space such that system Xε̄E,ε̄D,λ has at most 2 limit cycles in K, for each value (ε̄, E,D, λ) ∈ [0, ε̄0]×W. (2) (Slow-fast codimension 3 elliptic case) Let β2(E0, D0, λ0) < 0. Then there exists ε̄0 > 0, a neighborhood W of (E0, D0, λ0) in the (E,D, λ)-space and a neighborhood V of (x, y) = (0, 0) such that system Xε̄E,ε̄D,λ has at most 2 limit cycles in V, for each value (ε̄, E,D, λ) ∈ [0, ε̄0]×W. We prove the above theorem in Section 3.2.4. When β−1 = β0 = β1 = β2 = 0 and β3 6= 0, we deal with the slow-fast codimen- sion 4 saddle-node case. Theorem 2.9 (Slow-fast codimension 4 saddle-node case). Let Q1 and Q2 be smooth functions, β−1 = β0 = β1 = β2 = 0 and β3 6= 0 for some (E0, D0) ∈ R × Sm−1. For each compact set K in the (x, y)-space there exists ε̄0 > 0 and a neighborhood W of (E0, D0, λ0) in the (E,D, λ)-space such that system Xε̄E,ε̄D,λ has at most 2 limit cycles in K, for each value (ε̄, E,D, λ) ∈ [0, ε̄0]×W. The above theorem will be proved in Section 3.2.5. 3. Proof of Theorems 2.1–2.9 There are essentially two types of normal forms of (2.1) near the origin (x, y) = (0, 0) which turn out to be useful for proving the results stated in Section 2.2. One type is analytic Liénard normal form (3.1), used in the slow-fast Hopf region in the parameter space (see Section 3.1). The other type of normal form is smooth and EJDE-2020/90 FINITE CYCLICITY IN SLOW-FAST DARBOUX SYSTEMS 9 given in (3.11). We use it to prove the results in the other regions in the parameter space (see Section 3.2). 3.1. Proof of Theorems 2.1 and 2.4. 3.1.1. Slow-fast Hopf case. We consider slow-fast systems Xε̄,µ : { ẋ = y − F (x, µ) ẏ = ε̄G(x, µ), (3.1) where ε̄ ≥ 0, ε̄ ∼ 0, µ ∼ µ0 ∈ Rp and F and G are analytic. Following [12], we say that system Xε̄,µ for (ε̄, µ) = (0, µ0) has a slow-fast Hopf point (at (x, y) = (0, 0)) if F (0, µ0) = ∂F ∂x (0, µ0) = G(0, µ0) = 0, ∂ 2F ∂x2 (0, µ0) 6= 0 and ∂G ∂x (0, µ0) < 0. Suppose that system Xε̄,µ has a slow-fast Hopf point for (ε̄, µ) = (0, µ0). Then Theorem 1.2 in [12] implies that the slow-fast Hopf point in the analytic Liénard family Xε̄,µ has a finite cyclicity. More precisely, there exists ε̄0 > 0, a neighborhood V of (x, y) = (0, 0), a neighborhood W of µ0 in the µ-space and some N ∈ N such that Xε̄,µ has at most N limit cycles in V, for all (ε̄, µ) ∈ [0, ε̄0]×W. To prove Theorem 2.1.1 (resp. Theorem 2.4.1), it suffices to bring (non-Liénard analytic) system Xε̄E,ε̄D,λ, near (x, y) = (0, 0), into an analytic Liénard normal form of type (3.1) and to show that the obtained normal form has a slow-fast Hopf point at (x, y) = (0, 0) for (ε̄, E,D, λ) = (0, E0, D0, λ0) where (E0, D0) = (1, 0) (resp. (E0, D0) is introduced in Theorem 2.4.1). When (ε̄, E,D, λ) = (0, E0, D0, λ0), the linearized vector field of X0,0,λ0 = (y− x2)∂x + 0∂y at (x, y) = (0, 0) is of nilpotent type. This implies that the linear part of X0,0,λ0 at the origin is not radial, i.e. not of the form αx∂x + αy∂y. Using this and the fact that Q1 and Q2 are analytic we can find an analytic (ε̄, E,D, λ)-family of coordinate changes (x̄, ȳ) = Φε̄,E,D,λ(x, y) = ( Φε̄,E,D,λ1 (x, y),Φε̄,E,D,λ2 (x, y) ) , with (x, y) ∼ (0, 0), (ε̄, E,D, λ) ∼ (0, E0, D0, λ0) and Φ0,E0,D0,λ0(0, 0) = (0, 0), and a nowhere zero analytic function ψε̄,E,D,λ(x̄, ȳ) such that ψε̄,E,D,λ(x̄, ȳ) · ( ȳ − F (x̄, ε̄, E,D, λ) Ḡ(x̄, ε̄, E,D, λ) ) = DΦε̄,E,D,λ(x, y)Xε̄E,ε̄D,λ(x, y), (3.2) for some analytic functions F and Ḡ, F (0, 0, E0, D0, λ0) = Ḡ(0, 0, E0, D0, λ0) = 0. Thus, there exists a local analytic (ε̄, E,D, λ)-family of coordinate changes trans- forming Xε̄E,ε̄D,λ to an analytic (ε̄, E,D, λ)-family of Liénard equations, up to multiplication by a nowhere zero analytic function. See [16, Theorem 1]. First, let us show that Ḡ = O(ε̄). When ε̄ = 0, system Xε̄E,ε̄D,λ has the critical curve {y = x2}. Combining this fact with (3.2) we obtain Ḡ ( Φ0,E,D,λ 1 (x, x2), 0, E,D, λ ) = 0 (3.3) and Φ0,E,D,λ 2 (x, x2)− F (Φ0,E,D,λ 1 (x, x2), 0, E,D, λ) = 0. (3.4) If we differentiate (3.4) with respect to x, we obtain ∂Φ0,E0,D0,λ0 2 ∂x (0, 0)− ∂F ∂x (0, 0, E0, D0, λ0) ∂Φ0,E0,D0,λ0 1 ∂x (0, 0) = 0. (3.5) It follows from (3.5) that ∂Φ 0,E0,D0,λ0 1 ∂x (0, 0) 6= 0. Using this and (3.3) we obtain Ḡ(x̄, ε̄, E,D, λ) = ε̄G(x̄, ε̄, E,D, λ) where G is analytic and x̄ ∼ 0. 10 R. HUZAK EJDE-2020/90 In the rest of this section we prove that the analytic family ˙̄x = ȳ − F (x̄, ε̄, E,D, λ) ˙̄y = ε̄G(x̄, ε̄, E,D, λ) (3.6) has a slow-fast Hopf point at (x̄, ȳ) = (0, 0) for (ε̄, E,D, λ) = (0, E0, D0, λ0). We have Φ0,E0,D0,λ0 2 (x, 0) ≡ 0 because Φ0,E0,D0,λ0 preserves the line {y = 0}. Using this and the first component of (3.2) we obtain ψ0,E0,D0,λ0 ( Φ0,E0,D0,λ0(x, 0) ) F (Φ0,E0,D0,λ0 1 (x, 0), 0, E0, D0, λ0) = ∂Φ0,E0,D0,λ0 1 ∂x (x, 0)x2. (3.7) Since Φ0,E0,D0,λ0(0, 0) = (0, 0), ∂Φ 0,E0,D0,λ0 1 ∂x (0, 0) 6= 0 and ψ0,E0,D0,λ0(0, 0) 6= 0, (3.7) implies that F (0, 0, E0, D0, λ0) = ∂F ∂x (0, 0, E0, D0, λ0) = 0 and ∂2F ∂x2 (0, 0, E0, D0, λ0) = 2 ψ0,E0,D0,λ0(0, 0) ∂Φ 0,E0,D0,λ0 1 ∂x (0, 0) 6= 0. (3.8) On the other hand, as a consequence of the Implicit Function Theorem, the x- nullcline of Xε̄E,ε̄D,λ near the origin, defined in (2.1), is given by y = η(x, ε̄, E,D, λ) where η is analytic and η = x2 +O(ε̄). If we substitute the function η(x, ε̄, E,D, λ) for y in (3.2), divide the second component of (3.2) by ε̄ and let (ε̄, E,D, λ) → (0, E0, D0, λ0), we obtain ψ0,E0,D0,λ0 ( Φ0,E0,D0,λ0(x, x2) ) .G(Φ0,E0,D0,λ0 1 (x, x2), 0, E0, D0, λ0) = ∂Φ0,E0,D0,λ0 2 ∂y (x, x2)2xf(x,E0, D0, λ0) (3.9) where f is the slow dynamics defined in (2.2). Note that β−1(D0, λ0) = 0 and β0(E0, D0, λ0) < 0 for (E0, D0) = (1, 0) or for a parameter (E0, D0) satisfying the assumptions of Theorem 2.4.1 where β−1 and β0 are defined after (2.2). Since ∂Φ 0,E0,D0,λ0 2 ∂x (0, 0) = 0 (Φ0,E0,D0,λ0 2 (x, 0) ≡ 0), we have ∂Φ 0,E0,D0,λ0 2 ∂y (0, 0) 6= 0 and (3.9) implies G(0, 0, E0, D0, λ0) = 0 and ∂G ∂x (0, 0, E0, D0, λ0) = 2 ∂Φ 0,E0,D0,λ0 2 ∂y (0, 0)β0(E0, D0, λ0) ψ0,E0,D0,λ0(0, 0) ∂Φ 0,E0,D0,λ0 1 ∂x (0, 0) 6= 0. (3.10) Since the critical curve {y = x2} of X0,0,λ0 is concave up, we conclude that ∂Φ 0,E0,D0,λ0 2 ∂y (0, 0) > 0 (resp. ∂Φ 0,E0,D0,λ0 2 ∂y (0, 0) < 0) if the critical curve of (3.6), for (ε̄, E,D, λ) = (0, E0, D0, λ0), is concave up, i.e. (3.8) is positive (resp. con- cave down, i.e. (3.8) is negative). Now comparing expressions (3.8) and (3.10), we see that ∂G ∂x (0, 0, E0, D0, λ0) < 0. This completes the proof of Theorem 2.1.1 and Theorem 2.4.1. 3.1.2. Slow-fast saddle case. In this section we prove Theorems 2.1.2 and 2.4.2. Let us recall that β−1(D0, λ0) = 0 and β0(E0, D0, λ0) > 0 for (E0, D0) = (−1, 0) or for a parameter (E0, D0) satisfying the assumptions of Theorem 2.4.2. It can be easily seen that system Xε̄E,ε̄D,λ has a hyperbolic saddle near (x, y) = (0, 0), for EJDE-2020/90 FINITE CYCLICITY IN SLOW-FAST DARBOUX SYSTEMS 11 (ε̄, E,D, λ) ∼ (0, E0, D0, λ0) and ε̄ > 0. Thus we have no limit cycles near the origin in the phase space. On the other hand, there are no detectable canard limit cycles for (ε̄, E,D, λ) ∼ (0, E0, D0, λ0), ε̄ ≥ 0, because the slow dynamics (2.2) points from the repelling part of the critical curve to the attracting part of the critical curve near x = 0 (x′ = β0 +O(x) > 0 for (E,D, λ) = (E0, D0, λ0)). 3.2. Proof of Theorems 2.2, 2.6, 2.8 and 2.9. 3.2.1. Bringing Xε̄E,ε̄D,λ to a smooth normal form for generic nilpotent contact points. We consider slow-fast systems ẋ = y ẏ = −xy + ε̄g(x, ε̄, µ) + ε̄y2H(x, y, ε̄, µ), (3.11) where ε̄ ∼ 0, ε̄ ≥ 0, µ ∼ µ0 ∈ Rp and g and H are smooth functions. System (3.11) has a generic nilpotent contact point at the origin. The cyclicity of this nilpotent contact point has been studied in [8, 9, 17], depending on the singularity order at the contact point, i.e. the order of vanishing of g(x, 0, µ0) at x = 0. Suppose that Q1 and Q2 in (2.1) are smooth. We transform system Xε̄E,ε̄D,λ, near (x, y) = (0, 0), to a slow-fast system of type (3.11). Lemma 3.1. Consider a smooth slow-fast system Xε̄E,ε̄D,λ, defined in (2.1), with (E,D, λ) ∼ (E0, D0, λ0), ε̄ ∼ 0 and ε̄ ≥ 0. There exists a local smooth (ε̄, E,D, λ)- family of coordinate changes (x, y) 7→ Φε̄,E,D,λ(x, y), with Φ0,E,D,λ(0, 0) = (0, 0), bringing Xε̄E,ε̄D,λ in a smooth (ε̄, E,D, λ)-family (up to multiplication by a smooth strictly positive function) ẋ = y ẏ = −xy + ε̄g(x, ε̄, E,D, λ) + ε̄y2H(x, y, ε̄, E,D, λ), (3.12) with smooth functions g and H, and g(x, 0, E,D, λ) = xf( x√ 2 , E,D, λ), (3.13) where f is the slow dynamics of Xε̄E,ε̄D,λ. Proof. We can write the slow-fast system Xε̄E,ε̄D,λ as ẋ = y − x2 +O(ε̄) ẏ = ε̄ ( − 2Ex+ 2Exy + 〈D, Q̃(x, y, ε̄D, λ)〉 ) (3.14) where O(ε̄) is a smooth function in (x, y, ε̄, E,D, λ) and Q̃ is a smooth function defined in Section 2.1. Using the (smooth) coordinate change Y = y − x2 + O(ε̄) near (x, y) = (0, 0), the vector field (3.14) changes into ẋ = y ẏ = ε̄ ( − 2Ex+ 2Ex(y + x2) + 〈D, Q̃(x, y + x2, 0, λ)〉+O(ε̄) ) + (−2x+O(ε̄))y (3.15) 12 R. HUZAK EJDE-2020/90 where we denote Y again by y and where the O(ε̄)-terms are smooth functions in (x, y, ε̄, E,D, λ). Using the first-order Taylor expansion of the y-component in (3.15) w.r.t. y about y = 0, the vector field (3.15) can be written as ẋ = y ẏ = (−2x+O1(ε̄))y + ε̄ ( 2xf(x,E,D, λ) +O2(ε̄) ) +O(ε̄y2) (3.16) where f is defined in (2.2), O1(ε̄), O2(ε̄) and O(ε̄y2) are smooth functions and O1(ε̄) and O2(ε̄) are independent of y. After a translation x→ x+ ε̄α, with α smooth in (ε̄, E,D, λ), the vector field (3.16) becomes ẋ = y ẏ = −xl(x, ε̄, E,D, λ)y + ε̄ ( 2xf(x,E,D, λ) +O(ε̄) ) +O(ε̄y2) (3.17) where l is a smooth function, l(x, 0, E,D, λ) ≡ 2, O(ε̄) and O(ε̄y2) are smooth and O(ε̄) is independent of y. Since l is strictly positive near x = 0, there exists a strictly positive smooth function L(x, ε̄, E,D, λ) (L(x, 0, E,D, λ) ≡ 2) such that x2 2 L(x) = ∫ x 0 sl(s)ds. After differentiating this with respect to x and after division by x we obtain L(x) + x 2 L′(x) = l(x). (3.18) Using the coordinate change X = x √ L(x) (we write x = XL̃(X) where L̃ is a smooth function and L̃(x, 0, E,D, λ) ≡ 1√ 2 ), the expression (3.18) and multiplica- tion by √ L(x) l(x) > 0, the vector field (3.17) changes into ẋ = y ẏ = −xy + ε̄ ( 2x l(xL̃(x)) f(xL̃(x), E,D, λ) +O(ε̄) ) +O(ε̄y2) (3.19) where we denote X again by x and O(ε̄) and O(ε̄y2) are new smooth functions. It is clear now that the vector field (3.19) is of type (3.12) with g(x, ε̄, E,D, λ) = 2x l(xL̃(x)) f(xL̃(x), E,D, λ) +O(ε̄). This implies that g(x, 0, E,D, λ) = xf( x√ 2 , E,D, λ). � We use the normal form (3.12) when we study limit cycles of Xε̄E,ε̄D,λ near (x, y) = (0, 0) (see Sections 3.2.2–3.2.5). 3.2.2. Proof of Theorem 2.2. Suppose that Q1, Q2 are smooth and β−1(D0, λ0) 6= 0 for some D0 ∈ Sm−1. Since β−1(D0, λ0) 6= 0, (3.13) implies that the order of vanishing of g(x, 0, E,D0, λ0) at x = 0 is 0 for each E kept in a compact subset of R. Thus, the contact point of (3.12) (or Xε̄E,ε̄D,λ) is of jump type and there are no limit cycles of Xε̄E,ε̄D,λ. See e.g. [9, Section 3.4] for a detailed study of the jump point. (Note that system (3.12) (or Xε̄E,ε̄D,λ) has no singularities in a fixed neighborhood of (x, y) = (0, 0) for ε̄ ∼ 0, ε̄ > 0, (D,λ) ∼ (D0, λ0) and E kept in the compact subset of R.) EJDE-2020/90 FINITE CYCLICITY IN SLOW-FAST DARBOUX SYSTEMS 13 3.2.3. Proof of Theorem 2.6. Suppose that Q1 and Q2 are smooth, β−1(D0, λ0) = β0(E0, D0, λ0) = 0 and β1(D0, λ0) 6= 0 for some (E0, D0) ∈ R×Sm−1. Using (3.13) we have that the order of vanishing of g(x, 0, E0, D0, λ0) at x = 0 is 2. This implies that the contact point of (3.12) (or Xε̄E,ε̄D,λ) is of slow-fast Bogdanov-Takens type and we can apply the results of [8]. Following [8], system (3.12) has at most 1 (hyperbolic) limit cycle in an (ε̄, E,D, λ)-uniform neighborhood of (x, y) = (0, 0) for (ε̄, E,D, λ) ∼ (0, E0, D0, λ0) and ε̄ ≥ 0. On the other hand, there are no detectable canard limit cycles in Xε̄E,ε̄D,λ for (ε̄, E,D, λ) ∼ (0, E0, D0, λ0) and ε̄ ≥ 0 because the passage from the attracting part to the repelling part of the critical curve is not possible (for (E,D, λ) = (E0, D0, λ0), x ∼ 0 and x 6= 0, the slow dynamics (2.2) is given by x′ = x(β1(D0, λ0) +O(x))). This completes the proof of Theorem 2.6. 3.2.4. Proof of Theorem 2.8. Suppose thatQ1 andQ2 are smooth, β−1 = β0 = β1 = 0, β2 6= 0 and β3 6= 0 for some (E0, D0) ∈ R × Sm−1. Since β2 6= 0, (3.13) implies that the order of vanishing of g(x, 0, E0, D0, λ0) at x = 0 is 3, and the contact point of (3.12) is of slow-fast codimension 3 saddle (β2 > 0) or elliptic (β2 < 0) type studied in [9]. Following [9, 10, 15, 18], the number of limit cycles near (x, y) = (0, 0) depends on the higher order terms in g(x, 0, E0, D0, λ0) and when the (symmetry breaking) coefficient in front of the quartic term in g(x, 0, E0, D0, λ0) is nonzero (i.e. β3 6= 0), system (3.12) (or Xε̄E,ε̄D,λ) has at most 2 limit cycles in an (ε̄, E,D, λ)- uniform neighborhood of (x, y) = (0, 0) for (ε̄, E,D, λ) ∼ (0, E0, D0, λ0) and ε̄ ≥ 0. Moreover, in the saddle case detectable canard limit cycles of Xε̄E,ε̄D,λ are not possible because the slow dynamics of Xε̄E,ε̄D,λ points from the repelling part of the critical curve to the attracting part of the critical curve (x′ = x2(β2(E0, D0, λ0) + O(x)) > 0, for (E,D, λ) = (E0, D0, λ0), x ∼ 0 and x 6= 0). This completes the proof of Theorem 2.8. 3.2.5. Proof of Theorem 2.9. Suppose that Q1 and Q2 are smooth, β−1 = β0 = β1 = β2 = 0 and β3 6= 0 for some (E0, D0) ∈ R × Sm−1. Then the order of g(x, 0, E0, D0, λ0) at x = 0 is 4 and the contact point of (3.12) is of slow-fast codimension 4 saddle-node type studied in [17]. Following [17], system (3.12) (i.e. Xε̄E,ε̄D,λ) has at most 2 limit cycles in an (ε̄, E,D, λ)-uniform neighborhood of (x, y) = (0, 0) for (ε̄, E,D, λ) ∼ (0, E0, D0, λ0) and ε̄ ≥ 0. Large (canard) limit cycles of Xε̄E,ε̄D,λ are not possible because the passage from the attracting part of the critical curve to the repelling part of the critical curve is not possible. We have x′ = x3(β3(D0, λ0) + O(x)) for (E,D, λ) = (E0, D0, λ0), x ∼ 0 and x 6= 0. This completes the proof of Theorem 2.9. 4. Generalization of the slow-fast Darboux integrable system Xε We consider a slow-fast Darboux integrable system Yε :  ẋ = −P0(x, y) ∂P1 ∂y (x, y)− εP1(x, y) ∂P0 ∂y (x, y) ẏ = P0(x, y) ∂P1 ∂x (x, y) + εP1(x, y) ∂P0 ∂x (x, y), where ε ∼ 0 and P0 and P1 are smooth or analytic functions. System Yε has the first integral (of Darboux type) H = P ε0P1. (When P0 = y− x2 and P1 = 1− y, Yε becomes Xε defined in (1.1).) The fast subsystem of Yε is given by Y0. We assume that the vector field Yε satisfies the following conditions (see [3] or [1]). 14 R. HUZAK EJDE-2020/90 • There is a compact region Ō bounded by {P0 = 0} and {P1 = 0} and the fast subsystem Y0 has no singularities in the interior of Ō (i.e. ∇P1 6= (0, 0) in the interior of Ō). • The curve {P0 = 0} is transverse to Y0, i.e. 〈∇P0, (−∂P1 ∂y , ∂P1 ∂x )〉 6= 0 (nor- mally hyperbolic singularities), except for one point where we deal with a nilpotent singularity, i.e. 〈∇P0, (−∂P1 ∂y , ∂P1 ∂x )〉 = 0, ∇P0 6= (0, 0) and ∇P1 6= (0, 0). Moreover, we assume that the contact at the nilpotent singularity is quadratic. Like in Sections 1 and 2, we can try to find a suitable blow-up at (ε, δ) = (0, 0) and to study the cyclicity of the nilpotent contact point in smooth or analytic deformations of Yε, in different directions in the parameter space (ε, δ). In the integrable direction, polynomial deformations of Yε have been studied in [3]. References [1] M. Bobieński, L. Gavrilov; Finite cyclicity of slow-fast Darboux systems with a two-saddle loop, Proc. Amer. Math. Soc., 144 (2016), no. 10, 4205–4219. [2] M. Bobieński, P. Mardesic, D. Novikov; Pseudo-Abelian integrals: unfolding generic expo- nential, J. Differential Equations, 247 (2009), no. 12, 3357–3376. [3] M. Bobieński, P. Mardesic, D. Novikov; Pseudo-abelian integrals on slow-fast Darboux sys- tems, Ann. Inst. Fourier (Grenoble), 63 (2013), no. 2, 417–430. [4] P. De Maesschalck, F. Dumortier; Time analysis and entry-exit relation near planar turning points, J. Differential Equations, 215 (2005), no. 2, 225–267. [5] P. De Maesschalck, F. Dumortier; Canard cycles in the presence of slow dynamics with singularities, Proc. Roy. Soc. Edinburgh Sect. A , 138 (2008), no. 2, 265–299. [6] P. De Maesschalck, F. Dumortier; Singular perturbations and vanishing passage through a turning point, J. Differential Equations, 248 (2010), no. 9, 2294–2328. [7] P. De Maesschalck, F. Dumortier; Detectable canard cycles with singular slow dynamics of any order at the turning point, Discrete Contin. Dyn. Syst., 29 (2011), no. 1, 109–140. [8] P. De Maesschalck, F. Dumortier; Slow-fast Bogdanov-Takens bifurcations, J. Differential Equations, 250 (2011), no. 2, 1000–1025. [9] P. De Maesschalck, F. Dumortier, R. Huzak; Limit cycles in slow-fast codimension 3 saddle and elliptic bifurcations, J. Differential Equations, 255 (2013), no. 11, 4012–4051. [10] P. De Maesschalck, F. Dumortier, R. Huzak; Primary birth of canard cycles in slow-fast codi- mension 3 elliptic bifurcations, Communications on Pure and Applied Analysis, 13 (2014), no. 6, 2641–2673. [11] F. Dumortier, R. Roussarie; Canard cycles and center manifolds, Mem. Amer. Math. Soc., 121 (1996), no. 577, x+100, With an appendix by Li Chengzhi. [12] F. Dumortier, R. Roussarie; Birth of canard cycles, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), no. 4, 723–781. [13] F. Dumortier, R. Roussarie, C. Rousseau; Hilbert’s 16th problem for quadratic vector fields, J. Differential Equations,110 (1994), no. 1, 86–133. [14] L. Gavrilov; On the number of limit cycles which appear by perturbation of Hamiltonian two-saddle cycles of planar vector fields, Bull. Braz. Math. Soc. (N.S.), 42 (2011), no. 1, 1–23. [15] R. Huzak; Cyclicity of the origin in slow-fast codimension 3 saddle and elliptic bifurcations, Discrete Contin. Dyn. Syst. ,36 (2016), no. 1, 171–215. [16] R. Huzak; Normal forms of Liénard type for analytic unfoldings of nilpotent singularities, Proc. Amer. Math. Soc., 145 (2017), no. 10, 4325–4336. [17] R. Huzak; Regular and slow-fast codimension 4 saddle-node bifurcations, J. Differential Equa- tions, 262 (2017), no. 2, 1119–1154. [18] R. Huzak; Canard explosion near non-Liénard type slow-fast Hopf point, J. Dynam. Differ- ential Equations, 31 (2019), no. 2, 683–709. [19] M. Krupa, P. Szmolyan; Relaxation oscillation and canard explosion, J. Differential Equa- tions, 174 (2001), no. 2, 312–368. EJDE-2020/90 FINITE CYCLICITY IN SLOW-FAST DARBOUX SYSTEMS 15 [20] E. F. Mishchenko, Yu. S. Kolesov, A. Yu. Kolesov, N. Kh. Rozov; Asymptotic methods in singularly perturbed systems, Monographs in Contemporary Mathematics, Consultants Bureau, New York, 1994, Translated from the Russian by Irene Aleksanova. [21] S. Smale; Mathematical problems for the next century, Math. Intelligencer, 20 (1998), no. 2, 7–15. Renato Huzak Hasselt University, Campus Diepenbeek, Agoralaan Gebouw D, 3590 Diepenbeek, Bel- gium Email address: renato.huzak@uhasselt.be 1. Introduction 2. Slow dynamics and statement of results 2.1. The fast subsystem and slow dynamics 2.2. Statement of results 3. Proof of Theorems ??–?? 3.1. Proof of Theorems ?? and ?? 3.2. Proof of Theorems ??, ??, ?? and ?? 4. Generalization of the slow-fast Darboux integrable system X References