This is a preprint of: “Cyclicity of (1,3)-switching FF type equilibria”, Xingwu Chen, Jaume Llibre, Weinian Zhang, Discrete Contin. Dyn. Syst. Ser. B, vol. 24(12), 6541–6552, 2019. DOI: [10.3934/dcdsb.2019153] CYCLICITY OF (1, 3)-SWITCHING FF TYPE EQUILIBRIA XINGWU CHENA , JAUME LLIBREB AND WEINIAN ZHANGA Abstract. Hilbert’s 16th Problem suggests a concern to the cyclicity of pla- nar polynomial differential systems, but it is known that a key step to the answer is finding the cyclicity of center-focus equilibria of polynomial differ- ential systems (even of order 2 or 3). Correspondingly, the same question for polynomial discontinuous differential systems is also interesting. Recently, it was proved that the cyclicity of (1, 2)-switching FF type equilibria is at least 5. In this paper we prove that the cyclicity of (1, 3)-switching FF type equilibria with homogeneous cubic nonlinearities is at least 3. 1. Introduction and the main result A differential system of the form (1) ẋ = P (x, y), ẏ = Q(x, y), where the dot denotes derivative with respect to an independent variable t, and P and Q are both polynomials in the real variables x and y, is called a polynomial differential system on the plane R2 . The maximum of the degrees of P and Q is referred to the degree of system (1). Thus, a planar polynomial differential system of degree one is a linear differential system, and a planar polynomial differential system of degree two (or three) is called a quadratic (or cubic) differential system. A periodic orbit of a differential system which is isolated in the set of all periodic orbits of the system is called a limit cycle, which is one of main topics in the qualitative theory of differential equations in the plane (see [5, 10, 17, 19]). The rise of limit cycles near an equilibrium caused by the changes of its stability is called Hopf bifurcation (see [15]). The cyclicity of that equilibrium is the maximum number of limit cycles which can be bifurcated from that equilibrium with Hopf bifurcation in a given family of differential systems. Usually, we also call it Hopf cyclicity. Clearly, the cyclicity of linear differential systems is 0 because they do not have limit cycles. Bautin [1] proved that the cyclicity of a center-focus equilibrium in quadratic systems is 3. It is proved in [26] that the cyclicity of a center-focus equilibrium in cubic systems is at least 12, but the exact cyclicity of a center-focus equilibrium is still unknown for general polynomial differential systems of degree ≥ 3. In contrast to the above continuous differential systems, in recent decades switch- ing differential systems became attractive because nonsmooth dynamics and sliding 2010 Mathematics Subject Classification. 37G15, 37D45. Key words and phrases. Hopf bifurcation, cyclicity, discontinuous differential system, limit cycle. 1 2 XINGWU CHEN, JAUME LLIBRE AND WEINIAN ZHANG mode control were considered in mechanics, electronics and economics as shown in books by Bernardo et al [2], Filippov [12], Kunze [18] and Simpson [25], the survey [23] by Makarenkov and Lamb, and references therein. Accordingly, the cyclicity of center-focus equilibria is also interesting for discontinuous differential systems but some new difficulties are involved (see [8, 11, 16]) because the well-known fact that the coefficient of even order is generated by coefficents of lower orders in the polynomial ring is not true again and therefore all coefficients in the displacement function are needed. A general form of switching polynomial differential systems is the following ( ) { ẋ X+ (x, y) if y > 0, (2) = ẏ X− (x, y) if y < 0. This is a special class of discontinuous differential systems on R2 with a single switching line y = 0. We are interested in the case that the origin O : (0, 0) is an FF type equilibrium (see [11]), i.e., both systems (ẋ, ẏ) = X+ (x, y) and (ẋ, ẏ) = X− (x, y) have a center-focus equilibrium at O. Since the transformation y → −y in (2) does not change the number of limit cycles, for convenience we simply call (2) an (m, n)-switching differential system if both X+ and X− are of polynomial forms and m = min{deg(X+ ), deg(X− )}, n = max{deg(X+ ), deg(X− )}. It is known from [22] that the cyclicity of (1, 1)-switching FF type equilibria is 2. For (2, 2)-switching FF type equilibria, the cyclicity was proved to be at least 5 and 9 in the case of weak focus and the case of center respectively in [6, 8]. In 2001 Coll, Gasull and Prohens [11] considered a special (1, 2)-switching FF type equilibrium, which appears in (2) with X+ (x, y) := (λ + i)z + p20 z 2 + p11 z z̄ + p02 z̄ 2 and X− (x, y) := iz, where z = x + i y, 11 15 6 − 3a 11 5 p20 = − + 2b π − c + i, p11 = + c − 4b π + i, 6 32 8 12 8 37 5 6 − 3a p02 = + 2b π − c − i, 48 32 8 and λ, a, b, c ∈ R, and showed that the Hopf cyclicity is at least 4. In 2003 Gasull and Torregrosa in [13] generally considered the (1, 2)-switching FF type equilibrium O of system (2) with ( ) ( ) w1 x − y + x2 + (α + 7/5)xy + p02 y 2 −y X+ (x, y) := , X− (x, y) := , x + w1 y + x2 + q11 xy + q02 y 2 x where 17 3 99 32 16 3 3 p02 = − + α − w2 + w5 + αw5 + w4 − αw2 + 24w2 w5 − 8w3 , 50 20 40 25 5 2 2 13 6 1 3 q11 = + 2α − 32w3 , q02 = − − α + w2 , 10 5 2 4 and proved by using the Liapunov constants that the Hopf cyclicity is at least 5, Recently, the authors in [7] applied the averaging method up to 6-th order to the more general (1, 2)–switching FF type equilibrium of system (2) with ( ) ( ) λ1 x − y − λ3 x2 + (2λ2 + λ5 )xy + λ6 y 2 −y X+ (x, y) := , X− (x, y) := , x + λ1 y + λ2 x2 + (2λ3 + λ4 )xy + (λ7 − λ2 )y 2 x CYCLICITY OF FF TYPE EQUILIBRIA 3 where 7 ∑ 7 ∑ λ1 = λ1j ϵj , λi = λij ϵj , i = 2, ..., 7, j=1 j=0 and also obtained that the Hopf cyclicity is at least 5. In this paper, as an effort to higher degree, we further investigate the cyclicity problem for the system (1.2) with ( ) ( ) λ1 x − y + λ2 x3 + λ3 x2 y + λ4 xy 2 + λ5 y 3 −y X+ (x, y) := , X− (x, y) := . x + λ1 y + λ6 x3 + λ7 x2 y + λ8 xy 2 + λ9 y 3 x It is a (1, 3)-switching system (2) with homogeneous cubic nonlinear terms and obviously has an FF type equilibrium at O. For convenience we use HC(1,3) to label this system. We will prove the following results. Theorem 1. The averaging method up to k-th order provides an upper bound of cyclicity of system HC(1,3) at O to be at most k. Theorem 2. The averaging method up to 6-th order provides a lower bound of cyclicity of system HC(1,3) at O to be at least 3. The method up to 7-th order does not increase the bound. Theorems 1 and 2, providing an upper bound and a lower bound respectively, will be proved in sections 2 and 3 respectively. Remark that the result of lower bound, given in Theorem 2 for special cubic systems, actually implies a general result: the cyclicity of a general (1,3)-switching FF type equilibrium is at least 3. 2. Prelimilaries and proof of Theorem 1 In polar coordinates (r, θ) ∈ R × S1 , where x = r cos θ and y = r sin θ, system HC(1,3) can be rewritten as ( ) { ṙ Y+ (r, θ, λ) if θ ∈ [0, π], (3) = θ̇ Y− (r, θ, λ) if θ ∈ [π, 2π], where λ = (λ1 , ..., λ9 ) ∈ R9 , Y− (r, θ, λ) = (0, 1)T , Y+ (r, θ, λ) = (H(r, θ, λ), G(r, θ, λ))T , and H(r, θ, λ) = λ1 r + r3 (λ2 cos4 θ + (λ3 + λ6 ) cos3 θ sin θ + (λ4 + λ7 ) cos2 θ sin2 θ +(λ5 + λ8 ) cos θ sin3 θ + λ9 sin4 θ), G(r, θ, λ) = 1 + r2 (λ6 cos4 θ − (λ2 − λ7 ) cos3 θ sin θ −(λ3 − λ8 ) cos2 θ sin2 θ − (λ4 − λ9 ) cos θ sin3 θ − λ5 sin4 θ). Since we want to study the Hopf bifurcation at the origin of an FF-type equilibrium of system (3), introduce a parameter ε, make the rescaling r → rε, and express λi s in the form 9 ∑ λi := λij εj , i = 1, ..., 9, j=1 4 XINGWU CHEN, JAUME LLIBRE AND WEINIAN ZHANG with λ10 = 0. Then system (3) reduces to the differential equation  ∞    H(r, θ, λ, ε) ∑ i dr = ε Fi (θ, r, λ) if θ ∈ [0, π], (4) = G(r, θ, λ, ε) i=1 dθ    0 if θ ∈ (π, 2π], where each Fi is a polynomial in the variables r, sin θ, cos θ and λ. Then the problem on the cyclicity of the Hopf bifurcation at the origin of system (3) becomes the problem on the cyclicity of the Hopf bifurcation at the origin of the differential system (4), which is written in the normal form for applying the averaging theory of arbitrary order for studying its periodic orbits. The classical averaging theory (see for instance [3, 4, 24]) has been extended recently for computing periodic orbits of analytical differential equations of one variable with arbitrary order in the small parameter ε by Giné, Grau and Llibre [14]. Later on this theory was extended by Llibre, Novaes and Teixeira [20] to arbitrary order in ε for continuous differential systems in n variables. Using Llibre, Novaes and Teixeira’s arguments (see [21]) the formulas obtained for the averaged functions of arbitrary order in [14] or [20] also work for the discontinuous differential systems. Let φ(·, z) : [0, tz ] → Rn 1 be the solution of the unperturbed differential equation ′ r (θ) = 0 such that φ(0, z) = z. Then φ(θ, z) ≡ z. Using the notations and results of [20, 21], we see that the averaged function fi : (0, ∞) → R of order i = 1, 2, . . . , k is given by wi (π, z) (5) fi (z) = , i! where functions wi : R × (0, ∞) → R, i = 1, 2, . . . , k, are defined recurrently by ∫ t( wi (t, z) = i! Fi (s, φ(s, z)) + 0 i−1 ∑∑ ∏l ) 1 ∂L bj F i−l (s, φ(s, z)) wj (s, z) ds, b1 ! b2 !2!b2 · · · bl !l!bl ∂rL j=1 l=1 Sl where L = b1 + b2 + · · · + bl and Sl is the set of all l-tuples of non–negative integers (b1 , b2 , · · · , bl ) satisfying b1 +2b2 +· · ·+lbl = l. Observe that in (5) we have wi (π, z) instead of wi (2π, z) this is due to the fact that the contribution of the linear vector field on the half-plane y < 0 to the averaged function is zero, see for more details [21]. In subsection 4.1 of [20] are provided all the expressions of the function fi (r) for i = 1, . . . , 5. The result stated in Theorem A of [20] and [21] implies that if the first aver- aged function f1 (r) is not identically zero, then every positive simple zero of f1 (r) provides a limit cycle of the discontinuous differential equation (4) on the cylinder C = {(r, θ) ∈ (0, ∞) × S1 }. So it also provides a limit cycle of the discontinuous differential system HC(1,3) in R2 . Furthermore if f1 (r) is identically zero and f2 (r) is not identically zero, then every positive simple zero of f2 (r) provides a limit cycle of the discontinuous differential equation (4) on the cylinder C, and it also provides a limit cycle of the discontinuous differential system HC(1,3) in R2 . If f1 (r) and 1It seems that it should be → R. CYCLICITY OF FF TYPE EQUILIBRIA 5 f2 (r) are identically zero and f3 (r) is not identically zero, a similar result we have for the simple zeros of the function f3 (r), and so on. In short, we need to compute the averaged functions fi (r), and to study the positive simple zeros of the function fi (r) when the functions fk (r) are identically zero for k = 1, . . . , i − 1. Proof of Theorem 1. From (4) it follows easily that the functions Fi (θ, r, λ) is a polynomial in the variable r of degree 2i + 1 of the form rGi (r2 ), where the coeffi- cients of the polynomial Gi depend on θ and λ. Therefore looking at the definition of the averaged function of order i–th we obtain that fi (r) is also a polynomial of the form rgi (r2 ) of degree at most 2i+1. So the number of positive real roots of the polynomial fi (r) is at most i. Consequently the function fi (r) has at most i positive simple zeros, and it provides at most i limit cycles of system HC(1,3), and therefore from the averaging theory described just before this proves the theorem.  3. Proof of Theorem 2 Using the software Mathematica, we compute f1 (r) = λ11 πr, 1 f2 (r) = λ12 πr + (3λ20 + λ40 + λ70 + 3λ90 )πr3 , 8 1 f3 (r) = λ13 πr + (3λ21 + λ41 + λ71 + 3λ91 )πr3 , 8 (6) 1 f4 (r) = λ14 πr + (3λ22 + λ42 + λ72 + 3λ92 )πr3 8 1 − (λ20 λ30 + 3λ20 λ50 + 3λ20 λ60 + λ30 λ70 + λ50 λ70 + λ60 λ70 16 +λ20 λ80 + λ70 λ80 + 2λ30 λ90 + 2λ80 λ90 )πr5 . Of course the above expression for the function fk (r) has been computed when fi (r) ≡ 0 for all i ≤ k − 1. Moreover it is easy to find that both f2n (r) and f2n+1 (r) are polynomials with only monomials in r of odd degree and the degree of the polynomial f2n+1 (r) is less than or equal to 2n + 1. Thus, in order to find ℓ limit cycles, we need to compute averaged functions of order 2ℓ. This is why we can find 5 limit cycles for (1, 2)-switching system by the procedure of averaging up to order 6 but only get 3 limit cycles for our (1, 3)-switching system HC(1,3) up to the same order 6. We denote by #Z+ (fi ) the cardinal of the set Z+ (fi ) consisting of all positive simple zeros of the averaged function fi (r). #Z+ (f1 ) f1≡ 0 #Z+ (f2 ) f2≡ 0 #Z+ (f3 ) f3≡ 0 #Z+ (f4 ) f4 ≡ 0 #Z+ (f5 ) 0 λ11= 0 1 C2 1 C3 2 C41 2 C42 2 C43 2 Table 1. Numbers of positive simple zeros of f1 , ..., f5 6 XINGWU CHEN, JAUME LLIBRE AND WEINIAN ZHANG Lemma 3. The numbers #Z+ (fi ) for i = 1, ..., 5 are given in Table 1 under the conditions λ11 = 0, C2 , C3 , C41 , C42 and C43 , which are given in (7), (8), (9), (10) and (11), respectively. Proof. Clearly from the expression of the polynomial f1 (r) given in (6) it follows that it has no positive simple zeros. Note that f1 (r) ≡ 0 if and only if λ11 = 0. Furthermore from the expression of the polynomial f2 (r) given in (6) we obtain that f2 (r) has at most 1 positive simple zero, and that there are polynomials f2 (r) with 1 positive simple zero. Note that f2 (r) ≡ 0 if and only if (7) λ12 = 0, and λ40 = −3λ20 − λ70 − 3λ90 . These two conditions are denoted by C2 in Table 1. When (7) holds, we get that the expression of f3 (r) is given in (6). We obtain that f3 (r) has at most 1 positive simple zero, and that there are polynomials f3 (r) with 1 positive simple zero. Note that f3 (r) ≡ 0 if and only if (8) λ13 = 0, and λ41 = −3λ21 − λ71 − 3λ91 . These two conditions are denoted by C3 in Table 1. When (8) holds, we get that the expression of f4 (r) is given in (6). We obtain that f4 (r) has at most 2 positive simple zeros, and that there are polynomials f4 (r) with 2 positive simple zeros. Note that f4 (r) ≡ 0 if and only if either λ14 = 0, λ42 = −3λ22 − λ72 − 3λ92 , 3λ20 + λ70 ̸= 0, −1 (9) λ50 = (λ20 λ30 + 3λ20 λ60 + λ30 λ70 + λ60 λ70 3λ20 + λ70 +λ20 λ80 + λ70 λ80 + 2λ30 λ90 + 2λ80 λ90 ), or λ14 = 0, λ42 = −3λ22 − λ72 − 3λ92 , (10) 1 λ20 = − λ70 , λ30 = −λ80 , 3 or λ14 = 0, λ42 = −3λ22 − λ72 − 3λ92 , (11) 1 1 λ20 = − λ70 , λ90 = − λ70 , 3 3 denoted by C41 , C42 , C43 in Table 1, respectively. CYCLICITY OF FF TYPE EQUILIBRIA 7 When (9) holds, we get π f5 (r) =λ15 πr + (3λ23 + λ43 + λ73 + 3λ93 )r3 8 π − (3λ220 λ31 + 9λ220 λ51 + 9λ220 λ61 − 6λ21 λ80 λ90 16(3λ20 + λ70 ) − 2λ21 λ30 λ70 + 4λ20 λ31 λ70 + 6λ20 λ51 λ70 + 6λ20 λ61 λ70 + λ31 λ270 (12) + λ51 λ270 + λ61 λ270 + 2λ20 λ30 λ71 − 2λ21 λ70 λ80 + 2λ70 λ81 λ90 + 2λ20 λ71 λ80 + 3λ220 λ81 + 4λ20 λ70 λ81 + λ270 λ81 − 6λ21 λ30 λ90 + 6λ20 λ31 λ90 + 2λ31 λ70 λ90 − 2λ30 λ71 λ90 − 2λ71 λ80 λ90 + 6λ20 λ81 λ90 + 6λ20 λ30 λ91 + 2λ30 λ70 λ91 + 6λ20 λ80 λ91 + 2λ70 λ80 λ91 )r5 . We obtain that f5 (r) has at most 2 positive simple zeros, and that there are poly- nomials f5 (r) with 2 positive simple zeros. Similarly, when (10) (resp. (11)) holds, we get π f5 (r) =λ15 πr + (3λ23 + λ43 + λ73 + 3λ93 )r3 8 (13) π − (9λ21 λ50 + 9λ21 λ60 + 2λ31 λ70 + 3λ50 λ71 48 + 3λ60 λ71 + 2λ70 λ81 + 6λ31 λ90 + 6λ81 λ90 )r5 . (resp. π f5 (r) =λ15 πr + (3λ23 + λ43 + λ73 + 3λ93 )r3 8 (14) π − (λ21 λ30 + 3λ21 λ50 + 3λ21 λ60 + λ30 λ71 + λ50 λ71 16 + λ60 λ71 + λ21 λ80 + λ71 λ80 + 2λ30 λ91 + 2λ80 λ91 )r5 .) So we obtain that f5 (r) has at most 2 (resp. 2) positive simple zeros, and that there are polynomials f5 (r) with 2 (resp. 2) positive simple zeros. This completes the proof of Table 1.  condition for f4 ≡ 0 condition for f5 ≡ 0 #Z+ (f6 ) C41 C411 3 C42 C421 2 C422 2 C423 2 C43 C431 3 C432 2 C433 3 Table 2. Number of positive simple zeros of f6 . By the numbers #Z+ (fi ) for i = 1, ..., 5 given in Lemma 3 we get some lower bounds of the cyclicity of the Hopf bifurcation at the origin of the discontinuous differential system HC(1,3). In order to find a greater bound, we consider averaged functions of higher order. 8 XINGWU CHEN, JAUME LLIBRE AND WEINIAN ZHANG Lemma 4. The number #Z+ (f6 ) is given in Table 2 under the conditions C411 , C421 , C422 , C423 , C431 , C432 and C433 , which are given in (15), (16), (17), (18), (19), (20), (21), respectively. Proof. When λ11 = 0 and conditions C2 , C3 , C41 hold, f1 (r) ≡ f2 (r) ≡ f3 (r) ≡ f4 (r) ≡ 0 and the expression of f5 (r) is given in (12). It is easy to check that f5 (r) ≡ 0 if and only if λ15 = 0, λ43 = −3λ23 − λ73 − 3λ93 , 1 λ51 = (−3λ220 λ31 − 9λ220 λ61 + 2λ21 λ30 λ70 (3λ20 + λ70 )2 −4λ20 λ31 λ70 − 6λ20 λ61 λ70 − λ31 λ270 − λ61 λ270 − 2λ20 λ30 λ71 (15) +2λ21 λ70 λ80 − 2λ20 λ71 λ80 − 3λ220 λ81 − 4λ20 λ70 λ81 − λ270 λ81 +6λ21 λ30 λ90 − 6λ20 λ31 λ90 − 2λ31 λ70 λ90 + 2λ30 λ71 λ90 +6λ21 λ80 λ90 + 2λ71 λ80 λ90 − 6λ20 λ81 λ90 − 2λ70 λ81 λ90 −6λ20 λ30 λ91 − 2λ30 λ70 λ91 − 6λ20 λ80 λ91 − 2λ70 λ80 λ91 ), denoted by C411 in Table 2. Under C411 , we compute f6 (r) and obtain π f6 (r) = λ16 πr + (3λ24 + λ44 + λ74 + 3λ94)r3 8 π − (9λ320 λ32 + 27λ320 λ52 + 27λ320 λ62 + 6λ221 λ30 λ70 16(3λ20 + λ70 )2 −6λ20 λ22 λ30 λ70 − 6λ20 λ21 λ31 λ70 + 15λ220 λ32 λ70 + 27λ220 λ52 λ70 +27λ220 λ62 λ70 − 2λ22 λ30 λ270 − 2λ21 λ31 λ270 + 7λ20 λ32 λ270 + 9λ20 λ52 λ270 +λ20 λ62 λ270 + λ32 λ370 +λ52 λ370 +λ62 λ370 −6λ20 λ21 λ30 λ71 +2λ270 λ80 λ92 +6λ220 λ31 λ71 + 2λ21 λ30 λ70 λ71 + 2λ20 λ31 λ70 λ71 − 2λ20 λ30 λ271 +6λ220 λ30 λ72 + 2λ20 λ30 λ70 λ72 + 6λ221 λ70 λ80 − 6λ20 λ22 λ70 λ80 −2λ22 λ270 λ80 − 6λ20 λ21 λ71 λ80 + 2λ21 λ70 λ71 λ80 − 2λ20 λ271 λ80 +6λ220 λ72 λ80 + 2λ20 λ70 λ72 λ80 − 6λ20 λ21 λ70 λ81 − 2λ21 λ270 λ81 +6λ220 λ71 λ81 + 2λ20 λ70 λ71 λ81 + 9λ320 λ82 + 15λ220 λ70 λ82 7λ20 λ270 λ82 +λ370 λ82 + 18λ221 λ30 λ90 − 18λ20 λ22 λ30 λ90 − 18λ20 λ21 λ31 λ90 +18λ220 λ32 λ90 − 6λ22 λ30 λ70 λ90 − 6λ21 λ31 λ70 λ90 + 12λ20 λ32 λ70 λ90 +2λ32 λ270 λ90 + 12λ21 λ30 λ71 λ90 − 6λ20 λ31 λ71 λ90 − 2λ31 λ70 λ71 λ90 +2λ30 λ271 λ90 − 6λ20 λ30 λ72 λ90 − 2λ30 λ70 λ72 λ90 + 18λ221 λ80 λ90 −18λ20 λ22 λ80 λ90 − 6λ22 λ70 λ80 λ90 + 12λ21 λ71 λ80 λ90 + 2λ271 λ80 λ90 −6λ20 λ72 λ80 λ90 − 2λ70 λ72 λ80 λ90 − 18λ20 λ21 λ81 λ90 − 6λ21 λ70 λ81 λ90 −6λ20 λ71 λ81 λ90 − 2λ70 λ71 λ81 λ90 + 18λ220 λ82 λ90 + 12λ20 λ70 λ82 λ90 +2λ270 λ82 λ90 − 18λ20 λ21 λ30 λ91 + 18λ220 λ31 λ91 − 6λ21 λ30 λ70 λ91 +12λ20 λ31 λ70 λ91 + 2λ31 λ270 λ91 − 6λ20 λ30 λ71 λ91 − 2λ30 λ70 λ71 λ91 −18λ20 λ21 λ80 λ91 − 6λ21 λ70 λ80 λ91 − 6λ20 λ71 λ80 λ91 − 2λ70 λ71 λ80 λ91 +18λ220 λ81 λ91 + 12λ20 λ70 λ81 λ91 + 2λ270 λ81 λ91 + 18λ220 λ30 λ92 +12λ20 λ30 λ70 λ92 + 2λ30 λ270 λ92 + 18λ220 λ80 λ92 + 12λ20 λ70 λ80 λ92 )r5 CYCLICITY OF FF TYPE EQUILIBRIA 9 π − (λ20 + 2λ70 + 5λ90 )(9λ320 + λ20 λ230 + 3λ20 λ30 λ60 64(3λ20 + λ70 ) +6λ220 λ70 + λ230 λ70 + λ20 λ270 − λ20 λ30 λ80 + λ30 λ60 λ70 + 3λ20 λ60 λ80 +λ30 λ70 λ80 +λ60 λ70 λ80 −2λ20 λ280 + 9λ220 λ90 + 6λ20 λ70 λ90 + λ270 λ90 )r7 . We obtain that f6 (r) has at most 3 positive simple zeros, and that there are poly- nomials f6 (r) with 3 positive simple zeros by the same method used in the proof of Lemma 3.2 of [7]. When λ11 = 0 and conditions C2 , C3 , C42 hold, f1 (r) ≡ f2 (r) ≡ f3 (r) ≡ f4 (r) ≡ 0 and the expression of f5 (r) is given in (13). It is easy to check that f5 (r) ≡ 0 if and only if either λ15 = 0, λ43 = −3λ23 − λ73 − 3λ93 , −1 (16) λ31 = (9λ21 λ50 + 9λ21 λ60 2(λ70 + 3λ90 ) + 3λ50 λ71 + 3λ60 λ71 + 2λ70 λ81 + 6λ81 λ90 ), or λ15 = 0, λ43 = −3λ23 − λ73 − 3λ93 , (17) λ70 = −3λ90 , λ50 = −λ60 , or λ15 = 0, λ43 = −3λ23 − λ73 − 3λ93 , (18) λ70 = −3λ90 , λ50 = −3λ21 , denoted by C421 , C422 , C423 in Table 2 respectively. Under C421 we compute f6 (r) and obtain π f6 (r) = λ16 πr + (3λ24 + λ44 + λ74 + 3λ94)r3 8 π − (−27λ221 λ50 − 27λ221 λ60 + 18λ22 λ50 λ70 + 18λ21 λ51 λ70 96(λ70 + 3λ90 ) +18λ22 λ60 λ70 + 18λ21 λ61 λ70 + 4λ32 λ270 − 36λ21 λ50 λ71 − 36λ21 λ60 λ71 +6λ51 λ70 λ71 + 6λ61 λ70 λ71 − 9λ50 λ271 − 9λ60 λ271 + 6λ50 λ70 λ72 +6λ60 λ70 λ72 + 4λ270 λ82 + 54λ22 λ50 λ90 + 54λ21 λ51 λ90 + 54λ22 λ60 λ90 +54λ21 λ61 λ90 + 24λ32 λ70 λ90 + 18λ51 λ71 λ90 + 18λ61 λ71 λ90 + 18λ50 λ72 λ90 +18λ60 λ72 λ90 + 24λ70 λ82 λ90 + 36λ32 λ290 + 36λ82 λ290 − 54λ21 λ50 λ91 −54λ21 λ60 λ91 − 18λ50 λ71 λ91 − 18λ60 λ71 λ91 )r5 . We obtain that f6 (r) has at most 2 positive simple zeros, and that there are poly- nomials f6 (r) with 2 positive simple zeros by the same method used in the proof of Lemma 3.2 of [7]. Under C422 we compute f6 (r) and obtain π f6 (r) = λ16 πr + (3λ24 + λ44 + λ74 + 3λ94)r3 8 π − (λ21 λ31 + 3λ21 λ51 + 3λ21 λ61 + λ31 λ71 + λ51 λ71 + λ61 λ71 + λ21 λ81 16 +λ71 λ81 + 2λ31 λ91 + 2λ81 λ91 )r5 . 10 XINGWU CHEN, JAUME LLIBRE AND WEINIAN ZHANG We obtain that f6 (r) has at most 2 positive simple zeros, and that there are poly- nomials f6 (r) with 2 positive simple zeros by the same method used in the proof of Lemma 3.2 of [7]. Under C423 we compute f6 (r) and get π f6 (r) = λ16 πr + (3λ24 + λ44 + λ74 + 3λ94)r3 8 π + (2λ21 λ31 − 3λ22 λ50 − 3λ22 λ60 − λ50 λ72 − λ60 λ72 16 +2λ21 λ81 − 2λ31 λ91 − 2λ81 λ91 )r5 . We obtain that f6 (r) has at most 2 positive simple zeros, and that there are poly- nomials f6 (r) with 2 positive simple zeros by the same method used in the proof of Lemma 3.2 of [7]. When λ11 = 0 and conditions C2 , C3 , C43 hold, f1 (r) ≡ f2 (r) ≡ f3 (r) ≡ f4 (r) ≡ 0 and the expression of f5 (r) is given in (14). It is easy to check that f5 (r) ≡ 0 if and only if either λ15 = 0, λ43 = −3λ23 − λ73 − 3λ93 , −1 (19) λ50 = (λ21 λ30 + 3λ21 λ60 + λ30 λ71 3λ21 + λ71 ) + λ60 λ71 + λ21 λ80 + λ71 λ80 + 2λ30 λ91 + 2λ80 λ91 ), or λ15 = 0, λ43 = −3λ23 − λ73 − 3λ93 , (20) λ30 = −λ80 , λ71 = −3λ21 or λ15 = 0, λ43 = −3λ23 − λ73 − 3λ93 , (21) λ21 = λ91 , λ71 = −3λ21 , denoted by C431 , C432 , C433 in Table 2 respectively. Under C431 we compute f6 (r) and obtain π f6 (r) = λ16 πr + (3λ24 + λ44 + λ74 + 3λ94)r3 8 π − (3λ221 λ31 + 9λ221 λ51 + 9λ221 λ61 − 2λ22 λ30 λ71 16(3λ21 + λ71 ) +4λ21 λ31 λ71 + 6λ21 λ51 λ71 + 6λ21 λ61 λ71 + λ31 λ271 + λ51 λ271 +λ61 λ271 + 2λ21 λ30 λ72 − 2λ22 λ71 λ80 + 2λ21 λ72 λ80 + 3λ221 λ81 +4λ21 λ71 λ81 + λ271 λ81 − 6λ22 λ30 λ91 + 6λ21 λ31 λ91 + 2λ31 λ71 λ91 −2λ30 λ72 λ91 − 6λ22 λ80 λ91 − 2λ72 λ80 λ91 + 6λ21 λ81 λ91 +2λ71 λ81 λ91 + 6λ21 λ30 λ92 + 2λ30 λ71 λ92 + 6λ21 λ80 λ92 + 2λ71 λ80 λ92 )r5 π − (λ70 (λ30 + λ80 )2 (λ21 + 2λ71 + 5λ91 ))r7 . 96(3λ21 + λ71 ) Therefore f6 (r) has at most 3 positive simple zeros, and that there are polynomials f6 (r) with 3 positive simple zeros by the same method used in the proof of Lemma 3.2 of [7]. CYCLICITY OF FF TYPE EQUILIBRIA 11 Under C432 we compute f6 (r) and have π f6 (r) = λ16 πr + (3λ24 + λ44 + λ74 + 3λ94)r3 8 π − (−2λ21 λ31 + 3λ22 λ50 + 3λ22 λ60 + λ50 λ72 + λ60 λ72 16 −2λ21 λ81 + 2λ31 λ91 + 2λ81 λ91 )r5 . Hence f6 (r) has at most 2 positive simple zeros, and that there are polynomials f6 (r) with 2 positive simple zeros by the same method used in the proof of Lemma 3.2 of [7]. Under C433 we compute f6 (r) and get π f6 (r) = λ16 πr + (3λ24 + λ44 + λ74 + 3λ94)r3 8 π − (λ22 λ30 + 3λ22 λ50 + 3λ22 λ60 + λ30 λ72 + λ50 λ72 + λ60 λ72 16 +λ22 λ80 + λ72 λ80 + 2λ30 λ92 + 2λ80 λ92 )r5 π + (λ70 (λ30 + λ80 )(λ30 + 5λ50 + 5λ60 + λ80 ))r7 . 192 So f6 (r) has at most 3 positive simple zeros, and that there are polynomials f6 (r) with 3 positive simple zeros by the same method used in the proof of Lemma 3.2 of [7].  Proof of Theorem 2. By Lemmas 3 and 4, the averaging method up to 6-th order provides a lower bound of cyclicity of system HC(1,3) at the origin to be at least 3. We continue applying the averaging method up to 7-th order and do not find a higher bound than 3 because f7 (r) is odd and has degree less than or equal to 7.  Finally we note that the results of switching normal forms obtained in [9], which imply that our system HC(1,3) does not represent the general system with (1,3)- switching FF type equilibrium at the origin. So, the lower bound 3 obtained in Theorem 2 is expected to be improved, but more difficulties will be involved in the computation. Acknowledgements The first author is supported by NSFC #11871355. The second author has been partially supported by the Ministerio de Economı́a, Industria y Competitivi- dad, 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 third author is supported by NSFC #11726623 and #11771307. References [1] N.N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Math. Sbornik 30 (1952) 181–196 (in Russian); Transl. Amer. Math. Soc. 100 (1954) 397–413. 12 XINGWU CHEN, JAUME LLIBRE AND WEINIAN ZHANG [2] M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dy- namical Systems, Theory and Applications, Springer-Verlag, London, 2008. [3] N.N. Bogoliubov, On Some Statistical Methods in Mathematical Physics, Akademiya Nauk Ukrainskoǐ, 1945. [4] N.N. Bogoliubov and N. Krylov, The Application of Methods of Nonlinear Mechanics in the Theory of Stationary Oscillations, Ukrainian Academy of Science, 1934. [5] X. Cen, J. Llibre and M. Zhang, Periodic solutions and their stability of some higher-order positively homogenous differential equations, Chaos, Solitons & Fractals 106 (2018), 285– 288. [6] X. Chen and Z. Du, Limit cycles bifurcate from centers of discontinuous quadratic systems, Comput. Math. Appl. 59 (2010), 3836–3848. [7] X. Chen, J. Llibre and W. Zhang, Averaging approach to cyclicity of Hopf bifurcation in pla- nar linear-quadratic polynomial discontinuous differential systems, Discrete and Continuous Dynamical Systems-Series B 22 (2017), 3953–3965. [8] X. Chen, V.G. Romanovski and W. Zhang, Degenerate Hopf bifurcations in a family of FF-type switching systems, J. Math. Anal. Appl. 432 (2015), 1058–1076. [9] X. Chen and W. Zhang, Normal forms of planar switching systems, Disc. Cont. Dyn. Syst. 36 (2016), 6715–6736. [10] C. Christopher and C. Li, Limit cycles of differential equations, Advanced Courses in Math- ematics, CRM Barcelona, Birkhäuser Verlag, Basel, 2007. [11] B. Coll, A. Gasull and R. Prohens, Degenerate Hopf bifurcations in discontinuous planar systems, J. Math. Anal. Appl. 253 (2001), 671–690. [12] A.F. Filippov, Differential Equation with Discontinuous Right-Hand Sides, Kluwer Academic, Amsterdam, 1988. [13] A. Gasull and J. Torregrosa, Center-focus problem for discontinuous planar differential equa- tions, Int. J. Bifurc. Chaos 13 (2003), 1755–1765. [14] J. Giné, M. Grau and J. Llibre, Averaging theory at any order for computing periodic orbits, Physica D 250 (2013), 58–65. [15] J. K. Hale and H. Hoçak, Dynamics and Bifurcations, Springer-Verlag, Berlin, 1991. [16] M. Han and W. Zhang, On Hopf bifurcation in non-smooth planar systems, J. Diff. Equa. 248 (2010), 9: 2399–2416. [17] Yu. Ilyashenko, Centennial history of Hilbert’s 16th problem, Bull. (New Series) Amer. Math. Soc. 39 (2002), 301–354. [18] M. Kunze, Non-Smooth Dynamical Systems, Springer-Verlag, Berlin, 2000. [19] J. Li, Hilbert’s 16th problem and bifurcations of planar polynomial vector fields, Int. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003), 47–106. [20] J. Llibre, D. D. Novaes and M. A. Teixeira, Higher order averaging theory for finding periodic solutions via Brouwer degree, Nonlinearity 27 (2014), 563–583. [21] J. Llibre, D. D. Novaes and M. A. Teixeira, On the birth of limit cycles for non–smooth dynamical systems, Bull. Sci. math. 139 (2015), 229–244. [22] J. Llibre, D. D. Novaes and M. A. Teixeira, Maximum number of limit cycles for certain piecewise linear dynamical systems, Nonlinear Dynamics 82 (2015), 1159–1175. [23] O. Makarenkov and J.S.W. Lamb, Dynamics and bifurcations of nonsmooth systems: A survey, Physica D 241 (2012), 1826–1844. [24] P. Patou, Sur le mouvement d’un système soumis à des forces à courte période, Bull. Soc. Math. France 56 (1928), 98–139. [25] D.J.W. Simpson, Bifurcations in Piecewise–Smooth Continuous Systems, World Scientific Series on Nonlinear Science A, vol 69, World Scientific, Singapore, 2010. [26] P. Yu and Y. Tian, Twelve limit cycles around a singular point in a planar cubic-degree polynomial system, Comm. Nonlinear Sci. Nume. Simu. 19(2014), 2690–2705. A School of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P. R. China E-mail address: xingwu.chen@hotmail.com (Xingwu Chen), matzwn@126.com (Weinian Zhang) CYCLICITY OF FF TYPE EQUILIBRIA 13 B Department de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bel- laterra, Barcelona, Catalonia, Spain E-mail address: jllibre@mat.uab.cat (Jaume Llibre)