This is a preprint of: “Limit cycles of a class of discontinuous piecewise differential systems sep- arated by the curve y = xn via averaging theory”, Zhifei Guo, Jaume Llibre, Internat. J. Bifur. Chaos Appl. Sci. Engrg., vol. 32(12), 2250187, 2022. DOI: [10.1142/S0218127422501875] LIMIT CYCLES OF DISCONTINUOUS PIECEWISE QUADRATIC PERTURBATIONS OF A LINEAR CENTER SEPARATED BY THE CURVE y = xn ZHIFEI GUO1 AND JAUME LLIBRE2 Abstract. The study of the limit cycles of planar differential systems is one of the main problems in the qualitative theory of differential systems. These last years a big interest appeared for studying the limit cycles of the piecewise differential systems due to their many applications. Here we prove that the linear center ẋ = y, ẏ = −x, can produce at most 6 crossing limit cycles for n ≥ 4 even and at most 7 crossing limit cycles for n ≥ 5 odd using the averaging theory of first order, when it is perturbed by discontinuous piecewise differential systems formed by two pieces separated by the curve y = xn (n ≥ 4), and having in each piece a quadratic polynomial differential system. Using the averaging theory of second order the perturbed system can be chosen in such way that it has 0, 1, 2, 3, 4, 5, 6, 7 or 9 crossing limit cycles if 4 ≤ n ≤ 74 is even and 0, 1, 2, 3, 4, 5, 6, 7, 9 or 11 crossing limit cycles if n ≥ 76 is even. The averaging theory of second order produces the same number of crossing limit cycles as the averaging theory of first order if n ≥ 5 is odd. The main tools for proving our results are the new averaging theory developed for studying the crossing limit cycles of the discontinuous piecewise differential systems, and the theory for studying the zeros of a function using the extended Chebyshev systems. 1. Introduction and statement of the main results The method of averaging has been one of the effectively analytical methods to detect the existence of limit cycles of the nonlinear differential equations. For the s- mooth differential equations the averaging theory can be found in some monographs (see [18, 19, 39]). In recent years the classical averaging theory for computing pe- riodic solutions was developed rapidly (see [5, 16, 17, 27–29, 37]). Recently the averaging theory for computing periodic solutions has been developed for discon- tinuous piecewise differential systems (see [21, 26, 30, 42]). It is well-known that a limit cycle of a differential system is an isolated periodic solution in the set of all periodic solutions of the system. Limit cycles have been a significant research topic in the qualitative theory of planar differential systems. The second part of the well-known Hilbert 16th problem (see [12, 23]) asks for the maximum number and relative positions of the limit cycles for the polynomial systems of degree n. To find an upper bound for the maximum number of limit cycles is an open problem but the possible relative positions were solved in [31]. 2010 Mathematics Subject Classification. 34C29, 34C25, 34A36. Key words and phrases. Limit cycles, the method of averaging, discontinuous piecewise differ- ential systems. 1 2 ZHIFEI GUO AND JAUME LLIBRE Also the Hilbert problem can be extended to nonsmooth differential systems with discontinuities because such differential systems are widely used in various fields, for example, control systems, economy, neuron system, electrical circuits and mechanics (see [3, 8, 13, 35, 40, 41]). For a nonsmooth differential system we follow the Filippov’s convention to define the vector fields on its discontinuous boundary (see [15]). Given a piecewise smooth system ẋ = F± (x), x ∈ Σ± := Rn \ Σ, where Σ is called the set of discontinuity or switching boundary, Σ± are open sets and F± are continuous functions in Σ± respectively. The vector field at each point of Σ± is defined by F± , respectively. We call a point x0 ∈ Σ a crossing point if F± (x0 ) point into Σ+ or Σ− simultaneously. In this case an orbit of the system near x0 is a concatenation of the orbits of the two subsystems. The collection of all crossing points forms the crossing region. If a closed curve is formed by concatenating the orbits of the two subsystems and it intersects with Σ only at crossing points then we call the closed curve a crossing periodic orbit. The so-called crossing limit cycle is an isolated crossing periodic orbit in the set of all crossing periodic orbits of the system. There are many works studying the number of limit cycles of the discontinuous piecewise differential systems (see [6, 9–11, 14, 20, 25, 33, 43]). Recently (see [1, 2, 7, 24, 26, 32, 38]) attentions were paid to the maximum number of crossing limit cycles for the perturbation of the linear center system (ẋ, ẏ)T = (y, −x)T by discontinuous piecewise polynomial differential systems with two zones of R2 separated by a discontinuous boundary Σ, i.e.  m  X i ±    y + ε fi (x, y)  ẋ  i=1  (1) = m  if (x, y) ∈ Σ± := R2 \ Σ,  ẏ  X  −x + εi gi± (x, y) i=1 where fi± and gi± (i = 1, . . . , m) are real polynomials. In particular, interests were made to investigate system (1) with nonlinear switching boundary Σ. For Σ = {y = x2 } Llibre, Mereu and Novaes [26] studied the number of crossing limit cycles for system (1) with quadratic perturbations and obtained that the maximum number of crossing limit cycles is 6 using the averaging theory up to order 2. Later on Buzzi, Llibre and Novaes [2] proved that the maximum number of crossing limit cycles is 7 for system (1) with linear perturbations and Σ = {y = x3 } using the Melnikov functions up to order 2. For system (1) with quadratic perturbations and Σ = {y = xn } using the Melnikov function of order 1 Ramirez and Alves [38] obtained that the maximum number of crossing limit cycles is 4 if n = 2, and 6 if n ≥ 4 even and, moreover, they gave an upper bound estimate of crossing limit cycles for n odd and higher order perturbations. Recently Andrade, Cespedes, Cruz and Novaes [1] considered the case Σ = {y = xn } (n ≥ 4) for system (1) with linear perturbations applying the higher order Melnikov method, and obtained that H(2) ≥ 4, H(3) ≥ 8, H(n) ≥ 7, for n ≥ 4 even, and H(n) ≥ 9, for n ≥ 5 odd, where H(n) denotes an upper bound of the maximum number of crossing limit cycles for system (1) with linear perturbations and Σ = {y = xn }. In this paper we investigate the number of crossing limit cycles for system (1) with Σ = {y = xn } (n ≥ 4) using the averaging theory up to order 2 but with 3 quadratic perturbations in a small parameter ε, i.e. the following system  !   y + εA1 (x, y) + ε2 A2 (x, y)     −x + εB1 (x, y) + ε2 B2 (x, y) if hn (x, y) ≥ 0,  ẋ (2) = ! ẏ  y + εC1 (x, y) + ε2 C2 (x, y)     −x + εD (x, y) + ε2 D (x, y) if hn (x, y) ≤ 0, 1 2 where hn (x, y) := y − xn (n ≥ 4) and 2 X 2 X Ai := aijk xj y k , Bi := bijk xj y k , j+k=0 j+k=0 2 X 2 X Ci := cijk xj y k , Di := dijk xj y k , j+k=0 j+k=0 for i = 1, 2. Let Qe1 be the set of conditions a110 = −d101 , a111 = −b120 + c111 + d120 , b100 = d100 , (3) b101 = d101 , b102 = d102 , c110 = −d101 , Qo1 the set of conditions a100 = c100 , a110 = −b101 − c110 − d101 , a120 = c120 , (4) a111 = −b120 + c111 + d120 , a102 = −b111 + c102 + d111 , b100 = d100 , b102 = d102 , Qe2 the set of conditions a110 = b100 = b101 = b102 = c100 = c110 = c120 = d100 = d101 = d102 = 0, (5) a111 = −b120 , c101 = −d110 , c111 = −d120 , c102 = −d111 , and Qo2 the set of conditions a100 = a120 = b100 = b102 = c100 = c110 = c120 = d100 = d101 = d102 = 0 (6) a110 = −b101 , a111 = −b120 , a102 = −b111 , c101 = −d110 , c111 = −d120 , c102 = −d111 , where Qe2 and Qo2 are subsets of Qe1 and Qo1 , respectively. Our main results are the following. Theorem 1. For |ε| sufficiently small using the averaging theory of first order system (2) with n ≥ 4 even (resp. n ≥ 5 odd) has at most 6 (resp. 7) crossing limit cycles when the conditions Qe1 (resp. Qo1 ) do not hold. Moreover we can choose parameters aijk , bijk , cijk and dijk (i = 1, 2 and 0 ≤ j + k ≤ 2) such that system (2) with n ≥ 4 even (resp. n ≥ 5 odd) has exactly 0, 1, 2, 3, 4, 5 or 6 (resp. 0, 1, 2, 3, 4, 5, 6 or 7) crossing limit cycles. Theorem 2. For |ε| sufficiently small using the averaging theory of second order we can choose the parameters aijk , bijk , cijk and dijk (i = 1, 2 and 0 ≤ j + k ≤ 2) such that system (2) under the conditions Qe2 with 4 ≤ n ≤ 74 even (resp. n ≥ 76 even) has 0, 1, 2, 3, 4, 5, 6, 7 or 9 (resp. 0, 1, 2, 3, 4, 5, 6, 7, 9 or 11) crossing limit cycles. For n ≥ 5 odd the averaging theory of second order produces the same number of 4 ZHIFEI GUO AND JAUME LLIBRE crossing limit cycles as the averaging theory of first order when the conditions Qo2 hold. Note that as it is written in the proof of Theorem 2 we cannot determine whether the bound 8 for n ≥ 4 even and the bound 10 for n ≥ 76 even are reachable. Theorems 1 and 2 are proved in Section 3. 2. Preliminaries We state some necessary definitions and results, which are used in our proofs. Let A and D be open subsets of Rd and S1 = R/T , where T is a positive period. Let χA (t, x) denote the characteristic function, i.e. χA (t, x) = 1 (resp. 0) if (t, x) ∈ A (resp. 6∈ A). Let Sj be a finite sequence of open disjoint subsets of S1 × D for j=1 S j = S × D, as usual S j denotes the closure of Sj . j = 1, 2, . . . , M such that ∪M 1 Moreover we use Σ to denote the union of boundaries of all S j for j = 1, 2, . . . , M . Given the following discontinuous piecewise differential system (7) ẋ = εF1 (t, x) + ε2 F2 (t, x) + ε3 R(t, x, ε), where M X M X Fi := χS̄j (t, x)Fij (t, x) for i = 1, 2, R := χS̄j (t, x)Rij (t, x, ε), j=1 j=1 and Fij : S ×D → R , 1 d Rij : S × D × (−ε0 , ε0 ) → Rd with ε0 > 0, i = 1, 2 and 1 j = 1, . . . , M , are all continuous functions and are all T -periodic in the variable t. From [26] the averaged functions of orders one and two for system (7) are Z T Z T  ∂F1 (t, z) (8) f1 (z) = F1 (t, z)dt, f2 (z) = y1 (t, z) + F2 (t, z) dt, 0 0 ∂x Z t where y1 (t, z) = F1 (s, z)ds. 0 A point p ∈ Σ is called a generic point of discontinuity if there exists a neigh- borhood U of p such that Sp = U ∩ Σ is a C k embedded hypersurface. The crossing hypothesis (CH) for system (7) is the following. (CH) There exists an open bounded set C ⊂ D such that for each z ∈ C the curve {(t, z) : t ∈ S1 } reaches transversely the set Σ only at generic points of discontinuity. The assumption (CH) implies the following: Lemma 3 ([26, Proposition 2]). For |ε| =6 0 sufficiently small, every solution of system (7) starting in C reaches the set of discontinuity Σ only at its crossing region. The followings are two averaging theorems of order up to 2 for system (7), where the notation dB (fi , U, 0) (i = 1, 2) denote the Brouwer degree, see [4] or the Appendix A of [26] for a definition. 5 Theorem 4 ([26, Theorem A]). In addition to the crossing hypothesis (CH) assume the following conditions. (Ha1) For i = 1, 2 and j = 1, 2, . . . , M the continuous functions Fij and Rij are locally Lipschitz with respect to x, and T-periodic with respect to the time t. Furthermore, for j = 1, 2, . . . , M , the boundaries of Sj are piecewise C k embedded hypersurfaces with k ≥ 1. (Ha2) For a∗ ∈ C with f1 (a∗ ) = 0 there exists a neighborhood U ⊂ C of a∗ such that f1 (z) 6= 0 for all z ∈ U \{a∗ } and dB (f1 , U, 0) 6= 0. Then for |ε| =6 0 sufficiently small there exists a T -periodic solution x(t, ε) of system (7) such that x(0, ε) → a∗ as ε → 0. Theorem 5 ([26, Theorem B]). Suppose that f1 (z) ≡ 0. In addition to the crossing hypothesis (CH) assume the following conditions. (Hb1) For j = 1, 2, . . . , M the functions F1j (t, ·) are of class C 1 for all t ∈ R; for j = 1, 2, . . . , M the functions Dx F1j , F2j and R are locally Lipschitz with respect to x. Furthermore, for j = 1, 2, . . . , M , the boundaries of Sj are piecewise C k embedded hypersurfaces with k ≥ 1. (Hb2) If (t, z) ∈ Σ then (0, y1 (t, z)) ∈ T(t,z) Σ. (Hb3) For a∗ ∈ C with f2 (a∗ ) = 0, there exists a neighborhood U ⊂ C of a∗ such that f2 (z) 6= 0 for all z ∈ U \{a∗ } and dB (f2 , U, 0) 6= 0. Then for |ε| = 6 0 sufficiently small there exists a T -periodic solution x(t, ε) of system (7) such that x(0, ε) → a∗ as ε → 0. It is known that if a function f is C 1 then it is sufficient to check that the determinant of the Jacobian matrix D(f ) is non-zero in order to have that the Brouwer degree dB (f, U, 0) 6= 0, for more details see [34]. We shall need in the proofs of our results the theory on the ECT-systems. Let I denote a proper real interval of R. An ordered set of real-valued functions F = [f0 (t), f1 (t), . . . , fn (t)] defined in the interval I is called an Extended Chebyshev system or simply ET-system in the interval I if and only if any nontrivial linear combination of fi (i = 0, 1, . . . , n) has at most n zeros counting multiplicities. Furthermore F becomes an Extended Complete Chebyshev system, or simply an ECT-system in the interval I if and only if for any 0 ≤ k ≤ n, (f0 , f1 , . . . , fk ) is an ET-system. We can see the monograph [22] for more details. F is an ECT- system in the interval I if and only if W (f0 , f1 , . . . , fk )(t) 6= 0 in the interval I for 0 ≤ k ≤ n, where W (f0 , f1 , . . . , fk )(t) denotes the Wronskian of the functions (f0 , f1 , . . . , fk ) with respect to t, i.e. f0 (t) f1 (t) ··· fk (t) f00 (t) f10 (t) ··· fk0 (t) W (f0 , f1 , . . . , fk )(t) = .. .. .. .. . . . . . (k) (k) (k) f0 (t) f1 (t) · · · fk (t) The following two theorems give the maximum number of isolated zeros of the function Σni=0 αi fi (t) in the case that some of the Wronskians vanish, where αi are real numbers for i = 0, 1, . . . , n. Theorem 6 ([36, Corollary 1.4]). Let F = [f0 (t), f1 (t), . . . , fn (t)] be an ordered set of C ∞ functions defined in the interval [a, b]. Assume that all the Wronskians 6 ZHIFEI GUO AND JAUME LLIBRE W (f0 , f1 , . . . , fk )(t) do not vanish for k = 0, 1, . . . , n−1, except W (f0 , f1 , . . . , fn )(t), which has exactly one zero in the interval (a, b) and this zero is simple. Then any linear combination of the functions fi (i = 0, 1, . . . , n) has at most n + 1 zeros and for any set of m ≤ n + 1 zeros there exists a linear combination of m + 1 functions fi realizing it. Theorem 7 ([36, Theorem 1.2]). Let F = [f0 , f1 , . . . , fn ] be an ordered set of C ∞ functions defined in the interval [a, b]. Assume that all the Wronskians do not vanish except W (f0 , f1 , . . . , fn−1 )(t) and W (f0 , f1 , . . . , fn )(t), which have respectively k and l zeros in the interval (a, b) and these zeros are simple. Then there exists a linear combination of n + 1 functions fi (i = 0, 1, . . . , n) having exactly n + k + l simple zeros. Theorem 8 ([36, Theorem 1.1]). Let F = [f0 , f1 , . . . , fn ] be an ordered set of analytic functions defined in the interval [a, b]. Assume that all the νk zeros of the Wronskian W (f0 , f1 , . . . , fk )(t) are simple for k = 0, . . . , n. Then the number of isolated zeros for any linear combination of n + 1 functions fi (i = 0, 1, . . . , n) does not exceed n + νn + νn−1 + 2(νn−2 + · · · + ν0 ) + µn−1 + · · · + µ3 , where µi = min(2νi , νi−3 + · · · ν0 ), for i = 3, . . . , n − 1. 3. Proofs of the main results Proof of Theorem 1. In the polar coordinates x = r cos θ and x = r sin θ system (2) takes the form      ε A1 (x, y) cos θ + B1 (x, y) sin θ    +ε2 A (x, y) cos θ + B (x, y) sin θ ,          2 2    if h̃n (θ, r) ≥ 0,    −1 + ε B1 (x, y) cos θ − A1 (x, y) sin θ /r !      +ε2 B2 (x, y) cos θ − A2 (x, y) sin θ /r ṙ (9) =    θ̇     ε C1 (x, y) cos θ + D1 (x, y) sin θ    +ε2 C (x, y) cos θ + D (x, y) sin θ ,          2 2    if h̃n (θ, r) ≤ 0,    −1 + ε D1 (x, y) cos θ − C1 (x, y) sin θ /r    +ε2 D2 (x, y) cos θ − C2 (x, y) sin θ /r where h̃n (θ, r) := r sin θ − rn cosn θ with n ≥ 4. Taking θ as the new time variable system (9) is equivalent to the following discontinuous piecewise differential system ( P (θ, r) := εP1 (θ, r) + ε2 P2 (θ, r) + O(ε3 ) if h̃n (θ, r) ≥ 0, (10) ṙ = Q(θ, r) := εQ1 (θ, r) + ε2 Q2 (θ, r) + O(ε3 ) if h̃n (θ, r) ≤ 0, where Qi (θ, r) := Pi (θ, r)|aijk =cijk , bijk =dijk for i = 1, 2, j, k = 0, 1, 2 and 0 ≤ j +k ≤ 2, and Pi (θ, r) (i = 1, 2) are given in the Data.pdf. Note that each point (θ, r) on h̃n (θ, r) = 0 is determined by xn x r2 = x2 + x2n , sin θ = , cos θ = , r r 7 implying that tan θ = xn−1 . Thus there exists a unique θ1 := arctan(xn−1 ) ∈ (0, π/2) such that h̃n (θ1 , r) = 0. Let θ2 := π−(−1)n θ1 . Then θ2 satisfies h̃n (θ2 , r) = 0 and θ2 ∈ (π/2, π) (resp. ∈ (π, 3π/2)) if n is even (resp. odd). Furthermore one can check that h̃n (θ, r) > 0 (resp. < 0) if θ ∈ (θ1 , θ2 ) (resp. ∈ (0, θ1 ) ∪ (θ2 , 2π)). Hence the switching curve of system (10) is given by Σ e := {(θ1 , r) : r > 0}∪{(θ2 , r) : r > 0}. Computations show that D ED E ∇h̃n (θ1 , r), (1, P (θ1 , r)) ∇h̃n (θ1 , r), (1, Q(θ1 , r)) 2 (n − 1)2 x2n = x + nx2n−1 + + O(ε) > 0, x2 + x2n D ED E ∇h̃n (θ2 , r), (1, P (θ2 , r)) ∇h̃n (θ2 , r), (1, Q(θ2 , r)) 2 2 (xn + n(−x)n ) = x + (−1)n−1 nx2n−1 + + O(ε) > 0. x2 + x2n So the hypothesis (CH) holds for system (10). From (8) the averaged function of order one is Z 2π Z θ1 Z θ2 Z 2π dṙ f1 (r) = dθ = Q1 (θ, r)dθ + P1 (θ, r)dθ + Q1 (θ, r)dθ, 0 dε ε=0 0 θ1 θ2 whose computation gives k1e g1e + k2e g2e + k3e g3e + k4e g4e + k5e g5e + k6e g6e f1e (r) := f1 (r) = √ 6 x2 + x2n for n ≥ 4 even, and k1e g1o + k2e g2o + k3e g3o + k4e g4o + k5e g5o + k6e g6o + k7e g7o f1o (r) := f1 (r) = √ 6 x2 + x2n for n ≥ 5 odd, where g1e := x, g2e := x3 , g3e := xn+1 , g4e := x2n+1 , g5e := x2 + x2n , (11) g6e := (x2 + x2n ) arctan(xn−1 ), g1o := x, g2o := x3 , g3o := xn , g4o := xn+2 , g5o := x2n+1 , g6o := x3n , g7o := x2 + x2n , and k1e := −12(b100 − d100 ), k2e := −4(a111 + 2b102 + b120 − c111 − 2d102 − d120 ), k3e := 6(a110 + d101 − b101 − c110 ), k4e := −12(b102 − d102 ), k5e := −3π(a110 + b101 + c110 + d101 ), k6e := 6(a110 + b101 − c110 − d101 ), k1o := −12(b100 − d100 ), k2o := 4(−2b102 + 2d102 + c111 + d120 − a111 − b120 ), k3o := 12(a100 − c100 ), k4o := 12(a120 − c120 ), k5o := −12(b102 − d102 ), k6o := 4(a102 + 2a120 + b111 − c102 − 2c120 − d111 ), k7o := −3π(a110 + b101 + c110 + d101 ). From Theorem 4 every simple zero of the function f1 (r) corresponds to a crossing limit cycle of system (2). Compute the Wronskians W1 (g1e ) := x, W2 (g1e , g2e ) := 2x3 , W3 (g1e , g2e , g3e ) := 2n(n − 2)xn+2 , 8 ZHIFEI GUO AND JAUME LLIBRE W4 (g1e , . . . , g4e ) := 8n3 (n − 1)(n − 2)x3n , W5 (g1e , . . . , g5e ) := −8n3 (n − 2)(n − 1)2 (2n − 1)((2n − 3)x2n−2 + 1)x3n−2 , 32n3 (n − 2)(n − 1)4 (2n − 1)x6n Φe1 (ς) W6 (g1e , . . . , g6e ) := , (x2n + x2 )4 W1 (g1o ) := x, W2 (g1o , g2o ) := 2x3 , W3 (g1o , g2o , g3o ) := 2(n − 1)(n − 3)xn+1 , W4 (g1o , . . . , g4o ) := 4(n + 1)(n − 1)2 (n − 3)x2n , W5 (g1o , . . . , g5o ) := 16n(n + 1)2 (n − 3)(n − 1)4 x4n−3 , W6 (g1o , . . . , g6o ) := 192n2 (n − 3)(n − 1)7 (n + 1)2 (3n − 1)x7n−8 , W7 (g1o , . . . , g7o ) := 192n3 (n − 2)(n − 3)(n − 1)7 (n + 1)2 (2n − 1)(3n − 1)x7n−12 Φo1 (ς), where ς = xn−1 and Φe1 (ς) := − n(n − 2)(2n − 3)(3n − 2)ς 6 + (3n − 2)(12n3 − 45n2 + 62n − 32)ς 4 − (3n − 4)(2n3 − 13n2 + 12n − 4)ς 2 − (n − 2)(3n − 4)(3n − 2), Φo1 (ς) := n(2n − 3)ς 2 − (3n − 2). Note that neither Wi (g1e , . . . , gie ) (i = 1, . . . , 5) with n ≥ 4 even, nor Wj (g1o , . . . , gjo ) (j = 1, . . . , 6) with n ≥ 5 odd has zeros in the interval (0, +∞). Then we only need to discuss the zeros of W6 (g1e , . . . , g6e ) and W7 (g1o , . . . , g7o ), i.e. to discuss the zeros of Φe1 (ς) with n ≥ 4 even and the zeros of Φo1 (ς) with n ≥ 5 odd, respectively. Obviously Φo1 (ς) has only one zero in the interval (0, +∞), i.e. s 3n − 2 ς= , n(2n − 3) which is simple. From Theorem 6 the set of functions {g1o , . . . , g7o } is an ECT- system in the interval (0, +∞). Hence f1o (r) has at most 7 simple zeros in the interval (0, +∞) and thereby system (2) with n ≥ 5 odd has at most 7 crossing limit cycles. Let Φe e (ν) := Φe (ς) with ν = ς 2 . Then Φ e e (ν) is a polynomial of degree 3 in ν. 1 1 1 e e The discriminant of Φ1 (ν) with respect to ν is 6 ∆Φe e := 512(n − 1) (3n − 4)(3n − 2)Υ1 , 1 2 2  4 2 where Υ1 := (3n  − 16n + 26)n + 8n(n − 4) + 10 3n + 2n(37n − 142n + 204) + 96(n − 4) . One can check that Υ1 > 0 for n ≥ 4 even. Actually we have 3n2 − 16n + 26 > 0 and 37n2 − 142n + 204 > 0 for n ∈ N because their discriminants with respect to the variable n are −56 and −10028, respectively. Then Υ1 > 0 for n ≥ 4 even, implying that ∆Φ ee e e e > 0 for n ≥ 4 even. Thus Φ1 (ν) as well as Φ1 (ς) 1 e has only one zero in the interval (0, +∞). Furthermore the resultant of Φ1 (ς) and (Φe1 (ς))0 is 512n(n − 2)(2n − 3)(3n − 4)(n − 1)6 (3n − 2)2 (3n4 − 16n3 + 34n2 − 32n + 10)(3n4 + 74n3 − 284n2 + 504n − 384). 9 One can show that the resultant has no zeros for n ≥ 4 even, indicating that Φe1 (ς) and (Φe1 (ς))0 have no common zeros with respect to the variable ς for n ≥ 4 even. Thus the uniquely positive zero of Φe1 (ς) is simple. From Theorem 6 the set of functions {g1e , . . . , g6e } is an ECT-system in the interval (0, +∞), implying that f1e (r) has at most 6 simple zeros in the interval (0, +∞). Hence system (2) has at most 6 crossing limit cycles for n ≥ 4 even. Furthermore one can compute the determinants   0 0 −12 0 0 0  0 −4 0 0 −8 0    ∂(k1e , k2e , k3e , k4e , k5e , k6e )  6 0 0 −6 0 −6  det = det   0  ∂(a110 , a111 , b100 , b101 , b102 , c110 )  0 0 0 −12 0  −3π 0 0 −3π 0 −3π  6 0 0 6 0 −6 = −248832π, and ∂(k1o , k2o , k3o , k4o , k5o , k6o , k7o ) det ∂(a100 , a110 , a120 , a111 , a102 , b100 , b102 )   0 0 0 0 0 −12 0 0 0 0 −4 0 0 −8    12 0 0 0 0 0 0    = det  0 0 12 0 0 0 0  = 995328π, 0 0 0 0 0 0 −12    0 0 8 0 4 0 0  0 −3π 0 0 0 0 0 which are different from zero, implying that the coefficients {k1e , k2e , k3e , k4e , k5e , k6e } and {k1o , k2o , k3o , k4o , k5o , k6o , k7o } are linearly independent, respectively. Hence we can choose parameters aijk , bijk , cijk and dijk (i = 1, 2 and 0 ≤ j + k ≤ 2) such that system (2) with n ≥ 4 even (resp. n ≥ 5 odd) has exactly 0, 1, 2, 3, 4, 5 or 6 (resp. 0, 1, 2, 3, 4, 5, 6 or 7) crossing limit cycles. This proves Theorem 1.  Proof of Theorem 2. The averaged function of order two is computed when f1 (r) = 0. Solving f1 (r) = 0, i.e. f1e (r) = 0 for n ≥ 4 even and f1o (r) = 0 for n ≥ 5 odd, which are equivalent to k1e = k2e = k3e = k4e = k5e = k6e = 0 and k1o = k2o = k3o = k4o = k5o = k6o = k7o = 0 respectively, yields the conditions Qe1 given in (3) for n ≥ 4 even and the conditions Qo1 given in (4) for n ≥ 5 odd. Let   Z θi   dṙ Γi (r) := ∇h θi , r , s, dϕ , i = 1, 2, 0 dε (ε,θ)=(0,ϕ) where θ1 = arctan(xn−1 ), θ2 = π − (−1)n θ1 and x satisfies r2 = x2 + x2n . Compu- tations show that   Z θ1   Γ1 (r) = ∇h θ1 , r , s, Q1 (t, r)dt 0  n n  sin θ1 − nrn−1 cosn θ1 Ψ1 (r) = s nr tan θ1 cos θ1 + r cos θ1 + , 12 10 ZHIFEI GUO AND JAUME LLIBRE   Z θ1 Z θ2   Γ2 (r) = ∇h θ2 , r , s, Q1 (t, r)dt + P1 (t, r)dt 0 θ1   n n  sin θ2 − nrn−1 cosn θ2 Ψ1 (r) + Ψ2 (r) = s nr tan θ2 cos θ2 + r cos θ2 + , 12 where Ψ1 (r) := − 12c100 sin θ1 − 6r(c101 + d110 ) sin2 θ1 − 4r2 (c102 + d111 ) sin3 θ1 − 6c110 r(θ1 + sin θ1 cos θ1 ) + 4r2 (c111 + d120 )(cos θ1 − 1)(cos2 θ1 + cos θ1 + 1) − c120 r2 (9 sin θ1 + sin(3θ1 )) + 12d100 (cos θ1 − 1) + 3d101 r(sin(2θ1 ) − 2θ1 ) − 16d102 r2 (cos θ1 + 2) sin4 (θ1 /2), Ψ2 (r) := − 3((−1)n − 1)(4a100 + r2 (a102 + 3a120 + b111 )) sin(θ1 ) + r2 ((−1)n − 1)(a102 − a120 + b111 ) sin(3θ1 ) − 6r(a110 + b101 )(π − θ1 ((−1)n + 1)) + 3r((−1)n + 1)(a110 − b101 ) sin(2θ1 ) − (24b100 + 6r2 (a111 + 3b102 + b120 )) cos(θ1 ) − 2r2 (a111 − b102 + b120 ) cos(3θ1 ). In order to ensure that Γi (r) = 0 if and only if s = 0, we need to eliminate Ψi for i = 1, 2, equivalently all coefficients of Ψi are equal to zero, i.e. b100 = c100 = c110 = c120 = d100 = d101 = d102 = 0, (12) c101 + d110 = c102 + d111 = c111 + d120 = a110 + b101 = 0, a110 − b101 = a111 + 3b102 + b120 = a111 − b102 + b120 = 0 for n ≥ 4 even, and a100 = b100 = c100 = c110 = c120 = d100 = d101 = d102 = c101 + d110 = 0, (13) c102 + d111 = c111 + d120 = a102 + 3a120 + b111 = a102 − a120 + b111 = 0, a110 + b101 = a111 + 3b102 + b120 = a111 − b102 + b120 = 0 for n ≥ 5 odd. From (12) and f1e (r) = 0 we obtain the conditions Qe2 given in (5) for n ≥ 4 even. Similarly from (13) and f1o (r) = 0 we obtain the conditions Qo2 given in (6) for n ≥ 5 odd. From (8) the averaged function of order two is Z 2π ( Z θ ! ) ∂ 2 ṙ ∂ ṙ 1 ∂ 2 ṙ f2 (r) = dϕ + dθ 0 ∂r∂ε ε=0 0 ∂ε (ε,θ)=(0,ϕ) 2 ∂ε2 ε=0 Z θ1 ( Z ) ∂Q1 (θ, r) θ = Q1 (ϕ, r)dϕ + Q2 (θ, r) dθ 0 ∂r 0 Z θ2 ( Z θ1 Z θ ! ) ∂P1 (θ, r) + Q1 (ϕ, r)dϕ + P1 (ϕ, r)dϕ + P2 (θ, r) dθ θ1 ∂r 0 θ1 Z 2π ( Z θ1 Z θ2 Z θ ! ∂Q1 (θ, r) + Q1 (ϕ, r)dϕ + P1 (ϕ, r)dϕ + Q1 (ϕ, r)dϕ θ2 ∂r 0 θ1 θ2 ) + Q2 (θ, r) dθ. 11 Computations show that k̃1e g̃1e + k̃2e g̃2e + k̃3e g̃3e + k̃4e g̃4e + k̃5e g̃5e + k̃6e g̃6e + k̃7e g̃7e + k̃8e g̃8e f2e (r) := f2 (r) = √ 24 x2 + x2n for n ≥ 4 even, and k̃1o g1o + k̃2o g2o + k̃3o g3o + k̃4o g4o + k̃5o g5o + k̃6o g6o + k̃7o g7o f2o (r) := f2 (r) = √ 6 x2 + x2n for n ≥ 5 odd, where gio (i = 1, . . . , 6) are given in (11) and g̃1e := x, g̃2e := x3 , g̃3e := xn+1 , g̃4e := xn+3 , g̃5e := x2n+1 , g̃6e := x2 + x2n , π  g̃7e := (x2 + x2n )arctan(xn−1 ), g̃8e := x3n+1 − (x2 + x2n )2 − arctan(xn−1 ) , 2 k̃1e := −48(b200 − d200 ), k̃2e := −16(a101 b120 + a211 + b110 b120 + 2b202 + b220 − c211 − 2d202 − d220 ), k̃3e := 24(a100 b120 + a210 − b201 − c210 + d201 ), k̃4e := −6b120 (a102 − 5a120 + b111 ), k̃5e := −48(b202 − d202 ), k̃6e := −12π(a100 b120 + a210 + b201 + c210 + d201 ), k̃7e := 24(a100 b120 + a210 + b201 − c210 − d201 ), k̃8e := 6b120 (a102 + 3a120 + b111 ), k̃1o := −12(b200 − d200 ), k̃2o := −4(a101 b120 + a211 + b101 b111 + b110 b120 + 2b202 + b220 − c211 − 2d202 − d220 ), k̃3o := 12(a200 − c200 ), k̃4o := 12(a220 − b101 b120 − c220 ), k̃5o := −12(b101 b111 + b202 − d202 ), k̃6o := 4(a101 b111 + a202 + 2a220 − b101 b120 + b110 b111 + b211 − c202 − 2c220 − d211 ), k̃7o := −3π(a210 + b201 + c210 + d201 ). Note that f2o (r) has the same form as f1o (r). Then f2o (r) has at most 6 simple zeros. Moreover one can check that ∂(k̃1o , k̃2o , k̃3o , k̃4o , k̃5o , k̃6o , k̃7o ) det ∂(a200 , a210 , a211 , a202 , b200 , b202 , c220 )   0 0 0 0 −12 0 0 0 0 −4 0 0 −8 0    12 0 0 0 0 0 0    = det   0 0 0 0 0 0 −12  = −995328π 6= 0,  0 0 0 0 0 −12 0    0 0 0 4 0 0 −8  0 −3π 0 0 0 0 0 implying that the coefficients {k̃1o , k̃2o , k̃3o , k̃4o , k̃5o , k̃6o , k̃7o } are linearly independent. Hence the averaged function of order two produces the same result as the ones of order 1 for n ≥ 5 odd. 12 ZHIFEI GUO AND JAUME LLIBRE For n ≥ 4 even we compute the Wronskians W1 (g̃1e ) := x, W2 (g̃1e , g̃2e ) := 2x3 , W3 (g̃1e , g̃2e , g̃3e ) := 2n(n − 2)xn+2 , W4 (g̃1e , . . . , g̃4e ) := 4n2 (n2 − 4)x2n+2 , W5 (g̃1e , . . . , g̃5e ) := 16n4 (n − 2)2 (n2 + n − 2)x4n−1 , W6 (g̃1e , . . . , g̃6e ) := −16n4 (n + 2)(2n − 1)(n − 1)2 (n − 2)2 x4n−4 Φe2 (ς), 128n4 (n + 2)(2n − 1)(n − 1)4 (n − 2)2 x7n−1 Φe3 (ς) W7 (g̃1e , . . . , g̃7e ) := , (x2 + x2n )5 128n4 (n + 2)(2n − 1)(n − 1)5 (n − 2)2 x7n+6 Φe4 (ς) W8 (g̃1e , . . . , g̃8e ) := , (x2 + x2n )10 where ς = xn−1 and Φe2 (ς) := (n − 3)(2n − 3)ς 2 − (n + 1), Φe3 (ς) := n2 (n − 3)(n − 2)(2n − 3)(3n − 2)ς 8 − 2(3n − 2)(15n5 − 110n4 + 302n3 − 395n2 + 240n − 48)ς 6 + 2(3n − 4)(3n − 2)(5n4 − 33n3 + 83n2 − 87n + 38)ς 4 − 2(3n − 4)(n5 + 18n3 − 61n2 + 54n − 16)ς 2 + (n − 2)2 (n + 1)(3n − 4)(3n − 2), 4 Φe4 (ς) := Φe41 (ς) + 2Φe42 (ς) 1 + ς 2 arctan(ς), and Φe4i (ς) (i = 1, 2), given in the Data.pdf, are polynomials of degrees 22 and 14 in ς, respectively. For n ≥ 4 even Φe2 (ς) has exactly one positive zero s n+1 ς= , (n − 3)(2n − 3) which is simple. Let Φ e e (ν) := Φe (ς) with ν = ς 2 . Then Φ e e (ν) is a polynomial of 3 3 3 degree 4 in ν. We can prove that Φ e (ν) has exactly two positive zeros for 4 ≤ n ≤ 74 e 3 even, where the two positive zeros are simple. We present the isolation intervals for the two positive zeros of Φ e e (ν) with 4 ≤ n ≤ 74 even in Table 1. In order 3 to determine if Φ e e (ν) has multiple zeros we compute the resultant between the 3 polynomials Φ e e (ν) and (Φ e e (ν))0 with respect to the variable ν, which is 3 3 R = 49152n2 (n − 2)(n − 3)(2n − 3)(n − 1)12 (3n − 2)3 (3n − 4)2 (3n8 − 49n7 + 315n6 − 1084n5 + 2257n4 − 2973n3 + 2445n2 − 1146n + 228)(108n12 − 9972n11 + 161301n10 − 1155787n9 + 4608695n8 − 11264874n7 + 17422780n6 − 16171208n5 + 6010272n4 + 5255936n3 − 9284608n2 + 5511168n − 860160). e e (ν) and (Φ We obtain that R has no zeros for n ≥ 4 even, implying that Φ e e (ν))0 3 3 has no common zeros. Thus the two zeros of Φ e (ν) with 4 ≤ n ≤ 74 even are e 3 simple. 13 4 ≤ n ≤ 74 even e e (ν) The isolation intervals of positive zeros of Φ 3 n=4 [81/1024, 41/512], [25831/2048, 51663/4096] n=6 [609/4096, 305/2048], [51565/4096, 25783/2048] n=8 [49/64, 785/1024], [52347/4096, 13087/1024] n = 10 [215/256, 861/1024], [13259/1024, 53037/4096] n = 12 [449/512, 899/1024], [26787/2048, 53575/4096] n = 14 [921/1024, 461/512], [26997/2048, 53995/4096] n = 16 [937/1024, 469/512], [54327/4096, 6791/512] n = 18 [237/256, 949/1024], [54597/4096, 27299/2048] n = 20 [957/1024, 479/512], [54819/4096, 13705/1024] n = 22 [241/256, 965/1024], [55005/4096, 27503/2048] n = 24 [485/512, 971/1024], [27581/2048, 55163/4096] n = 26 [975/1024, 61/64], [27649/2048, 55299/4096] n = 28 [489/512, 979/1024], [55415/4096, 6927/512] n = 30 [491/512, 983/1024], [27759/2048, 55519/4096] n = 32 [985/1024, 493/512], [6951/512, 55609/4096] n = 34 [987/1024, 247/256], [55689/4096, 27845/2048] n = 36 [989/1024, 495/512], [55761/4096, 27881/2048] n = 38 [991/1024, 31/32], [27913/2048, 55827/4096] n = 40 [993/1024, 497/512], [13971/1024, 55885/4096] n = 42 [995/1024, 249/256], [27969/2048, 55939/4096] n = 44 [249/256, 997/1024], [27993/2048, 55987/4096] n = 46 [997/1024, 499/512], [56031/4096, 1751/128] n = 48 [499/512, 999/1024], [7009/512, 56073/4096] n = 50 [999/1024, 125/128], [28055/2048, 56111/4096] n = 52 [125/128, 501/512], [56145/4096, 28073/2048] n = 54 [125/128, 501/512], [56177/4096, 28089/2048] n = 56 [501/512, 251/256], [56207/4096, 3513/256] n = 58 [501/512, 251/256], [14059/1024, 56237/4096] n = 60 [251/256, 503/512], [28131/2048, 56263/4096] n = 62 [251/256, 503/512], [56287/4096, 1759/128] n = 64 [251/256, 503/512], [28155/2048, 56311/4096] n = 66 [503/512, 63/64], [14083/1024, 56333/4096] n = 68 [503/512, 63/64], [56353/4096, 28177/2048] n = 70 [503/512, 63/64], [14093/1024, 56373/4096] n = 72 [503/512, 63/64], [28195/2048, 56391/4096] n = 74 [63/64, 505/512], [7051/512, 56409/4096] Table 1. The isolation intervals for two positive zeros of Φ e e (ν) 3 with 4 ≤ n ≤ 74 even. For n ≥ 76 even we take four closed intervals Ii ⊂ (0, +∞) (i = 1, . . . , 4), where       1 1 1 1 I1 := 0, , I2 := , , I3 := , 1 , I4 := [1, 15] . 25 25 2 2 Note that Φe e (ν) is a polynomial of degree 4 in ν. Then Φe e (ν) has at most 4 zeros 3 3 with respect to ν. We claim that Φ e (ν) has exactly one positive zero in each interval e 3 14 ZHIFEI GUO AND JAUME LLIBRE Ii (i = 1, . . . , 4) for n ≥ 76 even. We prove that e e (ν) has the opposite signs at Φ 3 the endpoints of Ii (i = 1, . . . , 4) for n ≥ 76 even. More precisely we show that for n ≥ 76 even e e (0) > 0, Φ Φ e e (1/25) < 0, Φe e (1/2) > 0, e e (1) < 0, Φ e e (15) > 0, Φ 3 3 3 3 3 Obviously Φe e (0) = (n−2)2 (n+1)(3n−4)(3n−2) > 0 for n ≥ 76 even. Computations 3 show that Φe e (1/25) = − 16(2484n6 − 198427n5 + 1099723n4 − 1818787n3 3 + 303779n2 + 1348050n − 679700)/390625. We obtain that Φ e e (1/25) has exactly one zero with respect to the variable n in each 3 interval J1j (j = 1, . . . , 6), where       841 105 167 669 5461 2731 J11 := − ,− , J12 := , , J13 := , , 1024 128 256 1024 4096 2048       3661 7323 5927 741 18953 75813 J14 := , , J15 := , , J16 := , . 2048 4096 2048 256 256 1024 One can check that J1j ∩ [76, +∞) = ∅ for i = 1, . . . , 6. Then Φ e e (1/25) has no 3 e e zeros in the interval [76, +∞), implying that Φ3 (1/25) does not change sign in the interval [76, +∞). Note that e e3 (1/25)|n=76 = − 183080538143552 < 0. Φ 390625 Thus Φe e (1/25) < 0 for n ≥ 76 even. Similarly we have 3 e e3 (1/2) = 3(46n6 − 497n5 + 1689n4 − 2736n3 + 2756n2 − 1824n + 512)/16, Φ e e (1) = −16(5n5 − 29n4 + 65n3 − 85n2 + 66n − 20), Φ 3 e e3 (15) = 64(315n6 + 1236n5 − 24435n4 + 85508n3 − 112887n2 + 56232n − 8017). Φ We obtain that Φ e e (1/2), Φe e (1) and Φ e e (15) have exactly one zero in each interval 3 3 3 J2j (j = 1, . . . , 4), J3j (j = 1, 2, 3) and J4j (j = 1, . . . , 4) respectively, where       309 619 2759 5519 1071 8569 J21 := , , J22 := , , J23 := , , 512 1024 2048 4096 512 4096       6357 3179 321 643 701 5609 J24 := , , J31 := , , J32 := , , 1024 512 512 1024 512 4096       1417 5669 6277 50215 481 963 J33 := , , J41 := − ,− , J42 := , , 512 2048 512 4096 2048 4096     341 683 3033 6067 J43 := , , J44 := , . 512 1024 2048 4096 Note that Jij ∩ [76, +∞) = ∅ for i = 2, 3, 4 and j = 1, 2, 3 or 4. Then Φ e e (1/2), 3 e (1) and Φ Φ e e (15) have no zeros with respect to the variable n in the interval e 3 3 [76, +∞), indicating that Φe e (1/2), Φ e e (1) and Φ e e (15) do not change signs in the 3 3 3 interval [76, +∞) separately. Furthermore one can check that e e3 (1/2)|n=76 = 1436100761268 > 0, Φ e e3 (1)|n=76 = −187810740032 < 0, Φ 15 e e3 (15)|n=76 = 4035588498286528 > 0. Φ Thus Φe e (1/2) > 0, Φ e e (1) < 0 and Φ e e (15) > 0 for n ≥ 76 even, which together with 3 3 3 e e e e Φ3 (0) > 0 and Φ3 (1/25) < 0 implies that Φ e e (ν) has exactly one positive zero in 3 each interval Ii (i = 1, . . . , 4) for n ≥ 76 even. This proves the claim that Φ e e (ν) 3 has exactly one positive zero in each interval Ii (i = 1, . . . , 4) for n ≥ 76 even. From Theorem 7 there exist two linear combinations of functions {g̃1e , . . . , g̃7e } such that they have 9 positive zeros for 4 ≤ n ≤ 74 even and 11 positive zeros for n ≥ 76 even, respectively. Solving k̃8e = 0 we obtain that either b120 = 0 or b111 = −a102 − 3a120 . Choosing b111 = −a102 − 3a120 , where we do not take b120 = 0 because it results in k̃4e = 0, we compute the determinant ∂(k̃1e , k̃2e , k̃3e , k̃4e , k̃5e , k̃6e , k̃7e ) det ∂(a211 , b200 , b201 , b120 , b202 , c210 , d201 )   0 −48 0 0 0 0 0 −16 0 0 −16(a101 + b110 ) −32 0 0     0 0 −24 24a100 0 −24 24    = det   0 0 0 48a120 0 0 0    0 0 0 0 −48 0 0     0 0 −12π −12πa100 0 −12π −12π  0 0 24 24a100 0 −24 −24 = 48922361856πa120 , which is different from zero for fixed a120 6= 0. It implies that the coefficients {k̃1e , k̃2e , k̃3e , k̃4e , k̃5e , k̃6e , k̃7e } are linearly independent for fixed a120 6= 0. Hence in the case b111 = −a102 − 3a120 we obtain that f2e (r) has at most 9 positive zeros for 4 ≤ n ≤ 74 even and at most 11 positive zeros for n ≥ 76 even, where the bounds 9 and 11 are reachable from Theorem 7 but we do not know whether the bounds 0, 1, 2, 3, 4, 5, 6, 7, 8 for 4 ≤ n ≤ 74 even and the bounds 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 for n ≥ 76 even are reachable from Theorem 7. Next we give more information for the above unknown bounds. From k̃3e = k̃8e = 0 we obtain (14) b111 = −a102 − 3a120 , d201 = −a210 + b201 + c210 − a100 b120 . One can compute the Wronskians W1 (g̃1e ) := x, W2 (g̃1e , g̃2e ) := 2x3 , W3 (g̃1e , g̃2e , g̃4e ) := 2n(n + 2)xn+4 , W4 (g̃1e , g̃2e , g̃4e , g̃5e ) := 8n2 (n − 2)(n − 1)(n + 2)x3n+2 , W5 (g̃1e , g̃2e , g̃4e , g̃5e , g̃6e ) := −8n2 (n − 2)(n − 1)(n + 2)(2n − 1)x3n Θ1 (ν), 16n2 (n − 2)(n − 1)2 (n + 2)(2n − 1)x4n−6 Θ2 (ν) W5 (g̃1e , g̃2e , g̃4e , g̃5e , g̃6e , g̃7e ) := − , (x2n + x2 )4 where ν = x2n−2 and Θ1 (ν) := (n − 3)(2n − 3)ν + (n + 1), 16 ZHIFEI GUO AND JAUME LLIBRE Θ2 (ν) := n2 (n − 3)(n − 2)(2n − 3)ν 5 + n2 (n − 2)(12n3 − 54n2 + 53n − 1)ν 4 − 2(36n6 − 303n5 + 1009n4 − 1708n3 + 1504n2 − 624n + 96)ν 3 + 2(6n6 − 81n5 + 307n4 − 578n3 + 664n2 − 456n + 128)ν 2 + (n − 2)(16n4 − 41n3 + 11n2 + 56n − 32)ν + n2 (n − 2)(n + 1). Obviously Θ1 (x) has no positive zeros with respect to the variable ν for n ≥ 4 even. We claim that Θ2 (ν) has five zeros with respect to the variable ν for n ≥ 4 even. Note that Θ2 (ν) has at most five zeros because it is a polynomial of degree 5 in ν. For n ≥ 4 even we take the five intervals     ˜ ˜ 1 1 I1 := (−∞, −2] , I2 := −2, − , I3 := − , 0 , I˜4 := [0, 1] , I˜5 := [1, 6] . ˜ n n Since the leading coefficient of Θ2 (ν) with respect to the variable ν is n2 (n − 3)(n − 2)(2n − 3), which is positive for n ≥ 4 even, we have (15) lim Θ2 (ν) = −∞ ν→−∞ for each fixed n ≥ 4 even. We can prove that Θ2 (−2) > 0, Θ2 (−1/n) < 0, Θ2 (0) > 0, Θ2 (1) < 0, Θ2 (6) > 0 for n ≥ 4 even, which together with (15) implies that Θ2 (ν) has a unique zero with respect to the variable ν in each I˜i (i = 1, . . . , 5) for n ≥ 4 even and, moreover, the five zeros are simple because Θ2 (ν) is a polynomial of degree 5 in ν. This proves the claim. Hence Θ2 (ν) has two positive zeros with respect to the variable ν for n ≥ 4 even, where the two positive zeros are simple. Under (14) compute the determinant ∂(k̃1e , k̃2e , k̃4e , k̃5e , k̃6e , k̃7e ) det ∂(a211 , b200 , b201 , b120 , b202 , c210 )   0 −48 0 0 0 0 −16 0 0 −16(a101 + b110 ) −32 0     0 0 0 48a 0 0  = det   0 120   0 0 0 −48 0    0 0 −24π 0 0 −24π  0 0 0 48a100 0 −48 = 2038431744πa120 , which is different from zero for fixed a120 6= 0, implying that the coefficients {k̃1e , k̃2e , k̃4e , k̃5e , k̃6e , k̃7e } are linearly independent. From Theorem 6 we obtain that f2e (r) has at most 7 positive zeros for n ≥ 4 even under (14), where the bounds 0, 1, 2, 3, 4, 5, 6, 7 are reachable. From k̃5e = k̃8e = 0 we obtain (16) b111 = −a102 − 3a120 , d202 = b202 . Under (16) using the similar method we can prove that f2e (r) has at most 9 positive zeros for n ≥ 76 even. In fact one can compute the Wronskians W1 (g̃1e ) := x, W2 (g̃1e , g̃2e ) := 2x3 , W3 (g̃1e , g̃2e , g̃3e ) := 2n(n − 2)xn+2 , 17 W4 (g̃1e , g̃2e , g̃3e , g̃4e ) := 4n2 (n2 − 4)x2n+2 , W5 (g̃1e , g̃2e , g̃3e , g̃4e , g̃6e ) := 4n2 (n − 2)(n − 1)(n + 2)x2n Ξ1 (ν), 32n2 (n − 2)(n − 1)3 (n + 2)x5n+2 Ξ2 (ν) W6 (g̃1e , g̃2e , g̃3e , g̃4e , g̃6e , g̃7e ) := , (x2n + x2 )4 where ν = x2n−2 and Ξ1 (ν) := (n − 3)(2n − 3)(2n − 1)ν − (n + 1), Ξ2 (ν) := n2 (n − 3)(n − 2)(2n − 3)(2n − 1)ν 3 + (−32n6 + 248n5 − 747n4 + 1141n3 − 916n2 + 348n − 48)ν 2 + (12n6 − 108n5 + 377n4 − 641n3 + 594n2 − 304n + 64)ν − (n − 2)(n + 1)(3n − 4)(3n − 2). Clearly Ξ1 (ν) has a unique positive zero with respect to the variable ν for n ≥ 76 even. Note that Ξ2 (ν) is a polynomial of degree 3 in ν. Then Ξ2 (ν) has at most three zeros. We can prove that Ξ2 (ν) has three positive zeros for n ≥ 76 even, which lie in the intervals       1 1 1 1 0, , , , ,8 , 4 4 2 2 respectively, where the three zeros are simple. Under (16) we compute the deter- minant ∂(k̃1e , k̃2e , k̃3e , k̃4e , k̃6e , k̃7e ) det ∂(a211 , b200 , b201 , b120 , c210 , d201 )   0 −48 0 0 0 0 −16 0 0 −16(a + b ) 0 0   101 110   0 0 −24 24a100 −24 24   = det    0 0 0 48a120 0 0   0 0 −12π −12πa100 −12π −12π  0 0 24 24a100 −24 −24 = −1019215872πa120 , which is different from zero for fixed a120 6= 0, implying that the coefficients {k̃1e , k̃2e , k̃3e , k̃4e , k̃6e , k̃7e } are linearly independent. From Theorem 6 we obtain that f2e (r) has at most 9 positive zeros for n ≥ 76 even under (16), where the bound 9 is reachable. Here we cannot determine whether the bound 8 for n ≥ 4 even and the bound 10 for n ≥ 76 even are reachable. This proves Theorem 2.  Theorem 2 with n ≥ 4 even is obtained based on k̃8e = 0. When k̃8e 6= 0, we need to discuss the positive zeros of W8 (g̃1e , . . . , g̃8e ), equivalently to discuss the positive zeros of Φe4 (ς). However it is difficult because the transcendental function arctan(ς) is involved in Φe4 (ς). Once the maximum number of the positive zeros of Φe4 (ς) is obtained, where all zeros are simple, we can give an upper bound on the number of the positive zeros of f2e (r) as well as an upper bound for the number of the crossing limit cycles for system (2) with n ≥ 4 even. In fact, if Φe4 has at most ` positive zeros and they are simple, then from Theorem 8 and our Theorem 2 we obtain that f2e (r) has at most 11 + ` positive zeros for 4 ≤ n ≤ 74 even and at most 13 + ` positive zeros for n ≥ 76 even, implying that system (2) has at most 11 + ` crossing limit cycles for 4 ≤ n ≤ 74 even and at most 13 + ` crossing limit cycles for n ≥ 76 18 ZHIFEI GUO AND JAUME LLIBRE even, where we do not determine whether the bounds 11+` and 13+` are reachable from Theorem 8. Although it is difficult to give the exact number for the positive zeros of Φe4 (ς), we have the following. Proposition 9. Φe4 (ς) has at least two positive zeros for n ≥ 4 even and their multiplicities are not greater than 2. Proof. One can check that (17) Φe4 (0) = −3(n − 3)(n − 2)2 (n + 1)(2n − 3)(3n − 5)(3n − 4)(3n − 2)π < 0 for n ≥ 4 even. We claim that (18) Φe4 (1/2) > 0, lim Φe4 (ς) = −∞, ς→+∞ for n ≥ 4 even. In fact computations show that P(n) (19) Φe4 (1/2) = , 4194304 where P(n) := P1 (n, π, arctan(1/2)) − P2 (n, π, arctan(1/2)) and P1 (n, u, v) := 1134000000n12 v + 1441352448n12 + 3412800000n11 v + 121218048n11 + 60970837500n10 u + 583290525000n9 v + 509788601040n9 + 663611153125n8 u + 1999679006250n7 v + 889035231524n7 + 1396262796875n6 u + 294867027364n6 + 4021862818750n5 v + 2409126075000n4 u + 2540209625760n4 + 4139680025000n3 v + 1225019550000n2 u + 1192575320128n2 + 926299200000nv + 82454400000u + 76723261440, P2 (n, u, v) := 567000000n12 u + 1706400000n11 u + 121941675000n10 v + 104802757040n10 + 291645262500n9 u + 1327222306250n8 v + 1025258589540n8 + 999839503125n7 u + 2792525593750n6 v + 2010931409375n5 u + 1821531770900n5 + 4818252150000n4 v + 2069840012500n3 u + 2117692526960n3 + 2450039100000n2 v + 463149600000nu + 436788493312n + 164908800000v. Since each term in Pi (n, u, v) (i = 1, 2) is positive for u > 0, v > 0 and n ≥ 4 even, we obtain that Pi (n, u, v) (i = 1, 2) are monotonically increasing with respect to u and v, i.e. (20) Pi (n, u1 , v1 ) < Pi (n, u2 , v2 ), i = 1, 2, for 0 < u1 < u2 and 0 < v1 < v2 . It is not difficult to obtain that π ∈ (αl , αr ) and arctan(1/2) ∈ (βl , βr ), where 31415926535 31415926536 αl := , αr := , 10000000000 10000000000 4636476090 4636476091 βl := , βr := . 10000000000 10000000000 19 From (20) we have P1 (n, π, arctan(1/2)) > P1 (n, αl , βl ), P2 (n, π, arctan(1/2)) < P2 (n, αr , βr ), which together with (19) implies that P3 (n) (21) P(n) > P1 (n, αl , βl ) − P2 (n, αr , βr ) = , 3200000 where P3 (n) := 594706566447360n12 − 11703229107545088n11 + 96655398605897804n10 − 435202418619644304n9 + 1421356431555721857n8 − 4239701812023250196n7 + 10837146103965175535n6 − 20077862508308980156n5 + 25199140207829937432n4 − 21442988667971270224n3 + 12496425497233152368n2 − 4679486287651730432n + 829765849245441024. We find that P3 (n) has no zeros greater than 4 with respect to the variable n, indicating that P3 (n) does not change sign for n ≥ 4 even. Computations show that 728196387383804091 P3 (4) = > 0, 2500 indicating that P3 (n) > 0 for n ≥ 4 even. It together with (21) implies that P(n) > 0 for n ≥ 4 even, i.e. Φe4 (1/2) > 0 for n ≥ 4 even. Next we prove the second equality in (18). Since Φe42 (ς) is a polynomial in ς and its leading coefficient with respect to ς does not vanish for n ≥ 4 even, we have Φe42 (ς) 6= 0 for ς > 0 sufficiently large. Then we write Q2 (ς) Φe4 (ς) = , Q1 (ς) where 1 Φe41 (ς) (22) Q1 (ς) := , Q2 (ς) := + 2 arctan ς. (1 + ς 2 )4 Φe42 (ς) (1 + ς 2 )4 Φe42 (ς) Obviously limς→+∞ Q1 (ς) = 0. On the other hand, since both Φe41 (ς) and (1 + ς 2 )4 Φe42 (ς) are polynomials of degree 22 in ς, we have Φe41 (ς) lcoeff(Φe41 (ς), ς) lim e =  = −π, ς→+∞ (1 + ς 2 )4 Φ (ς) 42 lcoeff (1 + ς 2 )4 Φe42 (ς), ς where the notation lcoeff(f (x), x) denotes the leading coefficient of a polynomial f (x) with respect to the variable x. It together with limς→+∞ arctan ς = π/2 means that limς→+∞ Q2 (ς) = 0. From the l’Hospital rule we have Q2 (ς) Q02 (ς) Q4 (ς) lim = lim = lim , ς→+∞ Q1 (ς) ς→+∞ Q0 (ς) ς→+∞ Q3 (ς) 1 where Q4 (ς) := −32ςQ41 (ς)Q42 (ς) and Q3 (ς), Q41 (ς), Q42 (ς) are given in the Data.pdf. Note that lcoeff(Q3 (ς), ς) = 53909856000, lcoeff(Q4 (ς), ς) = −2922784594329600000 20 ZHIFEI GUO AND JAUME LLIBRE for n = 4, and that lcoeff(Q3 (ς), ς) = 33n3 (n − 2)(n − 3)(5n − 3)(4n − 3)(3n − 1)(2n − 1)(2n − 3)(3n − 2)(4n − 1), lcoeff(Q4 (ς), ς) = −32n5 (n − 4)(4n − 3)(3n − 1)(3n − 4)(2n − 1)(5n − 3)(3n − 5)(4n − 1)(n − 3)2 (2n − 3)2 (3n − 2)2 (n − 2)4 , for n ≥ 6 even. Moreover one can check that deg(Q4 (ς), ς) = 29 for n = 4 and deg(Q4 (ς), ς) = 31 for n ≥ 6 even, which together with deg(Q3 (ς), ς) = 14 indicates that   lcoeff(Q4 (ς), ς) lim ς 15 = − 596377600 lim ς 15 if n = 4,   Q4 (ς)  lcoeff(Q3 (ς), ς) ς→+∞ 11 ς→+∞ lim = ς→+∞ Q3 (ς)   lcoeff(Q4 (ς), ς) lim ς 17 = 32Υ2 (n) lim ς 17 if n ≥ 6 even,   lcoeff(Q3 (ς), ς) ς→+∞ 33 ς→+∞ where Υ2 (n) := −n2 (n − 3)(n − 4)(3n − 4)(2n − 3)(3n − 2)(3n − 5)(n − 2)3 . Clearly Υ2 (n) < 0 for n ≥ 6 even. Thus we have Q4 (ς) Q2 (ς) lim = −∞, i.e. lim = −∞, ς→+∞ Q3 (ς) ς→+∞ Q1 (ς) for n ≥ 4 even. This proves the claim. From (17) and (18) together with the continuity of Φe4 (ς) we obtain that Φe4 (ς) has at least two positive zeros with respect to the variable ς for n ≥ 4 even, one in the interval (0, 1/2) and the other in the interval (1/2, +∞). Finally we prove the second part of the proposition. Let ς0 be a positive zero of Φe4 (ς). First we show that Φe42 (ς0 ) 6= 0. In fact, if Φe42 (ς0 ) = 0 then we have Φe41 (ς0 ) = 0 from Φe4 (ς0 ) = 0. Note that  lcoeff Φe42 (ς), ς = 3n3 (n − 2)(n − 3)(5n − 3)(4n − 3)(3n − 1)(2n − 1)(2n − 3)(3n − 2)(4n − 1), which is different from zero for n ≥ 4 even. The resultant of the polynomials Φe41 (ς) and Φe42 (ς) with respect to the variable ς is  res Φe41 (ς), Φe42 (ς), ς = R1 (n)R22 (n)R23 (n), where R1 (n) := n23 (3n − 5)3 (4n − 1)5 (3n − 1)5 (5n − 3)5 (2n − 1)5 (4n − 3)5 (n + 1)5 (3n − 4)9 (2n − 3)10 (n − 3)10 (3n − 2)18 (n − 2)21 (n − 1)108 , R2 (n) := 108n12 − 9972n11 + 161301n10 − 1155787n9 + 4608695n8 − 11264874n7 + 17422780n6 − 16171208n5 + 6010272n4 + 5255936n3 − 9284608n2 + 5511168n − 860160, and we omit the long expression of R3 (n). Using the command “IsolatingInterval” together with “Root” in the software Mathematica we can prove that R2 (n) and R3 (n) have no zeros for n ≥ 4 even, which together with R1 (n) 6= 0 for n ≥ 4 even implies that Φe41 (ς) and Φe42 (ς) have no common zeros for n ≥ 4 even. Thus the positive zero ς0 of Φe4 (ς) satisfies Φe42 (ς0 ) 6= 0. 21 Assume that the positive zero ς0 of Φe4 (ς) is of multiplicity 3. Then ς0 is a positive zero of Q2 (ς) with multiplicity 3, where Q2 (ς) is given in (22). Therefore Q02 (ς) and Q002 (ς) have positive common zeros with respect to the variable ς for n ≥ 4 even. It is not difficult to check that the zeros of Q02 (ς) and Q002 (ς) are determined by two polynomials Q5 (ς) and Q6 (ς) respectively, where we omit the long expressions of Qi (ς) for i = 5, 6. Using the command “Resultant” together with the commands “IsolatingInterval” and “Root” in the software Mathematica we can prove that Q5 (ς) and Q6 (ς) have no positive common zeros with respect to the variable ς for n ≥ 4 even, a contradiction with the assumption. Hence the multiplicity of ς0 is at most 2. This proves the proposition.  Numerically we do not find that Φe4 (ς) has three positive zeros with respect to the variable ς for n ≥ 4 even. Moreover it is difficult to determine whether the two positive zeros of Φe4 (ς) obtained in Proposition 9 are simple because the transcendental function arctan(ς) is involved. For some fixed n ≥ 4 even we plot the graphs of Φe4 (ς) and (Φe4 (ς))0 and find that they have no common intersection points at the positive ς-axis. Acknowledgements The first author is partially supported by the grant CSC #202106240110 from the P. R. China. The second author is partially supported by the Agencia Estatal de Investigación grant PID2019-104658GB-I00, and the H2020 European Research Council grant MSCA-RISE-2017-777911. References [1] K. d. S. Andrade, O. A. R. Cespedes, D. R. Cruz, D. D. Novaes, Higher order Melnikov analysis for planar piecewise linear vector fields with nonlinear switching curve, J. Differential Equations 287 (2021), 1–36. [2] J. R. Bastos, C. A. Buzzi, J. Llibre, D. D. Novaes, Melnikov analysis in nonsmooth differential systems with nonlinear switching manifold, J. Differential Equations 267 (2019), 3748–3767. [3] M. di Bernardo, C. J. Budd, A. R. Champneys, P. Kowalczyk, Piecewise-Smooth Dynamical Systems, Theory and Applications, Springer-Verlag, London, 2008. [4] F. Browder, Fixed point theory and nonlinear problems, Bull. Amer. Math. Soc. 9 (1983), 1–39. [5] A. Buică, L. Llibre, Averaging methods for finding periodic orbits via Brouwer degree. Bull. Sci. Math. 128 (2004), 7–22. [6] C. A. Buzzi, J. C. Medrado, J. Torregrosa, Limit cycles in 4-star-symmetric planar piecewise linear systems, J. Differential Equations 268 (2020), 2414–2434. [7] P. T. Cardin, J. Torregrosa, Limit cycles in planar piecewise linear differential systems with nonregular separation line, Phys. D 337 (2016), 67–82. [8] H. Chen, S. Duan, Y. Tang, J. Xie, Global dynamics of a mechanical system with dry friction, J. Differential Equations 265 (2018), 5490–5519. [9] X. Chen, J. Llibre, W. Zhang, Averaging approach to cyclicity of Hopf bifurcation in planar linear-quadratic polynomial discontinuous differential systems, Discrete Con- tin. Dyn. Syst. Ser. B 22 (2017), 3953–3965. [10] X. Chen, J. Llibre, W. Zhang, Cyclicity of (1, 3)-switching FF type equilibria, Dis- crete Contin. Dyn. Syst. Ser. B 24 (2019), 6541–6552. 22 ZHIFEI GUO AND JAUME LLIBRE [11] X. Chen, V. G. Romanovski, W. Zhang, Degenerate Hopf bifurcations in a family of FF-type switching systems, J. Math. Anal. Appl. 432 (2015), 1058–1076. [12] C. Christopher, C. Li, Limit Cycles of Differential Equations, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser Verlag, Basel, 2007. [13] S. Coombes, Neuronal networks with gap junctions: a study of piecewise linear planar neuron models, SIAM J. Appl. Dyn. Syst. 7 (2008), 1101–1129. [14] L. P. C. Cruz, D. D. Novaes, J. Torregrosa, New lower bound for the Hilbert number in piecewise quadratic differential systems, J. Differential Equations 266 (2019), 4170–4203. [15] A. F. Filippov, Differential Equation with Discontinuous Right-Hand Sides, Kluwer Academic, Amsterdam, 1988. [16] J. Giné, M. Grau, J. Llibre, Averaging theory at any order for computing periodic orbits, Phys. D 250 (2013), 58–65. [17] J. Giné, J. Llibre, K. Wu, X. Zhang, Averaging methods of arbitrary order, periodic solutions and integrability, J. Differential Equations 260 (2016), 4130–4156. [18] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifur- cations of Vector Fields, Springer, New York, 1983. [19] J. K. Hale, Ordinary Differential Equations, 2nd ed., Wiley, New York, 1980. [20] M. Han, W. Zhang, On Hopf bifurcation in non-smooth planar systems, J. Differen- tial Equations 248 (2010), 2399–2416. [21] J. Itikawa, J. Llibre, D. D. Novaes, A new result on averaging theory for a class of discontinuous planar differential systems with applications, Rev. Mat. Iberoam. 33 (2017), 1247–1265. [22] S. J. Karlin, W. J. Studden, Tchebycheff systems: With Applications in Analysis and Statistics, Pure Appl. Math., Interscience Publishers, New York, London, Sidney, 1966. [23] J. Li, Hilbert’s 16th problem and bifurcations of planar polynomial vector fields, Int. J. Bifur. Chaos 13 (2003), 47–106. [24] T. Li, J. Llibre, Limit cycles in piecewise polynomial systems allowing a non-regular switching boundary, Phys. D 419 (2021), 132855. [25] J. Llibre, A. C. Mereu, Limit cycles for discontinuous quadratic differential systems with two zones, J. Math. Anal. Appl. 413 (2014), 763–775. [26] J. Llibre, A. C. Mereu, D. D. Novaes, Averaging theory for discontinuous piecewise differential systems, J. Differential Equations 258 (2015), 4007–4032. [27] J. Llibre, R. Moeckel, C. Simó, Central Configurations, Periodic Orbits, and Hamil- tonian Systems, Advanced Courses in Mathematics, CRM Barcelona, Birkhäuser 2015. [28] J. Llibre, D. D. Novaes, Improving the averaging theory for computing periodic solutions of the differential equations, Z. Angew. Math. Phys. 66 (2015), 1401–1412. [29] J. Llibre, D. D. Novaes, M. A. Teixeira, Higher order averaging theory for finding periodic solutions via Brouwer degree, Nonlinearity 27 (2014), 563–583. [30] J. Llibre, D. D. Novaes, M. A. Teixeira, On the birth of limit cycles for non-smooth dynamical systems, Bull. Sci. Math. 139 (2015), 229–244. [31] J. Llibre, G. Rodrı́guez, Configurations of limit cycles and planar polynomial vector fields, J. Differential Equations 198 (2004), 374–380. [32] J. Llibre, Y. Tang, Limit cycles of discontinuous piecewise quadratic and cubic poly- nomial perturbations of a linear center, Discrete Contin. Dyn. Syst. Ser. B 24 (2019), 1769–1784. [33] J. Llibre, X. Zhang, Limit cycles for discontinuous planar piecewise linear differential systems separated by one straight line and having a center, J. Math. Anal. Appl. 467 (2018), 537–549. [34] N. G. Lloyd, Degree Theory, Cambridge Tracts in Mathematics, vol. 73, Cambridge University Press, Cambridge-New York-Melbourne, 1978. 23 [35] O. Makarenkov, J. S. W. Lamb, Dynamics and bifurcations of nonsmooth systems: a survey, Phys. D 241 (2012), 1826–1844. [36] D. D. Novaes, J. Torregrosa, On extended Chebyshev systems with positive accuracy, J. Math. Anal. Appl. 448 (2017), 171–186. [37] D. D. Novaes, F. B. Silva, Higher order analysis on the existence of periodic solu- tions in continuous differential equations via degree theory, SIAM J. Math. Anal. 53 (2021), 2476–2490. [38] O. Ramirez, A. M. Alves, Bifurcation of limit cycles by perturbing piecewise non- Hamiltonian systems with nonlinear switching manifold, Nonlinear Anal. Real World Appl. 57 (2021), 103188. [39] J. A. Sanders, F. Verhulst, J. Murdock, Averaging Method in Nonlinear Dynamical Systems, Appl. Math. Sci., vol. 59, Springer, New York, 2007. [40] D. J. W. Simpson, Bifurcations in Piecewise-Smooth Continuous Systems, World Scientific Series on Nonlinear Science A, vol 69, World Scientific, Singapore, 2010. [41] A. Tonnelier, The McKean’s caricature of the FitzHugh-Nagumo model. I. The space- clamped system, SIAM J. Appl. Math. 63 (2002), 459–484. [42] L. Wei, X. Zhang, Averaging theory of arbitrary order for piecewise smooth dif- ferential systems and its application, J. Dynam. Differential Equations 30 (2018), 5579. [43] P. Yu, M. Han, X. Zhang, Eighteen limit cycles around two symmetric foci in a cubic planar switching polynomial system, J. Differential Equations 275 (2021), 939–959. 1 School of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P. R. China E-mail address: mathgzf@126.com 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra 08193, Barcelona, Catalonia, Spain E-mail address: jllibre@mat.uab.cat