This is a preprint of: “Limit cycles of the discontinuous piecewise differential systems on the cylinder”, Jie Li, Jaume Llibre, J. Appl. Anal. Comp., vol. 12(3), 952–963, 2022. DOI: [10.11948/20220031] LIMIT CYCLES OF THE DISCONTINUOUS PIECEWISE DIFFERENTIAL SYSTEMS ON THE CYLINDER JIE LI1 AND JAUME LLIBRE2 This paper is dedicated to Professor Jibin Li for his 80th birthday Abstract. In order to understand the dynamics of the differential systems the limit cycles play a main role, but in general their study is not easy. These last years an increasing interest appeared for studying the limit cycles of some classes of discontinuous piecewise differential systems, due to the rich applications of this kind of differential systems. Very few papers studied the limit cycles of the discontinuous piecewise differential systems in spaces different from the plane R2 . Here we study the limit cycles of a class of discontinuous piecewise differential systems on the cylinder. 1. Introduction and statement of the main Consider the following differential equation on the cylinder (r, θ) ∈ R × S1 dr (1) = a0 (θ) + a1 (θ)r + a2 (θ)r2 + ... + an (θ)rn . dθ All the functions ai (θ) are continuous and 2π-periodic in the variable θ. Equation (1) with n = 1 is a linear differential equation having at most one limit cycle, see for instance [4]. While for n = 2 it is a Riccati equation with at most two limit cycles, see [6]. For n = 3 it is an Abel equation. If a3 (θ) > 0 Pliss [9] proved that the Abel equation has at most three limit cycles (see also [3, 8]). For n ≥ 4 a constant sign in the leading coefficient an is not sufficient to bound uniformly the number of limit cycles (see [6, 8]). Lins Neto in [8] gave a example with at least 2010 Mathematics Subject Classification. 34C05, 37G15. Key words and phrases. Limit cycles, discontinuous piecewise smooth system, differential systems on the cylinder. 1 2 JIE LI AND JAUME LLIBRE n + 3 limit cycles for suitable functions a and f , for the Abel equation dx = εf (θ)x3 + a(θ)x2 + δx, dθ where |δ| is small, a(θ) is a trigonometric polynomial of degree 1, and f (θ) is a trigonometric polynomial of degree 2n. Calanchi and Ruf [2] proved that if in equation (1) n is odd, the leading term is fixed and the remaining terms are small enough, then the number of limit cycles is at most n. In [1] Bakhshalizadeh and Llibre considered the discontinuous piece- wise differential systems of the form ẋ = a0 (θ) + a1 (θ)x + · · · + an (θ)xn , if 0 ≤ θ ≤ π, (2) ẋ = b0 (θ) + b1 (θ)x + · · · + bm (θ)xm , if 0 ≤ θ ≤ 2π, where a0 (θ), a1 (θ), · · · , an (θ) and b0 (θ), b1 (θ), · · · , bm (θ) are 2π-periodic, and gave exact bounds for the maximum number of limit cycles. On the lines of discontinuity x = 0 and x = π of systems (2), the flow is defined following the rules of Filippov [5]. In the rest of the paper always the flow on the lines of discontinuity is defined according with Filippov. The objective of this paper is to extend the results on the maximum number of limit cycles obtained in [1] for the discontinuous piecewise differential systems on the cylinder with two straight lines of separation, to the discontinuous piecewise differential systems on the cylinder with an arbitrary number of lines of separation. Let C be the cylinder {(θ, x) ∈ S1 × R}. Consider the discontinuous piecewise differential systems on the cylinder P m1 ẋ = a1l (θ)xl , if 0 ≤ θ ≤ 2π/n, l=0 P m2 ẋ = a2l (θ)xl , if 2π/n ≤ θ ≤ 2 · 2π/n, l=0 .. . (3) P mk ẋ = akl (θ)xl , if 2π(k − 1)/n ≤ θ ≤ 2kπ/n, l=0 .. . P mn ẋ = anl (θ)xl , if 2π(n − 1)/n ≤ θ ≤ 2π, l=0 LIMIT CYCLES OF THE PIECEWISE DIFFERENTIAL SYSTEMS 3 where akl (θ), for k = 1, · · · n and l = 0, 1, · · · , mk , are 2π-periodic functions in the variable θ. Then H(m1 , · · · , mn ) denotes the maxi- mum number of limit cycles that the discontinuous piecewise differen- tial systems (3) can exhibit. Theorem 1. The discontinuous piecewise differential systems on the cylinder C of the form ẋ = a0 (θ) + a1 (θ)x, if 0 ≤ θ ≤ 2π/3, (4) ẋ = b0 (θ) + b1 (θ)x, if 2π/3 ≤ θ ≤ 4π/3, ẋ = c0 (θ) + c1 (θ)x, if 4π/3 ≤ θ ≤ 2π, where ai (θ), bi (θ) and ci (θ) for i = 0, 1 are 2π-periodic functions in the variable θ, have at most one limit cycle, i.e, H(1, 1, 1) = 1. Corollary 2. The discontinuous piecewise differential systems on the cylinder C of the form ẋ = a10 (θ) + a11 (θ)x, if 0 ≤ θ ≤ 2π/n, ẋ = a20 (θ) + a21 (θ)x, if 2π/n ≤ θ ≤ 2 · 2π/n, .. . ẋ = ak0 (θ) + ak1 (θ)x, if 2π(k − 1)/n ≤ θ ≤ 2kπ/n, .. . ẋ = an0 (θ) + an1 (θ)x, if 2π(n − 1)/n ≤ θ ≤ 2π, where ak0 (θ) and ak1 (θ), for k = 1, · · · n, are 2π-periodic functions in the variable θ, have at most one limit cycle, i.e., H(1, · · · , 1) = 1. Theorem 3. The discontinuous piecewise differential systems on the cylinder C of the form ẋ = a0 (θ) + a1 (θ)x + a2 (θ)x2 , if 0 ≤ θ ≤ π, (5) ẋ = b0 (θ) + b1 (θ)x + b2 (θ)x2 , if π ≤ θ ≤ 2π, where ai (θ) and bi (θ), for i = 0, 1, 2, are 2π-periodic functions in the variable θ, have at most two limit cycles, i.e., H(2, 2) = 2. 4 JIE LI AND JAUME LLIBRE Corollary 4. The discontinuous piecewise differential systems on the cylinder C of the form ẋ = a10 (θ) + a11 (θ)x + a12 x2 , if 0 ≤ θ ≤ 2π/n, ẋ = a20 (θ) + a21 (θ)x + a22 x2 , if 2π/n ≤ θ ≤ 2 · 2π/n, .. . ẋ = ak0 (θ) + ak1 (θ)x + ak2 x2 , if 2π(k − 1)/n ≤ θ ≤ 2kπ/n, .. . ẋ = an0 (θ) + an1 (θ)x + an2 x2 , if 2π(n − 1)/n ≤ θ ≤ 2π, where ak0 (θ), ak1 (θ) and ak2 (θ), for k = 1, · · · n, are 2π-periodic func- tions in the variable θ, have at most two limit cycles, i.e., H(2, · · · , 2) = 2. Theorem 5. The discontinuous piecewise differential systems on the cylinder C of the form ẋ = a0 (θ) + a1 (θ)x, if 0 ≤ θ ≤ 2π/3, 2 (6) ẋ = b0 (θ) + b1 (θ)x + b2 (θ)x , if 2π/3 ≤ θ ≤ 4π/3, ẋ = c0 (θ) + c1 (θ)x, if 4π/3 ≤ θ ≤ 2π, where ai (θ), bi (θ) and ci (θ), for i = 0, 1 or 2, are 2π-periodic functions in the variable θ, have at most two limit cycles, i.e., H(1, 2, 1) = 2. According to the above Theorems 1, 3, 5 and Corollaries 2, 4, we can conclude the following corollary. Corollary 6. The discontinuous piecewise differential systems on the cylinder C of the form (3) with max{m1 , · · · , mn } ≤ 2 have at most one limit cycle if m1 = · · · = mn = 1, i.e., H(1, · · · , 1) = 1, otherwise H(m1 , · · · , mn ) = 2. Theorem 7. For every positive integer k there are discontinuous piece- wise differential systems on the cylinder C of the form ẋ = a(θ)x, if 0 ≤ θ ≤ 2π/3, 2 3 (7) ẋ = b2 (θ)x + εb3 (θ)x , if 2π/3 ≤ θ ≤ 4π/3, ẋ = c(θ)x, if 4π/3 ≤ θ ≤ 2π, where a(θ), c(θ) and bi (θ) for i = 2, 3, are 2π-periodic functions in the variable θ, having at least k limit cycles on the cylinder, i.e., H(1, 3, 1) = +∞. According to the above Theorem 7, we can conclude the following corollary. LIMIT CYCLES OF THE PIECEWISE DIFFERENTIAL SYSTEMS 5 Corollary 8. The discontinuous piecewise differential systems on the cylinder C of the form (3) with max{m1 , · · · , mn } ≥ 3 have at least k limit cycles for any positive integer k, i.e., H(m1 , · · · , mn ) = +∞. 2. Proof of the main results In this section we will prove the main results as stated in Theorems 1, 3, 5, 7 and Corollaries 2, 4, 6. Proof of Theorem 1. Consider the discontinuous piecewise differential systems (4). The solution of the first equation of (4) satisfying x(0) = ρ is x1 (θ, ρ) = (I1 (θ) + ρ) eK1 (θ) , Z θ I1 (θ) = a0 (s)e−K1 (s) ds, Z0 s K1 (s) = a1 (w)dw. 0 The solution of the second equation of (4) satisfying x(2π/3) = x1 (2π/3, ρ) is x2 (θ, x1 (π/3, ρ)) = (I2 (θ) + x1 (π/3, ρ)) eK2 (θ) , Z θ I2 (θ) = b0 (s)e−K2 (s) ds, 2π Z3 s K2 (s) = b1 (w)dw, 2π 3 and the solution of the third equation of (4) satisfying x(4π/3) = x2 (4π/3, x1 (2π/3, ρ)) is x3 (θ, x2 (4π/3, x1 (2π/3, ρ))) = (I3 (θ) + x2 (4π/3, x1 (2π/3, ρ))) eK3 (θ) , Z θ I3 (θ) = c0 (s)e−K3 (s) ds, 4π Z3 s K3 (s) = c1 (w)dw. 4π 3 Define the function Π1 (ρ) = x3 (2π, x2 (4π/3, x1 (2π/3, ρ))) − ρ  = eK3 (2π) eK1 (2π/3)+K2 (4π/3) I1 (2π/3)+eK2 (4π/3) I2 (4π/3)+I3 (2π) +(eK1 (2π/3)+K2 (4π/3)+K3 (2π) − 1)ρ. Thus the periodic orbits of the discontinuous piecewise differential sys- tems (4) are associated with the zeros of the linear equation Π1 (ρ) = 0. 6 JIE LI AND JAUME LLIBRE Clearly there is at most one zero. Thus the discontinuous piecewise differential systems (4) have at most one limit cycle. □ The proof of Corollary 2 is similar to the proof of Theorem 1. Proof of Theorem 3. Consider the discontinuous piecewise differential systems (5). On the two bands of the cylinder with θ ∈ [0, π] and θ ∈ [π, 2π] we have a Riccati differential equation. Suppose that we have a periodic solution x(θ) = xp (θ)|θ∈[0,π] ∪xq (θ)|θ∈[π,2π] . Then doing the change of variable x → X1 where 1 X1 (θ) = , x(θ) − xp (θ) we write the first differential equation in (5) with θ ∈ [0, π] as dX1  (8) = −a2 (θ) − 2a2 (θ)xp (θ) + a1 (θ) X1 . dθ Then the solution of the linear equation (8) with θ ∈ [0, π] is written as X1 (θ) = (N1 (θ) + X1 (0)) eM1 (θ) , Z θ N1 (θ) = −a2 (s)e−M1 (s) ds, Z0 θ  M1 (θ) = − 2a2 (w)xp (w) + a1 (w) dw. 0 Undoing the change of variables we obtain that the solution of the first equation of (5) satisfying x(0) = ρ is A1 (θ) + B1 (θ)ρ x1 (θ, ρ) = , C1 (θ) + D1 (θ)ρ where A1 (θ) = xp (θ)eM1 (θ) (1 − xp (0)N1 (θ)) − xp (0), B1 (θ) = xp (θ)N1 (θ)eM1 (θ) + 1, C1 (θ) = eM1 (θ) (1 − xp (0)N1 (θ)) , D1 (θ) = eM1 (θ) N1 (θ). Similarly we write the second differential equation of (6) in θ ∈ [π, 2π] as dX2  (9) = −b2 (θ) − 2b2 (θ)xq (θ) + b1 (θ) X2 dθ doing the change 1 X2 (θ) = . x(θ) − xq (θ) LIMIT CYCLES OF THE PIECEWISE DIFFERENTIAL SYSTEMS 7 Both changes of variables the change of variables x → X1 when x ∈ [0, π] and the change of variables x → X2 when x ∈ [π, 2π] coincide on the periodic orbits intersection with the straight lines x = 0 and π, so the structure of the discontinuous piecewise differential systems (5) is preserved. Indeed, 1 1 X1 (0) = X2 (2π) = = , x(0) − xp (0) x(2π) − xq (2π) 1 1 X1 (π) = X2 (π) = = , x(π) − xp (π) x(π) − xq (π) because x(0) = x(2π), xp (0) = xq (2π) and xp (π) = xq (π) on the peri- odic orbits. Then the solution of the differential equation (9) is X2 (θ) = (N2 (θ) + X2 (π)) eM2 (θ) , Z θ N2 (θ) = −b2 (s)e−M2 (s) ds, Zπ s  M2 (s) = − 2b2 (w)xq (w) + b1 (w) dw. π Undoing the change of variables we get the solution of the second equa- tion of (5) satisfying x(π) = x1 (π, ρ) is A2 (θ) + B2 (θ)x1 (π, ρ) x2 (θ, ρ) = C2 (θ) + D2 (θ)x1 (π, ρ) C1 (π)A2 (θ) + A1 (π)B2 (θ) + (D1 (π)A2 (θ) + B1 (π)B2 (θ))ρ = , C1 (π)C2 (θ) + A1 (π)D2 (θ) + (B1 (π)D2 (θ) + D1 (π)D2 (θ))ρ where A2 (θ) = xq (θ)eM2 (θ) (1 − xq (π)N2 (θ)) − xq (π), B2 (θ) = xq (θ)N2 (θ)eM2 (θ) + 1, C2 (θ) = eM2 (θ) (1 − xq (π)N2 (θ)) , D2 (θ) = eM2 (θ) N2 (θ). Define the function Π2 (ρ) = x2 (2π, x1 (π, ρ)) − ρ, which is A2 C1 + A1 B2 + (A2 D1 + B1 B2 − C1 C2 − A1 D2 )ρ − (B1 D2 + D1 D2 )ρ2 , C1 C2 + A1 D2 + (B1 D2 + D1 D2 )ρ with A1 = A1 (π), B1 = B1 (π), C1 = C1 (π), D1 = D1 (π) and A2 = A2 (2π), B2 = B2 (2π), C2 = C2 (2π), D2 = D2 (2π). Thus the periodic orbits of the discontinuous piecewise differential systems (5) are asso- ciated with the zeros of the equation Π2 (ρ) = 0. It follows that the 8 JIE LI AND JAUME LLIBRE discontinuous piecewise differential systems (5) have at most two limit cycles. □ The proof of Corollary 4 is similar to the proof of Theorem 3. Proof of Theorem 5. Consider the discontinuous piecewise differential systems (6). On the second band of the cylinder, i.e., θ ∈ [2π/3, 4π/3], we have a Riccati differential equation. Suppose that there is a periodic solution x(θ) = xr (θ)|θ∈[0,2π/3] ∪ xs (θ)|θ∈[2π/3,4π/3] ∪ xt (θ)|θ∈[4π/3,2π] . In what follows for studying the limit cycles on the cylinder of systems (7) we will do three changes of variables, in each strip of cylinder defined by the straight lines θ = 0, θ = 2π/3 and θ = 4π/3. Later on we will show that these changes of variables coincide on the three straight lines θ = 0, θ = 2π/3 and θ = 4π/3. Then doing the change of variable x → Xr , where 1 Xr (θ) = , x(θ) − xr (θ) we write the first differential equation of (6) with θ ∈ [0, 2π/3] as dXr (10) = −a1 (θ)Xr . dθ Then the solution of the linear equation (10) with θ ∈ [0, 2π/3] with an initial value Xr (0) is written as Xr (θ, Xr (0)) = Xr (0)eK1 (θ) , Z θ K1 (θ) = (−a1 (s))ds. 0 Correspondingly we obtain the solution of the first differential equation of (6) with θ ∈ [0, 2π/3] satisfying x(0) = ρ is xr (θ)eK1 (θ) − xr (0) ρ x1 (θ, ρ) = K (θ) + . e 1 e 1 (θ) K Note that on the cylinder with θ ∈ [2π/3, 4π/3] we have a Riccati differential equation. Then doing the change of variable x → Xs , where 1 Xs (θ) = , x(θ) − xs (θ) we write the second differential equation of (6) with θ ∈ [2π/3, 4π/3] as dXs  (11) = −b2 (θ) − 2b2 (θ)xs (θ) + b1 (θ) Xs . dθ LIMIT CYCLES OF THE PIECEWISE DIFFERENTIAL SYSTEMS 9 The solution of the linear equation (11) with θ ∈ [2π/3, 4π/3] is written as Xs (θ) = (I2 (θ) + Xs (2π/3)) eK2 (θ) , Z θ I2 (θ) = −b2 (s)e−K2 (s) ds, 2π Z3 s  K2 (s) = − 2b2 (w)xs (w) + b1 (w) dw. 2π 3 Undoing the change of variables the solution of the second equation of (6) satisfying x(2π/3) = x1 (2π/3, ρ) is A(θ) + B(θ)ρ (12) x2 (θ, ρ) = , C(θ) + D(θ)ρ where  A(θ) = xs (θ)eK2 (θ) eK1 (2π/3) − I2 (θ)xr (0) − xr (0) B(θ) = I2 (θ)xs (θ)eK2 (θ) + 1,  C(θ) = eK2 (θ) eK1 (2π/3) − I2 (θ)xr (0) , D(θ) = I2 (θ)eK2 (θ) . Similarly, doing the change of variable x → Xt , where 1 Xt (θ) = , x(θ) − xt (θ) we write the third differential equation of (6) with θ ∈ [4π/3, 2π] as dXt (13) = −c1 (θ)Xt . dθ Then the solution of the linear equation (13) with an initial value Xt (4π/3) is written as Xt (θ, Xt (4π/3)) = Xt (4π/3)eK3 (θ) , Z θ K3 (θ) = (−c1 (s))ds. 0 Correspondingly the solution of the third differential equation of (6) with θ ∈ [4π/3, 2π] satisfying x(4π/3) = x2 (4π/3, ρ) is xt (θ)eK3 (θ) + x2 (4π/3, ρ) − xt (4π/3) (14) x3 (θ, ρ) = . eK3 (θ) Note that we need to check that the change of variables x → Xr when x ∈ [0, 2π/3], the change of variables x → Xs when x ∈ [2π/3, 4π/3], the change of variables x → Xt when x ∈ [4π/3, 2π] coincide on the periodic orbits intersection with the straight lines θ = 0, 2π/3 and 10 JIE LI AND JAUME LLIBRE 4π/3, and consequently the structure of the discontinuous piecewise differential systems (6) is preserved. Indeed, 1 1 Xr (0) = Xt (2π) = = , x(0) − xr (0) x(2π) − xt (2π) 1 1 Xr (2π/3) = Xs (2π/3) = = , x(2π/3) − xr (2π/3) x(2π/3) − xs (2π/3) 1 1 Xs (4π/3) = Xt (4π/3) = = , x(4π/3) − xs (4π/3) x(4π/3) − xt (4π/3) because x(0) = x(2π), xr (0) = xt (2π) and xr (2π/3) = xs (2π/3) and xs (4π/3) = xt (4π/3) on the periodic orbits. Define the function Π3 (ρ) = x3 (2π, ρ) − ρ. Then by (12) and (14) we obtain E(2π) + F (2π)ρ − eK3 (2π) D(4π/3)ρ2 Π3 (ρ) = , eK3 (2π) (C(4π/3) + D(4π/3)ρ) where E(θ) = (xt (θ)eK3 (θ) − xt (4π/3))C(4π/3) + A(4π/3), F (θ) = (xt (θ)eK3 (θ) − xt (4π/3))D(4π/3) + B(4π/3) − eK3 (θ) C(4π/3). Thus the periodic orbits of the discontinuous piecewise differential sys- tems (6) are associated with the zeros of the equation Π3 (ρ) = 0. Clearly there is at most two zeros, and therefore the discontinuous piecewise differential systems (6) have at most two limit cycles. □ Proof of Theorem 7. We consider the discontinuous piecewise differen- tial systems ẋ = a(θ)x, if 0 ≤ θ ≤ 2π/3, 2 (15) ẋ = b2 (θ)x , if 2π/3 ≤ θ ≤ 4π/3, ẋ = c(θ)x, if 4π/3 ≤ θ ≤ 2π. The solution of the first differential equation of (15) with θ ∈ [0, 2π/3] satisfying x(0) = ρ is Z θ J1 (θ) x1 (θ, ρ) = ρe , J1 (θ) = a(s)ds. 0 On the other hand the solution of the second differential equation of (15) with θ ∈ [2π/3, 4π/4] satisfying x(2π/3) = x1 (2π/3, ρ) is Z θ ρeJ1 (2π/3) x2 (θ, ρ) = , J2 (θ) = b2 (s)ds. 1 − ρeJ1 (2π/3) J2 (θ) 2π 3 LIMIT CYCLES OF THE PIECEWISE DIFFERENTIAL SYSTEMS 11 Eventually the solution of the third differential equation of (15) with θ ∈ [4π/3, 2π] satisfying x(4π/3) = x2 (4π/3, ρ) is Z θ ρeJ1 (2π/3)+J3 (θ) x3 (θ, ρ) = , J3 (θ) = c(s)ds. 1 − ρeJ1 (2π/3) J2 (4π/3) 4π 3 Define the function ρeJ1 (2π/3)+J3 (2π) Π4 (ρ) = x3 (2π, ρ) − ρ = − ρ. 1 − ρeJ1 (2π/3) J2 (4π/3) Then Π4 (ρ) ≡ 0 if we assume (16) J1 (2π/3) + J3 (2π) = J2 (4π/3) = 0, where we choose the functions a(θ), b2 (θ) and c(θ) in order that the equalities (16) hold. We obtain that the discontinuous piecewise dif- ferential systems (15) has a continuum of periodic solutions in the neighborhood of ρ = 0. In what follows we consider the discontinuous piecewise differen- tial systems (7) with a small parameter ε. The solution of the first differential equation of (7) with θ ∈ [0, 2π/3] satisfying x(0) = ρ is x1 (θ, ρ) given in (12). Let x2 (θ, ρ, ε) denote the solution of the second differential equation with initial value x2 (2π/3, ρ, ε) = x1 (2π/3, ρ) = ρeJ1 (2π/3) = x̄1 . Then the solution x2 (θ, x̄1 , ε) can be expanded with respect to ε as follows x2 (θ, x̄1 , ε) = x20 (θ, x̄1 ) + x21 (θ, x̄1 )ε + O(ε2 ), where x20 (θ, x̄1 ) = x2 (θ, x̄1 , ε)|ε=0 and x21 (θ, x̄1 ) = ∂x2 (θ, x̄1 , ε)/∂ε|ε=0 . Similarly the solution x3 (θ, ρ, ε) of the third differential equation of sys- tems (7) with θ ∈ [4π/3, 2π] satisfying x3 (4π/3, ρ, ε) = x2 (4π/3, x̄1 , ε) = x̄2 is x3 (θ, x̄2 , ε) = x2 (4π/3, x̄1 , ε)eJ3 (θ) . Then we similarly obtain a function Πε4 (ρ, ε) = x3 (2π, x̄2 , ε) − ρ = x2 (4π/3, x̄1 , ε)eJ3 (2π) − ρ  = x20 (4π/3, x̄1 )eJ3 (2π) − ρ + x21 (4π/3, x̄1 )eJ3 (2π) ε + O(ε2 ). 12 JIE LI AND JAUME LLIBRE In order to find the number of zeros of the equation Πε4 = 0, asso- ciated with the number of limit cycles of the discontinuous piecewise differential systems (7), first we have x˙2 (θ, x̄1 , ε) = ẋ20 (θ, x̄1 ) + ẋ21 (θ, x̄1 )ε + O(ε2 ) 2 = b2 (θ) x20 (θ, x̄1 ) + x21 (θ, x̄1 )ε + O(ε2 ) + εb3 (θ) 3 x20 (θ, x̄1 ) + x21 (θ, x̄1 )ε + O(ε2 ) + O(ε2 ). From this equality we get (17) ẋ20 (θ, x̄1 ) = b2 (θ)x220 (θ, x̄1 ), with the initial value x20 (2π/3, x̄1 ) = ρeJ1 (2π/3) and (18) ẋ21 (θ, x̄1 ) = 2b2 (θ)x20 (θ, x̄1 )x21 (θ, x̄1 ) + b3 (θ)x320 (θ, x̄1 ), with an initial value x21 (2π/3, x̄1 ) = 0, where ẋ2i (θ, x̄1 ) = ∂x2i (θ, x̄1 )/∂θ for i = 0, 1. Integrating the differential equation (17) in the interval [2π/3, 4π/3], by (16) we get ρeJ1 (2π/3) x20 (4π/3, ρ) = = ρeJ1 (2π/3) . 1 − ρeJ1 (2π/3) J2 (4π/3) On the other hand we get ẋ20 (θ, x̄1 ) b2 (θ)x20 (θ, x̄1 ) = , x20 (θ, x̄1 ) from the differential equation (17). Substituting the previous equality in (18), we have   ∂ x21 (θ, x̄1 ) = b3 (θ)x20 (θ, x̄1 ). ∂θ x220 (θ, x̄1 ) Integrating this differential equation in the interval [2π/3, 4π/4] and combining the assumption (16), we obtain Z 4π 3 2 x21 (4π/3, x̄1 ) = x20 (4π/3, x̄1 ) b3 (θ)x20 (θ, x̄1 )dθ 2π 3 Z 4π 3 3J1 (2π/3) 3 b3 (θ) = ρe dθ. 2π 3 1− ρe 1 (2π/3) J2 (θ) J We further obtain  Πε4 (ρ, ε) = x20 (4π/3, x1 )eJ3 (2π) − ρ + x21 (4π/3, x1 )eJ3 (2π) ε + O(ε2 ) = ερ3 e2J1 (2π/3) M (ρ) + O(ε2 ), where Z 4π 3 b3 (θ) M (ρ) = dθ, 2π 3 1− ρe 1 (2π/3) J2 (θ) J LIMIT CYCLES OF THE PIECEWISE DIFFERENTIAL SYSTEMS 13 is a Melnikov function. It follows from the Implicit Function Theorem that the simple zeros of M (ρ) which are non-zero can be associated with the simple zeros of the function Πε4 (ρ, ε) distinct from zero. More con- cretely, if ρ = ρ0 ̸= 0 satisfying M (ρ0 ) = 0 and M ′ (ρ0 ) ̸= 0, then there exists a differential function ϕ such that ϕ(0) = ρ0 and Πε4 (ϕ(ε), ε) ≡ 0 for a small enough ε. For any n ∈ N we choose b2 (θ) = cos(eJ1 (2π/3) θ), b3 (θ) = P (sin(eJ1 (2π/3) θ)), where P is a polynomial of degree n in the variable sin(eJ1 (2π/3) θ). We introduce the family of analytic functions Z 4π 3 sink t Ik (ρ) = dt, 2π 3 1 − ρ sin t where t = eJ1 (2π/3) θ and k is a positive integer. By Theorem A of [7] the maximum number of zeros of M (ρ) is the same as the degree n of the polynomial P (sin t). □ Acknowledgements The first author is partially supported by the China Scholarship Council (CSC) No. 202006240222. 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] A. Bakhshalizadeh, J. Llibre, Limit cycles of piecewise differential equations on the cylinder, Bull. Sci. Math. 170 (2021), 103013, 13 pp. [2] M. Calanchi, B. Ruf, On the number of closed solutions for polynomial ODE’s and a special case of Hilbert’s 16th problem, Adv. Diff. Equ., 7, n.2, (2002), 197-216. [3] B. Coll, A. Gasull, J. Llibre, Some theorems on the existence, uniqueness, and nonexistence of limit cycles for quadratic systems, J. Differential Equations 67 (1987): 372-399. [4] W. A. Coppel, Dichotomies in Stability Theory, vol. 629 of Lecture Notes in Mathematics, Springer, 1978. [5] A.F. Filippov, Differential equations with discontinuous right-hand sides, translated from Russian. 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