This is a preprint of: “Limit cycles of piecewise differential equations on the cylinder”, Ali Bakhshal- izadeh, Jaume Llibre, Bull. Sci. Math., vol. 170, 103013:1–13, 2021. DOI: [10.1016/j.bulsci.2021.103013] LIMIT CYCLES OF PIECEWISE DIFFERENTIAL EQUATIONS ON THE CYLINDER ALI BAKHSHALIZADEH1 AND JAUME LLIBRE2 Abstract. We consider the piecewise differential equations of the form  dx  1 (  dt  a0 (t) + a1 (t)x1 + .... + an (t)xn1 , if 0 6 t 6 π,  = 2 , if π 6 t 6 2π, dx2 b0 (t) + b1 (t)x2 + .... + bm (t)xm dt where a0 (t), a1 (t), ..., an (t) and b0 (t), b1 (t), ..., bm (t) are 2π-periodic functions in the variable t, and we study the number of limit cycles of such equations on the cylinder. In this way we give exact bounds for the maximum number of limit cycles that the piecewise differential equations have in function of n and m. 1. Introduction and statements of the main results Pugh proposed the following problem (see [7]). Let a0 , a1 , ..., an : [0, 2π] → R be analytic functions and consider the differential equation dx (1) = a0 (t) + a1 (t)x + .... + an (t)xn , 0 6 t 6 2π. dt A solution x(t) of (1) is called a closed solution or a periodic solution if it is defined in the interval [0, 2π] and x(0) = x(2π). The adjectives closed and periodic are motivated by the case where a0 , ..., an are 2π-periodic, in which (1) can be considered in the cylinder and the closed solutions really correspond to closed orbits in the cylinder. Closed orbits in polynomial planar differential systems can be isolated or belong to an annulus of periodic orbits. In the isolated case they are called limit cycles. So the problem is this: Is there a bound on the number of limit cycles of (1)? We note that the differential equation (1) with n = 1 (resp. n = 2) is a linear equation (resp. a Riccati equation), and when n = 3, (1) is called an Abel equation. It is well known that linear (resp. Riccati) equations have either a continuum of periodic solutions or at most 1 (resp. 2) periodic solutions. For the Abel equation it was proved that for any k there exist equations (1) with ai (t) trigonometric 2π-periodic polynomials, having at least k limit cycles. A similar result holds for n > 3, see for more details [7, 9, 1]. In this paper we deal with the piecewise differential equation   dx1 (  dt  a0 (t) + a1 (t)x1 + .... + an (t)xn1 , if 0 6 t 6 π, (2)  = 2 , if π 6 t 6 2π, dx2 b0 (t) + b1 (t)x2 + .... + bm (t)xm dt 2010 Mathematics Subject Classification. 37G15. Key words and phrases. Limit cycles, piecewise smooth system, Hilbert number, differential equations on the cylinder. 1 2 ALI BAKHSHALIZADEH AND JAUME LLIBRE where a0 (t), a1 (t), ..., an (t) and b0 (t), b1 (t), ..., bm (t) are 2π-periodic functions in the variable t. If we consider that x1 (t) = φ1 (t, ρ) is the solution of dx1 = a0 (t) + a1 (t)x1 + .... + an (t)xn1 , dt such that φ1 (0, ρ) = ρ, and let x2 (t) = φ2 (t, ρ) be the solution of dx2 = b0 (t) + b1 (t)x2 + .... + bm (t)xm 2 , dt such that φ2 (2π, ρ) = ρ, then a solution of the piecewise differential equation (2) satisfying φ1 (π, ρ) = φ2 (π, ρ) is called a closed solution or a periodic solution. Like in the smooth case the adjectives closed and periodic are motivated by the case where a0 , a1 , ..., an and b0 , b1 , ..., bm are 2π-periodic, in which (2) can be considered in the cylinder and the closed solutions really correspond to periodic orbits in the cylinder. Then the function Π(ρ) := φ1 (π, ρ) − φ2 (π, ρ) plays a great role to find the number of closed orbits. Thus the zeros of Π(ρ) correspond to closed orbits of (2), and their isolated zeros provide initial conditions for the limit cycles of equation (2). Then the problem is this: Is there a bound on the number of limit cycles of (2)? We note that the problem here studied is an extension of the Pugh’s problem to piecewise differential equations on the cylinder. Then H(n, m) denotes the maximum number of limit cycles that some piecewise differen- tial equations (2) can exhibit. The number H(n, m) is usually called the Hilbert number of piecewise differential equation (2). The Hilbert number for polynomial differential systems in the plane is the main objective of the well known 16th Hilbert problem, see [4, 5, 6]. The Hilber number also has been considered for piecesise differential linear centers in the plane, see for instance [8]. The main results of this paper are stated in the following theorems. Theorem 1. The piecewise differential equation   dx1 (  dt  a0 (t) + a1 (t)x1 , if 0 6 t 6 π, (3)  = dx2 b0 (t) + b1 (t)x2 , if π 6 t 6 2π, dt where a0 (t), a1 (t), b0 (t) and b1 (t) are 2π-periodic functions in the variable t, has at most one limit cycle on the cylinder. In particular, H(1, 1) = 1. Theorem 2. The following piecewise differential equations   dx1 (  dt  a0 (t) + a1 (t)x1 , if 0 6 t 6 π, (4)  = dx2 b0 (t) + b1 (t)x2 + b2 (t)x2 , if π 6 t 6 2π, 2 dt where ai (t) and bi (t) are 2π-periodic functions in the variable t, have at most two limit cycles on the cylinder. In particular, H(1, 2) = 2. Theorem 3. There are piecewise differential equations of the form   dx1 (  dt  a1 (t)x1 , if 0 6 t 6 π, (5)  = dx2 b2 (t)x2 + εb3 (t)x2 , if π 6 t 6 2π, 2 3 dt LIMIT CYCLES OF PIECEWISE DIFFERENTIAL EQUATIONS 3 where ai (t) and bi (t) are 2π-periodic functions in the variable t, having at least k limit cycles on the cylinder for all positive integer k. In particular, H(1, 3) = ∞. We present a corollary of Theorem 3 in the following. Corollary 4. The piecewise differential equation (2) with max{n, m} ≥ 3 has H(n, m) = ∞. 2. Proof of the main results Here we will prove the main results of the paper that are stated in Theorems 1-3 and Corollary 4. Proof of Theorem 1. Consider the piecewise differential equation (3). The solution of dx1 /dt = a0 (t) + a1 (t)x1 satisfying x1 (0) = ρ is Z t  Z s −K1 (s) K1 (t) x1 = φ1 (t, ρ) = a0 (s)e ds + ρ e , K1 (s) = a1 (w)dw, 0 0 and the solution of dx2 /dt = b0 (t) + b1 (t)x2 satisfying x2 (2π) = ρ is Z t  Z s −K2 (s) K2 (t) x2 = φ2 (t, ρ) = b0 (s)e ds + ρ e , K2 (s) = b1 (w)dw. 2π 2π Therefore we have the function Π(ρ) = φ1 (π, ρ) − φ2 (π, ρ) = m1 − m2 + (n1 − n2 )ρ, where Z π m1 =e K1 (π) a0 (s)e−K1 (s) ds, n1 = eK1 (π) , Z0 π m2 =eK2 (π) b0 (s)e−K2 (s) ds, n2 = eK2 (π) . 2π Then the initial conditions for periodic orbits of equation (3) correspond to the zeros ρ of the equation Π(ρ) = 0. This equation has 0, 1 or a continuum of solutions according to the values of m1 , n1 , m2 and n2 . We conclude that piecewise differential equations (3) have at most 1 limit cycle. It is easy to construct examples of equations (3) with 1 limit cycle. For example, the piecewise differential equation   dx1 (  dt  2 sin t + sin t x1 , if 0 6 t 6 π, (6)  = dx2 2 sin t − x2 , if π 6 t 6 2π, dt has a unique limit cycle. In this case the solution of ẋ1 = 2 sin t+sin t x1 satisfying x1 (0) = ρ is x1 = φ1 (t, ρ) = −2 + (2 + ρ)e1−cos t , and the solution of ẋ2 = 2 sin t − x2 satisfying x2 (2π) = ρ is x2 = φ2 (t, ρ) = sin t − cos t + (1 + ρ)e2π−t . Then the solution of Π(ρ) = x1 (π) − x2 (π) = 0 gives us eπ−2 + 3 e−2 − 2 ρ=− . eπ−2 − 1 4 ALI BAKHSHALIZADEH AND JAUME LLIBRE We know that the number of simple zeros of Π(ρ) is equivalent with the number of limit cycles of the piecewise differential equation (6).  Proof of Theorem 2. Consider the piecewise differential equation (4). On the half cylinder with t ∈ [0, π] we have a linear differential system and on the half cylinder with t ∈ [π, 2π] we have a Riccati differential system. Suppose that we have a periodic solution x(t) = x1q (t)|t∈[0,π] ∪ x2q (t)|t∈[π,2π] . Then doing the change of variable x2 → y where 1 y= , x2 − x2q (t) the differential equation (4) with t ∈ [π, 2π] is written as dy = −b2 (t) − (2b2 (t)x2q (t) + b1 (t))y, dt which is a linear differential equation. By computing its solution and undoing the change of variables we see that the solution of (4) with t ∈ [π, 2π] satisfying x2 (2π) = ρ is written as A(t) + B(t)ρ x2 (t) = φ2 (t, ρ) = , C(t) + D(t)ρ where A(t) = N (t) x2 q (t) x2 q (2 π) + e−M (2 π) x2 q (t) − e−M (t) x2 q (2 π) , B(t) = −N (t) x2 q (t) + e−M (t) , Z 2π −M (2 π) C(t) = N (t) x2 q (2 π) + e , D(t) = −N (t), N (t) = −b2 (s)e−M (s) ds, t Z s M (s) = −(2b2 (z)x2q (z) + b1 (z))dz. 0 The solution of (4) with t ∈ [0, π] satisfying x1 (0) = ρ is x1 (t) = φ1 (t, ρ) = H(t) + ρL(t), where Z Z t s −K1 (s) K1 (t) H(t) = a0 (s)e ds, L(t) = e , K1 (s) = a1 (w)dw. 0 0 Therefore the zeros of ρ that are obtained from the equation a + bρ Π(ρ) = φ1 (π, ρ) − φ2 (π, ρ) = h + ρ` − = 0, c + dρ correspond to the periodic orbits of the piecewise differential equation (4), where h = H(π), ` = L(π), a = A(π), b = B(π), c = C(π), d = D(π). This equation has 0, 1, 2 or a continuum of solutions according to the values of the parameters h, `, a, b, c, d. We conclude that the piecewise differential equation (4) has at most 2 limit cycles. Now we construct an example with two limit cycles for this case. Consider the following piecewise differential equation   dx1 (   cos t, if 0 6 t 6 π, (7)  dt  = dx2 1 + sin t + cos t − cos t − (1 + 2 sin t)x2 + x2 , if π 6 t 6 2π. 2 2 dt LIMIT CYCLES OF PIECEWISE DIFFERENTIAL EQUATIONS 5 The solution of equation ẋ1 = cos t with t ∈ [0, π] satisfying x1 (0) = ρ is x1 (t) = ρ + sin t. The solution of ẋ2 = 1 + sin t + cos t − cos2 t − (1 + 2 sin t)x2 + x22 , with x2 (2π) = ρ is −et sin t + (et sin t − e2 π sin t − e2 π ) ρ x2 (t) = . −et + (et − e2 π ) ρ Therefore the zeros of equation ρ e2 π x1 (π) − x2 (π) = ρ + = 0, −eπ + (eπ − e2 π ) ρ are ρ = 0, 1. It shows that the piecewise differential equation (7) has two limit cycles. In fact the limit cycles of piecewise differential equation (7) are x(t) = sin t|t∈[0,2π] and x(t) = 1 + sin t|t∈[0,2π] .  Before starting to prove Theorem 3 we give some notations that will be used in the paper, for more details, [2, 3] is referred. Definition 5. Let f0 , f1 , ..., fn−1 be real analytic functions on some open interval I of R. Then (i) (f0 , f1 , ..., fn−1 ) is called a Chebyshev system (T-system) on I if any nontrivial linear combination α0 f0 (x) + α1 f1 (x) + ... + αn−1 fn−1 (x) has at most n − 1 isolated zeros on I. (ii) (f0 , f1 , ..., fn−1 ) is called a complete Chebyshev system (CT-system) on I if (f0 , f1 , ..., fk−1 ) is a T-system for all k = 1, 2, ..., n. (iii) (f0 , f1 , ..., fn−1 ) is called an extended complete Chebyshev system (ECT-system) on I if for all k = 1, 2, ..., n, any nontrivial linear combination α0 f0 (x) + α1 f1 (x) + ... + αk−1 fk−1 (x) has at most k − 1 isolated zeros on I counting multiplicity. Definition 6. Let f0 , f1 , ..., fk−1 be real analytic functions on some open interval I of R. The Wronskian of (f0 , f1 , ..., fk−1 ) at x ∈ I is f0 f1 ... fk−1 f00 f10 ... 0 fk−1 W [f0 , f1 , ..., fk−1 ](x) = .. .. .. .. . . . . . (k−1) (k−1) (k−1) f0 f1 . . . fk−1 The following relation between an extended complete Chebyshev space and their contin- uous Wronskians is well known. Lemma 7. The set {f0 , f1 , ..., fn−1 } is an extended complete Chebyshev space on I if and only if, for each k = 1, 2, ..., n, W [f0 , f1 , ..., fk−1 ](x) 6= 0 f or all x ∈ I. The next result can be found in Theorem A in [2] that we will use this theorem to prove our results. 6 ALI BAKHSHALIZADEH AND JAUME LLIBRE Theorem 8. For any n ∈ N and any α ∈ R\Z− , the ordered set of functions (I0,α , I1,α , ..., In,α ), defined by Z b g k (t) Ik,α (y) := α dt, a (1 − yg(t)) is an ECT-system on J. When α ∈ Z− it is an ECT-system on J if and only if n 6 −α. In particular, when this set of functions is an ECT-system, any non-trivial function of the form X n (8) Φα (y) := ak Ik,α (y), k=0 with ak ∈ R, has at most n zeros in J counting multiplicities. Proof of Theorem 3. Consider the piecewise differential equation   dx1 (  dt  a1 (t)x1 , if 0 6 t 6 π, (9)  = dx2 b2 (t)x22 , if π 6 t 6 2π. dt We compute the solution of dx1 = a1 (t)x1 , dt satisfying x1 (0) = ρ and we get Z t A2 (t) x1 = φ1 (t, ρ) = ρ e , where A2 (t) = a1 (s)ds. 0 On the other hand the solution of dx2 (10) = b2 (t)x22 , dt satisfying x2 (2π) = ρ is Z t ρ x2 = φ20 (t, ρ) = , where B2 (t) = b2 (s)ds. 1 − ρB2 (t) 2π Imposing that A2 (π) = B2 (π) = 0 we obtain that the piecewise differential equation (9) has a continuum of periodic solutions in the neighborhood of ρ = 0. In order to find an appropriate equation with at least k limit cycles we perturb (9) as follows   dx1 (  dt  a1 (t)x1 , if 0 6 t 6 π, (11)  = dx2 b2 (t)x22 + ε b3 (t)x32 , if π 6 t 6 2π, dt where ε is small parameter. Let φ2 (t, ρ, ε) be the solution of dx2 (12) = b2 (t)x22 + εb3 (t)x32 , dt with initial condition φ2 (2π, ρ, ε) = ρ. Then the solution (12) can be expressed for small ε > 0 in the form φ2 (t, ρ, ε) = φ20 (t, ρ) + εψ(t, ρ) + O(ε2 ), where ψ(t, ρ) = ∂φ2 (t, ρ, ε)/∂ε|ε=0 . For simplicity we write φ2 (t, ρ, ε) = φ2 = φ20 + εψ + O(ε2 ), b2 = b2 (t), and b3 = b3 (t). LIMIT CYCLES OF PIECEWISE DIFFERENTIAL EQUATIONS 7 Then we have ∂  2 3 φ20 + εψ + O(ε2 ) =b2 φ20 + εψ + O(ε2 ) + εb3 φ20 + εψ + O(ε2 ) ∂t  =b2 φ220 + 2 εφ20 ψ + O(ε2 ) + εb3 (φ320 + O(ε)) =b2 φ220 + ε(2b2 φ20 ψ + b3 φ320 ) + O(ε2 ). Therefore ψ 0 = 2b2 φ20 ψ + b3 φ320 , where ψ 0 = ∂ψ(t, ρ)/∂t. Using that φ20 is a solution of (10), this differential equation is written as ψ 0 = 2φ020 ψ/φ20 +b3 φ320 , or equivalent (ψ/φ220 )0 = b3 φ20 . Solving this differential equation we have Z t Z t 2 2 ρb3 (s) ψ(t, ρ) = φ20 (t, ρ) b3 (s)φ20 (s, p)ds = φ20 (t, ρ) ds. 2π 2π 1 − ρB2 (s) Recall that the solution starting at ρ is a limit cycle of the perturbed differential equation if it is an isolated zero of function D(ρ, ε) := φ1 (π, ρ) − φ2 (π, ρ, ε). This equation is Z π 3 b3 (t) D(ρ, ε) := ερ dt + O(ε2 ) = 0. 2π 1 − ρB 2 (t) The function Z π b3 (t) M (ρ) := dt, 2π 1 − ρB2 (t) is known as a Melnikov function associated to the problem. By the Implicit Function Theo- rem applied to D(ρ, ε)/ε it follows that the simple zeros of M (ρ) lead to simple zeros of the function D(ρ, ε). More specifically if ρ = ρ̄ satisfies M (ρ̄) = 0, M 0 (ρ̄) 6= 0 then there exists a differentiable function g such that g(0) = ρ̄ and for ε small enough D(g(ε), ε) ≡ 0. In other words what we have seen is that each of the simple non-zero roots of M (ρ) gives a limit cycle of the Abel differential equation (11). Therefore we reduced the proof of the theorem to find functions b2 and b3 such that the corresponding function M (ρ) has at least k simple zeros. For any k ∈ N we take b2 (t) = cos t and b3 (t) = P (sin t), where P is a polynomial of degree k in the variable sin t. Since Z π P (sin t) M (ρ) := dt, 2π 1 − ρ sin t can be written into the form (8), with Z π sinj t Ij,1 (ρ) = dt, for j = 0, 1, ..., k. 2π 1 − ρ sin t Then from Theorem 8 the maximum number of zeros of M (ρ) is k, and k can be reached taking conveniently the coefficients of the polynomial P (sin t).  Proof of Corollary 4. Similar to the proof of Theorem 3 we consider the piecewise differential equation   dx1 (  dt  a1 (t)x1 , if 0 6 t 6 π, (13)  = dx2 b2 (t)x22 , if π 6 t 6 2π. dt The solution of dx1 = a1 (t)x1 , dt 8 ALI BAKHSHALIZADEH AND JAUME LLIBRE satisfying x1 (0) = ρ is Z t A2 (t) x1 = φ10 (t, ρ) = ρ e , where A2 (t) = a1 (s)ds, 0 and the solution of dx2 (14) = b2 (t)x22 , dt satisfying x2 (2π) = ρ is Z t ρ x2 = φ20 (t, ρ) = , where B2 (t) = b2 (s)ds. 1 − ρB2 (t) 2π Imposing that A2 (π) = B2 (π) = 0 we obtain that the piecewise differential equation (13) has a continuum of periodic solutions in the neighborhood of ρ = 0. Now we perturb (13) as follows (15)   dx1 (  dt  a1 (t)x1 + ε2 (a0 (t) + a2 (t)x21 + ... + an (t)xn1 ) , if 0 6 t 6 π,  = dx2 b2 (t)x2 + ε b3 (t)x2 + ε (b0 (t) + b4 (t)x2 + ... + bm (t)x2 ) , if π 6 t 6 2π, 2 3 2 4 m dt where ε is small parameter and max{n, m} ≥ 3. The solution of the differential equations dx1  = a1 (t)x1 + ε2 a0 (t) + a2 (t)x21 + ... + an (t)xn1 , dt and dx2  = b2 (t)x22 + ε b3 (t)x32 + ε2 b0 (t) + b4 (t)x42 + ... + bm (t)xm 2 , dt can be expressed for small ε > 0 in the form φ1 (t, ρ, ε) = φ10 (t, ρ) + εψ1 (t, ρ) + O(ε2 ), and φ2 (t, ρ, ε) = φ20 (t, ρ) + εψ2 (t, ρ) + O(ε2 ), respectively, where ψi (t, ρ) = ∂φi (t, ρ, ε)/∂ε|ε=0 for i = 1, 2. Then we have Z t Z t 2 2 ρb3 (s) ψ1 (t, ρ) = 0, ψ2 (t, ρ) = φ20 (t, ρ) b3 (s)φ20 (s, p)ds = φ20 (t, ρ) ds. 2π 2π 1 − ρB2 (s) Therefore the simple zeros of the function Z π 3 b3 (t) D(ρ, ε) := φ1 (π, ρ, ε) − φ2 (π, ρ, ε) = ερ dt + O(ε2 ) = ερ3 M (ρ) + O(ε2 ) = 0, 2π 1 − ρB2 (t) correspond to limit cycles of piecewise differential equation (15). Now Taking again b2 (t) = cos t and b3 (t) = P (sin t), where P is a polynomial of degree k in the variable sin t, we can apply Theorem 8 to function M (ρ) and obtain that the maximum number of simple zeros of the function M (ρ) is k. This completes the proof of Corollary 4.  LIMIT CYCLES OF PIECEWISE DIFFERENTIAL EQUATIONS 9 Acknowledgments The first author is partially supported by Iranian Ministry of Science, Research and Tech- nology and Iran’s National Elites Foundation. The second author is partially supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. References [1] A. Gasull, De les equacions diferencials d’Abel al problema XVI de Hilbert, Bull. Soc. Catalana de Math., 28 (2013), 123–146. [2] A. Gasull, C. Li, and J. Torregrosa, A new Chebyshev family with applications to Abel equations, J. Diff. Equ., 252 (2012), 1635–1641. [3] M. Grau, F. Manosas and J. Villadelprat, A Chebyshev criterion for Abelian integrals, Trans. Amer. Math. Soc., 363 (2010), 10–129 . [4] D. 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Soc., 2 (1975), 1–10. 1 Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, Iran, 84156-83111. Email address: a.bakhsh@math.iut.ac.ir 2 Departament de Matematiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat