This is a preprint of: “A survey on algebraic and explicit non-algebraic limit cycles in planar differential systems”, Jaume Llibre, Xiang Zhang, Exposition. Math., vol. 39, 48–61, 2021. DOI: [10.1016/j.exmath.2020.03.001] A SURVEY ON ALGEBRAIC AND EXPLICIT NON-ALGEBRAIC LIMIT CYCLES IN PLANAR DIFFERENTIAL SYSTEMS JAUME LLIBRE1 AND XIANG ZHANG2 Abstract. In the qualitative theory of differential equations in the plane one of the most difficult objects to study is the exis- tence of limit cycles. There are many papers dedicated to this subject. Here we will present a survey mainly dedicated to the algebraic and explicit non-algebraic limit cycles of the polynomial differential systems in R2 and of the discontinuous piecewise dif- ferential systems in R2 formed by two linear differential systems separated by a straight line. For this class of discontinuous piece- wise differential systems the study of their algebraic and explicit non-algebraic limit cycles just is starting. Here we provide the first explicit non-algebraic limit cycle for the discontinuous piece- wise linear differential systems. Additionally we recall seven open questions related with these types of limit cycles. 1. Introduction We start by recalling the definition of the two classes of differential systems whose algebraic and explicit non-algebraic limit cycles we will study. Let P (x, y) and Q(x, y) be real polynomials in the variables x and y. Then the differential system ẋ = P (x, y), (1) ẏ = Q(x, y), where as usual the dot denotes the derivative with respect to the inde- pendent variable t, is a polynomial differential system. The maximum of the degrees of the polynomials P and Q is the degree of the polyno- mial differential system (1). 2010 Mathematics Subject Classification. 34C29, 34C25. Key words and phrases. Algebraic limit cycle, non-algebraic limit cycle, explicit limit cycle, polynomial differential system, piecewise linear differential system. 1 2 JAUME LLIBRE AND XIANG ZHANG Here we consider the discontinuous piecewise differential systems (2) X ± : (ẋ, ẏ) = (f ± (x, y), g ± (x, y)), defined in the half-planes Σ± = {(x, y) ∈ R2 : ±x > 0}. On the straight line Σ = {x = 0} the differential system is bivaluated. The straight line Σ is called the straight line of discontinuity when the two vector fields X ± do not coincide on it. We use the Filippov conventions for defining the discontinuous piecewise differential system on Σ, see [33]. If f + (0, y)f − (0, y) > 0 at the point (0, y) ∈ Σ we say that (0, y) is a crossing point. If a periodic orbit of a discontinuous piecewise differential system (2) has exactly two crossing points we say that it is a crossing periodic orbit. A limit cycle (respectively crossing limit cycle) of system (1) (respec- tively (2)) is an isolated periodic orbit in the set of all periodic orbits (respectively crossing periodic orbits) of system (1) (respectively (2)). Here if h(x, y) is a real polynomial irreducible in the ring R[x, y] of all real polynomials in the variables x and y, the zero set {h(x, y) = 0} is an algebraic curve. An algebraic limit cycle is a limit cycle contained in an algebraic curve of the plane, otherwise such a limit cycle is called non–algebraic. The degree of an algebraic limit cycle is the degree of the irreducible polynomial which defines the algebraic curve contain- ing the limit cycle. It is well known that the orbits of a polynomial differential system (1) are contained in analytic curves, which usually are not algebraic curves. In general it is a difficult problem to distinguish if a limit cycle is algebraic or not. The proof that the famous limit cycle exhibited in the van der Pol equation in 1926 (see [80]) was not algebraic arrived in 1995, see Odani [76]. The differential equation of van der Pol can be written as a polynomial differential system (1) of degree 3, but we do not know explicitly its limit cycle. More precisely, we do not know the explicit expression of the analytic curve which contains the non- algebraic limit cycle of van der Pol equation. We remark that all the algebraic limit cycles are explicit because we only know them when we provide the algebraic curve containing the limit cycle. A crossing limit cycle of the piecewise differential system (2) is al- gebraic if all its points, with the exception of the ones which are in Σ, are contained in algebraic curves of the half-planes Σ± . The degree (n− , n+ ) of an algebraic crossing limit cycle is formed by the degree of the irreducible polynomials defining the algebraic curves which con- tain the crossing limit cycle, thus n− (respectively n+ ) is the degree of 3 the irreducible polynomial defining the algebraic curve in Σ− (respec- tively Σ+ ) which contains the piece of the crossing limit cycle in Σ− (respectively Σ+ ). 2. Algebraic limit cycles of polynomial differential systems 2.1. Some general results on algebraic limit cycles. A nice result on algebraic limit cycles is the following one. Theorem 1 (Bautin–Christopher–Dolov–Kuzmin Theorem). Assume that f (x, y) = 0 is a non-singular algebraic curve of degree m, and cx + dy + e = 0 a straight line which does not intersect any bounded component of the algebraic curve f (x, y) = 0. Let a and b real numbers such that ac + bd 6= 0. Consider the polynomial differential system (3) ẋ = af − (cx + dy + e)fy , ẏ = bf + (cx + dy + e)fx , of degree m. Then every bounded component of the algebraic curve f (x, y) = 0 is a hyperbolic algebraic limit cycle of the polynomial dif- ferential system (3). Furthermore the differential system (3) has no other limit cycles. Bautin in [5] proved a result similar to Theorem 1. But Bautin’s paper contains a mistake which was solved in [26], and this result was generalized in [27]. The present statement of Theorem 1 is due to Christopher, see [24] where also there is a proof of it. Let f = f (x, y) be a real polynomial in the variables x and y. The algebraic curve f (x, y) = 0 of R2 is an invariant algebraic curve of the polynomial differential system (1) if for some polynomial K ∈ R[x, y] we have ∂f ∂f (4) P +Q = Kf. ∂x ∂y From (4) we note that the gradient (∂f /∂x, ∂f /∂y) of the curve on the points of the algebraic curve f (x, y) = 0 is orthogonal to the vector field (P, Q), and consequently the vector field (P, Q) is tangent to the curve f (x, y) = 0 in all its points. In other words the curve f (x, y) = 0 is formed by orbits of the vector field (P, Q). This explains the name of invariant algebraic curve given to the algebraic curve f (x, y) = 0, because the curve f (x, y) = 0 is invariant under the flow of the vector field (P, Q). 4 JAUME LLIBRE AND XIANG ZHANG Consider a polynomial differential system having a unique irreducible invariant algebraic curve, then the following result provides the maxi- mum number of algebraic limit cycles that such differential system can have in function of the degree of that algebraic curve, for a proof see for instance [55]. Theorem 2. Suppose that the algebraic curve f (x, y) = 0 of degree m is the unique irreducible invariant algebraic curve of a polynomial vector field (P, Q). Then the vector field (P, Q) can have at most [(m−1)(m− 2)/2] + 1 algebraic limit cycles. Moreover choosing that f (x, y) = 0 has the maximal number of ovals for the irreducible algebraic curves of degree m, there exist vector fields X of degree m having exactly [(m − 1)(m − 2)/2] + 1 algebraic limit cycles. 2.2. Configurations of limit cycles via algebraic limit cycles. Hilbert in 1900 and in the second part of his 16th problem (see [43]) proposed to provide a uniform upper bound for the maximum number of limit cycles of all polynomial differential systems of a given degree, additionally he also proposed to study the possible configurations or distributions of limit cycles in the plane for all polynomial differential systems. This last question has been solved in [64] using algebraic limit cycles. A finite set C = {C1 , . . . , Cn } of disjoint simple closed curves in the plane satisfying Ci ∩ Cj = ∅ for all i 6= j is called a configuration of limit cycles. Let C = {C1 , . . . , Cn } and C 0 = {C10 , . . . , Cm 0 } be two configurations of limit cycles we say that they are (topologically) equivalent if there 0 is a homeomorphism h : R2 → R2 such that h (∪ni=1 Ci ) = (∪m i=1 Ci ). Of course, in order that two equivalent configurations of limit cycles C and C 0 be equivalent we need that n = m. The configuration of limit cycles C is realized by a polynomial vector field (P, Q) if the set of all limit cycles of (P, Q) is equivalent to C. In 2004 and in [64] it was proved the following result. Theorem 3. Let C = {C1 , . . . , Cn } be an arbitrary configuration of limit cycles. Then the configuration C is realizable with algebraic limit cycles by a polynomial vector field. Inspired in the proof of Theorem 3 it is possible to provide an alter- native proof using the Bautin–Christopher–Dolov–Kuzmin Theorem. 5 In 2013 Coll, Dumortier and Prohens in [25] proved that any config- uration of limit cycles can be realized for a convenient Liénard polyno- mial differential system. 2.3. The 16th Hilbert problem restricted to algebraic limit cycles. The original 16th Hilbert problem states: give an upper bound of the maximum number of limit cycles that the polynomial differential systems of degree n can exhibit in function of n, see more details on this famous problem in [43, 47, 50]. In this subsection we consider this problem restricted to the algebraic limit cycles. If a set of irreducible algebraic curves, fj = fj (x, y) = 0 for j = 1, . . . , k, satisfies the following five conditions it is called generic. (i) The curve fj = 0 is a non-singular algebraic curve, i.e. there are no points (x,y) satifying simultaneously fj = 0, ∂fj /∂x = 0 and ∂fj /∂y = 0. (ii) The homogeneous terms of higher order of fj have no repeated factors. (iii) If two distinc curves fi = 0 and fj = 0 intersect in a point of the affine plane, they intersect transversally at that point. (iv) For every point of the affine plane at most pass two curves fj = 0. (v) The highest order homogeneous terms of two distinct polyno- mials fi and fj do not have common factors. In the paper [62] it was proved the following result. Theorem 4. Assume that the set of all irreducible invariant algebraic curves of a polynomial differential system of degree n ≥ 2 is generic. Then the maximum number of algebraic limit cycles of this system is at most 1 + (n − 1)(n − 2)/2 if n is even, and (n − 1)(n − 2)/2 if n is odd. And there are polynomial differential systems of degree n reaching these upper bounds. Related with the results of Theorem 4 are the results of the paper [63]. In the papers [52, 55] appear the following open problem very related with Theorem 4. Open problem 1. Every quadratic polynomial differential system has at most one algebraic limit cycle. 6 JAUME LLIBRE AND XIANG ZHANG Partial results in the sense that the open problem 1 will have a positive answer are given in the papers [70, 71, 72]. From Theorem 4 it follows that if all the irreducible invariant alge- braic curves of a cubic polynomial differential system are generic, then the system has at most 1 algebraic limit cycle. But there are cubic polynomial differential systems in [62] with two algebraic limit cycles, of course the invariant algebraic curves of those cubic polynomial differ- ential systems are not generic. Thus the cubic polynomial differential system ẋ = 2y(10 + xy), ẏ = 20x + y − 20x3 − 2x2 y + 4y 3 , has two algebraic limit cycles contained in the invariant algebraic curve 2x4 − 4x2 + 4y 2 + 1 = 0, see Proposition 19 of [52]. Another cubic polynomial differential system having the two limit cycles x2 + y 2 = r2 and (x − a)2 + y 2 = r2 is ẋ = y(a2 − r2 − 3ax + 3x2 − ay + 2xy + y 2 ), ẏ = −a2 x + 3ax2 − 2x3 − r2 y + axy − x2 y + y 3 , when r < a/2. These cubic polynomial differential systems exhibiting two algebraic limit cycles motivated that in the paper [55] appeared the next open problem, which also can be found in [52]. Open problem 2. Every cubic polynomial differential system has at most two algebraic limit cycles. Clearly the open questions 1 and 2 hold for the quadratic and cubic polynomial differential systems if all their invariant algebraic curves are generic. But they remain open for those polynomial differential systems having non-generic invariant algebraic curves. These two previous open problems where extended in [62] to the following most general open problem. Open problem 3. The maximum number of algebraic limit cycles that a polynomial differential system of degree n can have is 1 + (n − 1)(n − 2)/2. The open problem 3 was proved for planar polynomial differential systems having only nodal invariant algebraic curves by Zhang in [86] On the other hand the 16th Hilbert problem restricted to algebraic limit cycles formed by circles has been solved in [61]. 7 2.4. Algebraic limit cycles of quadratic polynomial differen- tial systems. The first algebraic limit cycle found in the quadratic polynomial differential systems is due to Qin [82, 83] and to Liu [53], they in 1957 and 1958 respectively proved that such systems can have algebraic limit cycles of degree 2, and that if a quadratic polynomial differential system has an algebraic limit cycle then it is the unique limit cycle of the system. Later on it was proved that the quadratic polynomial differential systems cannot have algebraic limit cycles of degree 3 by Evdokimenco in [30, 31, 32], more recently this result has been proved in a shorter and clear way in [21, 54], and also in the projective quadratic polynomial differential systems by Zhang [87]. The first family of algebraic limit cycles of degree 4 in the quadratic polynomial differential systems was found in 1966 by Yablonskii [81]. In 1973 Filiptsov [34] found a second family of algebraic limit cycles of degree 4. In [21] two new families of algebraic limit cycles of degree 4 has been found, and the authors of [23] shown that the quadratic polynomial differential systems has no more families of algebraic limit cycles of degree 4. Moreover in [20] was proved that when a quadratic polynomial differential system has an algebraic limit cycle of degree 4 this is the unique limit cycle of the system. See [65, 66] for some other results on the algebraic limit cycles of the quadratic polynomial differential systems. In [23] using birational transformation of the plane some families of algebraic limit cycles of degree 4 of the quadratic polynomial differen- tial systems were transformed in new families of algebraic limit cycles of degrees 5 and 6 also for quadratic polynomial differential systems. These results were improved in [1] where a new family of algebraic limit cycles of degrees 5 was found for the quadratic polynomial differential systems. Open problem 4. Provide the maximum degree that can reach the algebraic limit cycles of the quadratic polynomial differential systems. Of course until now we know that the quadratic polynomial differen- tial systems have algebraic limit cycles of degree 6, but it is unknown if these systems can have algebraic limit cycles of degree higher than 6. 8 JAUME LLIBRE AND XIANG ZHANG 3. Algebraic limit cycles of discontinuous piecewise linear differential systems This is a very new area of research. At this moment we only know the paper of Buzzi, Gasull and Torregrosa [17] dedicated to this problem. In what follows we summarize the main results of [17]. For the family of piecewise linear differential systems (2) being f ± (x, y) and g ± (x, y) polynomials of degree 1 defined in Σ± the following state- ments hold. (i) If a piecewise linear differential system (2) has an algebraic crossing limit cycle, then every linear differential system either has no equilibrium points, or has a unique equilibrium point of type: center, saddle, or node with distinct eigenvalues. More- over, if the equilibrim point is a saddle or a node with distinct eigenvalues the quotient of both eigenvalues is a rational num- ber. (ii) If a piecewise linear differential system (2) has an algebraic crossing limit cycle, all the other limit cycles of the system are also algebraic, and all of them are nested. If Γ1 and Γ2 are two limit cycles, they are nested if either Γ1 is contained in the region limited by Γ2 , or vice versa. In a similar way any number of limit cycles are said to be nested if every two of them are nested. (iii) Does not exist an algebraic crossing limit cycles Γ of a piecewise linear differential system (2) such that the algebraic curve in Σ− and the algebraic curve in Σ+ containing Γ are defined by the same polynomial. (iv) There are piecewise linear differential system (2) having exactly two algebraic crossing limit cycles. (v) There are piecewise linear differential system (2) with exactly one semistable algebraic crossing limit cycle. (vi) For every pair of positive integers (m, n) with m ≥ 2 and n ≥ 2 there exists a piecewise linear differential system (2) with an algebraic crossing limit cycle of degree (m, n). In [17] the authors stated the following open problem. Open problem 5. Can the piecewise linear differential system (2) exhibit more than two algebraic crossing limit cycles? 9 4. Explicit non-algebraic limit cycles of polynomial differential systems Recently, since 2006 up to now, many articles have been showing explicit non-algebraic limit cycles in polynomial differential systems, i.e. in those articles the authors provided the explicit expression of the analytic curve containing the limit cycle. In this direction Gasull, Giacomini and Torregrosa [39] provided as far as we know the first explicit non-algebraic limit cycle for a polynomial differential system of degree 5. Clearly, if we multiply the right hand part of that quintic polynomial differential system by the expression (ax + by + c)n being n any given positive integer, and we choose the straight line ax+by+c = 0 in such a way that it does not intersect the explicit non-algebraic limit cycle of the differential system, we obtain a new polynomial differential system of degree 5 + n exhibiting an explicit non-algebraic limit cycle for all integer n ≥ 1. One year after the article of Gasull, Giacomini and Torregrosa and inspired in this article appeared the paper of Al-Dosary [2], where the author exhibits another explicit non-algebraic limit cycle for a distinct polynomial differential system of degree 5. In 2006 Giné and Grau [41] shown the simultaneous existence of two limit cycles one algebraic and an explicit non-algebraic in a polynomial differential system of degree 9. Note that the paper [39] is quoted in [41]. The first paper providing an explicit non-algebraic limit cycle for polynomial differential systems of degree less than 5 was given by Ben- terki and Llibre [11] in 2012 for a polynomial differential system of degree 3, and of course it can be extended to any degree larger than 3. Later on many other papers have been published providing explicit non-algebraic limit cycles for several polynomial differential systems, but in all these papers the explicit non-algebraic limit cycles are for polynomial differential systems of degree larger than or equal to 3, see [6, 7, 8, 9, 10, 11, 12, 13, 15, 16], the authors of these papers are Aziza, Bendjeddou, Benterki, Benyoucef, Berbache, Boukoucha, Boukoucha, Cheurfa, Grazem and Salhi. Of course the polynomial differential systems of degree 1, i.e. the linear differential systems, cannot have limit cycles, because when they have a periodic orbit this forms part of a continuum of periodic orbits surrounding a center. In 2012 and in the paper [11] it was stated the following open problem, which remains open until now. 10 JAUME LLIBRE AND XIANG ZHANG Open problem 6. Provide an explicit non–algebraic limit cycle for a polynomial differential system of degree 2. We want to mention the paper of Garcı́a [38] which provides a method for proving the non-algebraicity of limit cycles for particu- lar polynomial systems under the generic assumption that the line at infinity is invariant. 5. Explicit non-algebraic limit cycles for the discontinuous piecewise linear differential systems A discontinuous piecewise linear differential system with two pieces separated by a straight line in the plane R2 after a linear change of variables can be written into the form    −    −  ẋ a11 a− 12 x b1 = + in x < 0; ẏ a− 21 a − 22 y b− 2 and    +    +  ẋ a11 a+12 x b1 = + in x > 0. ẏ a+ 21 a + 22 y b+ 2 That is, without loss of generality we can assume that the discontinuity straight line is x = 0. In 1930’s Andronov, Vitt and Khaikin studied these discontinuous piecewise linear differential systems in their seminal book [3], where these differential systems appear in a natural way when they studied electrical, mechanical or control problems. The work of Levinson [48] is another important milestone in the history of the piecewise differential systems because it allowed to understand the complicated dynamics of the forced van der Pol equation, and this work inspired Smale [79] to find the horseshoe dynamics. Nowadays the piecewise linear systems continue receiving the attention of many researchers, see the books of di Bernardo [14] and Simpson [78], the survey of Makarenkov and Lamb [75], and the hundreds of references quoted in these last three works. During these last twenty years many authors have been studied the limit cycles of the discontinuous piecewise linear differential systems (5), see for instance the papers [4, 18, 22, 19, 29, 35, 36, 37, 40, 42, 44, 45, 46, 51, 56, 57, 58, 60, 67, 68, 69, 74, 77, 84, 85]. The authors of these articles are Artés, Braga, Buzzi, Castillo, Ting Chen, Xiaoyan Chen, Euzébio, Freire, Giannakopoulos, Gouveia, Han, Chuangxia Huang, Lihong Huang, Wentao Huang, Liping Li, Llibre, Medrado, Mello, No- vaes, Ordóñez, Pessoa, Pliete, Ponce, Rodrigo, Shui, Teixeira, Torre- grosa, Torres, Verduzco, Wang, Jiazhong Yang, Xiao-Song Yang, Yu, 11 Cheng Zhang, Weinian Zhang and Xiang Zhang. In all these papers at most 3 crossing limit cycles were found for the discontinuous piecewise linear differential systems (5), and from 2012 it remains the following open problem. Open problem 7. Is 3 the maximum number of crossing limit cycles that a discontinuous piecewise linear differential systems with a straight line of separation can have? In all these articles dedicated to study the crossing limit cycles of the discontinuous piecewise linear differential systems, do not appear explicit non-algebraic limit cycles, they proved their existence using different methods as the Poincaré map, the Melnikov function, the averaging theory, the Newton-Kantorovich Theorem, the first integrals, ... In what follows we provide a new result showing an explicit non- algebraic limit cycle for the discontinuous piecewise linear differential system (2) defined by f − (x, y) = αx + y, g − (x, y) = −x + αy, (5) f + (x, y) = −αx + y − α, g + (x, y) = −x − αy − 1, with α 6= 0. It is easy to check that differential system (5) has the explicit non- algebraic crossing periodic orbit given by the analytic curves (x− (t), y − (t)) in the half-plane x < 0 for t ∈ [0, π], and by (x+ (t), y + (t)) in the half- plane x > 0 for t ∈ [π, 3π/2] where πα x− (t) = −eαt− 2 sin t, πα y − (t) = −eαt− 2 cos t,  (6) πα x+ (t) = e−αt cos t − e− 2 sin t − 1, πα y + (t) = e−αt −e− 2 cos t − sin t . This non-algebraic crossing periodic orbit is drawn Figure 1 and has period 3π/2. We shall provide the expression of the analytic curve containing the non-algebraic crossing periodic orbit parametrized by the time t, but using the first integrals of both linear differential sys- tems and knowing that the non-algebraic crossing periodic orbit passes through the point (0, −e−πα/2 ) when t = 0 and through the point (0, eαπ/2 ) when t = π. Thus these expressions are p y πα √ x2 + y 2 eα arctan( x ) = e 2 e−πα , 12 JAUME LLIBRE AND XIANG ZHANG in x < 0, and   − πα 2 x+αy+1 4α arctan( α+αx−y 4α arctan 1−e − παα e ) e α+e 2 = , ((x + 1)2 + y 2 )2 (e−πα + 1)2 in x > 0. Figure 1. The non-algebraic limit cycle (6) of the piece- wise linear differential system (5) for α = −1. Now we shall prove that this non-algebraic crossing periodic orbit is a limit cycle. For a smooth differential system (1) in the plane a periodic solution (x(t), y(t)) of period T such that Z T  ∂P ∂Q (7) + (x(t), y(t))dt 6= 0, 0 ∂x ∂y is a limit cycle, usually called a hyperbolic limit cycle. This is a well known result, for a proof see for instance Theorem 1.23 of [28]. The formula (7) can be extended to the discontinuous piecewise linear dif- ferential systems here considered. Then for the discontinuous piecewise linear differential system (2)-(5) we have that Z π Z 3π/2 2αdt + (−2α)dt = απ 6= 0, 0 π hence the non-algebraic crossing periodic orbit (6) is a limit cycle. 13 Acknowledgements The first 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 Univer- sitaris i de Recerca grant 2017SGR1617, and the H2020 European Re- search Council grant MSCA-RISE-2017-777911. The second author is partially supported by National Natural Science Foundation of China grants 11671254 and 11871334. References [1] M. Alberich-Carramiñana, A. Ferragut and J. Llibre, Quadratic pla- nar differential systems with algebraic limit cycles via quadratic plane Cremona maps, preprint, 2019. [2] K.I.T. Al-Dosary, Non–algebraic limit cycles for parametrized planar poly- nomial systems, Int. J. of Math. 18 (2007), 179–189. [3] A. Andronov, A. Vitt and S. Khaikin, Theory of Oscillations, Pergamon Press, Oxford, 1966 (Russian edition ≈ 1930). [4] J.C. Artés, J. Llibre, J.C. Medrado and M.A. Teixeira, Piecewise linear differential systems with two real saddles, Math. Comput. Simul. 95 (2013), 13–22. [5] N.N. Bautin, Estimation of the number of algebraic limit cycles of the system x0 = P (x, y), y 0 = Q(x, y) with algebraic right-hand sides, (Russian) Differ- entsial’nye Uravneniya 16 (1980), no. 2, 362–383. [6] A. Bendjeddou, B. Aziza and R. Cheurfa, Exact algebraic and non- algebraic limit cycles for a class of integrable quintic and planar polynomial differential systems, An. ?tiin?. Univ. Al. I. Cuza Ia?i. Mat. (N.S.) 64 (2018), no. 2, 253–259. [7] A. Bendjeddou, R. Benterki and T. Salhi, Explicit non-algebraic limit cycle for polynomial systems of degree seven, Appl. Math. Sci. (Ruse) 3 (2009), no. 13–16, 613–622. [8] A. Bendjeddou and A. Berbache, A class of differential systems of odd degree with explicit non algebraic limit cycle. Int. Electron. J. Pure Appl. Math. 9 (2015), no. 4, 243–254. [9] A. Bendjeddou and R. Boukoucha, Explicit limit cycles of a cubic poly- nomial differential systems, Stud. Univ. Babe?-Bolyai Math. 61 (2016), no. 1, 77–85. [10] A. Bendjeddou and M. Grazem, A class of quintic Kolmogorov systems with explicit non-algebraic limit cycle, Zh. Sib. Fed. Univ. Mat. Fiz.12 (2019), no. 3, 285–297. [11] R. Benterki and J. Llibre, Polynomial differential systems with explicit non-algebraic limit cycles, Electron. J. Differential Equations 2012, No. 78, 6 pp. [12] S. Benyoucef and A. Bendjeddou, Kolmogorov system with explicit hy- perbolic limit cycle, Zh. Sib. Fed. Univ. Mat. Fiz. 10 (2017), no. 2, 216–222. 14 JAUME LLIBRE AND XIANG ZHANG [13] A. Bendjeddou and A.A. Berbache, A class of differential systems of odd degree with explicit non algebraic limit cycle, Int. Electron. J. Pure Appl. Math. 9 (2015), no. 4, 243–254. [14] M. di Bernardo, C.J. Budd, A.R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems: Theory and Applications, Appl. Math. Sci., vol. 163, Springer-Verlag, London, 2008. [15] R. Boukoucha, Explicit expression for hyperbolic limit cycles of a class of polynomial differential system, Tr. Inst. Mat. Mekh. 23 (2017), no. 3, 300–307. [16] R. Boukoucha, Explicit limit cycles of a family of polynomial differential systems, Electron. J. Differential Equations 2017, No. 217, 7 pp. [17] C.A. Buzzi, A. Gasull and J. Torregrosa, Algebraic limit cycles in piece- wise linear differential systems, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 28(3) (2018), 1850039 (14 pages). [18] C. Buzzi, C. Pessoa and J. Torregrosa, Piecewise linear perturbations of a linear center, Discrete Contin. Dyn. Syst. 33 (2013), 3915–3936. [19] J. Castillo, J. Llibre and F. Verduzco, The pseudo–Hopf bifurcation for planar discontinuous piecewise linear differential systems, Nonlinear Dynam. 90 (2017), 1829–1840. [20] J. Chavarriga, H. Giacomini and J. Llibre, Uniqueness of algebraic limit cycles for quadratic systems, J. Math. Anal. and Appl. 261 (2001), 85–99. [21] J. Chavarriga, J. Llibre and J. Sorolla, Algebraic limit cycles of degree 4 for quadratic systems, J. Differential Equations 200 (2004), 206–244. [22] T. Chen, L. Huang, P. Yu and W. Huang, Bifurcation of limit cycles at infinity in piecewise polynomial systems, Nonlinear Anal. Real World Appl. 41 (2018), 82–106. [23] C. Christopher, J. Llibre and G. Świrszcs, Invariant algebraic curves of large degree for quadratic system, J. Math. Anal. Appl. 303 (2005), 450–461. [24] C. Christopher, Polynomial vector fields with prescribed algebraic limit cy- cles, Geometria Dedicata 88 (2001), 255–258. [25] B. Coll, F. Dumortier and R. Prohens, Configurations of limit cycles in Liénard equations, J. Differential Equations 255 (2013), no. 11, 4169–4184. [26] M.V. Dolov and R.V. Kuzmin, On limit cycles of a class of systems, Dif- ferential Equations 29(9) (1993), 1282–1285. [27] M.V. Dolov and R.V. Kuzmin, Limit cycles of systems with a given par- ticular integral, Differential Equations 30(7) (1994), 1044–1050. [28] F. Dumortier, J. Llibre and J.C. Artés, Qualitative Theory of Planar Differential Systems, Springer Verlag, New York, 2006. [29] R.D. Euzébio and J. Llibre, On the number of limit cycles in discontinuous piecewise linear differential systems with two pieces separated by a straight line, J. Math. Anal. Appl. 424 (2015), 475–486. [30] R.M Evdokimenco, Construction of algebraic paths and the qualitative in- vestigation in the large of the properties of integral curves of a system of dif- ferential equations, Differential Equations 6 (1970), 1349–1358. [31] R.M Evdokimenco, Behavior of integral curves of a dynamic system, Differ- ential Equations 9 (1974), 1095–1103. [32] R.M Evdokimenco, Investigation in the large of a dynamic system with a given integral curve, Differential Equations 15 (1979), 215–221. 15 [33] A.F. Filippov, Differential equations with discontinuous right–hand sides, translated from the Russian. Mathematics and its applications (Soviet Series), vol. 18, Kluwer Academic Publishers Group, Dordrecht, 1988. [34] V.F. Filiptsov, Algebraic limit cycles, Differential Equations 9 (1973), 983– 986. [35] E. Freire, E. Ponce, F. Rodrigo and F. Torres, Bifurcation sets of continuous piecewise linear systems with two zones, Int. J. Bifurcation and Chaos 8 (1998), 2073–2097. [36] E. Freire, E. Ponce and F. Torres, Canonical discontinuous planar piece- wise linear systems, SIAM J. Applied Dynamical Systems 11 (2012), 181–211. [37] E. Freire, E. Ponce and F. Torres, A general mechanism to generate three limit cycles in planar Filippov systems with two zones, Nonlinear Dy- namics 78 (2014), 251–263. [38] I.A. Garcı́a, Transcendental limit cycles via the structure of arbitrary degree invariant algebraic curves of polynomial planar vector fields, Rocky Mountain J. Math.35 (2005), no. 2, 501–515. [39] A. Gasull, H. Giacomini and J. Torregrosa, Explicit non-algebraic limit cycles for polynomial systems, J. Comput. Appl. Math. 200 (2007), 448–457. [40] F. Giannakopoulos and K. Pliete, Planar systems of piecewise linear differential equations with a line of discontinuity, Nonlinearity 14 (2001), 1611– 1632. [41] J. Giné and M. Grau, Coexistence of algebraic and non–algebraic limit cy- cles, explicitly given, using Riccati equations, Nonlinearity 19 (2006), 1939– 1950. [42] M. Han and W. Zhang, On Hopf bifurcation in non–smooth planar systems, J. of Differential Equations 248 (2010), 2399–2416. [43] D. Hilbert, Mathematische Problem (lecture), Second Internat. Congress Math. Paris, 1900, Nachr. Ges. Wiss. Göttingen Math.–Phys. Kl. 1900, pp 253–297. [44] S.M. Huan and X.S. Yang, On the number of limit cycles in general planar piecewise systems, Discrete Cont. Dyn. Syst., Series A 32 (2012), 2147–2164. [45] S.M. Huan and X.S. Yang, Existence of limit cycles in general planar piece- wise linear systems of saddle–saddle dynamics, Nonlinear Anal. 92 (2013), 82–95. [46] S.M. Huan and X.S. Yang, On the number of limit cycles in general planar piecewise linear systems of node–node types, J. Math. Anal. Appl. 411 (2014), 340–353. [47] Yu.S. Ilyashenko, Centennial history of Hilbert’s 16th problem, Bull. Amer. Math. Soc. 39 (2002), 301–354. [48] N. Levinson, A second order differential equation with singular solutions, Annals of Mathematics, Second Series, Vol. 50, No. 1 (Jan., 1949), 127–153. [49] L. Li, Three crossing limit cycles in planar piecewise linear systems with saddle-focus type, Electron. J. Qual. Theory Differ. Equ. 2014, No. 70, pp. 14. [50] J. Li, Hilbert’s 16th problem and bifurcations of planar polynomial vector fields, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13 (2003), 47–106. [51] D. Liu, The cubic systems with cubic curve solution y 2 = ax3 + bx2 + cx + d, Northeast Math. J. 5 (1989), 427–447. 16 JAUME LLIBRE AND XIANG ZHANG [52] M.J. Liu and C. Zhang, Comment: Algebraic limit cycles of cubic sys- tems with two quadratic algebraic curve solutions, J. Math. Res. Exposition 15 (1995), Suppl., 37–39, by Zhang (Chinese); J. Math. Res. Exposition 19 (1999), suppl., 266–268. [53] Y.C. Liu, On differential equation with algebraic limit cycle of second degree dy/dx = (a10 x + a01 y + a30 x3 + a21 x2 y + a12 xy 2 + a03 y 3 )/(b10 x + b01 y + b30 x3 + b21 x2 y + b12 xy 2 + b03 y 3 ), Advancement in Math 4 (1958), 143–149. [54] J. Llibre, Integrability of polynomial differential systems, Handbook of Differ- ential Equations, Ordinary Differential Equations, Eds. A. Cañada, P. Drabek and A. Fonda, Elsevier, Vol. 1, 2004, pp 437–533. [55] J. Llibre, Open problems on the algebraic limit cycles of planar polynomial vector fields, Bul. Acad. Stiinte Repub. Mold. Mat. 1 (2008), 19–26. [56] J. Llibre, D.D. Novaes and M.A. Teixeira, Limit cycles bifurcating from the periodic orbits of a discontinuous piecewise linear differential center with two zones, Int. J. Bifurcation and Chaos 25 (2015), 1550144, pp. 11. [57] J. Llibre, D.D. Novaes and M.A. Teixeira, Maximum number of limit cy- cles for certain piecewise linear dynamical systems, Nonlinear Dyn. 82 (2015), 1159–1175. [58] J. Llibre, M. Ordóñez and E. Ponce, On the existence and uniqueness of limit cycles in a planar piecewise linear systems without symmetry, Nonlinear Anal. Series B: Real World Appl. 14 (2013), 2002–2012. [59] J. Llibre and C. Pantazi, Counterexample to a conjecture on the algebraic limit cycles of polynomial vector fields, Geom. Dedicata 110 (2005), 213–219. [60] J. Llibre and E. Ponce, Three nested limit cycles in discontinuous piecewise linear differential systems with two zones, Dyn. Cont. Disc. Impul. Syst., Series B 19 (2012), 325–335. [61] J. Llibre, R. Ramı́rez, V. Ramı́rez and N. Sadovskaia, The 16th Hilbert problem restricted to circular algebraic limit cycles, J. Differential Equations 260 (2016), no. 7, 5726–5760. [62] J. Llibre, R. Ramı́rez and N. Sadovskaia, On the 16th Hilbert problem for algebraic limit cycles, J. Differential Equations 248 (2010), no. 6, 1401–1409. [63] J. Llibre, R. Ramı́rez and N. Sadovskaia, On the 16th Hilbert problem for limit cycles on nonsingular algebraic curves, J. Differential Equations 250 (2011), no. 2, 983–999. [64] J. Llibre and G. Rodrı́guez, Configurations of limit cycles and planar polynomial vector fields, J. Differential Equations 198 (2004), 374–380. [65] J. Llibre and G. Swirszcz, Relationships between limit cycles and algebraic invariant curves for quadratic systems, J. Differential Equations 229 (2006), 529–537. [66] J. Llibre and G. Swirszcz, Classification of quadratic systems admitting the existence of an algebraic limit cycle, Bull. Sci. Math. 131 (2007), 405–421. [67] J. Llibre and M.A. Teixeira, Piecewise linear differential systems without equilibria produce limit cycles?, Nonlinear Dyn. 88 (2017), 157–164. [68] J. Llibre and M.A. Teixeira, Periodic orbits of continuous and discontin- uous piecewise linear differential systems via first integrals, São Paulo J. Math. Sci. 12 (2018), 121–135. 17 [69] J. Llibre, M.A. Teixeira and J. Torregrosa, Lower bounds for the max- imum number of limit cycles of discontinuous piecewise linear differential sys- tems with a straight line of separation, Int. J. Bifurcation and Chaos 23 (2013), 1350066, pp. 10. [70] J. Llibre and C. Valls, On the uniqueness of algebraic limit cycles for quadratic polynomial differential systems with two pairs of equilibrium points at infinity, Geom. Dedicata 191 (2017), 37–52. [71] J. Llibre and C. Valls, Algebraic limit cycles on quadratic polynomial dif- ferential systems, Proc. Edinb. Math. Soc. (2) 61 (2018), no. 2, 499–512. [72] J. Llibre and C. Valls, Algebraic limit cycles for quadratic polynomial dif- ferential systems, Discrete Contin. Dyn. Syst. Ser. B 23 (2018), no. 6, 2475– 2485. [73] J. Llibre and X. Zhang, On the algebraic limit cycles of Liénard systems, Nonlinearity 21 (2008), 2011–2022. [74] J. Llibre and 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. [75] O. Makarenkov and J.S.W. Lamb, Dynamics and bifurcations of non- smooth systems: a survey, Phys. D 241 (2012), 1826–1844. [76] K. Odani, The limit cycle of the van der Pol equation is not algebraic, J. Differential Equations 115 (1995), 146–152. [77] S. Shui, X. Zhang and J. Li, The qualitative analysis of a class of planar Filippov systems, Nonlinear Anal. 73 (2010), 1277–1288. [78] D.J.W. Simpson, Bifurcations in Piecewise-Smooth Continuous Systems, World Sci. Ser. Nonlinear Sci. Ser. A, vol. 69, World Scientific, Singapore, 2010. [79] S. Smale, Finding a horseshoe on the beaches of Rio, The Mathematical In- telligencer 20 (1998), 39–44. [80] B. van der Pol, On relaxation-oscillations, The London, Edinburgh and Dublin Phil. Mag. & J. of Sci. 2(7) (1926), 978–992. [81] A.I. Yablonskii, Limit cycles of a certain differential equations, Differential Equations 2 (1966), 335–344 (in Russian). [82] Y. Qin, On algebraic limit cycles of degree 2 of the differential equation P i j dx 0≤i+j≤2 aij x y = P i j , dt 0≤i+j≤2 bij x y Sci. Record (N.S.) 1 (1957), no. 2, 1–3. [83] Y. Qin, On the algebraic P limit cycles P of second degree of the differential equation dy/dx = 0≤i+j≤2 aij xi y j / 0≤i+j≤2 bij xi y j , Acta Math. Sinica 8 (1958), 23–35. [84] J. Wang, X. Chen and L. Huang, The number and stability of limit cycles for planar piecewise linear systems of node-saddle type, J. Math. Anal. Appl. 469 (2019), 405–427. [85] J. Wang, C. Huang and L. Huang, Discontinuity-induced limit cycles in a general planar piecewise linear system of saddle-focus type, Nonlinear Anal. Hybrid Syst. 33 (2019), 162–178. [86] X. Zhang, The 16th Hilbert problem on algebraic limit cycles, J. Differential Equations 251 (2011), no. 7, 1778–1789. 18 JAUME LLIBRE AND XIANG ZHANG [87] X. Zhang, Invariant algebraic curves and rational first integrals of holomor- phic foliations in CP (2), Sci. China Ser. A 46 (2003), 271–279. 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat 2 School of Mathematical Sciences and MOE–LSC, Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China Email address: xzhang@sjtu.edu.cn