On some open problems in planar differential systems and Hilbert’s 16th problem

Chaos, Solitons and Fractals 31 (2007) 1118–1134 www.elsevier.com/locate/chaos

On some open problems in planar differential systems and HilbertÕs 16th problem Jaume Gine´

1

Universitat de Lleida, Departament de Matema`tica, Avda. Jaume II, 69, 25001 Lleida, Spain Accepted 20 October 2005

Communicated by Prof. M.S. El Naschie

Abstract This review paper contains a brief summary of topics and concepts related with some open problems of planar differential systems. Most of them are related with 16th Hilbert problem which refers to the existence of a uniform upper bound on the number of limit cycles of a polynomial system in function of its degree. These open problems are proposed as open questions throughout the text. Finally, an extensive bibliography, which does not intend to be exhaustive, is also given. Ó 2005 Elsevier Ltd. All rights reserved.

1. Introduction and historical precedents The history of science and technology is founded on the human necessities to explain natural phenomena, to objectively predict the future and to actively take part in it. Undoubtedly, mathematics is the language of science and it takes an important part in the attainment of these necessities. Since the introduction of differential calculus, the problems of the dynamic character have been expressed by means of differential equations. Isaac Newton (1643–1727) emphasized the utility of finding the solutions of differential equations. Today, it is difficult to imagine how the various problems in pure and applied sciences could be solved or even expressed without differential equations. Joseph Liouville (1809–1882) showed, in the middle of the 19th century, the insolubility of certain differential equations by means of quadratures, that is, the impossibility of expressing its general solution by a combination of elementary functions or Liouville functions. An elementary function is an element of the mathematical closure of the set of polynomial functions, exponential functions, logarithmic functions, trigonometrical functions and inverse trigonometrical functions by the arithmetic operations and composition applied a finite number of times. A Liouville function (or a function which can be expressed by means of quadratures) is a function constructed from rational functions by using algebraic operations, composition, exponentials and integration applied a finite number of times. LiouvilleÕs works at 1 The author is partially supported by a DGICYT grant number MTM2005-06098-C02-01, by a CICYT grant number 2005SGR 00550 and by a DURSI of Government of CataloniaÕs ‘‘Distincio´ de la Generalitat de Catalunya per a la promocio´ de la recerca universita`ria’’. E-mail address: [email protected].

0960-0779/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.10.057

J. Gine´ / Chaos, Solitons and Fractals 31 (2007) 1118–1134

1119

the end of the 19th century showed a new perspective in the study of differential equations known as Qualitative theory of ordinary differential equations. The fundamental idea of this theory is the study and determination of the properties of the solution directly from the differential equation instead of seeking the solution itself since, in general, such solution is not known. The qualitative theory was born from Poincare´ [101,102] and LiapunovÕs works [80] at the end of the 19th century and the beginning of the 20th century. Its main purpose consists in the qualitative description of significant mathematical objects to understand properly the solutionsÕ structure of a differential equation or of a system of differential equations. Poincare´, in [101], writes, on page 375 (translated into English by the author himself): . . .It is thus necessary to study the functions defined by differential equations themselves and without seeking to bring them back to simpler functions, in the same way as we made for the algebraic functions, which someone had sought to bring them back to radicals and that now are studied directly, in the same way as we made for the algebraic differentials integrals, that one endeavoured for a long time to express in finished terms. These thoughts induced Poincare´ to tackle the study of differential equations beyond an essentially different point of view from that of his predecessors. His study provokes a conceptual change in the understanding of differential equations. These equations are, according to him, no longer purely formal objects subject only to calculus rules but more importantly they do have a geometrical interpretation. Nowadays, due to the great advances in computational science, the qualitative theory of differential equations has been revived not only in mathematics but also in related subjects. This theory does not only use methods of the classical theory of differential equations but also takes advantage of the results and techniques of functional analysis, algebraic and differential geometry and algebraic topology. In particular, the qualitative theory has a wide development for systems in the plane. From a topological viewpoint, the main result may be the Poincare´–Bendixson Theorem, which has no generalization to higher dimension. This theorem states that any bounded solution not tending to a singular point must necessarily be a periodic orbit, or tend to a limit cycle or tend to a graphic (see definitions below). The algebraic theory of differential equations on the plane was born in the last third of the 19th century with the ideas of Darboux [44], Poincare´ [103], Painleve´ [97] and Autonne [5]. The main goal of this theory is to characterize the differential equations which have a rational first integral or a first integral generated by rational functions. In the last few years, as this theory has been adopted again, new and interesting results have been obtained.

2. Definitions and preliminary concepts Let us consider a first order differential equation dy Qðx; yÞ ¼ ; dx P ðx; yÞ

ð1Þ

where P and Q are functions of class Ck ðUÞ, k P 1, and where U is an open subset of R2 . Following Poincare´Õs ideas, we introduce a parameter t, which is usually known as time, and which allows us to express the former differential equation as a system of first order differential equations in the plane: dx ¼ P ðx; yÞ; dt

dy ¼ Qðx; yÞ. dt

ð2Þ

If P and Q are polynomials with real coefficients, i.e., P ; Q 2 R½x; y, we will denote system (2) by polynomial system. We define the degree of the polynomial system (2) by m = max{deg P, deg Q}. Let /(t, (x0, y0)) be the solution curve of Eq. (2) passing through the point ðx0 ; y 0 Þ 2 U. Let us recall that, by the existence and uniqueness theorem, the solution curve of system (1) with initial condition y(x0) = y0 exists if Q(x, y)/ P(x, y) is a continuous function in a neighborhood of the point (x0, y0) and is unique if Q(x, y)/P(x, y) satisfies the Lipschitz condition with respect to the variable y in a neighborhood of the point (x0, y0). The Lipschitz condition states that for any pair of points (x, y1), (x, y2) belonging to U, the inequality jQ(x, y1)/P(x, y1)  Q(x, y2)/P(x, y2)j 6 Ljy1  y2j is verified, where L is a positive constant. We call orbit of system (2) passing through the point ðx0 ; y 0 Þ 2 U the set C ¼ fðx; yÞ 2 Ujðx; yÞ ¼ /ðt; ðx0 ; y 0 ÞÞ; t 2 Rg. We say that a point p is an x-limit point of the solution curve /(t, (x, y)) of system (2) if there exists a succession ðtn Þn2N with limn!1 tn = 1 such that limn!1/(tn, (x, y)) = p. Analogously, we say that a point q is an a-limit point of the solution curve /(t, (x, y)) of system (2) if there exists a succession ðtn Þn2N with limn!1tn = 1 such that limn!1/ (tn, (x, y)) = q. Let us notice that an a-limit point or an x-limit point does not necessarily belong to the solution curve.

J. Gine´ / Chaos, Solitons and Fractals 31 (2007) 1118–1134

1120

The set of all x-limit points of a trajectory is called x-limit set of this curve and the set of all a-limit points of a trajectory is called a-limit set of this solution curve. We say that a point (x0, y0) is singular point, also called critical point or equilibrium point, for system (2) if both P(x0, y0) = 0 and Q(x0, y0) = 0. We call periodic orbit, any closed orbit which does not contain singular points. We call homoclinic orbit an orbit such that its a-limit and its x-limit meet at the same singular point. We call heteroclinic orbit an orbit whose a-limit and x-limit are two distinct singular points. We call separatrix cycle a closed curve given by the union of two heteroclinic orbits and the singular points which connect them. The finite union of separatrix cycles with an adequate orientation is called a graphic. A limit cycle is an isolated periodic orbit. In general, the solution curves and the orbits for system (2) do not need to be expressible by means of elementary functions. The first question to be asked about the solution curves of a polynomial system (2) of degree m is if these can be implicitly described by f(x, y) = 0, where f is a polynomial with coefficients in C. The equation f(x, y) = 0, where f(x, y) is a polynomial with complex coefficients of degree n, can be seen as an affine representation of an algebraic curve of degree n. Let us assume that (2) has a solution curve, contained in an algebraic curve f(x, y) = 0. It is obvious that the derivative of f with respect to t must vanish on the algebraic curve f(x, y) = 0. This condition can be expressed by of of f_ ¼ P þ Q ¼ Kf ; ox oy

ð3Þ

where Kðx; yÞ 2 C½x; y is a polynomial of degree less than or equal to m  1, and m is the degree of system (2). We call invariant algebraic curve of system (2) to any algebraic curve given by f(x, y) = 0, such that f(x, y) verifies (3). It is easy to see that if f(x, y) = 0 is an invariant algebraic curve and f ðx; yÞ ¼ f1m1 ðx; yÞ    frmr ðx; yÞ, with mi 2 N, is a factorization in irreducible elements of C½x; y, then each of the factors fi(x, y) = 0 for i = 1, . . . , r is also an invariant algebraic curve, see for instance [35,36]. We say that the algebraic curve f(x, y) = 0 with f 2 C½x; y is an algebraic solution for system (2) if, and only if, it is an invariant algebraic curve and f(x, y) is an irreducible polynomial in C½x; y. Given a planar algebraic curve with degree n, we can always write it as f(x, y) = fr(x, y) + fr+1(x, y) +    + fn(x, y) where fi(x, y) are homogeneous polynomials of degree i with fr(x, y) f 0. We say that a point p belongs to the curve f = 0 if f(p) = 0. By means of a translation we can always assume that the point p is the origin. Therefore, if r = 0 the curve does not contain the point. If r = 1, we say thatQ p is a simple point of f and if r > 1 we say that p is a multiple point of f of r-order. In the case r > 1 we have that fr ¼ ki¼1 lsi i , with s1 + s2 +    + sk = r, where each li is a distinct factor of fr which is called a tangent line of f = 0 at the point p and si is the multiplicity of this tangent line. We say that the origin is an ordinary multiple point if the multiplicity of all the tangents is 1. If not, it is called a non-ordinary multiple point. In particular, for r = 2, we say that the singular point is double; we call a node a double point with different tangent lines and we call a cusp a double point with equal tangent lines. We say that a point p is a dicritical singular point of the vector field if there is an infinite number of solutions of the vector field which have the point p as a-limit or x-limit. Let U be an open set of R2 , we say that the function H 2 Ck(U), not identically constant on U, is a strong first integral for system (2) if H is constant on each solution of system (2) defined on U. Here, k P 0 means that k = 0, 1, 2, . . . , 1, x, that is, k = 0 means that H is continuous, k = 1, 2, . . . , 1 means that H is of class Ck and k = x means that H is analytic. If k P 1 then the previous definition of integrability implies that the derivative of H in the direction given by the vector field X = P o/ox + Q o/oy is zero, that is, XH = 0 in U. This definition of strong first integral is the usual definition of first integral given in most of the books on differential equations, as quoted in [4] or [114]. We notice that with this definition, the linear system x_ ¼ x, y_ ¼ y defined in R2 has no continuous strong first integral. If this system has a first integral H defined in an open neighborhood U of the origin, since this function has a constant value for each straight line through the origin, then H would be constant on U, in contradiction with the definition. To avoid this problem, we say that a function H 2 Ck(U), not constant on U, with k P 0, where U is an open set of R2 n R is a weak first integral of system (2) if H is constant on each solution of the system (2) contained in U. With this definition, the function H(x, y) = xy/(x2 + y2) is a weak first integral of the linear system x_ ¼ x, y_ ¼ y, where R = {(0, 0)}, see [14]. We say that a function R 2 Ck(U) with k P 1, not identically null in U, is an integrating factor of system (2) in U if oðRP Þ oðRQÞ ¼ . ox oy In this case a first integral H associated to this integrating factor R is given by Z H ðx; yÞ ¼ Rðx; yÞP ðx; yÞ dy þ f ðxÞ; where oH/ox = RQ.

ð4Þ

ð5Þ

J. Gine´ / Chaos, Solitons and Fractals 31 (2007) 1118–1134

1121

Let U be an open set of R2 , we say that the function V 2 Ck(U) with k P 1, not identically null in U, is an inverse integrating factor of system (2) in U if it satisfies the following linear equation in partial derivatives:   oV oV oP oQ P þQ ¼ þ V. ð6Þ ox oy ox oy We notice that this function V is a particular solution of system (2). The expression oP/ox + oQ/oy is called divergence of system (2). This function V is very important because R = 1/V defines in Un{V = 0} an integrating factor of system (2) which let us determine a first integral for system (2) in Un{V = 0}. In [65] it is proved that this function V must be null over all the limit cycles contained in U.

3. Invariant algebraic curves The knowledge of algebraic solutions of system (2) allows, in general, a better understanding of the dynamics of the system and can be used to show topological properties of the system. Algebraic solutions and integrability are narrowly related as shown in DarbouxÕs works. The development of the qualitative theory, founded by Poincare´, has relegated to a second position the works by Darboux for many years. Nowadays, this theory has been rediscovered and revised by several authors. DarbouxÕs works give a different point of view based on algebraic solutions of differential equations extended to the complex projective plane CP2 . Darboux showed in his ‘‘Me´moire sur les e´quations diffe´rentielles alge´briques du premier ordre et du premier degre´’’ (1878) the way of constructing first integrals of polynomial systems of differential equations with a number of algebraic solutions high enough. In particular, Darboux showed that a polynomial system of degree m with at least m(m + 1)/2 + 1 invariant algebraic curves has a first integral which can be expressed by means of these algebraic curves. DarbouxÕs idea consists on looking for a first integral of system (2) with the form H ðx; yÞ ¼

q Y

fiki ðx; yÞ;

i¼1

where ki 2 C and fi(x, y) = 0 are invariant algebraic curves of system (2). The former first integral is called a Darboux first integral. In general, a Darboux first integral is a weak first integral. Jouanoulou [77] showed in 1979 that if the number of algebraic solutions for a polynomial system of degree m is at least m(m + 1)/2 + 2, then the system has a rational first integral and all the solutions of system (2) are algebraic. Prelle and Singer [104] demonstrated in 1983 that if a polynomial system has an elementary first integral, then this integral can be computed using the algebraic solutions of the system; in particular they showed that this polynomial system admits an integrating factor R such that RN, with N 2 Z, is a rational function with coefficients in C, see [104]. This result is known as the Prelle-Singer procedure and has been applied to an infinity of problems, see [31] for instance and references therein. Later on, in 1992 Singer [112] showed that if a polynomial system has a Liouvillian first integral then the system has an integrating factor of the form ! Z ðx;yÞ Rðx; yÞ ¼ exp U ðx; yÞ dx þ V ðx; yÞ dy ; ð7Þ ðx0 ;y 0 Þ

where U and V are rational Q functions which verify oU/oy = oV/ox. We call generalized Darboux functions to the functions of the form H ¼ eg fiki where g is a rational function, fi are polynomials and ki 2 C. In 1999, Colin Christopher [33] showed that the integrating factor (7) can be written as a generalized Darboux integrating factor. Finally, the contribution of Chavarriga et al. [30] must be noted since they showed that if a polynomial system has certain singular points, the number of algebraic solutions necessary to have a first integral expressible by means of them can be reduced. From all these works we can infer that given a polynomial system of degree m either the system has a finite number of algebraic solutions strictly lower than m(m + 1)/2 Pq + 2ni or the system has an infinite number of algebraic solutions and admits a rational first integral of the form H ¼ i¼1 fi , where ni 2 Z. Moreover, the degree of each algebraic solution is P P bounded by N ¼ maxf ni >0 ni deg fi ; ni <0 jni j deg fi g. Other improvements and related results to the integrability theory of Darboux have been published by several authors, for instance Christopher and Kooij [79], Christopher and Llibre [35,36], Garcı´a, Giacomini and Gine´ [59– _ ßdek [132]. 61,68], Gasull [62] and Zoła The cited works [104,105] give a criterion for when a polynomial system has an elementary first integral or a Liouvillian first integral. An important fact following from those results is that invariant algebraic curves and exponential factors play a distinguished role in these criteria. Moreover, this characterization is expressed in terms of an inverse integrating factor. Non-algebraic invariant curves with polynomial cofactor can also be used in order to find a first

J. Gine´ / Chaos, Solitons and Fractals 31 (2007) 1118–1134

1122

integral for a system. This observation permits the generalization of the DarbouxÕs integrability theory given in [59,60] where a new kind of first integrals, not only the Liouvillian ones, is described. In [63] it is showed that a transformation method relating planar differential systems to second order linear equations is an effective tool for finding non-Liouvillian first integrals. The problem of finding a first integral for system (1) and the functional class that it belongs to is what we call the integrability problem. Open problem 1. Characterize the integrability class of system (1) in terms of some type of transformations so as to get more simple differential equations. One way to find algebraic solutions of a given polynomial system (2) consists in computing the s-extactic curve. Let v1, v2, . . . , vl be a basis of the polynomials of C½x; y with degree 6s. We infer that l = (s + 1)(s + 2)/2. The s-extactic curve associated to system (2) is 2 3 v1 v2  vl 6 v_ v_2  v_l 7 6 1 7 es :¼ det 6 .. 7 .. .. 6 .. 7; 4 . . 5 . . ðl1Þ

v1

ðl1Þ

v2



ðl1Þ

vl

ðjÞ

where v_i ¼ P ovi =ox þ Qovi =oy for i = 1, . . . , l and vi ¼ v_i ðj1Þ for i = 1, . . . , l and j = 2, . . . , l  1. The definition of extactic curve is independent from the chosen basis. It is easy to prove that if f(x, y) = 0 is an algebraic solution of a degree lower than or equal to s of system (2), then f(x, y) must be a factor of the s-extactic curve. In case that the system has a rational first integral, it is easy to see that from a certain order all the extactic curves are identically null. This gives an algorithmic method which allows the computation of invariant algebraic curves and one which characterizes the polynomial systems with a rational first integral. We must point out that this method is computationally inefficient. This method has been rediscovered by Pereira [98], although historically Lagutinskii was the first to show it. We must recognize the remarkable historical work due to DobrovolÕskii, Lokot and Strelcyn [46] on this Russian mathematician. The integrability problem for polynomial differential systems in R3 is explored in [90,129]. One open problem stated for first time by Poincare´ [103] and afterwards by Prelle and Singer [104] is Open problem 2. Give a method to find an upper bound N to the degree of the algebraic solutions for a fixed polynomial system (2) of degree m P 2. Partial answers to this problem were given by Cerveau and Lins Neto [11] who showed that if an algebraic solution contains only singular points of node type then N(m) 6 m + 2; Carnicer [10] proved that if the system has no dicritical singularities then N(m) 6 m + 2; Chavarriga and Llibre [27] proved that if an algebraic solution has no multiple points then N(m) 6 m + 1 and if N(m) = m + 1 then the system has a rational first integral. It should be noted that for a fixed degree m P 2 there does not exist a uniform upper bound N(m) for the degree of an irreducible invariant algebraic curve depending only on the degree m of the system as shown in the following examples. For m = 1, the system x_ ¼ px, y_ ¼ qy, where p and q are positive integers, with p 5 q, has the rational first integral H(x, y) = xqyp and has algebraic solutions of arbitrarily high degree of the form yp  xq = 0. Just considering different values for p and q, the degree of the algebraic solution may be any value. For m = 2, the system 2 x_ ¼ y þ  xy; n

y_ ¼ x þ

nþ2 2 ðx  y 2 Þ; n

has the rational first integral " # 2 n2 nþ2 nþ2 2 2 ; 1 x H ðx; yÞ ¼ 1 x  y n n n and has algebraic solutions of degree n + 2, see [88]. On the other hand, until recently, there were not known families of polynomial systems of degree m with algebraic solutions of arbitrary degree without a rational first integral. Therefore another question, that appeared in a natural way, was to prove the existence of a uniform upper bound N(m) of the degree of the algebraic solutions of all the polynomial systems (2) for a fixed degree m P 2 that do not admit a rational first integral. Independently, Moulin-Ollagnier [94] and Christopher and Llibre [37] have demonstrated that this N(m) does not exist. In [37], they give the following example: x_ ¼ xð1  xÞ;

y_ ¼ ky þ Ax2 þ Bxy þ y 2 ;

J. Gine´ / Chaos, Solitons and Fractals 31 (2007) 1118–1134

1123

where k = c  1, A = ab(c  a)(c  b)/c2 and B = a + b  1  2ab/c. If, for any non-negative integer number k, it is chosen a = 1  k, b P a and c 62 Q, then, the previous quadratic system does not have any rational first integral and moreover, it has the irreducible invariant algebraic curve   ab y  x F ða; b; c; xÞ þ xð1  xÞF 0 ða; b; c; xÞ ¼ 0; c of degree k. Here, F(a, b, c; x) is an Hypergeometric function. Additionally, this system has a Darboux first integral and a Darboux integrating factor. In the light of this result, the following question is made. Open problem 3. Is there some number N(m) for which any polynomial system of degree m which has an algebraic solution of degree greater than M(m) has a Darboux or generalized Darboux first integral or integrating factor? Chavarriga and Grau [24] have studied the quadratic systems which admit algebraic solutions linear in one variable. Inside these systems they have found the examples given by Moulin-Ollagnier [94] and Christopher and Llibre [37] and also the following family of systems: x_ ¼ 1;

y_ ¼ 2n þ 2xy þ y 2 ;

ð8Þ

depending only on the parameter n. For n 2 N, the system (8) has an unique irreducible invariant algebraic curve. This curve has degree n + 1 and cofactor 2x + y, and is h(x, y) = Hn(x)y + 2n Hn1(x), where Hn(x) is the Hermite polynomial of degree n. System (8) has no Darboux first integral nor Darboux integrating factor for any value of n 2 N. In particular, system (8) has no rational first integral for any n 2 N. System (8) has no generalized Darboux first integral and has the general2 ized Darboux integrating factor V ðx; yÞ ¼ ex =hðx; yÞ2 . The Memoir of Darboux [44] deals with differential equations in the complex projective plane CP2 . Let us recall that the space CP2 is defined as fC3  fð0; 0; 0Þgg=  where [X, Y, Z]  [X 0 , Y 0 , Z 0 ] if, and only if, there exists a non-null k 2 C such that [X 0 , Y 0 , Z 0 ] = k[X, Y, Z]. Similarly, we can define RP2 . A differential equation of first order and degree m in the projective plane is an expression ðMZ  NY Þ dX þ ðNX  LZÞ dY þ ðLY  MX Þ dZ ¼ 0;

ð9Þ

where L, M, N are homogeneous polynomials of degree m in the variables X, Y, Z, and without common divisors. A polynomial system (2) of degree m can be extended to a differential equation in the complex projective plane just defining L(X, Y, Z) = ZmP(X/Z, Y/Z), M(X, Y, Z) = ZmQ(X/Z, Y/Z) and N(X, Y, Z)  0. Darboux showed that the number of singular points in CP2 of a differential equation (9) of degree m in the projective space is m2 + m + 1 (counted with their multiplicity). To show this result and other ones of some importance he stated a lemma that it turned out to be false. Recently, Chavarriga et al. [28] have proved a correct version of this lemma. The notion of algebraic particular solution of a given system (2) can be extended to a differential equation in the projective plane (9). Let F(X, Y, Z) = 0 be a homogeneous polynomial of degree n. In this case F(X, Y, Z) = 0 is an algebraic particular solution of system (9) if there exists a homogeneous polynomial K(X, Y, Z) of degree m  1 such that oF oF oF Lþ Mþ N ¼ KF . oX oY oZ A polynomial system (2) has a rational first integral when it has a first integral which is the quotient of two polynomials H(x, y) = f(x, y)/g(x, y). If the system has a rational first integral, then all its trajectories are algebraic and are described by f(x, y)  cg(x, y) = 0 where c is any constant. We define the degree p of H as the maximum of the degrees of f and g. Poincare´ showed that if a system has a rational first integral it is always possible to find a minimal integral, that is, a rational function H(x, y) = f(x, y)/g(x, y) verifying (f, g) = 1, deg f = deg g = p with p minimum over all the degrees of the rational first integrals of the system. This notion can be naturally extended to differential equations in the projective plane. In this case H(X, Y, Z) = F(X, Y, Z)/G(X, Y, Z) where F and G are homogeneous polynomials with the same degree p, that is H is a homogeneous function of degree 0. Given a differential equation in the projective plane with m2 + m + 1 distinct singular points then it is easy to prove that the eigenvalues associated to a singular point are all different from zero. Under these assumptions, a necessary condition for the differential equation to have a rational first integral is that the quotient of the eigenvalues related to each singular point is a rational number and that there are no logarithmic singular points, that is points whose associated Jordan matrix is not diagonal. Under these hypothesis, Poincare´ denotes by node a singular point whose quotient of eigenvalues k is a positive rational number and saddle if this quotient k is a negative rational number. In particular, if this quotient k is equals 1, then the node is called dicritical. We notice that if we consider a polynomial

J. Gine´ / Chaos, Solitons and Fractals 31 (2007) 1118–1134

1124

system (2) in the affine plane, the former conditions must be verified for all the singular points, real or complex, finite or not. Darboux [44], in his memory, studies the quadratic projective differential equations with one or several invariant algebraic curves with degree lower than or equal to three. Druzhkova [47] in 1968 gives, in function of the coefficients of the quadratic system the sufficient and necessary conditions for the existence and uniqueness of an invariant conic. Evdokimenco [54–56] in 1970 makes the following step in the study of the quadratic systems with a particular solution. In these works the conditions for the existence of a cubic invariant are given, in the particular case that its cofactor has the form K(x, y) = ax + by. The works of Evdokimenco tend to establish the existence of algebraic limit cycles given by an invariant cubic. In fact, he proves, except for some errors posteriorly corrected, that there are no invariant cubics with a connex component being a limit cycle of a quadratic system. Chavarriga et al. [28] have proved the same result by a more simple reasoning.

4. Limit cycles and Hilbert’s 16th problem The study of limit cycles has been one of the early lines of research in dynamical systems theory starting with Poincare´ in 1882. From the results obtained in the first decades of the last century we must highlight the work of Dulac [49] in 1923 on the finiteness of the number of limit cycles for a given polynomial system, recently corrected by Ilyashenko [75] and Ecalle [53]; the works by van de Pol [117] in 1926 proving the existence of limit cycles by using graphical methods and finally, the works of Lie´nard [82] in 1928 on nonlinear oscillators. During the 20s, the Italian biologist U. DÕAncona, with the help of the mathematician V. Volterra, were after a scientific explanation of the number of selaciens fish captured in FiumeÕs harbor (Italy) between 1914 and 1923. The answer given by Volterra [119] was a planar differential equation giving the relationship between preys and predators. Afterwards, the works of A. Kolmogorov [78] in 1934 gave a remarkable headstart to these prey–predator models. There is a relationship between the theory of limit cycles and invariant algebraic curves for planar polynomial systems as was suggested by Hilbert [73] when stating his famous 16th problem. This problem is divided into two parts: (a) about the topology of real algebraic curves and, in particular, the number of ovals they may have, (b) about the maximum number of limit cycles H(m) that a system (1) of degree m may have. The present statement of the 16th Hilbert problem, restated by S. Smale in his proposal of problems for the 21st century [113], is the following: 13th Problem of Smale. 16th Problem of Hilbert part (b). Consider the differential equation in R2 dx ¼ P ðx; yÞ; dt

dy ¼ Qðx; yÞ; dt

where P and Q are polynomials. Is there a bound K on the number of limit cycles of the form K 6 dq where d is the maximum of the degrees of P and Q, and q is a universal constant? This problem is one of the most difficult problems to be solved of the list given by Hilbert at the beginning of the last century. In fact, part (b) is not solved even for quadratic systems. From the famous HilbertÕs list, only two problems remain unsolved, the eighth (RiemmanÕs hypothesis about the zeros of the zeta function) and the 16th. In spite of the correct proof of DulacÕs Theorem regarding the finiteness of limit cycles, the existence of an upper bound for the number of limit cycles for a system (2) in function of m is still unknown. There is an extensive program developed by Dumortier, Roussarie and Rousseau among others, to show that this upper bound exists for quadratic systems. This program is based on the study of the bifurcations of limit cycles from all the graphics that can be realized by quadratic systems. Let L be the collection of two-dimensional autonomous differential systems (2), where P and Q are polynomials. For integers n > 1, let Lm be the subset of L consisting of those systems with P and Q of degree at most m. For S 2 L, we define p(S) to be the number of limit cycles of S, and define H ðmÞ ¼ supfpðSÞ; S 2 Lm g. By using different methods, lower bounds on the number of limit cycles for a system (2) have been found. We remark the latest bound given by Christopher and Lloyd [39] who show that H ðmÞ P m2 log m, where m is the degree of the system. We notice that the study of invariant algebraic curves let us better understand the existence and uniqueness of limit cycles. For instance, a quadratic system m = 2 with two invariant straight lines has no limit cycles. A quadratic system with one invariant straight line has at most one limit cycles. The first result was given by Bautin using Bendixson–Dulac criterion to ensure the non-existence of limit cycles, see [42,124]. One way to weaken the 16th HilbertÕs problem, and strengthening the hypothesis, is the study of the algebraic limit cycles, that is, algebraic curves f(x, y) = 0 which are particular solutions of (2) with a real closed oval which is a limit cycle. Several mathematicians have studied algebraic limit cycles getting results like that there are quadratic systems

J. Gine´ / Chaos, Solitons and Fractals 31 (2007) 1118–1134

1125

with algebraic limit cycles of degrees 2 and 4 but not of degree 3, see [57,108,122,123,126]. In [26] a summary of these results can be found with a new and shorter proof of the non-existence of algebraic limit cycles of degree 3. Chavarriga et al. [29] have found all the cases of algebraic limit cycles of degree 4 for quadratic polynomial systems and Chavarriga, Giacomini and Llibre [16] have proved their uniqueness. Christopher et al. [38] have given examples of quadratic systems with an algebraic limit cycle of degree 5 and 6. The fact that these limit cycles are hyperbolic is proved in [64]. A related open problem is: Open problem 4. Find, if they exist, polynomial systems of a fixed degree m with algebraic limit cycles of arbitrarily high degree. On the other hand, only few mathematicians have worked with non-algebraicity. Odani [96] initiated in 1995 the study of the existence of invariant algebraic curves for Lie´nard systems dx ¼ x_ ¼ y; dt

dy ¼ y_ ¼ f ðxÞy  gðxÞ; dt

ð10Þ

where the polynomials f and g have degrees m and n respectively. He showed that these systems do not have invariant algebraic curves if the following conditions are satisfied: (i) f, g 5 0, (ii) deg f P deg g and (iii) g(x)/f(x) 5 c, where c is a constant. In particular, these conditions imply that the famous limit cycle of Van der PolÕs equation dx ¼ x_ ¼ y; dt

dy ¼ y_ ¼ x þ kðx2  1Þy; dt

ð11Þ

where k is a parameter, is not algebraic. One year later, in 1996, Hayashi [72] studied the invariant algebraic curves of _ ßdek [135], who studied the cases system (10) when n = m + 1. A remarkable study of all these curves was made by Zoła n > m. He proved that for each pair (m, n) with n > m there is always a system with an invariant algebraic curve. Moreover, he showed that in the case n > m + 1 > 2 and (m, n) 5 (2, 4) there are always systems with an algebraic limit cycle. _ ßdek states, the problem of existence of algebraic limit cycles is still open. However, for the case (m, n) = (1, 3), as Zoła Dumortier et al. in [51,52] showed that these systems have, at most, one limit cycle and, in case it exists, it is hyperbolic. The Lie´nard systems have been extensibly studied in the last decades, see for instance [70,83,85] and references therein. Finally, Giacomini and Neukirch [66] found a powerful method to approximate non-algebraic limit cycles using algebraic curves with ovals tending to the limit cycles. This was an empirical method and, at the moment, there is no solid rigorous foundation for it. Open problem 5. To give to a theoretical base to the algorithm of algebraic approach of limit cycles presented in [66] and to generalize it to other type of systems that are not of Lie´nard type.

5. The center problem Poincare´ in [101] defined the notion of center for a real system of differential equations in the plane. We consider a system of differential equations of class Ck ðU Þ, with k P 1 and U an open set of R2 x_ ¼ P ðx; yÞ;

y_ ¼ Qðx; yÞ.

ð12Þ

A center of the system (12) is an isolated singular point O 2 U, that is, P(O) = Q(O) = 0 with a neighborhood V  U such that any p 2 V n fOg verifies P2(p) + Q2(p) 5 0 and the integral curve which passes through p is closed around O. Let us consider system (2) where P and Q are polynomials of degree m with an isolated singular point which we translate to the origin. Let us consider the case where the linear part of (2) has pure imaginary eigenvalues k = ±ix with x 2 R and x 5 0. In this case, we can consider a linear change of variable and a reescalation of time and system (2) reads for m X x_ ¼ y þ X s ðx; yÞ; s¼2

y_ ¼ x þ

m X

ð13Þ

Y s ðx; yÞ;

s¼2

with Xs and Ys homogeneous polynomials of degree s. Poincare´ showed in [102] that the origin of system (13) is a center if, and only if, there exists an open neighborhood of the origin of system (13) which has an analytic first integral. Liapunov [80] extended the previous result to analytic systems.

J. Gine´ / Chaos, Solitons and Fractals 31 (2007) 1118–1134

1126

By using polar coordinates x = r cos h, y = r sin h in (13), we call r(t, r0, h0) and h(t, r0, h0) the solution of the nonlinear system (13) which passes through the point (r0, h0). We say that the origin is a stable focus if there exists d > 0 such that for each 0 < r0 < d and h0 2 R we have that r(t, r0, h0) ! 0 and jh(t, r0, h0)j ! 1 when t ! 1. We say that the origin is an unstable focus if there exists d > 0 such that for each 0 < r0 < d and h0 2 R we have that r(t, r0, h0) ! 0 and jh(t, r0, h0)j ! 1 when t ! 1. Any trajectory verifying the former conditions is said to spiral around the origin. The center problem consists in distinguishing when the origin of system (13) is a center or a focus. The first analytic technique to approach this problem was introduced by Poincare´ in [102]. This technique consists in finding a formal power series H(x, y) = (x2 + y2)/2 + H3 + H4 +    such that each Hi is an homogeneous polynomial of degree i, with i P 2, verifying oH oH x_ þ y_ ¼ H_ ¼ ox oy

1 X

V i ðx2 þ y 2 Þiþ1 ;

i¼1

where H_ is the derivative of the function H with respect to the parameter t. Developing the former expression, we have !  !    m m X X oH 3 oH 4 oH 3 oH 4 xþ þ þ  y þ þ þ  xþ X s ðx; yÞ þ y þ Y s ðx; yÞ ox ox oy oy s¼2 s¼2 ¼ V 1 ðx2 þ y 2 Þ2 þ V 2 ðx2 þ y 2 Þ3 þ    We group together the terms by the powers of x and y, then to each order we get successive linear systems whose unknowns are the coefficients of Hi. These systems can be compatible or can be solved by choosing a certain constant Vi. These Vi are called Poincare´–Liapunov constants. Poincare´ in [102] demonstrated that if Vi = 0 for all value of i then a convergent series H(x, y) can be constructed and the origin is a center. The modern statement of the Poincare´Õs theorem is: Non-degenerate Center Theorem. System (13) has a center at the origin if and only if there exists a local analytic first integral of the form H = (x2 + y2) + F(x, y) defined in a neighborhood of the origin, where F starts with terms of order higher than 2. The convergence radius of the series H(x, y) is thought to be limited by the solution curves which have as a-limit set or x-limit set other singular points of system (13). Leaving out the convergence, the construction of the series H(x, y) is very useful since we can compute H until encountering the first term with Vi 5 0 and the sign of Vi gives the stability of the origin. Taking polar coordinates x = r cos h, y = r sin h in (13), we get r_ ¼ Oðr2 Þ; h_ ¼ 1 þ OðrÞ.

ð14Þ

Let r(t, r0, h0), h(t, r0, h0) be the solution with initial conditions r(0, r0, h0) = r0 and h(0, r0, h0) = h0. Then, for r0 > 0 sufficiently small h(t, r0, h0) is an strictly increasing function of t. Let t(h, r0, h0) be the inverse of the former strictly increasing function. For a certain h0 we define the function L(r0) = r(t(h0 + 2p, r0, h0), r0, h0). This function is an analytic function of r0, for r0 sufficiently small and it is known as the Poincare´ map. We consider thep Poincare ffiffiffiffiffiffiffiffiffiffiffiffiffiffi´ map associated to system (13) over the points on the axe OX+ and r < r0, we develop it in power series of r ¼ x2 þ y 2 and we subtract r. We obtain a power series in r such as LðrÞ  r ¼ L1 r þ L2 r2 þ L3 r3 þ    Frommer in 1934 showed that if V1 = V2 =    = Vp1 = 0 and Vp 5 0 then L1 = L2 =    = L2p = 0 and L2p+1 5 0 and, therefore, we have that if Vi = 0 for all i, then the origin is a center, that is, there is a neighborhood foliated by periodic orbits because the function L(r)  r  0 for all r < r0 and, therefore, the Poincare´ map is the identity. In fact, it can be proved that L2p+1 = 2pVp modulus the annulation of the previous constants, see [58]. On the other hand these Vi obtained by this process are polynomials on the coefficients of system (13). The infinite number of conditions Vi = 0 to ensure the existence of a center instead of a focus has a finite number of generators by HilbertÕs basis Theorem since the ring of the polynomials with variables the coefficients of the system is Noetherian. Therefore, the number of conditions to show the existence of a center instead of a focus for a given polynomial system (13) is finite. A way of obtaining such generators is still an open problem. This problem is a problem of computational algebra since the Poincare´–Liapunov constants are polynomials with rational coefficients and there are recursive methods to obtain them. The development of environments for technical computing has allowed the computation of the first constants. One of the

J. Gine´ / Chaos, Solitons and Fractals 31 (2007) 1118–1134

1127

main difficulties is that there are no efficient algorithms to determinate simple sets of generators. Basically, the main difficulty comes from the factorization of integers into prime numbers. Open problem 6. Given a system of the form (13) find the generators of the ideal generated by the Poincare´–Liapunov constants Vi of this system. From the study and annulation of the Poincare´–Liapunov constants we obtain distinct families with a center at the origin. Thanks to the works of Dulac [48] in 1908, Kapteyn [76] in 1911, Frommer [58] in 1934 and Bautin [6] in 1952 we know that the systems (13) with m = 2 have three independent Poincare´–Liapunov constants. The first integrals for the four center cases were given by Lunkevich and Sibirskii˘ [88] in 1982. In a linear center perturbed by homogeneous cubic _ ßdek [130] in 1994 and systems there are five independent constants as proven by Blows and Lloyd [8] in 1984 and by Zoła the first integrals for the three center cases were given by Lunkevich and Sibirskii˘ [89] in 1984 and by Chavarriga [12] in 1994 in polar coordinates. Some center cases are known for families of systems (13) as for instance, the systems of the form x_ ¼ y þ P s ðx; yÞ;

x_ ¼ x þ Qs ðx; yÞ;

ð15Þ

where Ps and Qs are homogeneous polynomials of degree s, see [17,18]; and the characterization of the centers for a _ ßdek cubic system, that is, x_ ¼ y þ P 2 þ P 3 , y_ ¼ x þ Q2 þ Q3 is not still complete. We remark the works by Zoła [131,134] in this direction. Only some subfamilies have been completely characterized, as for instance, the cubic systems (15) with degenerate infinity, see [84,19]. Frommer observed the relationship between Poincare´–Liapunov constants and local limit cycles around the origin, also called infinitesimal limit cycles. In 1952 Bautin [6] showed the existence of at most three infinitesimal limit cycles for a quadratic system when the origin is a focus point. In 1955 Petrovskii and Landis [99] wrongly stated that the maximum number of limit cycles for a quadratic system is three, that is H(2) = 3. In 1967 Petrovskii and Landis themselves recognized an error in a lemma which invalidated their result. In 1979, Shi [109] and simultaneously Chen and Wang [32] found examples which showed that H(2) P 4. In 1981 Shi [110] proved that for a system (13) the maximum number of infinitesimal limit cycles is M(n), where M(n) is the minimum number of generators of the ideal generated by Poincare´–Liapunov constants when these ones verify certain hypothesis. The proof basically consists in truncating the formal power series given by H(x, y), and applying Poincare´–BendixsonÕs theory to the rings formed by their level curves. Until recently it was known that H(3) P 11. By 1987 Li Jibin and Huang Qiming [81] found a cubic system with eleven _ ßdek [133] limit cycles and, more recently, different distributions of the eleven limit cycles are given in [127]. In 1995 Zoła showed that the perturbations of a given cubic system under certain hypothesis may give eleven infinitesimal limit cycles in a unique singular point. In [125] it was shown that H(3) P 12 and is the best result, so far, on the number of limit cycles for cubic systems. In the last few years different results have been obtained, for instance H(4) P 15, H(5) P 23 and H(6) P 35, see [115,120,121,128] and references therein. Another way to approach the center problem is by using normal forms. Using polar coordinates it can be shown that for any order ‘ there is a change of coordinates which transforms system (13) to its polar form r_ ¼ v3 r3 þ v5 r5 þ    þ Oðr‘ Þ;

h_ ¼ 1 þ p2 r2 þ p4 r4 þ    þ Oðr‘ Þ;

ð16Þ

where v2k+1 is the k-Poincare´–Liapunov constant. If v3 = v5 =    = v2k1 = 0 and v2k+1 5 0, system (13) is said to have a weak focus of order k. If all the Poincare´–Liapunov constants are null, then the origin is a center, since Poincare´ [102] showed that in this case there is an analytic change of variables which transforms system (13) into the following: r_ ¼ 0;

h_ ¼ 1 þ p2 r2 þ p4 r4 þ   

whose solution curves verify that the radius is constant. The constants pi have a fundamental role in the problem explained in the next section. Poincare´ showed in [102] that the system (13) has a center at the origin if, and only if, it has an analytic first integral defined in a neighborhood of the origin. Therefore, if we can find an analytic first integral defined in a neighborhood of the origin or an integrating factor well defined and different from zero in a neighborhood of the origin for a system (13), then the origin is a center. Recently, and due to the relations between integrability and invariant algebraic curves it is clear the important role of these invariant algebraic curves in the center-focus problem, see [9]. During these last years, interesting results relating algebraic solutions and Poincare´–Liapunov constants have been published. For instance, Cozma and S ß uba˘ showed in [43] that a weak focus of a polynomial system of degree m P 3 with the first Poincare´–Liapunov constant equal to zero and m(m + 1)/2  2 algebraic solutions has a Darboux integral or a Darboux integrating factor. Generalizing the previous result, Shube´ [111] proved that if v2k+1 = 0 for k = 1, . . . , [(m  1)/2] and the system has m(m + 1)/2  [(m + 1)/2] algebraic solutions, then the origin is a center. In [13] it is shown that if a polynomial

J. Gine´ / Chaos, Solitons and Fractals 31 (2007) 1118–1134

1128

system of degree m with an arbitrary linear part has a center and admits m(m + 1)/2  [(m + 1)/2] algebraic solutions, then this polynomial system has a Darboux integrating factor. In fact, many similar relations can be established between the problem of periodic solutions and that of algebraic solutions for a polynomial system. If the system admits an infinite number of periodic solutions then it has a C1 first integral [93], and is analytic when the linear part is not non-degenerated. In case the system admits an infinite number of algebraic solutions, then the system has a rational first integral. Finally, in case the number of periodic solutions is finite (limit cycles) we come up with part (b) of HilbertÕs problem, and if the number of algebraic solutions is finite we find the DarbouxÕs and JouanoulouÕs results but we do not have a uniform upper bound for the degree of such solutions. The center problem for a system with nilpotent linear part or with null linear part is more complicated. The difficulty resides in the fact that the problem is algebraically insoluble and it is not known whether it is analytically soluble, see [3,74]. The center problem for systems with a nilpotent part is algebraically soluble an the algorithm to detect them has been found, see for instance [1,7,15,95]. Partial results are known for centers of systems with null linear part, see for instance [3,67]. Open problem 7. To find an algorithm for the study of degenerate centers, i.e., centers of systems with null linear part.

6. Isochronous centers We observe periodic movements in any scientific field or in real life, for instance, the movement of the Moon around the Earth, the clock pendulum movement, the alternating current, the human brain waves, and the periodic movements in economy. Such movements are characterized by a periodic variation in the space of coordinates x, y, called phase space. If a planar differential system has a singular point O of center type, in a neighborhood V foliated by closed orbits around this singular point we can define a function T : V n fOg ! Rþ called the period function of the center and its value in each point is the period of the orbit passing through it. Our focus is to analyse the case when this period function is constant; in this case the center is called an isochronous center. The isochronicity problem was considered by Galileo Galilei in the 16th century when he studied the classical pendulum. Probably, the study of isochronous systems started before the development of the differential calculus. The basic problem was to find which curve in the vertical plane has the following property: A body under the gravity action and starting from rest such that from any point of the curve and moving through it uses the same time to arrive at the lower point (tautochronous property). The curve verifying this property is the cycloid. This problem was first solved by Huygens and published in his 1673 treatise on the theory of pendulum clocks (the first nonlinear example of isochronous oscillation such that the particle is moving over the cycloid and the period of the movement is independent from the amplitude) obtaining isochronous oscillations in opposition to the period monotony of the usual pendulum. Huygens applied this result to the construction of clocks. The cycloid curve is also known as braquistochronous because among all the possible curves this is the one with the quickest descensus of the material particle under the gravity law. Poincare´ [102] showed that for a non-degenerated center of an analytic system there is always a local analytic change of coordinates of the form u = x + o(j(x, y)j), v = y + o(j(x, y)j) and an analytic function w which transforms system (13) to u_ ¼ vð1 þ wðu2 þ v2 ÞÞ; v_ ¼ uð1 þ wðu2 þ v2 ÞÞ. It is known that the isochronicity problem only appears for non-degenerated centers, that is, centers whose linear part has pure imaginary eigenvalues, see [34]. There is always a local analytic change of coordinates for an isochronous center of an analytic system of the form u = x + o(j(x, y)j), v = y + o(j(x, y)j) and a constant k which transforms system (13) to u_ ¼ kv;

v_ ¼ ku.

ð17Þ

By a change to polar coordinates (x, y) = (r cos h, r sin h), system (13) writes as dr Rðr; hÞ ¼ ; dh 1 þ Hðr; hÞ

ð18Þ

J. Gine´ / Chaos, Solitons and Fractals 31 (2007) 1118–1134

1129

P P where Rðr; hÞ ¼ ms¼2 P s ðhÞrs , Hðr; hÞ ¼ ms¼2 Qs ðhÞrs1 being Ps and Qs trigonometrical homogeneous polynomials of degree s + 1. If we define r(h, q) the periodic solution of (18) such that r(0, q) = q with q sufficiently small, the period function is given by Z 2p Z 2p dh dh T ðqÞ ¼ ¼ . 1 þ Hðrðh; qÞ; hÞ h_ 0 0 It is known that the period function is analytic and, therefore, can be expressed by a convergent power series X T ðqÞ ¼ 2p þ T k qk ;

ð19Þ

kP1

where the coefficients Tk are called the period constants. A center is isochronous if, and only if, it is locally linearizable, that is, there is a local analytic change of variable which transforms system (13) to its linearization (y, x) around the singular point. This property gives an algorithm to find the conditions that imply the isochronicity of a center. This algorithm is based in the iterative construction of a change of variable in system (17). This process is used in many works to compute the isochronicity constants. Should we write system (13) as x_ ¼ y þ f ðx; yÞ, y_ ¼ x þ gðx; yÞ, where f(x, y) and g(x, y) are polynomials representing the nonlinear terms, we state that, x_y  y x_ ¼ x2 þ y 2 þ xgðx; yÞ  yf ðx; yÞ. On the other hand, if we use polar coor_ From the previous expressions we deduce that if xg(x, y)  y f(x, y)  0 then, when dinates we see that xy_  y x_ ¼ r2 h. the origin of the system is a center, it is isochronous. These centers are called trivial isochronous centers because in polar coordinates they verify the condition of constant angular speed h_ ¼ 1. The geometrical sense of the previous condition is that the terms is of order higher than one of the vector field (f, g) are perpendicular to the linear part (y, x). We notice that the isochronicity does not necessarily imply that the flow generated by the autonomous system in the plane behaves like the planar rotation of a rigid body, that is, the angular speed h_ does not need to be constant. The condition to be verified is Z T _ hðrðtÞ; hðtÞÞ dt ¼ 2p; 0

where (r(t), h(t)) is an arbitrary periodic solution of the center and T is the common period to all the solutions in a neighborhood of the origin. There is a simple method to generate isochronous centers for Hamiltonian vector fields. This method is based on the so-called Jacobian couples which are two polynomials P(x, y) and Q(x, y) such that P(0, 0) = Q(0, 0) = 0 and which verify that the determinant of the Jacobian of the application ðx; yÞ ! ðP ðx; yÞ; Qðx; yÞÞ is constant and different from zero. In this case, the Hamiltonian system associated to H ðx; yÞ ¼ 12 ðP 2 þ Q2 Þ is linearizable by the canonic change of coordinates u = P(x, y), v = Q(x, y). The behavior of the period function around a center has been studied in several works, particularly its connection with some particular classes of vector fields. For instance, the planar systems coming from a differential equation €x þ gðxÞ ¼ 0 have been studied by Urabe in [116], where necessary and sufficient conditions for isochronicity on the oscillations of these systems are given. Chicone and Jacobs [40] showed that for a complete comprehension of the properties of the period function in a certain class of systems, it is necessary to consider the bifurcations of the critical points of the period. The isochronous centers are used in [41] since the method to study the bifurcations of limit cycles needs the knowledge of the period function. The period function has been studied by different authors and many works in the last few years have been devoted to it. The characterization of isochronous centers has been treated by several authors. However, there is a low number of families of polynomial systems for which there is a complete classification of their isochronous centers. The quadratic systems with an isochronous center were classified by Loud [86] and the homogeneous cubic systems by Pleshkan [100]. The isochronous centers of a Kukles system, x_ ¼ y, y_ ¼ Qðx; yÞ, were studied in [34]. The class of complex systems of the form z_ ¼ iP ðzÞ where z = x + iy and the reversible cubic systems with h_ ¼ 1 have been studied in [92]. The isochronous centers of certain cubic systems with four complex invariant straight lines are found in [91]. The isochronous centers with degenerate infinity are studied in [84] and in [22]. The isochronous centers for systems with a center-type linear part perturbed by homogeneous polynomials have been studied in [21,23]. The study of the coexistence of isochronous centers and non-isochronous centers given in [45] must also be noticed. On the other hand, Lukasˇevicˇ in [87] showed that if P(x, y) and Q(x, y) are harmonic conjugate functions, that is, they satisfy Cauchy–Riemann equations Px = Qy and Py = Qx, then any center of the vector field (P, Q) is isochronous. Using the previous result, Villarini in [118] showed that if P(x, y) and Q(x, y) are harmonic conjugated functions, then the local flow of the vector field X = (P, Q) and his orthogonal vector field X ? ¼ ðQ; P Þ commute, in the sense that the Lie commutator ½X ; X ?  is null. In a later work, Sabatini [105] states that VillariniÕs proof works without great

1130

J. Gine´ / Chaos, Solitons and Fractals 31 (2007) 1118–1134

changes when changing orthogonal condition to transversal condition [X, XT] = 0 getting, in this way, an important method for the study of the isochronicity via the dynamic interpretation of the Lie product. In some works, such as [20], the importance of invariant algebraic curves when looking for transversal vector fields was studied. A method which uses an algebraic solution f = 0 of the vector field X to find a transversal field XT which commutes with X is developed. An remarkable survey of all the results on isochronicity was given in [25]. The isochronicity can be extended to other singular points through the concept of isochronous section. Adapting the two different techniques usually used for isochronous centers (the linearization and commutation), we can study isochronous singular points and isochronous limit cycles, see [69,106,107]. In summary, for non-degenerated singular points of linear part of center type the equivalence between linearization and existence of a commuting vector field has been shown. Therefore, we can consider the following problem. Open problem 8. How could one characterize the linearization and integrability problems of system (2) through the existence of non-trivial Lie symmetries, i.e., the existence of a commuting vector field or a normalizer.

7. Some fundamental problems The most important questions which are still open for planar differential equations are the following: Firstly, under which conditions do the original and the linearized system have the same qualitative behavior and the same topological structure around a singular point? This problem was first studied by Bendixon, Poincare´ and Liapunov at the end of the 19th century. A partial answer is given by the Hartman–Grobman Theorem [71], the Andreev Theorem [2] for nilpotent singular points and Dumortier Theorems [50] for degenerated singular points. Only the cases not included in these theorems must be studied, particularly, the problem of distinguishing a center from a focus for all kind of singular points remains unsolved. Secondly, which are the conditions for a system of differential equations to have a first integral defined in a neighborhood of a singular point with a certain type (Ck, k = 0, 1, 2, . . . , 1, x)? This problem is known as the local integrability problem. Thirdly, which are the conditions for a system of differential equations to have a first integral defined in the whole plane with a certain type (Ck, k = 0, 1, 2, . . ., 1, x)? This problem is known as the global integrability problem. Fourthly, what are the conditions for a system of differential equations to have a first integral of a certain type (polynomial, rational, Darboux, elementary, Liouville, etc.) defined in a neighborhood of a singular point? This problem is called formal integrability problem and the role of the invariant algebraic curves is very important as we have previously seen. These problems seem to be closely related to each others and some partial results confirm such supposition.

References [1] Amelkin VV, Lukasˇevicˇ NA, Sadovskii˘ AP. Nonlinear oscillations in second-order systems (Russian). Minsk: Beloruss Gos Univ; 1982. [2] Andreev A. Investigation of the behavior of the integral curves of a system of two differential equations in the neighborhood of a singular points. Trans Amer Math Soc 1958;8:187–207. [3] Anosov DV, Aranson SKh, Arnold VI, Bronshtein IU, Grines VZ, IlÕyashenko YuS. Ordinary differential equations and smooth dynamical systems. Berlin: Springer-Verlag; 1997. ´ quations diffe´rentielles ordinaires. Moscow: Mir; 1974. [4] Arnold V. E ´ cole Polytech 1891;61:35–122; [5] Autonne L. Sur la the´orie des e´quations diffe´rentielles du premier ordre et du premier degre´. J lÕE ´ cole Polytech 1892;62:47–180. Autonne L. Sur la the´orie des e´quations diffe´rentielles du premier ordre et du premier degre´. J lÕE [6] Bautin NN. On the number of limit cycles which appears with the variation of coefficients from an equilibrium position of focus or center type. Mat Sb 1952;30(72):181–96; Bautin NN. On the number of limit cycles which appears with the variation of coefficients from an equilibrium position of focus or center type. Trans Amer Math Soc 1954;100:397–413. [7] Berthier M, Moussu R. Re´versibilite´ et classification des centres nilpotents. Ann Inst Fourier (Grenoble) 1994;44:465–94. [8] Blows TR, Lloyd NG. The number of limit cycles of certain polynomial differential equations. Proc Roy Soc Edinburgh Sect A 1984;98:215–39. [9] Cairo´ L, Feix MR, Llibre J. Integrability and algebraic solutions for planar polynomial differential systems with emphasis on the quadratic systems. Resenhas 1999;4:127–61. [10] Carnicer MM. The Poincare´ problem in the nondicritical case. Ann Math 1994;140(2):289–94.

J. Gine´ / Chaos, Solitons and Fractals 31 (2007) 1118–1134

1131

[11] Cerveau D, Lins Neto A. Holomorphic foliations in CP(2) having an invariant algebraic curve. Ann Inst Fourier (Grenoble) 1991;41:883–903. [12] Chavarriga J. Integrable systems in the plane with a center type linear part. Appl Math (Warsaw) 1994;22:285–309. [13] Chavarriga J, Giacomini H, Gine´ J. An improvement to Darboux integrability theorem for systems having a center. Appl Math Lett 1999;12:85–9. [14] Chavarriga J, Giacomini H, Gine´ J, Llibre J. On the integrability of two-dimensional flows. J Differential Equations 1999;157: 163–82. [15] Chavarriga J, Giacomini H, Gine´ J, Llibre J. Local analytic integrability for nilpotent centers. Ergodic Theor Dynam Syst 2003; 23:417–28. [16] Chavarriga J, Giacomini H, Llibre J. Uniqueness of algebraic limit cycles for quadratic systems. J Math Anal Appl 2001;261: 85–99. [17] Chavarriga J, Gine´ J. Integrability of a linear center perturbed by fourth degree homogeneous polynomial. Publ Mat 1996;40: 21–39. [18] Chavarriga J, Gine´ J. Integrability of a linear center perturbed by fifth degree homogeneous polynomial. Publ Mat 1997;41: 335–56. [19] Chavarriga J, Gine´ J. Integrability of cubic systems with degenerate infinity. Differential Equations Dynam Syst 1998;6:425–38. [20] Chavarriga J, Gine´ J, Garcı´a IA. Isochronous centers of a linear center perturbed by homogeneous polynomials. In: Chavarriga J, Gine´ J, editors. Proceedings of the 3rd Catalan days on applied mathematics. Fundacio´ Publica Institut dÕEstudis Ilerdencs; 1996. p. 65–80. [21] Chavarriga J, Gine´ J, Garcı´a IA. Isochronous centers of a linear center perturbed by fourth degree homogeneous polynomial. Bull Sci Math 1999;123:77–96. [22] Chavarriga J, Gine´ J, Garcı´a IA. Isochronous centers of cubic systems with degenerate infinity. Differential Equations Dynam Syst 1999;7:49–66. [23] Chavarriga J, Gine´ J, Garcı´a IA. Isochronous centers of a linear center perturbed by fifth degree homogeneous polynomial. J Comput Appl Math 2001;126:351–68. [24] Chavarriga J, Grau M. Invariant algebraic curves linear in one variable for planar real quadratic systems. Appl Math Comput 2003;138:291–308. [25] Chavarriga J, Sabatini M. A survey of isochronous centers. Qual Theor Dyn Syst 1999;1:1–70. [26] Chavarriga J, Llibre J. On the algebraic limit cycles of quadratic systems. In: Garcı´a C, Olive´ C, Sanroma` M, editors. Proceedings of the IV Catalan days of applied mathematics, February 11–13, 1998. p. 17–24. [27] Chavarriga J, Llibre J. Invariant algebraic curves and rational first integral for planar polynomial vector fields. J Differential Equations 2001;196:1–16. [28] Chavarriga J, Llibre J, Moulin-Ollagnier J. On a result of Darboux. LMS J Comput Math 2001;4:197–210 (electronic). [29] Chavarriga J, Llibre J, Sorolla J. Algebraic limit cycles of degree 4 for quadratic systems. J Differential Equations 2004;200: 206–44. [30] Chavarriga J, Llibre J, Sotomayor J. Algebraic solutions for polynomial systems with emphasis in the quadratic case. Exposit Math 1997;15:161–73. [31] Chandrasekar VK, Pandey SN, Senthilvelan M, Lakshmanan M. Application of extended Prelle–Singer procedure to the generalized modified Emden type equation. Chaos, Solitons & Fractals 2005;26:1399–406. [32] Chen LS, Wang MS. The relative position and number of limit cycles of a quadratic differential system. Acta Math Sin 1979;22:751–8 (in Chinese). [33] Christopher CJ. Liouvillian first integrals of second order polynomials differential equations. Electron J Differential Equations 1999;1999:7 (electronic). [34] Christopher CJ, Devlin J. Isochronous centres in planar polynomial systems. SIAM J Math Anal 1997;28:162–77. [35] Christopher CJ, Llibre J. Algebraic aspects of integrability for polynomial systems. Qual Theor Dyn Syst 1999;1:71–95. [36] Christopher CJ, Llibre J. Integrability via invariant algebraic curves for planar polynomial differential systems. Ann Differential Equations 2000;16:5–19. [37] Christopher CJ, Llibre J. A family of quadratic polynomial differential systems with invariant algebraic curves of arbitrarily high degree without rational first integrals. Proc Amer Math Soc 2002;130:2025–30. [38] Christopher CJ, Llibre J, S´wirszcz G. Invariant algebraic curves of large degree for quadratic systems. J Math Anal Appl 2005; 303:450–61. [39] Christopher CJ, Lloyd NG. Polynomial systems: a lower bound for the Hilbert numbers. Proc R Soc Lond A 1995;450:219–24. [40] Chicone C, Jacobs M. Bifurcation of critical periods for plane vector fields. Trans Amer Math Soc 1989;312:433–86. [41] Chicone C, Jacobs M. Limit cycle bifurcations from quadratic isochronous. J Differential Equations 1991;91:268–327. [42] Coppel WA. A survey of quadratic systems. J Differential Equations 1966;2:293–304. [43] Cozma D, S ß uba˘ A. Partial integrals and the first focal value in the problem of the centre. NoDEA 1995;2:21–34. [44] Darboux G. Me´moire sur les e´quations diffe´rentielles alge´briques du premier ordre et du premier degre´ (Me´langes). Bull Sci Math 1878;32:60–96. p. 123–44, 151–200. [45] Devlin J. Coexisting isochronous and non-isochronous centers. Bull London Math Soc 1996;28:495–500. [46] DobrovolÕskii VA, Lokot NV, Strelcyn JM. Mikhail Nikolaevich Lagutinskii (1871–1915): Un Mathe´maticien Me´connu. Historia Math 1998;25:245–64.

1132

J. Gine´ / Chaos, Solitons and Fractals 31 (2007) 1118–1134

[47] Druzhova TA. The algebraic integrals of certain differential equations. Differential Equations 1968;4:736–9. [48] Dulac H. De´termination et inte´gration dÕune certaine classe dÕe´quations diffe´rentielles ayant pour point singulier un cente. Bull Sci Math Se´r 1908;32(2):230–52. [49] Dulac H. Sur les cycles limites. Bull Soc Math France 1923;51:45–188. [50] Dumortier F. Singularities of vector fields on the plane. J Differential Equations 1977;23:53–106. [51] Dumortier F, Rousseau C. Cubic Lie´nard equations with linear damping. Nonlinearity 1990;3:1015–39. [52] Dumortier F, Li C. On the uniqueness of limit cycles surrounding one or more singularities for Lie´nard equations. Nonlinearity 1996;9:1489–500. [53] Ecalle J. Introduction aux fonctions analysables et preuve constructive de la conjecture de Dulac. In: Actualite´s Mathe´matiques. Paris: Hermann; 1992. [54] Evdokimenco RM. Construction of algebraic paths and the qualitative investigation in the large of the properties of integral curves of a system of differential equations. Differential Equations 1970;6:1349–58. [55] Evdokimenco RM. Behavior of integral curves of a dynamic system. Differential Equations 1974;9:1095–103. [56] Evdokimenco RM. Investigation in the large of a dynamic system with a given integral curve. Differential Equations 1979;15: 215–21. [57] Filiptsov VF. Algebraic limit cycles. Differential Equations 1973;9:983–8. ¨ ber das Auftreten von Wirbeln und Strudeln (geschlossener und spiraliger Integralkurven) in der Umgebung [58] Frommer M. U rationaler Unbestimmheitssellen. Math Annalen 1934;109:395–424. [59] Garcı´a IA, Gine´ J. Generalized cofactors and nonlinear superposition principles. Appl Math Lett 2003;16:1137–41. [60] Garcı´a IA, Gine´ J. Non-algebraic invariant curves for polynomial planar vector fields. Discrete Contin Dyn Syst 2004;10:755–68. [61] Garcı´a IA, Giacomini H, Gine´ J. Generalized nonlinear superposition principles for polynomial planar vector fields. J Lie Theory 2005;15:89–104. [62] Gasull A. On polynomial systems with invariant algebraic curves. Equadiff 91, International Conference on Differential Equations, vol. 2. World Scientific; 1993. p. 531–7. [63] Giacomini H, Gine´ J, Grau M. Integrability of planar polynomial differential systems through linear differential equations. Rocky Mountain J Math, in press. [64] Giacomini H, Grau M. On the stability of limit cycles for planar differential systems. J Differential Equations 2005;213:368–88. [65] Giacomini H, Llibre J, Viano M. On the nonexistence, existence, and uniqueness of limit cycles. Nonlinearity 1996;9:501–16. [66] Giacomini H, Neukirch S. Number of limit cycles for the Lie´nard equation. Phys Rev E 1997;56:3809–13. [67] Gine´ J. Sufficient conditions for a center at completely degenerate critical point. Int J Bifur Chaos Appl Sci Eng 2002;12:1659–66. [68] Gine´ J, Giacomini H. An algorithmic method to determine integrability for polynomial planar vector fields. Eur J Appl Math, in press. [69] Gine´ J, Grau M. Characterization of isochronous foci for planar analytic differential systems. Proc Roy Soc Edinburgh Sect A 2005;135(5):985–98. [70] Han M, Bi P, Xiao D. Bifurcation of limit cycles and separatrix loops in singular Lienard systems. Chaos, Solitons & Fractals 2004;20:529–46. [71] Hartman P. Ordinary differential equations. New York: John Wiley & Sons; 1964. [72] Hayashi M. On polynomial Lie´nard systems which have invariant algebraic curves. Funkcial Ekvac 1996;39:403–8. [73] Hilbert D. Mathematische problem (lecture). In: Second Internat. Congress Math. Paris 1900, Nachr Ges Wiss Go¨ttingen MathPhys Kl 1900. p. 253–97. [74] IlÕyashenko YuS. Algebraic nonsolvability and almost algebraic solvability of the center-focus problem. Funct Anal Appl 1972;6: 30–7. [75] IlÕyashenko YuS. Finiteness theorems for limit cycles. Russian Math Surveys 1990;40:143–200. [76] Kapteyn W. On the centre of the integral curves which satisfy differential equations of the first order and the first degree. Konikl, Akademie von Wetenschappen te Amsterdam. Proceedings of the section of science 1911;13(2):1241–52; 1911;14(2):1185–95. ´ quations de Pfaff alge´briques. Lectures notes in mathematics, vol. 708. Berlin: Springer-Verlag; 1979. [77] Jouanolou JP. E [78] Kolmogorov AN. Sulla teoria di Volterra della lutta per lÕesistenza. Giorn Inst Ital Attuari 1936;7:74–80. [79] Kooij RE, Christopher CJ. Algebraic invariant curves and the integrability of polynomial systems. Appl Math Lett 1993;6:51–3. [80] Liapunov MA. Proble`me ge´ne´ral de la stabilite´ du mouvement. Ann Math Stud 17, Princeton University Press, 1947. [81] Li Jibin, Huang Qiming. Bifurcations of limit cycles forming compound eyes in the cubic system. Chinese Ann Math Ser B 1987;8:391–403. ´ tude des oscilations entretenues. Rev Ge´n lÕE´lectr 1928;XXIII-21:901–12; [82] Lie´nard A. E ´ tude des oscilations entretenues. Rev Ge´n lÕE´lectr 1928;XXIII-22:946–54. Lie´nard A. E [83] Lins A, de Melo W, Pugh CC. On Lie´nardÕs equation. Lecture notes in Mathematics, vol. 597. Berlin: Springer-Verlag; 1977. p. 335–57. [84] Lloyd NG, Christopher J, Devlin J, Pearson JM, Yasmin N. Quadratic like cubic systems. Differential Equations Dynam Syst 1997;5:329–45. [85] Lo´pez JL, Lo´pez-Ruiz R. The limit cycles of Lie´nard equations in the strongly nonlinear regime. Chaos, Solitons & Fractals 2000;11:747–56. [86] Loud WS. Behavior of the period of solutions of certain plane autonomous systems near centers. Contribut Differential Equations 1964;3:21–36.

J. Gine´ / Chaos, Solitons and Fractals 31 (2007) 1118–1134

1133

[87] Lukasˇevicˇ NA. The isochronism of a center of certain systems of differential equations (Russian). DifferencialÕnye Uravnenija 1965;1:295–302. [88] Lunkevich VA, Sibirskii˘ KS. Integrals of a general quadratic differential system in cases of the center (Russian). DifferentsialÕnye

[89]

[90] [91] [92] [93] [94] [95] [96] [97] [98] [99]

[100]

[101]

[102]

[103]

[104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119]

Uravneniya 1982;18:786–92; Lunkevich VA, Sibirskii˘ KS. Integrals of a general quadratic differential system in cases of the center (Russian). Differential Equations 1982;18:563–8. Lunkevich VA, Sibirskii˘ KS. Integrals of a system with homogeneous nonlinearities of the third degree in cases of a center (Russian). DifferentsialÕnye Uravneniya 1984;20:1360–5; Lunkevich VA, Sibirskii˘ KS. Integrals of a system with homogeneous nonlinearities of the third degree in cases of a center (Russian). Differential Equations 1984;20:1000–5. Maciejewski AJ. About algebraic integrability and non-integrability of ordinary differential equations. Chaos, Solitons & Fractals 1998;9:51–65. Mardesic P, Moser-Jauslin L, Rousseau C. Darboux linearization and isochronous centers with a rational first integral. J Differential Equations 1997;134:216–68. Mardesic P, Rousseau C, Toni B. Linearization of isochronous centers. J Differential Equations 1995;121:67–108. Mazzi L, Sabatini M. A characterization of centres via first integrals. J Differential Equations 1988;76(2):222–37. Moulin-Ollagnier J. About a conjecture on quadratic vector fields. J Pure Appl Algebra 2001;165:227–34. Moussu R. Syme´trie et forme normale des centres et foyers de´ge´ne´re´s. Ergodic Theor Dynam Syst 1982;2:241–51. Odani K. The limit cycle of the van der Pol equation is not algebraic. J Differential Equations 1995;115:146–52. ´ cole Normal Superi, 3 serie 1891;8:9–59, 103– Painleve´ P. Me´moire sur les e´quations diffe´rentielles du premier ordre. Ann Sci lÕE 40, 201–26 and 276–84; 1892;9:9–30, 101–44 and 283–308. Pereira JV. Vector fields, invariant varieties and linear systems. Ann Inst Fourier (Grenoble) 2001;51:1385–405. Petrovskii IG, Landis EM. On the number of limit cycles of the equation dy/dx = P(x, y)/Q(x, y), where P and Q are polynomials of the second degree. Amer Math Soc Transl Ser 1958;10(2):177–221; Petrovskii IG, Landis EM. On the number of limit cycles of the equation dy/dx = P(x, y)/Q(x, y), where P and Q are polynomials of the second degree. ‘‘Corrections’’, Mat Sb 1959;48:253–5. Pleshkan I. A new method of research on the isochronism of a system of two differential equations (Russian). DifferencialÕnye Uravnenija 1969;5:1083–90; Pleshkan I. A new method of research on the isochronism of a system of two differential equations (Russian). Differential Equations 1969;5:796–802. Poincare´ H. Me´moire sur les courbes de´finies par les e´quations diffe´rentielles. J Math Pures Appl 1881;7(3):375–422; Poincare´ H. Me´moire sur les courbes de´finies par les e´quations diffe´rentielles. J Math Pures Appl 1882;8:251–96; Poincare´ H. Me´moire sur les courbes de´finies par les e´quations diffe´rentielles. Oeuvres de Henri Poincare´, vol. I. Paris: GauthierVillars; 1951. pp. 3–84. Poincare´ H. Me´moire sur les courbes de´finies par les e´quations diffe´rentielles. J Math Pures Appl 1885;1(4):167–244; Poincare´ H. Me´moire sur les courbes de´finies par les e´quations diffe´rentielles. Oeuvres de Henri Poincare´, vol. I. Paris: GauthierVillars; 1951. pp. 95–114. Poincare´ H. Sur lÕinte´gration alge´brique des e´quations diffe´rentielles du premier ordre et du premier degre´ (I and II). Rendiconti del circolo matematico di Palermo 1891;5:161–91; Poincare´ H. Sur lÕinte´gration alge´brique des e´quations diffe´rentielles du premier ordre et du premier degre´ (I and II). Rendiconti del circolo matematico di Palermo 1897;11:193–239. Prelle MJ, Singer MF. Elementary first integrals of differential equations. Trans Amer Math Soc 1983;279:215–29. Sabatini M. Characterizing isochronous centres by Lie brackets. Differential Equations Dynam Syst 1997;5:91–9. Sabatini M. Non periodic isochronous oscillations in plane differential systems. Ann Math Pure Appl 2003;182(4):487–501. Sabatini M. Isochronous sections via normalizers. Int J Bifur Chaos Appl Sci Eng 2005;15(9):3031–7. Shen BQ. A sufficient and necessary condition of the existence of quartic curve limit cycle and separatrix cycle in a certain quadratic system. Ann Differential Equations 1991;7:282–8. Shi SL. A Concrete example of the existence of four limit cycles for plane quadratic systems. Sci Sinica Ser A 1980;23:153–8. Shi SL. A method of constructing cycles without contact around a weak focus. J Differential Equations 1981;41:301–12. Shube´ AS. Partial integrals, integrability and the center problem. Differential Equations 1996;32:884–92. Singer MF. Liouvillian first integrals of differential equations. Trans Amer Math Soc 1992;333:673–88. Smale S. Mathematical problems for the next century. Math Intell 1998;20:7–15. Sotomayor J. Lic¸o´es de ecuac¸o´es diferenciais ordina´rias. IMPA, Rio de Janeiro, 1979. Tang M, Hong X. Fourteen limit cycles in a cubic Hamiltonian system with nine-order perturbed term. Chaos, Solitons & Fractals 2002;14:1361–9. Urabe M. The potential force yielding a periodic motion whose period is an arbitrary continuous function of the amplitude of the velocity. Arch Rat Mech Anal 1962;11:27–33. ´ lectr 1927;XXII:489–90. Van der Pol B. Sur les oscillations de relaxation. Rev Ge´n lÕE Villarini M. Regularity properties of the period function near a centre of planar vector fields. Nonlinear Anal 1992;19:787–803. Volterra V. Leons sur la the´orie mathe´matique de la lutte pour la vie (French). Paris: Gauthier-Villars; 1931.

1134

J. Gine´ / Chaos, Solitons and Fractals 31 (2007) 1118–1134

[120] Wang S, Yu P. Bifurcation of limit cycles in a quintic Hamiltonian system under a sixth-order perturbation. Chaos, Solitons & Fractals 2005;26:1317–35. [121] Wu Y, Han M, Liu X. On the study of limit cycles of a cubic polynomials system under Z4-equivariant quintic perturbation. Chaos, Solitons & Fractals 2005;24:999–1012. [122] Yablonskii AI. On limit cycles of certain differential equations. Differential Equations 1966;2:164–8. [123] Yablonskii AI. Algebraic integrals of a differential-equation system. Differential Equations 1970;6:1326–33. [124] Ye Yan-Quian et al. Theory of limit cycles. Trans Math. Monographs 66, Amer Math Soc, Providence, 1986. [125] Yu P, Han M. Small limit cycles bifurcating from fine focus points in cubic order Z2-equivariant vector fields. Chaos, Solitons & Fractals 2005;24:329–48. P P [126] ChÕin Y-S. On algebraic limit cycles of degree 2 of the differential equation dy=dx ¼ 06iþj62 aij xi y j = 06iþj62 bij xi y j . Sci Sinica 1958;7:934–45; P P ChÕin Y-S. On algebraic limit cycles of degree 2 of the differential equation dy=dx ¼ 06iþj62 aij xi y j = 06iþj62 bij xi y j . Acta Math Sinica 1958;8:23–35. [127] Zhang T, Zang H, Han M. Bifurcations of limit cycles in a cubic system. Chaos, Solitons & Fractals 2004;20:629–38. [128] Zhang T, Han M, Zang H, Meng X. Bifurcations of limit cycles for a cubic Hamiltonian system under quartic perturbations. Chaos, Solitons & Fractals 2004;22:1127–38. [129] Zhou T, Lei J. The characteristic function method of finding first integrals of ODEÕs. Chaos, Solitons & Fractals 2003;18:709–20. _ ßdek H. On certain generalization of the BautinÕs Theorem. Nonlinearity 1994;7:233–79. [130] Zoła _ ßdek H. The classification of reversible cubic systems with center. Topol Methods Nonlinear Anal 1994;4:79–136. [131] Zoła _ ßdek H. On algebraic solutions of algebraic Pfaff equations. Stud Math 1995;114:117–26. [132] Zoła _ ßdek H. Eleven small limit cycles in a cubic vector field. Nonlinearity 1995;8:843–60. [133] Zoła _ ßdek H. Remarks on: ‘‘The classification of reversible cubic systems with center’’. Topol Methods Nonlinear Anal 1996;8: [134] Zoła 335–42. _ ßdek H. Algebraic invariant curves for the Lie´nard equation. Trans Amer Math Soc 1998;350:1681–701. [135] Zoła