The period function of the generalized Lotka-Volterra centers

THE PERIOD FUNCTION OF THE GENERALIZED LOTKA-VOLTERRA CENTERS J. VILLADELPRAT Abstract. The present paper deals with the period function of the quadratic centers. In the literature it is used different terminology to classify these centers, but essentially there are four families: Hamiltonian, reversible QR 3,codimension four Q4 and generalized Lotka-Volterra systems QLV 3.According to Chicone’s conjecture [2] the reversible centers have at most two critical periods, and the centers of the three other families have a monotonic period function. With regard to the second part of this conjecture, until now it has been proved the monotonicity in the Hamiltonian and Q4families [11, 35]. Concerning the QLV 3family, no substantial progress has been made since the middle 80s, when several authors showed independently the monotonicity of the classical Lotka-Volterra centers [23, 28, 33]. By means of the first period constant one can easily conclude that the period function of the centers in the QLV 3family is monotonous increasing near the inner boundary of its period annulus (i.e., the center itself). Thus, according to Chicone’s conjecture, it should be also monotonous increasing near the outer boundary, which in the Poincar´e disc is a polycycle. In this paper we show that this is true. In addition we prove that, except for a zero measure subset of the parameter plane, there are no bifurcation of critical periods from the outer boundary. Finally we show that the period function is globally (i.e., in the whole period annulus) monotonous increasing in two other cases different from the classical one. 1. Setting of the problem and results This paper concerns with the period function of centers. A critical point p of a planar differential system is a center if it has a punctured neighbourhood that consists entirely of periodic orbits surrounding p. The largest punctured neighbourhood with this property is called the period annulus of 1The author was partially supported by the CONACIT through the grant 2001SGR- 00173 and by the DGES through the grant BFM2002-04236-C02-2. 1 2 J. VILLADELPRAT the center and, in what follows, it will be denoted by P.The period function of the center assigns to each periodic orbit in Pits period. Questions related to the behaviour of the period function have been extensively studied. Let us quote, for instance, the problems of isochronicity (see [8, 10, 18]), monotonicity (see [3, 4, 29]) or bifurcation of critical periods (see [7, 26, 32]). In this setting the most studied polynomial family is the quadratic one. Taking a complex coordinate z=x+iy and using the terminology from [38], the list of quadratic centers at z= 0 is ˙z=−iz −z2+ 2|z|2+ (b+ic)¯z2,Hamiltonian (QH 3) ˙z=−iz +az2+ 2|z|2+b¯z2,reversible (QR 3) ˙z=−iz+ 4z2+ 2|z|2+(b+ic)¯z2,|b+ic|= 2,codimension four (Q4) ˙z=−iz +z2+ (b+ic)¯z2,generalized Lotka-Volterra (QLV 3) ˙z=−iz + ¯z2,Hamiltonian triangle where a, b and cstand for arbitrary real constants. There has been a substantial amount of work devoted to understand the behaviour of the period function of these centers. The quadratic isochronous (i.e., those centers with constant period function) were classified by Loud [17]. Coppel and Gavrilov [11] proved that the period function of any Hamiltonian quadratic center is monotonous and, more recently, Zhao [35] showed that the codimension four centers have the same property. Recall that the period function of a center is monotonous increasing (respectively, decreasing) if for any pair of periodic orbits inside P,say γ1and γ2with γ1⊂Int(γ2), we have that the period of γ2is greater (respectively, smaller) than the one of γ1.(Here by Int(γ) we mean the bounded connected component of R2\{γ}.) The period function of the quadratic reversible centers is not monotonous in general. The first example of non-monotonic reversible center is due to Chicone and Dumortier [5]. In this setting it is to be referred the results in [20], where are determined some regions in the parameter space of QR 3 for which the corresponding period function has at least one or two critical periods. It is important to note that the period function is defined on the set of periodic orbits in P.So usually the first step is to parametrize this set, let us say {γs}s∈(0,1),and then one can study the qualitative properties of the period function by means of the map s7−→ period of γs,which is smooth on (0,1).The critical periods are the critical points of this function and its number, character (maximum or minimum) and distribution does not depend on the particular parametrization of the set of periodic orbits used. THE PERIOD FUNCTION OF THE GENERALIZED LOTKA-VOLTERRA CENTERS3 Concerning the bifurcation of critical periods, there are three situations to study [20]. Indeed, compactifying R2to the Poincar´e disc, the boundary of Phas two connected components, the center itself and a polycycle. We call them respectively the inner and outer boundary of the period annulus. We have thus: (a) Bifurcations of critical periods from the inner boundary of the period annulus. (b) Bifurcations of critical periods from the interior of the period annulus. (c) Bifurcations of critical periods from the outer boundary of the period annulus. Chicone and Jacobs [7] described completely the bifurcation of critical periods from the inner boundary for the whole quadratic family. However the bifurcation from the outer boundary is far from being solved (see [19, 20] for results about the reversible family). The reason for this lack of results is twofold. The first one is that, contrary to the situation in the inner boundary, the period function does not extend smoothly to the outer boundary. The second one is that, in order to prove that a parameter is not a bifurcation value, one needs an asymptotic development which is uniform with respect to the parameters. This is not easily achieved because the shape of the polycycle at the outer boundary changes as the parameters vary. The study of the bifurcations from the interior of the period annulus seems out of reach for the moment because there are not specific tools to investigate them. Concerning the period function of the generalized Lotka-Volterra centers, apart from its behaviour near the inner boundary, almost nothing is known. The purpose of the present paper is to study the bifurcation of critical periods from the outer boundary and the monotonicity problem. By means of a rotation of axes (see [30] for instance), any generalized Lotka-Volterra system can be brought (not exhaustively) to ½˙x=−y−bx2−cxy +by2, ˙y=x+Axy. It is proved in [15] that if A= 0 then the period function of the above center is monotonous increasing. Hence the most interesting stratum is A6= 0,which can be brought to A= 1 by means of a rescaling, i.e., to the system (1) ½˙x=−y−bx2−cxy +by2, ˙y=x+xy. 4 J. VILLADELPRAT Figure 1. The curves Γ1and Γ2in Theorem A. Henceforth we shall use the notation µ:= (b, c) for the parameters. It is to be pointed out that the above family intersects the reversible one at c= 0.Furthermore the center at the origin is isochronous if and only if µ= (−1/2,0).Note on the other hand that the first period constant [7, 14] of the center of system (1) is given by π 12 ¡c2+ (2b+ 1)2¢.Therefore, except for the isochronous, the period function of (1) is increasing near the inner boundary of P(i.e., for periodic orbits near the center). In fact Chicone and Jacobs [7] conjectured that the period function of these centers is (globally) monotonous increasing. Define κ(µ):= c2+4b(b+1).Then, see Figure 1, the conic c2+4(b+1 2)2= 1 is given by κ(µ) = 0.We also consider Γ1:= ©µ∈R2:b(b+ 1)κ(µ) = 0ª and the segment Γ2:= (−1,0)×{0}.With this notation we can now state the main result of the present paper: Theorem A. Let {Xµ, µ ∈R2}be the family of vector fields in (1) and consider the period function of the center at the origin. Then, except for the isochronous at µ= (−1/2,0),the period function is monotonous increasing near the outer boundary of the period annulus. In addition, (a)There are no bifurcation of critical periods from the outer boundary of the period annulus at the parameters inside R2\©Γ1∪Γ2ª. (b)If µ∈Γ1then the period function of Xµis (globally)monotonous increasing. Let us now comment on the results in Theorem A in detail. Aside from reinforce Chicone’s conjecture, the fact that the period function is increasing near the outer boundary of Pimplies that either the center of system (1) is isochronous or has a finite number of critical periods (see [6, 27] for related THE PERIOD FUNCTION OF THE GENERALIZED LOTKA-VOLTERRA CENTERS5 results on this question). Indeed, to show this consider an analytic parametrization of the period function, taking for instance an analytic transverse section from the inner to the outer boundary of P.It is well known that this parametrization extends analytically to the inner boundary and, consequently, the critical periods can not accumulate there. Although it is not possible to extend it analytically to the outer boundary (not even smoothly), Theorem A shows that the critical periods do not accumulate there either. (Clearly if there exists a finite critical point at the outer boundary of P then the period function tends to infinity. However, even in this case, one can not assert in general that the period function is monotonous increasing near the outer boundary.) Roughly speaking the assertion in (a) guarantees that if we move slightly µ /∈Γ1∪Γ2then the critical periods of Xµcan not emerge or disappear at the outer boundary of Pµ.(That this neither occurs at the inner boundary is an easy consequence, except for the isochronous at µ= (−1/2,0),of the fact that the first period constant is positive.) Let us precise this. To this end we first parametrize the set of periodic orbits near the outer boundary of the period annulus Pµof Xµby means of a family of transverse sections Σµ. More concretely, Σµ=ξµ¡[0, δ]¢,where ξµ: [0, δ]−→ R2is a continuous family of analytic functions so that: (1) ξµ(0) belongs to the outer boundary of Pµ, (2) ξµ(s)∈ Pµand ξ0 µ(s) is transverse to Xµ¡ξµ(s)¢for all s∈(0, δ). Then, for each s∈(0, δ),we denote the period of the periodic orbit of Xµ passing through the point ξµ(s) by P(s;µ).Now the statement in (a) means by definition that for any µ0/∈Γ1∪Γ2there exist ε > 0 and a neighbourhood Uof µ0such that P0(s;µ)6= 0 for all s∈(0, ε) and µ∈U. (Of course this property does not depend on the particular transverse section used.) As we mentioned before, one of the difficulties to prove (a) is that the shape of the outer boundary of Pµchanges as µvaries. Let us advance that the parameters in Γ1correspond to bifurcations in the phase portrait of Xµthat affect Pµ(see Figure 2). It is worth noting that the result in (a) is the first step to prove the existence of a uniform bound for the number of critical periods of the family {Xµ, µ ∈R2}.More precisely, that there exists some n∈Nso that if µ6= (−1/2,0) then the center of Xµhas at most ncritical periods. According to Chicone’s conjecture, n= 0. Let us mention that it is also possible to derive global properties of the period function from the result in (a).To illustrate this let us first quote the work of Zhao [36, 37]. In these papers the author studies the period function of several families of quadratic centers that intersect with (1). Thus from the results in [36] and [37] respectively it follows the monotonicity for µ= (−3/2,0) and µ= (−2,0).Then, since at these parameters there are no 6 J. VILLADELPRAT bifurcation of critical periods from the boundary of P,we can conclude that the period function is monotonous increasing not only for µ= (−3/2,0) and µ= (−2,0),but in a small neighbourhood of them. Let us finally comment on the results in (b).The fact that the period function is monotonous increasing for b= 0 is not a new result. It corresponds to the classical Lotka-Volterra system and its monotonicity was proved by Rothe [23], Schaaf [28] and Waldvogel [33] independently. However, as far as we know, the monotonicity for b=−1 and the conic c2+ 4(b+1 2)2= 1 constitutes a new result. The paper is organized as follows. In Section 2 we first describe the different phase portraits of (1). In short, Xµhas three invariant straight lines in case that κ(µ) is positive and only one otherwise. We shall refer to them as the real and complex case respectively. In both cases there are situations in which the period annulus is unbounded and then its outer boundary has vertices, usually saddles, at infinity. It is therefore necessary a result about the time function associated to the passage through a saddle at infinity. In Section 2 we recall the tools developed in [19] to study this passage and introduce the related definitions. Sections 3 and 4 are devoted respectively to the complex and real case, that will cover the whole parameter plane except for µ∈Γ1.Finally in Section 5 we show the monotonicity for the parameters in Γ1and we collect all the results to prove Theorem A. 2. Tools and definitions Recall that the family of vector fields {Xµ, µ ∈R2}under consideration is given by Xµ:= ¡−y−bx2−cxy +by2¢∂x+x¡1 + y¢∂y, where µ= (b, c),and that κ(µ) := c2+ 4b(b+ 1).Let us begin with the following property (see [30, 31]). Remark 2.1 The multiple-valued function Hµ(x, y) = (1 + y)2b√κf+(x, y)√κ−cf−(x, y)√κ+c, where f±(x, y) = (c±√κ)x−2by + 2,is a first integral of Xµ.¤ Therefore the differential system (1) has three invariant straight lines, namely f±(x, y) = 0 (that may not be real) and y+ 1 = 0.In what follows we use the notation `±:= ©2by −(c±√κ)x= 2ª,which correspond to real straight lines in case that κ(µ)>0,i.e., µis outside the conic c2+4(b+1 2)2= 1.Another important property of the vector fields Xµis the following: Remark 2.2 The transformation (x, y, t)7−→ (−x, y, −t) brings X(b,c)to X(b,−c).¤ THE PERIOD FUNCTION OF THE GENERALIZED LOTKA-VOLTERRA CENTERS7 Figure 2. Phase portraits of the generalized Lotka- Volterra systems in the Poincar´e disc Using this property one can easily obtain the phase portrait of X(b,c)by means of the one corresponding to X(b,−c).Taking this into account, Figure 2 provides the phase portraits of (1) in the Poincar´e disc. The dotted curve in the diagram corresponds to bifurcations in the phase portrait that affect the period annulus of the center at the origin. This type of bifurcation does not occur on c= 0.However it is to be pointed out that on the segment (−1,0)×{0}the focus in the lower half-plane becomes a center. (This is so due to the symmetry explained in Remark 2.2.) Accordingly there are two centers coexisting on that segment. The bifurcation diagram of the phase portraits of all the quadratic centers can be found in [31]. Let us describe now the different type of period annuli in the family under consideration. Let us focus first in those parameters outside the conic κ(µ) = 0.In this case there are three different situations, namely b > 0, b∈(−1,0) and b < −1.If b > 0 then the polycycle at the outer boundary of Pµis a triangle made up with segments of the straight lines `+,{y+ 1 = 0} and `−.The critical points at the vertices of this triangle are hyperbolic saddles. For b < −1 and b∈(−1,0) the outer boundary of Pµis a triangle as well, but replacing one finite straight line by the line at infinity, say `∞. More concretely, if b < −1 then the triangle is made up with `+, `∞and `−, whereas in case that b∈(−1,0) it is made up with `+, `∞and {y+ 1 = 0} 8 J. VILLADELPRAT for c > 0 and `−, `∞and {y+ 1 = 0}for c < 0.As before the vertices are hyperbolic saddles, but now two of them are at infinity. Finally, for those parameters inside the conic κ(µ) = 0,the outer boundary of Pµis a bicycle with both vertices at infinity, hyperbolic saddles too. Remark 2.3 If we denote the period of the periodic orbit of Xµpassing through the point (0, y)∈R2with y > 0 by P(y;µ),then from Remark 2.2 it follows that P¡y; (b, c)¢=P¡y; (b, −c)¢.¤ It is clear at this point that to study the period function of the center at the origin we have to consider the time function associated to the passage through a hyperbolic saddle. If this saddle is at infinity then we must first perform a convenient compactification, and this yields to a meromorphic vector field. The rest of this section is devoted to introduce Proposition 2.8, which is the main result of a previous paper [19]. In short, it provides the asymptotic development of the time function associated to the passage through a hyperbolic saddle of a family of meromorphic vector fields. This result and the related definitions constitute the main ingredient in the proof of Theorem A. Let Wbe an open set of Rmand let {e Yµ, µ ∈W}be an analytic family of vector fields defined on some open set Vof R2. Assume that each vector field e Yµhas a hyperbolic saddle pµas the unique critical point inside V and let Sµand Tµbe its stable and unstable manifolds. We consider an analytic family of meromorphic vector fields Yµproportional to e Yµwith a pole of order n > 0 along Tµ. We can take a coordinate system (u, v, µ) on V×W⊂R2+msuch that pµ= (0,0, µ), Sµ={(u, v, µ) : u= 0}and Tµ={(u, v, µ): v= 0}. In these coordinates the family {Yµ, µ ∈W}writes as (2) Yµ(u, v) = 1 vn¡uP(u, v;µ)∂u+vQ(u, v;µ)∂v¢, where Pand Qare analytic functions such that P(u, 0; µ)>0 and Q(0, v;µ)<0 for any (0, v, µ)∈ Sµand (u, 0, µ)∈ Tµ. Moreover, by hypothesis, we have that λ(µ):= −Q(0,0; µ) P(0,0; µ)>0. The family {Yµ, µ ∈W}can be thought of as a single vector field Y defined on V×W⊂R2+mwhose trajectories lie on the submanifolds {µ= const}. Let σ:I×W−→ Σσand τ:I×W−→ Στbe two analytic transverse sections to Ydefined by σ(s;µ) = ¡σ1(s;µ), σ2(s;µ); µ¢and τ(s;µ) = ¡τ1(s;µ), τ2(s;µ); µ¢ THE PERIOD FUNCTION OF THE GENERALIZED LOTKA-VOLTERRA CENTERS9 Figure 3. Definition of Tin Proposition 2.8. such that σ(0; µ)∈ Sµand τ(0; µ)∈ Tµ. (Here Istands for a small interval of Rcontaining 0.) We denote the Dulac and time mappings between the transverse sections Σσand Στby Rand Trespectively. More precisely (see Figure 3), if ϕ¡t, (u0, v0); µ¢is the solution of Yµpassing through (u0, v0) at t= 0,for each s > 0 we define R(s;µ) and T(s;µ) by means of the relation (3) ϕ¡T(s;µ), σ(s); µ¢=τ¡R(s;µ)¢. Definition 2.4 We say that {Yµ, µ ∈W}verifies the family linearization property (FLP in short) if there exist an open set U⊂R2containing the origin and an analytic local diffeomorphism Φ : U×W→V×Wof the form Φ(x, y;µ) = ¡x+ h.o.t., y + h.o.t., µ¢such that Yµ= Φ∗µ1 f(x, y;µ)¡x∂x−λ(µ)y∂y¢¶, where fis an analytic function on U×W. ¤ Remark 2.5 It is easy to show that the family of meromorphic vector fields {Yµ, µ ∈W}defined in (2) verifies FLP if it has a Darboux first integral Hµ(x, y) = f1(x, y;µ)β1(µ)···fk(x, y;µ)βk(µ), where fj∈ Cω(U×W) for some open set U⊂R2containing the origin and βj∈ Cω(W).¤ Definition 2.6 Let Wbe an open subset of Rm. We denote by I(W) the set of germs of analytic functions h(s;µ) defined on (0, ε)×Wfor some ε > 0 such that lim s→0h(s;µ) = 0 and lim s→0s∂h(s;µ) ∂s = 0 16 J. VILLADELPRAT the transverse sections Σσ:= φ(Σ1) and Στ:= φ(Σ2).One can verify that Σσand Στare given respectively by σ(s) = µs s+α,1 s+α¶and τ(s) = µs+ 1 1+(α+ 1)s,1 1+(α+ 1)s¶. We will apply Proposition 2.8 to study T(s;µ) near s= 0.However it is first necessary to check that the FLP is fulfilled. To this end we first note that, from Remark 2.1, the single-valued real function Hµ(x, y) = = (y+ 1)³¡cx + 2(1 −by)¢2+|κ|x2´1 2b exp³c b|κ|1/2arctan³|κ|1/2x 2(by−1)−cx ´´ is a first integral of Xµfor all µ∈U. The exponential factor above is analytic everywhere except at the straight line 2(by −1) −cx = 0.Thus e Hµ(u, v):= Hµ¡φ−1(u, v)¢is a Darboux first integral of e Xµnear the saddle at (u, v) = (0,0) in case that c6= 0.(If c= 0 then 2(by −1) −cx = 0 cuts the saddle at infinity.) Therefore from Remark 2.5 it follows that ©e Xµ, µ ∈U\ {c= 0}ªverifies the FLP. On the other hand, since the exponential factor in e Hµ(u, v) disappears for c= 0,by Remark 2.5 again we can also assert that the 1-parameter family ©e Xµ, µ ∈U∩ {c= 0}ªfulfils the FLP. As a result of the above discussion, in order to apply Proposition 2.8 we must split ©Xµ, µ ∈Uªinto two subfamilies, the ones given by U\{c= 0} and U∩{c= 0}.This affects the properties that we can guarantee in the remainder term of the asymptotic development of T(s;µ) at s= 0.However, as we will see, it has no implications in the computation of the coefficients, that will be denoted by e ∆i(µ).Thus, for the sake of shortness, when we study the case c6= 0 we shall also compute the coefficients for c= 0.Define λ(µ):= −Q(0,0) P(0,0) =−b b+ 1, and note that then the subsets U1, U2and U3correspond to those µ∈U such that λ(µ)>1, λ(µ)<1 and 1/2< λ(µ)<2 respectively. Let us prove first the assertions concerning the case c6= 0.To this end define b Ui=Ui\ {c= 0}.Hence, taking (4) into account and applying Proposition 2.8, we can assert that if µ∈b U1then P(s;µ) = T¡s; (b, c)¢+T¡s; (b, −c)¢ =e ∆0(b, c) + e ∆0(b, −c) | {z } ∆0(µ) +¡e ∆1(b, c) + e ∆1(b, −c)¢ | {z } ∆1(µ) s+sI¡b U1¢. THE PERIOD FUNCTION OF THE GENERALIZED LOTKA-VOLTERRA CENTERS17 It is to be pointed out that here we apply Proposition 2.8 twice, to the vector fields e X(b,c)and e X(b,−c),and that we use that λ(µ) does not depend on c. Exactly the same way, if µ∈b U2then P(s;µ) = e ∆0(b, c) + e ∆0(b, −c) | {z } ∆0(µ) +¡e ∆2(b, c) + e ∆2(b, −c)¢ | {z } ∆2(µ) sλ+sλI¡b U2¢. Finally, if µ∈b U3then P(s;µ) = e ∆0(b, c) + e ∆0(b, −c) | {z } ∆0(µ) +¡e ∆3(b, c) + e ∆3(b, −c)¢ | {z } ∆3(µ) sω(s;λ) + +¡e ∆4(b, c) + e ∆4(b, −c)¢ | {z } ∆4(µ) s+sI¡b U3¢. So far we have proved the assertions with regard to the asymptotic development of the period function. It remains to compute the coefficients. The first one is given by e ∆0(µ) = Z0 1/α α dv b−α(2b+c)v+α2(c−1) v2 =2 |κ|1/2µarctanµ2(b+ 1) −c |κ|1/2¶−arctanµ2b+c |κ|1/2¶¶ =2 |κ|1/2µarctanµ|κ|1/2 c¶+χ(c)π¶,where χ(c) = ½0 if c⩾0, 1 if c < 0. Let us note that the second equality above follows from direct integration and the third one by applying Lemma 3.4. Therefore ∆0(b, c) = e ∆0(b, c) + e ∆0(b, −c) = 2π|κ(µ)|−1/2, where recall that by definition κ(µ) = c2+ 4b(b+ 1). The “higher order” coefficients are more complicated to compute. The key point to simplify them will be that T(s;µ) does not depend on α. Indeed, this implies that, although e Xµand the transverse sections depend on α, the coefficients e ∆i(µ) do not (cf. the computation of e ∆0(µ) above). Let us study first e ∆1(µ).By applying (a) in Proposition 2.8, some easy computations (which are not included here for the sake of brevity) show 18 J. VILLADELPRAT that e ∆1(µ) = −1 α(b+ 1) + (b+ 1) 1 2bexpµc b|κ|1/2arctanµ2(b+ 1) −c |κ|1/2¶¶ Z1 0¡(α+ 1)(2b+c+ 1) −2αc¢u+αc −2b−1 α¡(1 −c)u2+ (2b+c)u−b¢1 2b+2 · ·expµc b|κ|1/2arctanµ2(c−1)u−2b−c |κ|1/2¶¶ du u1/λ . Now, since in fact e ∆1(µ) does not depend on α, we can make α−→ +∞to compute it. This yields to e ∆1(µ) = (b+ 1) 1 2bexpµc b|κ|1/2arctanµ2(b+ 1) −c |κ|1/2¶¶ Z1 0 (2b−c+ 1)u+c ¡(1 −c)u2+ (2b+c)u−b¢1 2b+2 · ·expµc b|κ|1/2arctanµ2(c−1)u−2b−c |κ|1/2¶¶ du u1/λ . Our next goal is to simplify the above expression by means of the equalities in Lemma 3.4. To this end note first that if ξ1=2(b+ 1) −c |κ|1/2and ξ2=2(c−1)u−2b−c |κ|1/2, then ξ1+ξ2= 2|κ|−1/2(c−1)(u−1) >0 for µ∈Uand u∈(0,1).On the other hand 1−ξ1ξ2= 2(1 −c)¡2(b+ 1) −c¢u+c |κ|1/2. If c⩾0 then one can easily show that 1 −ξ1ξ2>0 on the region under consideration. If c < 0 then, setting u?:= c c−2(b+1) ,it follows that 1 −ξ1ξ2 is negative for u∈(0, u?) and positive for u∈(u?,1).Therefore, since ξ1+ξ2 1−ξ1ξ2 =|κ|1/2(1 −u) c(1 −u) + 2(b+ 1)u, THE PERIOD FUNCTION OF THE GENERALIZED LOTKA-VOLTERRA CENTERS19 in case that c⩾0 we obtain e ∆1(µ) = (b+ 1) 1 2bZ1 0 (2b−c+ 1)u+c ¡(1 −c)u2+ (2b+c)u−b¢1 2b+2 · ·expÃc b|κ|1/2arctanÃ|κ|1/2 c+ 2(b+ 1) u 1−u!! du u1/λ = (b+ 1) 1 2bZ+∞ 0 (2b+ 1)v+c ¡(b+ 1)v2+cv −b¢1 2b+2 · ·expµc b|κ|1/2arctanµ|κ|1/2 c+ 2(b+ 1)v¶¶ dv v1/λ =Z+∞ 0 F(v;µ)dv v1/λ . The first equality above follows from applying Lemma 3.4, and to obtain the second one we perform the change v=u 1−u.In case that c < 0,exactly the same way but taking u? 1−u?=−c 2(b+1) into account, we get e ∆1(µ) = expµcπ b|κ|1/2¶Z−c 2(b+1) 0 F(v;µ)dv v1/λ +Z+∞ −c 2(b+1) F(v;µ)dv v1/λ . 20 J. VILLADELPRAT Consider finally some µ= (b, c) with c⩾0.Then, using that F¡u; (b, −c)¢= −F¡−u; (b, c)¢,we obtain ∆1(µ) = e ∆1(b, c) + e ∆1(b, −c) =Z+∞ 0 F¡v; (b, c)¢dv v1/λ + expµ−cπ b|κ|1/2¶Zc 2(b+1) 0 F¡v; (b, −c)¢dv v1/λ +Z+∞ c 2(b+1) F¡v; (b, −c)¢dv v1/λ =Z+∞ 0 F(v;µ)dv v1/λ −expµ−cπ b|κ|1/2¶Zc 2(b+1) 0 F(−v;µ)dv v1/λ − Z+∞ c 2(b+1) F(−v;µ)dv v1/λ =Zc 2(b+1) 0µF(v)−F(−v) expµ−cπ b|κ|1/2¶¶ dv v1/λ + Z+∞ c 2(b+1) ¡F(v)−F(−v)¢dv v1/λ . This shows the validity of the expression of ∆1in the statement. Let us turn now to the computation of e ∆2(µ) by means of (b) in Proposition 2.8. To this end we note first that some easy simplifications yield to σ0 1(0)λσ2(0)L(0)λ= =λ1 2(b+1) αλ+1 expµc|κ|−1/2 b+ 1 µarctanµc−2(b+ 1) |κ|1/2¶+ arctanµ2b+c |κ|1/2¶¶¶ =λ1 2(b+1) αλ+1 expµ−c|κ|−1/2 b+ 1 µarctanµ|κ|1/2 c¶+χ(c)π¶¶, where χ(c) = ½0 if c⩾0, 1 if c < 0. (Here the second equality follows from applying Lemma 3.4.) On the other hand, after the change of variable given by v=αu 1−uin the integral, the THE PERIOD FUNCTION OF THE GENERALIZED LOTKA-VOLTERRA CENTERS21 second factor in e ∆2(µ) becomes τ1(0)−λ Q(0,0) +Zτ1(0) 0µM(u) P(u, 0) −M(0) P(0,0)¶du uλ+1 = =α b+ +α λ+1 b+ 1kZ+∞ 0õ1+ cv −bv2 b+ 1 ¶−2b+1 2(b+1) expµ−c|κ|−1/2 b+ 1 Sα (v;µ)¶−³1+ v α´λ−1 !dv vλ+1, where Sα(v;µ):= arctanµ(2bv −c)α+cv + 2(b+ 1) |κ|1/2(v+α)¶+ arctanµαc −2(b+ 1) α|κ|1/2¶. Recall at this point that e ∆2(µ) does not depend on α. Thus, by making α−→ +∞in the product of the two factors above, we conclude that e ∆2(µ) = =λ1 2(b+1) b+ 1 expµ−c|κ|−1/2 b+ 1 µarctanµ|κ|1/2 c¶+χ(c)π¶¶Z+∞ 0 G(v;µ) uλ+1 dv. (Here we apply the Dominate Convergence Theorem to commute the limit with the integration.) Consider finally some µ= (b, c) with c⩾0.Then, since one can verify that G¡u; (b, −c)¢=G¡−u; (b, c)¢,we obtain ∆2(µ) = e ∆2(b, c) + e ∆2(b, −c) =λ1 2(b+1) b+ 1 expµ−c|κ|−1/2 b+ 1 arctanµ|κ|1/2 c¶¶ Z+∞ 0µG(v) + G(−v) expµcπ|κ|−1/2 b+ 1 ¶¶ dv vλ+1 . This proves the validity of the expression of ∆2in the statement. It remains to compute ∆3(µ) in case that λ(µ) = 1,i.e., for those µ= (b, c) such that b=−1/2.To this end we use that ∆3(µ) = e ∆3(b, c) + e ∆3(b, −c) and apply (c) in Proposition 2.8. Fix some µ0= (−1/2, c0).Then some computations show that e ∆3(µ0) = −4c0expÃ4c0 p1−c2 0 arctanÃc0−1 p1−c2 0!!. 22 J. VILLADELPRAT Therefore ∆3(µ0) = e ∆3(−1/2, c0) + e ∆3(−1/2,−c0) = 4c0"expÃ4c0 p1−c2 0 arctanÃc0+ 1 p1−c2 0!!−expÃ4c0 p1−c2 0 arctanÃc0−1 p1−c2 0 !!# = 4c0expÃ4c0 p1−c2 0 arctanÃc0−1 p1−c2 0!!ÃexpÃ2πc0 p1−c2 0!−1!. In the last equality above we use that arctan(%)+arctan(1/%) = π/2 for % > 0 taking %=c0+1 √1−c2 0 .This concludes the proof of the assertions concerning the case c6= 0. Finally, in order to study the case c= 0 we apply Proposition 2.8 to the family ©e Xµ, µ ∈(−1,0)×{0}ª.Clearly, since λdoes not depend on c, the development is exactly the same as before but replacing b Uiby Ui∩ {c= 0}in the remainder term. Recall in addition that when we computed the coefficients for c6= 0 we also contemplated the case c= 0.Thus the coefficients in this case follow from the substitution c= 0 in that ones. This concludes the proof of the result. Lemma 3.5. Fix some µ= (b, c)in U1with c⩾0and define u?=c 2(b+1) . Then the following hold: (a)F(u;µ)−F(−u;µ) exp³−cπ b|κ(µ)|1/2´<0for all u∈(0, u?). (b)F(u;µ)−F(−u;µ)<0for all u∈(u?,+∞). Proof. Note that if µ∈U1then b∈(−1,−1/2) and that, by assumption, c⩾0.This easily implies that (5) (2b+ 1)u+c −(2b+ 1)u+c<1 and (b+ 1)u2−cu −b (b+ 1)u2+cu −b<1 for all u > 0. In the second inequality above we also use that c2+4b(b+1) = κ(µ)<0 for all µ∈U. Since F(−u;µ)>0 for all u > 0,the assertion in (a) is equivalent to prove that (6) F(u;µ) F(−u;µ)exp µcπ b|κ|1/2¶<1 for all u∈(0, u?). In fact this holds for all u > 0.Indeed, the function on the left of the above inequality writes as (2b+ 1)u+c −(2b+ 1)u+cµ(b+ 1)u2−cu −b (b+ 1)u2+cu −b¶1 2b+2 exp µc b|κ|1/2¡A(u;µ) + π¢¶, THE PERIOD FUNCTION OF THE GENERALIZED LOTKA-VOLTERRA CENTERS23 where A(u;µ):= arctanµ|κ|1/2 2(b+ 1)u+c¶+ arctanµ|κ|1/2 2(b+ 1)u−c¶. Thus, on account of the inequalities in (5) and due to 1 2b+ 2 >0, in order to show (6) it suffices to verify that A(u;µ) + π > 0.However this is clear because arctan x > −π/2 for all x. Let us prove next the assertion in (b).In this case, using again that F(−u;µ)>0 for all u > 0,it is equivalent to show that (7) F(u;µ) F(−u;µ)<1 for all u∈(u?,+∞). Now the function on the left writes as (2b+ 1)u+c −(2b+ 1)u+cµ(b+ 1)u2−cu −b (b+ 1)u2+cu −b¶1 2b+2 exp µc b|κ|1/2A(u;µ)¶, and therefore, taking account of (5) again, it suffices to show that A(u;µ)> 0 for all u > u?.However this is once again clear because 2(b+ 1)u+c > 0 for all u > 0 and 2(b+1)u−c > 0 for all u > u?.Hence the result is proved. Lemma 3.6. Fix some µ∈U2with c⩾0.Then G(u;µ) + G(−u;µ) exp³cπ|κ(µ)|−1/2 b+1 ´<0for all u > 0. Proof. Define g(u;µ) := G(u;µ) + G(−u;µ) exp³cπ|κ(µ)|−1/2 b+1 ´.We claim that g0(u;µ)<0 for all u > 0.Due to g(0; µ) = 0,it is clear that the result will follow once we prove this. To this end we first note that G0(u;µ) = f(u;µ)¡G(u;µ)+1¢with f(u;µ):= b b+ 1 (2b+ 1)u−c −bu2+cu +b+ 1. Accordingly g0(u;µ) = =G0(u;µ)−G0(−u;µ) expµcπ|κ(µ)|−1/2 b+ 1 ¶ =f(u;µ)¡G(u;µ)+1¢−f(−u;µ)¡G(−u;µ)+1¢expµcπ|κ(µ)|−1/2 b+ 1 ¶. Recall at this point that c⩾0 and that, since µ∈U2, b ∈(−1/2,0).Taking this into account and using that c2+ 4b(b+ 1) = κ(µ)<0 for all µ∈U, it easily follows that f(−u;µ)>0 for all u > 0.Consequently, since it is 24 J. VILLADELPRAT obvious that G(u;µ) + 1 >0 for all u∈R,the claim is equivalent to prove that f(u;µ) f(−u;µ) G(u;µ) + 1 G(−u;µ)+1expµ−cπ|κ(µ)|−1/2 b+ 1 ¶<1 for all u > 0. Straightforward manipulations show that the function on the left of the above inequality writes as c−(2b+ 1)u c+ (2b+ 1)uµb+ 1 −cu −bu2 b+1+cu −bu2¶4b+3 2(b+1) expµ−c|κ(µ)|−1/2 b+ 1 ¡B(u;µ) + π¢¶, (8) where B(u;µ):= arctanµ2bu −c |κ|1/2¶+ arctanµ2bu +c |κ|1/2¶. Now, using that µ∈U2and c⩾0 by assumption, one can verify that c−(2b+ 1)u c+ (2b+ 1)u<1 and b+ 1 −cu −bu2 b+1+cu −bu2<1 for all u > 0. These inequalities, due to 4b+3 2(b+1) >0 for b∈(−1/2,0),imply that the two first factors in (8) are smaller than one. However this is also the case for the last one. Indeed, B(u;µ) + π > 0 because arctan x > −π/2 for all x. This concludes the proof of the result. Proof of Proposition 3.2 In view of Lemma 3.3 it only remains to prove the assertions with regard to the signum of the coefficients. It suffices to study them on c⩾0 because we claim that ∆i(b, −c) = ∆i(b, c) for i= 1,2,3.Indeed, note that P(s;µ) and ∆0(µ) are even functions with respect to cby Lemma 3.3 and, on the other hand, λ(µ) depends only on b. This shows the claim because, for instance, ∆2(µ) = lim s−→0 P(s;µ)−∆0(µ) sλ(µ). The expressions of ∆1,∆2and ∆3for c⩾0 are given in Lemma 3.3. Taking them into account, Lemmas 3.5 and 3.6 show respectively that ∆1(µ)<0 for all µ∈U1∩{c⩾0}and that that ∆2(µ)<0 for all µ∈U2∩{c⩾0}. The fact that ∆3(−1/2, c)>0 for c∈(0,1) is clear from its expression. This concludes the proof of the result. 4. The real case This section is devoted to study the period function of the center at the origin in case that the straight lines `±are real. More concretely, those THE PERIOD FUNCTION OF THE GENERALIZED LOTKA-VOLTERRA CENTERS25 parameters inside V:= {µ∈R2:κ(µ)>0 and b(b+ 1) 6= 0}.We shall prove the following: Theorem 4.1. Consider the center at the origin of system (1) for the parameters inside V. Then its period function is monotonous increasing near the outer boundary of the period annulus. Moreover there are no bifurcation of critical periods from the outer boundary. To prove this result we must first take a parametrization of the period function. With this aim in view note that, on account of Remark 2.3, it suffices to study W:= V1∪V+ 2∪V3,where V1:= ©µ∈V:b < −1ª, V + 2:= ©µ∈V:b∈(−1,0) and c > 0ªand V3:= ©µ∈V:b > 0ª. Taking this subset of Venables us to parametrize the period function in a simple way. Recall, see Figure 2, that the polycycle at the outer boundary of Pµis a triangle for all µ∈V. If we consider only µ∈Wthen the triangle has always one side on the straight line `+.Furthermore, the point `+∩{y= 0}= (qµ,0),with qµ:= −2 c+κ1/2,is at the outer boundary of Pµfor all µ∈W. We take advantage of the fact that the segment that joins the origin with (qµ,0) is always inside Pµto parametrize the period function. Thus, for any s∈(0,1) and µ∈W, we denote by P(s;µ) the period of the periodic orbit of Xµpassing through the point ¡(1 −s)qµ,0¢∈R2.Notice that s≈0 corresponds to periodic orbits near the outer boundary of Pµ. Theorem 4.1 is an easy corollary of the following result, which provides the asymptotic expansion of P(s;µ) near s= 0. Proposition 4.2. For any µ∈Wit holds P(s;µ)=∆0(µ) ln s+ ∆1(µ) + I(W),where ∆iare analytic functions on V. Moreover ∆0(µ)<0for all µ∈W. The main part of this section is devoted to prove Proposition 4.2. For the reader’s convenience we prefer to postpone its proof and show Theorem 4.1 first. Proof of Theorem 4.1 By Remark 2.3 it is enough to consider those parameters inside W. Hence let us fix some µ?∈Wand note that then, by applying Proposition 4.2, sP0(s;µ)−→ ∆0(µ?) as (s, µ)−→ (0, µ?). (Here we took Definition 2.6 into account.) Since ∆0is negative, this implies that there exist a neighbourhood U?of µ?in Wand ε > 0 such that P0(s;µ)<0 for all s∈(0, ε) and µ∈U?.Accordingly there are no bifurcation of critical periods from the outer boundary at the parameter µ?. 32 J. VILLADELPRAT period annulus, it is monotonous increasing for b∈(−1/2,0) and decreasing for b∈(−1,−1/2).The center at the origin is isochronous for (D, F) = (−1/2,1/2) and hence system (1) with µ= (−1/2,0) has two isochronous centers coexisting. References [1] M. Abramowitz and I.A. Stegun, “Handbook of mathematical functions with formulas, graphs, and mathematical tables”, Dover Publications, New York, 1992. [2] C. Chicone, review in MathSciNet, ref. 94h:58072. [3] C. 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