The period function for second-order quadratic ODEs is monotone

The period function for second-order quadratic ODEs is monotone∗ Armengol Gasull (1), Antoni Guillamon (2) and Jordi Villadelprat (3) (1) Dept. de Matem`atiques, Universitat Aut`onoma de Barcelona, 08193 Bellaterra, Barcelona, Spain. (2) Dept. de Matem`atica Aplicada I, Universitat Polit`ecnica de Catalunya, Dr. Mara˜n´on 44-50, 08028 Barcelona, Spain. (3) Dept. d’Enginyeria Inform`atica i Matem`atiques, Universitat Rovira i Virgili, Av. dels Pa¨ısos Catalans 26, 43007 Tarragona, Spain. Dedicated to Professor Jorge Sotomayor on the occasion of his 60th birthday. Abstract. Very little is known about the period function for large families of centers. In one of the pioneering works on this problem, Chicone [?] conjectured that all the centers encountered in the family of second-order differential equations ¨x=V(x, ˙x), being Va quadratic polynomial, should have a monotone period function. Chicone solved some of the cases but some others remain still unsolved. In this paper we fill up these gaps by using a new technique based on the existence of Lie symmetries and presented in [?]. This technique can be used as well to reprove all the cases that were already solved, providing in this way a compact proof for all the quadratic second-order differential equations. We also prove that this property on the period function is no longer true when Vis a polynomial which nonlinear part is homogeneous of degree n>2. MSC: Primary: 37C-27; Secondary: 34C-25, 34C-14, 34A-26. 1 Introduction Let p0∈R2be a center of a planar system of differential equations. The period annulus of p0, that we denote by P,is defined as the greatest punctured neighborhood of p0foliated by periodic orbits. We take a parameterization of the set of periodic orbits in P,say s→ γs,and we consider the period function,s→ T(s),that assigns to each sthe period of the periodic orbit γs. ∗Partially supported by the DGES grant number BFM2002-04236 and CONACIT grant number 2001SGR-00173. 1 Several problems around the period function in planar vector fields have been studied in the last half-century, starting from the works of Urabe ([?]), Loud ([?]) and Pleshkan ([?]) on isochronicity (constant period function) in specific families of planar vector fields. Later on, the problem of the monotonicity of the period function attracted the attention of Coppel and Gavrilov (for potential systems, see [?]) and Waldwogel (for the Lotka-Volterra systems, see [?]), to quote significant examples, among others. Maybe, the most interesting contributions come from Chicone, see [?], [?], [?](with Jacobs), who studied not only the monotonicity but also computed the so-called period constants, and used them to study bifurcations of the period function in parametric families as well as boundary value problems. Coming back to the problem of isochronicity, in the early nineties, Villarini ([?]) and Sabatini ([?]) related the problem of isochronicity to the existence of Lie symmetries (see [?] for a survey on isochronicity) and, more recently, these ideas have been applied to the study of the monotonicity ([?]). In this paper we study the period function of the centers of quadratic systems that come from second-order ODEs ¨x=V(x, ˙x),that is, (1) ˙x=y, ˙y=−x+ax2+bxy +cy2, with a2+b2+c2=0. It is well known (see [?] for instance) that any center in this family can be brought, by means of a coordinate transformation into one of the following two forms: ˙x=y, ˙y=−x+ax2+bxy −ay2; (2) ˙x=y, ˙y=−x+ax2+cy2, (3) where a,band care arbitrary real numbers. For many subcases, which will be revisited in Section ??, Chicone gave a proof of the monotonicity of the period function (see [?,?]). As far as we know there are, though, some that were notprovedyet: Case I System (??)witha=0, Case II System (??)withac < 0. As we will see, the system of Case I can be easily brought to a system which is already known to have a center with monotonic period function. Thus, essentially, just one case remains unsolved. By means of a new technique for showing the monotonicity of the period function, we solve this last case and we give shorter proofs for the others. We can state therefore: 2 Theorem 1.1. If the origin is a center for system (??),then the associated period function is increasing. Chicone [?] has conjectured that if a quadratic system has a center with a period function which is not monotonic then, by an affine transformation and a constant rescaling of time, it can be brought to the Loud normal form ˙x=−y+Bxy, ˙y=x+Dx2+Fy2, and that the period function of these centers has at most two critical periods. In view of Theorem ??, with a rescaling, this conjecture is reduced to the case B=1.On the other hand, it is well known (see [?]) that the centers of ¨x=−x+˜ V(x, ˙x),with ˜ Vbeing a cubic polynomial without constant and linear terms, may have a non monotonic period function. In addition, we prove the following result: Proposition 1.2. For any m≥3, there are (reversible)centers of the form (4) ˙x=y, ˙y=−x+Vm(x, y), where Vmis a homogeneous polynomial of degree m, with a non monotonic period function. The paper is organized in the following way. In Section ?? we present the techniques used to prove Theorem ?? and we give new proofs of the cases already solved. Sections ?? and ?? are devoted respectively to show the monotonicity in Case I and Case II. Finally, in Section ?? we prove Proposition ??. 2 Previous results The mentioned techniques to ensure the monotonicity of the period function are based on the following result, which is proved in [?]. Theorem 2.1. Let pbe a center of a C1vector field Xand let Pdenote its period annulus. Let U be C1vector field on P∪{p},transversal to Xon P,and such that [X,U]=µX on Pfor some C1 function µon P∪{p}.Then, if ψ(s)is a trajectory of U, for any s0such that ψ(s0)∈Pit holds T(s0)=T(s0) 0 µx(t;s0),y(t;s0)dt, where x(t;s0),y(t;s0)is the periodic orbit of Xsuch that x(0; s0),y(0; s0)=ψ(s0)and T(s0) is its period. 3 For Theorem ?? to be useful we need to be able to compute µand control its integral. It is already known the existence of pairs (U, µ) satisfying [X,U]=µX for sufficiently regular vector fields Xwith a non-degenerate center. From a geometrical point of view, the vector field Uis the infinitesimal generator of the Lie group of symmetries of X. We will also take advantage of the next remark. Remark 2.2 Let Ube a vector field transversal to Xsuch that [X,U]=µX for some C1function µ. If we consider U=U+gX, where gis any C1function, then Uis also transversal to Xand [X,U]=µXwith µ=µ+(∇g)t·X.  The usefulness of Remark ?? lies in the fact that once we now a pair (µ, U) for a given vector field X, we can generate other pairs by adding “multiples” of Xto Uand modifying µconcordantly. In the rest of this section, for the sake of completeness and to show the efficiency of the method just presented, we apply it to reprove some cases already solved by Chicone. 2.1 System (??) In order to prove the monotonicity of the period function of the center of (??) we consider the following four cases: a= 0 and c=0 (3.1) a= 0 and c=0 (3.2) ac > 0(3.3) ac < 0(3.4) Chicone solved the first and second cases in [?], and the third one in [?]. As we already mentioned, in Section ?? we shall solve the fourth case. In order to reprove the case (3.1), note first that by means of the coordinate transformation {x1=2cx, y1=2cy}it is enough to consider only the case c=1/2.Then, renaming the variables as {x, y}, an additional change of variables, {w= log(1 + x−y2/2),z=y}, brings the system into X:= ˙z=1−ew, ˙w=z. Now one can verify that the vector field U:= ˙z=1 2z, ˙w=1−w ew−1 satisfies that [X,U]=µXwith µ(z,w):= 1−e2w+2wew 2(e2w−2ew+1). 4 Proposition 11.a in [?] shows that the integral of µalong the orbits of Xis always positive and so, from Theorem ??, it follows that the period function is increasing. In the case (3.2) the change of variables {x1=−ax, y1=−ay}allows to consider only the case a=−1.Renaming the variables as {x, y},one can show that the new vector field, say X,is transversal to U:= ˙x=3x+2x2 6(1+x), ˙y=1 2y, and that [X,U]=µXwith µ(x, y)=−x(2 + x) 6(1 + x)2. Then Proposition 11.b in [?] shows that the integral of µalong the orbits of Xis always positive and so, again from Theorem ??, the period function is increasing. The proof of the case (3.3) using Theorem ?? presents the same kind of difficulties than for the case (3.4), which is solved in Section ??, and so, for the sake of shortness, we prefer to avoid it in this paper. 2.2 System (??) In order to prove the monotonicity of the period function of the center of (??) we consider the following two cases: a=0 (2.1) a=0 (2.2) Chicone proved in [?] the monotonicity of the period function in the case (2.1). In order to reprove it by means of Theorem ?? we proceed as follows. The coordinate transformation {x1=−bx, y1= −by}allows to consider only the case b=−1.Then, renaming the variables as {x, y}, another change of variables, {z=x, w = log(1 + y),τ =−t}, brings the system into X:= ˙z=1−ew, ˙w=z. This is the same vector field that we obtained in the case (3.1) and so the result follows. Next section is devoted to prove the case (2.2). 5 3 Proof of the monotonicity in Case I By means of the coordinate transformation {x1=ax, y1=ay}system (??) can also be reduced to the case a= 1 without loss of generality. We therefore consider (5) ˙x=y, ˙y=−x+x2+bxy−y2. One can verify that the change of variables {z=−x−(b+b2+1)y,w =−x−(b−b2+1)y} brings system (??)to (6) ˙z=−w(1 + z)(√b2+1+b), ˙w=z(1 + w)(√b2+1−b). This is a Lotka-Volterra system and it is well-known (see [?] for instance) that the centers of these systems have an increasing period function. For the sake of compactness, let us point out that this fact was also proved in [?] directly from Theorem ??. 4 Proof of the monotonicity in Case II 4.1 Reduction of the problem through Theorem ?? We first note that, rescaling the variables, we can assume, without loss of generality, that a=1 and c<0. We consider therefore (7) ˙x=y, ˙y=−x+x2+cy2. A computation shows that H(x, y)=A(x)+C(x)y2,with A(x):= e−2cx 4c22cx2+2(1−c)x−1+1/c+c−1 4c3and C(x):= 1 2e−2cx, is a first integral of (??), and one can verify that κ(x):= −e−2cx is its corresponding integrating factor (see [?] for details). We shall take advantage of this fact to perform a coordinate transformation that brings (??) to a potential system. This follows from the next result. Lemma 4.1. Suppose that a given planar differential system has a first integral of the form H(x, y)=A(x)+B(x)y+C(x)y2and that its corresponding integrating factor, say κ, depends only on x. Then, if κ(x)C(x)=0,the coordinate transformation given by u=f(x):= x 0 κ(s) 2C(s)ds and v=2C(x)y+B(x) 2C(x) 6 brings the system to ˙u=−v, ˙v=gf−1(u), where g(x):= B(x)B(x)C(x)−2C(x)B(x) 2κ(x)2C(x)3+2C(x)A(x) κ(x). Proof. Note first of all that we can write the given differential system as (8)        ˙x=−Hy(x, y) κ(x), ˙y=Hx(x, y) κ(x), since, by hypothesis, His a first integral and κis its corresponding integrating factor. It is clear moreover that we can assume without loss of generality that C(x)>0.In this case we can rewrite the first integral as H(x, y)=1 22C(x)y+B(x) 2C(x)2 +4A(x)C(x)−B(x)2 4C(x). In order to obtain the desired coordinate transformation, we define v:= 2C(x)y+B(x) 2C(x) and then find u(x, y) such that  uxuy vxvy=κ(x) for all x. The simplest way to achieve this is by choosing u(x, y)=f(x) so that f(x)2C(x)=κ(x),which yields to u=f(x):= x 0 κ(s) 2C(x)ds. Finally, some computations show that this coordinate transformation brings (??) to the potential system given in the statement. By applying Lemma ?? to system (??) it turns out that the coordinate transformation u=e−cx −1 c,v=ye−cx 7 brings it to the potential system (9) ˙u=−v, ˙v=(1+cu) ln(1 + cu)c+ ln(1 + cu)/c2, that is, a Hamiltonian system with Hamiltonian function H(u, v)=1 2v2+F(u), where F(u):= 1 4c32ln 2(1 + cu)+4uc ln2(1 + cu)+2u2c2ln2(1 + cu)−2 ln(1 + cu) −4uc ln(1 + cu)−2u2c2ln(1 + cu)+2cu +u2c2+2cln(1 + cu) +4uc2ln(1 + cu)+2u2c3ln(1 + cu)−2uc2−u2c3. The above change of coordinates already appears in [?]. It is easy to check that if Xis the potential vector field associated to H(u, v)=v2+F(u),then [X,U]=µX, where Uis the vector field associated to the system ˙u=F(u)/F(u), ˙v=v/2and µ(u)=F(u) F(u)−1/2. In our case a computation shows that µ(u)=−c+2 6u+O(u2). Thus, since we look for some µ>0,we shall take advantage of Remark ?? to remove the linear term in µ. The choice g(u, v)=(c+2)v/6 (in the notation of Remark ??) provides µ(u):= µ(u)+c+2 6F(u)=2c2+5c+5 6u2+O(u3). Note in particular that 2c2+5c+5>0 for any c. Let us denote by (uL,u R) the projection of the period annulus of the center at the origin of system (??) onto the u-axis.Itisclear,onaccountofTheorem??, that to show the monotonicity of its period function it is enough to verify that µ(u)>0 for all u∈(uL,u R)\{0}.To do so it is first necessary to study the ranges of (uL,u R) for the different values of c. It is easy to show that (10) uL=e−c−1 cand uR<−1 cin case that c∈(−1,0), uL=1 ce1−c−√c2−1 2−1and uR=−1 cin case that c≤−1. Now, in order to simplify the formulae that we shall obtain, we perform the change of variable u=(e−x−1)/c, which one can verify that drives to (11) µ2(x):= µu(x)=e2x 12c2x2(c−x)2 6  i=0 Ci(x)xi, 8 with C0(x)=3c3(c−1)(e−2x−1),C 1(x)=3c2(c−1)(c+ce−2x−2e−2x+2), C2(x)=3c2(1 −c)(1 + 3 e−2x),C 3(x)=2c2(3 e−2x−c2e−3x−2ce−3x), C4(x)=6c2e−3x(c+2),C 5(x)=−6ce−3x(c+2), C6(x)=2e−3x(c+2). Consequently, taking (??) into account, we must prove that µ2(x)>0 for all x∈(c,+∞)\{0}in case that c∈(−1,0), µ2(x)>0 for all x∈c−1+√c2−1 2,+∞\{0}in case that c≤−1. The following two subsections are devoted to study these cases. 4.2 The case c∈(−1,0) We study the cases x∈(c, 0) and x∈(0,+∞) separately. 4.2.1 The study of µ2on x∈(0,+∞) We shall prove the following result. Proposition 4.2. If c∈(−1,0) then µ2(x)>0for all x>0. Note that, for the values of cunder consideration, C5(x)>0andC6(x)>0 for all x>0. Hence, on account of the expression of µ2given in (??), Proposition ?? will follow if we show that T(x):= C0(x)+C1(x)x+C2(x)x2+C3(x)x3+C4(x)x4 is positive on (0,+∞).To do so we proceed as follows. One can verify that T(x)=c2D0(x)+D1(x)c+D2(x)c2, where D0(x)=−6x+3x2+(6x+9x2+6x3)e−2x+12x4e−3x, D1(x)=3+3x−3x2−(3 + 9 x+9x2)e−2x+(−4x3+6x4)e−3x, D2(x)=−3+3x+(3+3x)e−2x−2x3e−3x. 9 4.3.2 The study of µ2on x∈(−1,0). Since one can check that D0(x)+D1(x)b+D2(x)b2=(4+6b+4b2)x4+O(x5),it is clear that, on account of the decomposition of µ2givenin(??), the result will follow once we prove these two lemmas. Lemma 4.7. D3(x)>0and D4(x)>0for all x∈(−1,0). Lemma 4.8. D1(x)2−4D0(x)D2(x)<0for all x∈(−1,0). Proof of Lemma ?? We shall prove in fact that  D3(x):= D3(−x)and  D4(x):= D4(−x)are positive on (0,1).To this end we will use that, for all x∈(0,1),0<m(x)<e x<M(x)with m(x)=1+x+1 2x2+1 6x3and M(x)=1+x+1 2x2+1 2x3. Consider first  D3,which can be written as  D3(x)=P1(x)e2x+P2(x)e3x+P3(x) where P1(x):= 21−51x+45x2−12x3,P2(x):= −2x3(x+2)(x−1)2and P3(x):= −21+9x+15x2. Since one can easily verify that P1(x)>0andP2(x)<0 for all x∈(0,1),by using the upper and lower bounds of exintroduced before it follows that  D3(x)>P 1(x)m(x)2+P2(x)M(x)3+P3(x) =−1 4x15 −3 4x14 −3 2x13 −3x12 −9 4x11 −3 4x10 +5 3x9+27 4x8+61 12 x7+16 3x6+39 4x5+17 4x4 =−x4 12 (9x6−51) + (27x6−117)x+(36x6−64)x2 +(18x6−61)x3+(9x6−81)x4+(3x6−20)x5 which is clearly positive on (0,1). Let us turn now to study  D4,which can be written as  D4(x)=P4(x)e2x+P5(x)e3x+P6(x), where P4(x):= 6(1−x)3,P 5(x):= 2x3(x−1)3and P6(x):= 6x2+6x−6.Hence, P4(x)>0and P5(x)<0on(0,1).Consequently  D4(x)>P 4(x)m(x)2+P5(x)M(x)3+P6(x) =x4 12 3x11 +9x9−9x7−35x5−6x4+6x3−4x2+66x+42 , and it is easy to check (for instance, from its Sturm sequence) that this polynomial takes only positive values on (0,1). 16 Proof of Lemma ?? We will show that ∆(x):= D1(−x)2−4D0(−x)D2(−x) is negative on (0,1). Since ∆(x)=−28 x8+O(x9),it is clear that the result follows if we prove that d8 dx 8∆(x)e−3x<0 for all x∈[0,1). To this end let us first note that d8 dx 8∆(x)e−3x=P0(x)+P2(x)e2x+P4(x)e4x+P5(x)e5x+P6(x)e6xe−3x, where P0(x) = 59049 x4−39366 x3−1869885 x2+ 4028454 x−1156923, P2(x)=−18 x4+ 792 x3−11394 x2+ 62856 x−109962, P4(x)=9x4+ 414 x3+ 5715 x2+ 28242 x+ 41157, P5(x) = 3072 x6+ 46080 x5+ 125952 x4−774144 x3−4230144 x2−5999616 x−2322432, P6(x)=−78732 x8−1679616 x7−13611888 x6−53187840 x5, −104509440 x4−95800320 x3−28304640 x2+ 5806080 x+ 2419200. It can be shown moreover that P2and P5are negative on [ 0,1) and that P4is positive. On the other hand, for x∈(0,1),we have that m2(x)<e 2x,m 5(x)<e 5x,e 4x<M 4(x)andm6(x)<e 6x<M 6(x), where m2(x)=1+2x+2x2+4/3x3+2/3x4+4/15 x5+4/45 x6 m5(x)=1+5x+25/2x2+ 125/6x3+ 625/24 x4+ 625/24 x5+ 3125/144 x6 m6(x)=1+6x+18x2+36x3+54x4+ 324/5x5+ 324/5x6 M4(x)=1+4x+8x2+32/3x3+32/3x4+ 128/15 x5+ 2816/9x6 M6(x)=1+6x+18x2+36x3+54x4+ 324/5x5+ 130896/5x6. We obtained these bounds by using the Taylor’s expansion of the corresponding functions. Then, for those x∈[0,1) such that P6(x)>0,we have that d8 dx 8∆(x)e−3x≤P0(x)+P2(x)m2(x)+P4(x)M4(x)+P5(x)m5(x)+P6(x)M6(x)e−3x =−10305703872 5x14 −219880525104 5x13 −5346920434616 15 x12 −6966948024432 5x11 −41103195192544 15 x10 −12590764760928 5x9 −3775600126304 5x8+699698439904 5x7+280999239968 5x6 −16332290304 5x5−1090396800 x4−218725632 x3 −14708736 x2+ 6773760 x−1128960e−3x 17 and, by applying again Sturm’s algorithm, we can assert that this polynomial is negative on [ 0,1). Finally, for those x∈[0,1) such that P6(x)<0 it follows that d8 dx 8∆(x)e−3x≤P0(x)+P2(x)m2(x)+P4(x)M4(x)+P5(x)m5(x)+P6(x)m6(x)e−3x =−25509168 5x14 −569704752 5x13 −14926114808 15 x12 −22105379952 5x11 −165175393504 15 x10 −81925377888 5x9 −79806672224 5x8−58413037856 5x7−34880542432 5x6 −16332290304 5x5−1090396800 x4−218725632 x3 −14708736 x2+ 6773760 x−1128960e−3x and then, again by means of Sturm’s algorithm, it can be shown that this polynomial is negative on the interval [ 0,1).This concludes the proof of the result. Remark 4.9 In Sections ?? and ?? we considered the cases c∈(−1,0) and c∈(−∞,−1) respectively. So it remains to study c=−1.In this case, from (??), one can check that µ2(x)= e2x 12c2x2(c−x)22x3(x+1) 3e−3x+6(x+1) 3e−2x−6−6x+6x2. This is precisely the function D4introduced at the beginning of Section ??, and the combination of Lemmas ?? and ?? show that it is positive on (−1,+∞)\{0}. The proof of Case II, which clearly contrasts with the simplicity of the statement, is then finished. Of course, the choice of gwe have made at the beginning of this section to obtain µ2is the best we have been able to get, but it does not eliminate the possibility of finding another g that makes µ2>0 more evident. This could provide, then, a shorter and more understandable proof. 5 Proof of Proposition ?? For any m≥3,the family of systems with homogeneous nonlinearities (13)    ˙x=y, ˙y=−x+ [m/2]  i=0 αixm−2iy2i, 18 has a reversible center at the origin. It is worth to mention that in [?] the authors conjecture that the centers of (??) are those systems which are invariant under the changes either (x, t)→ (−x, −t) or (y,t)→ (−y,−t).System (??) corresponds to the second case and, in polar coordinates (after changing the sign of the time for convenience), it writes as (14) ˙ R=−f(θ)Rm, ˙ θ=1−g(θ)Rm−1, where f(θ)=h(θ)sinθand g(θ)=h(θ)cosθwith h(θ)=         m/2  k=0 β2kcos(2kθ),if mis even, [m/2]  k=0 β2k+1 cos(2k+1)θ,if mis odd. The coefficients βi∈Rabove can be easily obtained from the coefficients αiof the initial system. A classical tool to simplify the study of (??) is to transform it into an Abel equation (see [?]) by means of the change r=Rm−1 1−g(θ)Rm−1. In our situation one can verify that the Abel equation that we obtain is dr dθ =(1−m)f(θ)g(θ)r3+g(θ)−(m−1)f(θ)r2, which, in terms of h, writes as dr dθ =A(θ)r3+B(θ)r2, where A(θ):= (1−m)sin(θ) cos(θ)h2(θ)andB(θ):= cos(θ)h(θ)−msin(θ)h(θ).Note that if r(θ;ρ) is a solution of this equation with initial condition r(0; ρ)=ρ, then r(θ;ρ)=ρ+∞  k=2 uk(θ)ρk for some functions ukthat can be obtained recursively. For instance, u2(θ)=θ 0 B(ψ)dψ and u3(θ)=θ 0A(ψ)+2B(ψ)u2(ψ)dψ. Now, from the second equation in (??) and using variables (r, θ) again, we obtain the following expression for the period function of the center at the origin of (??): T(ρ)=2π 0 dθ 1−g(θ)Rm−1=2π 01+g(θ)rdθ =2π+2π 0 cos(θ)h(θ)ρ+ k≥2 uk(θ)ρkdθ. We conclude therefore that T(ρ)=2π+i≥1Tiρiwith T1=2π 0 cos(θ)h(θ)dθ and Tk=2π 0 cos(θ)h(θ)uk(θ)ρkdθ, for k≥2. 19 In order to show that there are parameters for which the period function of the center at the origin of (??) is not monotonic, we study the cases meven and modd separately. In the first case, i.e., m=2nwith n≥2,we take h(θ)=acos(2θ)+cos(mθ).Let us assume first that n≥3.Then, by using the above formulas, some tedious computations show that T1=0, T2=−1 3 π(2a2n2−3a2n−3n−2a2) 1+2n, T3|T2=0 =0, T4|T2=0 =πn(32n6−102n5−59n4+ 396n3−299n2−132n+2) 8(n−2)2(n+ 2)(2 n+1) 3. It can be checked that the polynomial 32n6−102n5−59n4+ 396n3−299n2−132n+ 2 has all its real roots smaller than 3 and, consequently, for all n≥3,we can assert that T4>0 when T2=0. To show that there exist critical periods we first observe that T2vanishes at a±=±3n 2n2−3n−2 and that, on the other hand, ∂T2 ∂a a=a± =∓2πn(n−2) 3(2n−1). Note that the bifurcation values a±and their respective derivatives are well defined for all n≥3. Thus, taking for instance aa+,we will have that T2<0andT4>0.Therefore, for this parameter, the corresponding period function has at least one local minimum. Let us study next the case n= 2 (i.e., m= 4). One can verify that in this case T1=0,T 2=π2 5+a 2,T 3|T2=0 = 0 and T4|T2=0 =−66 625 π. Taking a−4/5 we will have that T2>0andT4<0 so that the period function has at least one local maximum. Finally, for an odd m, i.e., m=2n+ 1 with n≥1,we choose h(θ)=acos θ+ cos(mθ). In this case one can verify that T1=aπ and T2|T1=0 =π(2n+1)4(n+1)>0. It is clear then that, for a0,T 1<0andT2>0 so that the period function has at least one local minimum. This concludes the proof of Proposition ??. References [1] J. Chavarriga, M. Sabatini A survey of isochronous centers, Qual. Theory Dyn. Syst. 1 (1999), 1–70. 20 [2] L.A. Cherkas. Number of limit cycles of an autonomous second-order system. Differ. Eq. 5 (1976), 666–668. [3] C. Chicone. The monotonicity of the period function for planar Hamiltonian vector fields,J. Differential Equations 69 (1987), 310–321. [4] C. Chicone. Geometric methods for two-point nonlinear boundary value problems, J. Differential Equations 72 (1988), 360–407. [5] C. Chicone. Review in MathSciNet, ref. 94h:58072. [6] C. Chicone, M. Jacobs. Bifurcations of critical periods for plane vector fields, Trans. Amer. Math. Soc. 312 (1989), 433–486. [7] W.A. Coppel, L. Gavrilov. The period function of a Hamiltonian quadratic system, Differential Integral Equations 6 (1993), 1357–1365. [8] E. Freire, A. Gasull, A. Guillamon. First derivative of the period function with applications, Preprint (2001). [9] W.S. Loud. Behavior of the period of solutions of certain plane autonomous systems near centers, Contributions to Differential Equations 3 (1964), 21–36. [10] V. Lunkevich, K. Sibirskii. Integrals of a general quadratic differential system in cases of a center, Differential Equations 18 (1982), 563–568. [11] I.I. Pleshkan. A new method of investigating the isochronicity of a system of two differential equations, J. Differential Equations, 5 (1969) 796–802. [12] C. Rousseau, B. Toni. Local bifurcations of critical periods in the reduced Kukles system, Canad. J. Math. 49 (1997), 338–358. [13] M. Sabatini. Characterizing isochronous centres by Lie brackets, Diff. Eq. Dyn. Sys., 5, 91–99 (1997). [14] D. Schlomiuk, Algebraic and geometric aspects of the theory of polynomial vector fields, in “Bifurcations and periodic orbits of vector fields” (Montreal, PQ, 1992), 429–467, Kluwer Acad. Publ., Dordrecht, 1993. [15] M. Urabe. Potential forces which yield periodic motions of a fixed period,J.Math.Mech.10 (1961), 569–578. [16] M. Villarini. Regularity properties of the period function near a centre of a planar vector field, Nonlinear Analysis T.M.A. 19 (1992), 787–803. [17] E. P. Volokitin and V. V. Ivanov. Isochronicity and commutation of polynomial vector fields, Sibirsk. Mat. Zh. 40 (1999), 30–48. (translation in Siberian Math. J. 40 (1999), 23–38.) [18] J. Waldvogel. The period in the Lotka-Volterra system is monotonic, J. Math. Anal. Appl. 114 (1986), 178–184. 21