GLOBAL PHASE PORTRAITS OF SOME REVERSIBLE CUBIC CENTERS WITH NONCOLLINEAR SINGULARITIES

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1 This is a preprint of: Global phase portraits of some reversible cbic centers with noncollinear singlarities, Magalena Cabergh, Joan Torregrosa, Internat. J. Bifr. Chaos Appl. Sci. Engrg., vol. 23(9), (30 pages), DOI: [ /S ] GLOBAL PHASE PORTRAITS OF SOME REVERSIBLE CUBIC CENTERS WITH NONCOLLINEAR SINGULARITIES M. CAUBERGH AND J. TORREGROSA Abstract. The reslts in this paper show that the cbic vector fiels ẋ = y + M(x, y) y(x 2 + y 2 ), ẏ = x + N(x, y) + x(x 2 + y 2 ), where M, N are qaratic homogeneos polynomials, having simltaneosly a center at the origin an at infinity, have at least 61 an at most 68 topologically ifferent phase portraits. To this en the reversible sbfamily efine by M(x, y) = γxy, N(x, y) = (γ λ)x 2 + α 2 λy 2 with α, γ R an λ 0, is stie in etail an it is shown to have at least 48 an at most 55 topologically ifferent phase portraits. In particlar, there are exactly 5 for γλ < 0 an at least 46 for γλ > 0. Frthermore, the global bifrcation iagram is analyze. 1. Introction This paper finishes the topological classification of the global phase portraits on the Poincaré isc of the 6-parameter family of cbic ifferential eqations ẋ = y + ax 2 + bxy + cy 2 y(x 2 + y 2 ), ẏ = x + ex 2 + fxy + gy 2 + x(x 2 + y 2 ), that have simltaneosly a center in the origin an at infinity for a, b, c, e, f, g R. This sty is starte in [8, 9]. Recently relate analysis are one in [3, 4, 5, 7, 11, 12]. In particlar, in [12], the so-calle SIS-moel is consiere, that is se in the sty of infectios iseases. The papers [3, 4, 5, 7, 12] contribte to the classification of planar qaratic ifferential systems; e to the 6-imensional parameter an the richness of phase portraits, its bifrcation iagram also is stie for intrinsic sbclasses recing its imension, an similarly these sb-bifrcation iagrams then are analyze by slicing an imbeing in projective planes. Frthermore another generalization can be fon in [11], where a global topological classification is stie for a 1-parameter cbic Hamiltonian planar ifferential system in which a finite center is linke to singlarities at infinity. In [6] the ifferential systems (1) are characterize by a Hamiltonian class an a reversible class, that is symmetric with respective to straight lines. In [8] an [9] respectively the classification is obtaine for the fll Hamiltonian class an part of the reversible class, i.e the ones having infinitely many singlarities or all singlarities on the line of symmetry. The classification of the remaining vector fiels of (1) is sbject (1) Mathematics Sbject Classification. Primary 34C23; Seconary 34C25, 34C37, 37G10, 37C27, 34C29, 34C05. Key wors an phrases. reversible planar vector fiels, cbic vector fiels, global classification of phase portraits, bifrcation iagram. Partially spporte by the MINECO/FEDER grant nmber MTM an by the AGAUR grant nmber 2009SGR 410. The first athor also is spporte by the Ramón y Cajal grant nmber RYC

2 2 M. CAUBERGH AND J. TORREGROSA of this paper. These can be represente by the 3-parameter family Y (α,γ,λ) given by ẋ = y γxy y(x 2 + y 2 ), ẏ = x + (γ λ)x 2 + α 2 λy 2 + x(x 2 + y 2 ), where α, γ, λ R an λ 0. These vector fiels are reversible becase they are invariant with respect to the transformation (x, y, t, α, γ, λ) (x, y, t, α, γ, λ) an hence their phase portraits are symmetric with respect to the horizontal axis when reversing time. It is fon in Section 2 that the analysis can be restricte to the semi-algebraic set P = {(α, γ, λ) R 3 : α > 0, γ > 0, λ 0}. (3) The systematic classification here obtaine for (2) with respect to the nmber of singlarities is smmarize in the following theorem. Recall that isolate perioic orbits are so-calle limit cycles. Theorem 1. The sbfamily of cbic reversible vector fiels Y (α,γ,λ), given in (2) with α, γ, λ R an λ 0, has at most 55 topologically ifferent global phase portraits of which at least 48 are realize, an that are rawn in Figres 1, 15, 16, 17, 20, 21, 22, 24, 25 an 28. More precisely, if the lower bon resp. pper bon for the nmber of topologically ifferent phase portraits having i singlarities is enote by L i resp. U i, i N, then 7 i=1 L i = 48 an 7 i=1 U i = 55, where L = (1, 1, 4, 3, 10, 8, 21) an U = (1, 1, 4, 6, 13, 9, 21). In particlar, the nmber n of singlarities is finite with n {1, 2,..., 7} an there are no limit cycles. Notice that the figres mentione in previos theorem only show the phase portraits that constitte the lower bon. Frthermore these phase portraits are grope first accoring to the nmber of singlarities an secon accoring to local bifrcation regions. Moreover the specific orer an composition of the phase portraits respects the sbseqent global bifrcation phenomena that happen between ajacent phase portraits an is etaile for each case in the corresponing section. This also is the reason why some topologically eqivalent phase portraits appear. Aing reslts from [9] to Theorem 1 to eal with the case λ = 0 leas to the classification of the fll 3-parameter family Y (α,γ,λ), by aing one more phase portrait that has infinitely many singlarities. Corollary 2. The cbic reversible vector fiels Y (α,γ,λ), given in (2) with α, γ, λ R, has at most 56 topologically ifferent global phase portraits of which at least 49 are realize. In particlar, the nmber n of singlarities is finite if an only if λ 0 or λ = 0, γ 2. Frthermore, none of these phase portraits has limit cycles. Denote by F the qotient space of all phase portraits of (1) with respect to topological eqivalence. Consier the sbsets F H F an F R F that correspon to the phase portraits of the Hamiltonian an reversible class of (1) respectively. Write S to enote the carinal nmber of S, an hence corresponing to the topologically ifferent phase portraits. Then, combining Theorem 1 an the reslts from [8, 9] leas to the following classification for the fll family (1). Corollary 3. The 6-parameter family (1) having simltaneosly a center at the origin an at infinity inces 61 F 68 topologically ifferent phase portraits. In particlar, F H = 22, 53 F R 60 an F H F R = 14. Frthermore, none of these phase portraits present limit cycles an phase portraits in S have maximally n P A (S) (2)

3 REVERSIBLE CUBIC CENTERS 3 isjoint perio annli, where n P A (F) = n P A (F H ) = 7, n P A (F R ) = n P A (F H F R ) = 6 an n P A (F R \ F H ) = 4. In stying the topologically ifferent global phase portraits a great role is playe by the bifrcation iagram of the global phase portraits in terms of the parameter (α, γ, λ) P. To simplify the reaing we se, whenever no confsion is possible, the notation ζ = (α, γ, λ). Throghot this paper we shortly speak of the local (resp. global) bifrcation iagram referring to the bifrcation iagram of the local (resp. global) phase portraits with respect to topological eqivalence in terms of the parameter ζ. In this, local means near the singlarities. As sch local (resp. global) bifrcation srfaces an crves are the bifrcation srfaces an crves ealing with local (resp. global) bifrcation phenomena. First, in Section 3, we sty the local bifrcation iagram. Next the global bifrcation iagram is obtaine by aing global bifrcation srfaces to it, that are etermine by crossing of separatrices. In the transition from λ < 0 to λ > 0 the global center of the harmonic oscillator is istrbe by a circle of singlarities at 1+γx+x 2 +y 2 = 0 (see [9] an Proposition 29). De to their ifferent natre, the cases λ < 0 an λ > 0 are treate separately. To sty this 3-imensional bifrcation iagram systematically we consier slices of P for fixe λ, that we enote by P λ = {ζ : α, γ > 0}. Then characteristic slices of the global bifrcation iagram in P λ are stie in Sections 6 an 7 for λ < 0 an λ > 0 respectively. With increasing λ sbseqently more bifrcation srfaces are encontere. Frthermore, oppose to the case λ > 0, in case λ < 0 there are no Hamiltonian phase portraits an the local bifrcation phenomena etermine the global ones. This is illstrate by the bifrcation iagrams in Figre 2 for λ < 0 an in Figres 7, 32 an 33 for λ > 0. In Section 3 it is seen that the local bifrcation iagram is etermine by the following semi-algebraic sbsets of P = P P +, where P = P {λ < 0} an P + = P {λ > 0} : F = {γ > λ + 2}, E = {γ > λ 2}, F = {γ = λ + 2}, E = {γ = λ 2}, F = {γ < λ + 2}, E = {γ < λ 2}, G l = {α 2 γ 2 4(α 2 + 1) < 0}, H l = {2α 2 λ γ < 0}, G = {α 2 γ 2 4(α 2 + 1) = 0}, H = {2α 2 λ γ = 0}, G r = {α 2 γ 2 4(α 2 + 1) > 0}, H r = {2α 2 λ γ > 0}. Frthermore we se the notation E λ, F λ an H λ for the slices obtaine by intersecting the corresponing bifrcation srfaces with P λ for fixe λ 0. In Lemma 6 it is fon that the Hamiltonian reversible vector fiels Y ζ correspon with the parameter ζ H. A characteristic slice of H λ is non-empty only for λ > 0 an its projection in the (α, γ)-plane is presente in Figre 6. The global bifrcation iagram restricte to H is etaile in Section 5 sing reslts from [8]; in particlar it is shown in Figre 9 an the corresponing phase portraits are shown in Figre 8. Using the sbsets introce in (4) we now state the main reslts of this paper, istingishing between λ < 0 (i.e., in P ) an λ > 0 (i.e., in P + ). Theorem 4. The family (2) restricte to P has exactly 5 topologically ifferent global phase portraits, that are rawn in Figre 1. Frthermore the global phase portrait of Y ζ is niqely etermine by the nmber n of singlarities; in particlar, p to topological (4)

4 4 M. CAUBERGH AND J. TORREGROSA eqivalence, the phase portrait is eqal to n with n {1, 2, 3, 5, 7}. For fixe λ < 0 characteristic slices of the global bifrcation iagram are shown in Figre Figre 1. Phase portraits of (2) in P, see Theorem 4. λ 2 2 < λ < 0 Figre 2. Characteristic slices of the local an global bifrcation iagram for (2) in P λ for fixe λ < 0 (see also Figre 1). Theorem 5. The family (2) restricte to P + has at most 53 topologically ifferent global phase portraits of which at least 46 are realize, an that are rawn in Figres 15, 16, 17, 20, 21, 22, 24, 25 an 28. More precisely, if the lower bon resp. pper bon for the nmber of topologically ifferent phase portraits having i singlarities is enote by L + i resp. U + i, i N, then 7 i=1 L+ i = 46 an 7 i=1 U + i = 53, where L + = (1, 1, 4, 3, 9, 8, 20) an U + = (1, 1, 4, 6, 12, 9, 20). For λ > 0 the bifrcation iagram as well as the notation for the global phase portraits is more involve than for λ < 0. Typically the notation then is of the form n p q where n refers to the total nmber of singlarities an the symbols that appear in the inices p an q ifferent from 0, +,, 1, refer to the local bifrcation phenomena (i.e. appearance an isappearance of the singlarities an their local behavior). The fll notation is explaine in Section 3.2 relate to the local bifrcations an in Section 7 relate to the global ones. The paper is organize as follows. The parameter space is rece to P in Section 2 an the local bifrcation iagram is analyze in Section 3. In Section 4 it is shown that (2) has no limit cycles an conitions for homoclinic or heteroclinic orbits to exist are fon. In Sections 5, 6 an 7 we systematically sty the global bifrcation iagram for (2) restricte to P H, P an P + respectively. In Section 5 the global

5 REVERSIBLE CUBIC CENTERS 5 bifrcation iagram for the Hamiltonian reversible sbfamily is obtaine. Notice that we alreay ealt with this problem in [8] presenting the topological classification of the phase portraits bt not the bifrcation iagram. Here we recall this reslt an rewrite it in terms of ζ in Theorem 17, aing the bifrcation iagram in Figre 9. In particlar nfoling the Hamiltonian vector fiels in (2) is one of the key tools se to prove the existence of nearby global phase portraits of (2) for λ > 0. Next in Sections 6 an 7 the cases λ < 0 an λ > 0 are ealt with respectively an in particlar we prove Theorems 4 an 5. The case λ > 0 is ealt with systematically epening on the nmber of singlarities that are present. First we obtain rather irectly the classification for the cases of one, two an three singlarities, that is state in Proposition 19. De to increasing complexity the cases of seven, five, six an for singlarities are sbseqently ealt with in Sections 7.2, 7.3, 7.4 an 7.5. Section 7.2 treats the case when the nmber of singlarities is maximal, inepenently analyzing all possible global phase portraits, which are liste in Proposition 20. Then the other cases each time are consiere as bifrcation of previosly treate cases an the key in proving the existence of possible phase portraits is base on argments of continity. This is the reason why first the cases of seven an five singlarities are stie an then sbseqently the cases of six an for singlarities. Finally in Section 8 we combine Theorems 4 an 5 an exten them to the bonary of P ± sing reslts from [8, 9]. In this way, reslts for the fll parameter space are obtaine that prove Theorem 1 an Corollaries 2 an 3. The sty is mainly one sing classical techniqes in qalitative theory of ifferential eqations in the plane (see [2, 10]). To be sre to cover all global phase portraits, the parameter space is examine systematically by the possible bifrcations. Althogh some of these bifrcations reslt in topologically eqivalent ifferential systems, the corresponing phase portraits are explicitly rawn to stress on the bifrcation phenomena. Besies local also global bifrcations take place. In particlar a rigoros sty is one of how the separatrices starting from singlarities mtally cross an bifrcation methos are se to analyze near-hamiltonian cases. We en this introction by exposing part of the complexity of the problem consiere in this paper. From the qantitative part of view, the nmber of topologically ifferent phase portraits is srprisingly small, compare to the nmber that one obtains by combinatorial comptation after a mere sty near the singlarities. For instance consiering phase portraits having seven singlarities there are 8 attracting an 8 repelling separatrices that can connect. Compting their possibilities to connect one fins 8! = ifferent phase portraits. Using the reversibility property an a concise analysis of the α- an ω-limit sets of the separatrices of the sale points reces this nmber to exactly 1 for P an 20 for P +. The total nmber of possible topologically ifferent phase portraits is rece to 5 for P an at most 53 for P +. Aing, in the case of P +, a carefl nmerical sty of the relative positions of the separatrices in case of for singlarities this total nmber frther reces to 50. Only 46 of these 50 phase portraits are trace when for a concrete seqence of λ-vales the parameter plane (α, γ) is ran throgh. In particlar in each of these case sties the nmber of phase portraits with i singlarities is N i, where 7 i=1 N i an N = L + (see Theorem 5). Frthermore the sty evelope in this paper of which the theoretic an nmerical reslts also are confirme by the compter software package P 4 (see [10]), reinforces the iea that the other 4 phase portraits o not occr whatever the vale of the parameter. Notice that as a conseqence, incling the nmerical sty mentione above for the

6 6 M. CAUBERGH AND J. TORREGROSA case of for singlarities, the 6-parameter family (1) can have at most 64 topologically ifferent phase portraits. Frthermore base on the case sties it seems that the lower bon in Corollary 3 is the exact total nmber of topologically ifferent phase portraits. 2. Parameter space As mentione before the ifferential eqations left to classify efine a sbfamily of the reversible vector fiels in (1). This sbfamily is fon to have the following normal form: { ẋ = y + (ξ 2g)xy y(x X(g,ξ,e) R 2 + y 2 ), ẏ = x + ex 2 + gy 2 + x(x 2 + y 2 (5) ), where the parameter coorinates ξ, g, e R correspon to the notation se in [8] an to the respective parameter coorinates a, b, c se in [9]; notice however that these coorinates a, b, c o not correspon to the ones that appear in (1). In [8] the vector fiel (5) is enote by X(g,ξ,e) R while in [9] by X (a,b,c). Here we recall the notation of [8] since we want to etail on a reslt abot the Hamiltonian reversible vector fiels that have been state there. However throghot this paper we work with another representation that simplifies both the statement of the reslts as well as the calclations. The global phase portraits of (5) having only collinear singlarities or infinitely many are classifie in [9], as is the case for parameter vales satisfying (2g ξ e)g 0 or 2g ξ = 0. For (2g ξ e)g > 0 an 2g ξ 0 the vector fiel X(g,ξ,e) R has at most seven singlarities that are generally not collinear, bt sprea over at most three lines passing throgh the origin. For a systematic sty of the corresponing phase portraits, it is convenient to introce the new parameter coorinates (α, γ, λ) an to sty Y (α,γ,λ) which is in one-to-one corresponence with X(g,ξ,e) R in case (2g ξ e)g > 0. In fact sing the transformations ( g ) T (g, ξ, e) = ξ 2g + e, ξ + 2g, ξ + 2g e for (ξ 2g + e)g < 0, an (6) Q(α, γ, λ) = (α 2 λ, 2α 2 λ γ, γ λ) for α > 0, λ 0, one has the following relations between the ifferent representations for (2), Y (α,γ,λ) = X R Q(α,γ,λ) an XR (g,ξ,e) = Y T (g,ξ,e) an X (a,b,c) = Y T (b,a,c), (7) that are of interest in Sections 3.1 an 5 where reslts are se from [9] an [8] respectively. The geometric meaning of the parameter (α, γ, λ) becomes clear in the sty of the singlarities of Y (α,γ,λ) in polar coorinates, efine by x = r cos θ, y = r sin θ. Then the vector fiel Y (α,γ,λ) is transforme into the form r = r 2 A(θ), θ = 1 + B(θ)r + r 2, for some cbic homogeneos trigonometric polynomials A an B with coefficients epening on the parameter (α, γ, λ); in particlar, A an B rea as A(θ) = λ sin θ(α sin θ cos θ)(α sin θ + cos θ), B(θ) = cos θ[(α 2 λ + γ) sin 2 θ + (γ λ) cos 2 θ]. From (8) an (9) the geometric meaning of the parameter α is clear. It represents the symmetric rays along which singlarities of Y (α,γ,λ) can be carrie. (8) (9)

7 REVERSIBLE CUBIC CENTERS 7 By (7) it follows that Y (0,γ,λ) = X R (0, γ,γ λ), Y ( α,γ,λ) = Y (α,γ,λ), Y (α,0,λ) = X R (α 2 λ,2α 2 λ, λ) an Y (α,γ,0) = X R (0, γ,γ). Hence to sty the phase portraits of Y (α,γ,λ) having finitely many noncollinear singlarities, it sffices to sty the vector fiels Y (α,γ,λ) for α > 0, γ 0 an λ 0. (10) Lemma 6. Let Y (α,γ,λ) be efine in (2). Then we have (1) Y (α,γ,λ) = Y ( α,γ,λ), for all α, γ, λ R. (2) Y (α,γ,λ) is invariant with respect to the transformations (x, y, t, α, γ, λ) ( x, y, t, α, γ, λ), (11) (x, y, t, α, γ, λ) (x, y, t, α, γ, λ). (12) The invariance property (12) is the one of reversibility: the phase portrait of Y (α,γ,λ) is symmetric with respect to the horizontal axis when reversing time. (3) Y (α,γ,λ) is Hamiltonian if an only if γ = 2α 2 λ. By (10) an (11) we can assme throghot this paper that α, γ > 0, it is to say (α, γ, λ) P. In fact, since the phase portraits of Y (α,γ,λ) with γ < 0 are linearly eqivalent to Y (α, γ, λ), the reslts in Theorems 4 an 5 are evenly vali when replacing P an P + respectively by {(α, γ, λ) : γλ < 0} an {(α, γ, λ) : γλ > 0}. 3. Singlarities In this section we sty the possible location an type of singlarities of Y ζ. From (8) an (9) caniate singlarities (x, y) = (r cos θ, r sin θ) of Y ζ satisfy sin θ = 0 or cot θ = ±α. Frthermore r is soltion of the qaratic eqation 1 + B(θ)r + r 2 = 0. Since α > 0 we can take the angle θ 1 for which 0 < θ 1 < π/2 an cot θ 1 = α. (13) In particlar cos(π θ 1 ) = α/ α an sin(π θ 1 ) = 1/ α 2 + 1; therefore B(0) = γ λ, A (π θ 1 ) = 2αλ α2 + 1 an B(π θ 1) = αγ α (14) Clearly, for ζ fixe, it follows from the reversibility property an (14) that the origin is a center of Y ζ an all the other singlarities appear along the rays R 0 an R ±, that are efine as R 0 = {(x, 0) : (λ γ)x > 0}, (15) R ± = {(r, θ) : θ = π θ 1, r > 0} = {(x, y) : y = x/α, x < 0}. Notice that none of the rays contain the origin an by the choice γ > 0 the rays R ± are containe in the negative x-plane. From [8] we know that there are no singlarities at infinity. These observations are smmarize in the following lemma. Lemma 7. Let ζ P an R 0, R ± be efine in (15). Then the following properties hol. (1) The vector fiel Y ζ has at most seven singlarities of which one is the center at the origin an all singlarities are finite. (2) There are at most three rays that carry singlarities of Y ζ, namely the rays R 0 an R ±. Each of them carry at most two singlarities; in particlar the singlarities on R + an R are symmetric with respect to the horizontal axis.

8 8 M. CAUBERGH AND J. TORREGROSA (3) Let the configrations of singlarities on the rays R 0 an R ± be enote by (i, j), 0 i, j 2 if Y ζ has 1 + i + 2j singlarities in the global phase portrait, of which i lie along R 0 an j along R + (as many as along R ). The following configrations are possible: 1 + i + 2j (i, j) (0, 0) (1, 0) (2, 0), (0, 1) (1, 1) (2, 1), (0, 2) (1, 2) (2, 2) The precise location an topological type of the singlarities along R 0 (resp. along R ± ) are escribe in Proposition 8 (resp. Propositions 9, 10 an 11) Singlarities along R 0. Recall that R 0 is the horizontal ray as efine in (15) an that in [9] singlarities along R 0 are completely analyze for X (a,b,c) efine in (7). Then the topological type of the singlarities of Y ζ along R 0 epens only on γ λ an is state in the proposition below. Proposition 8. Let ζ P. Then the local phase portrait of (2) along R 0 is sketche in Figres 3 an 4 an is escribe as follows: (1) If ζ E F, then there are no singlarities along R 0. (2) If ζ E F, then there is only a csp singlarity at s 0 = ( sgn(γ λ), 0). Its irection is etermine by the sign of λ. (3) If ζ E F, then there are two singlarities, given by s 0 ± = ( sgn(γ λ)r±, 0 0), where r± 0 is the fnction of γ λ : r± 0 = γ λ ± (γ λ) Frthermore, s 0 σ 1 is a sale an s 0 σ 1 is a center where σ 1 = sgn(λ)sgn(γ λ). We concle that 5 ifferent possibilities for the local phase portraits along the horizontal axis are istingishe; restricting to P only 3 of them are encontere since P {γ > 0}. This is presente in Figres 3 an 4 in which frthermore the relative position of the separatrices at the singlarities of sale or csp type with respect to R 0 is shown. To istingish ifferent separatrices at a singlarity s along R 0 of sale or csp type, we enote the stable an nstable manifols at s by W(s) an U(s) respectively. Moreover we introce the sbsets W I (s) W(s) \ {s} an U I (s) U(s) \ {s} for the respective sbsets having their germ at s incle in {(x, y) : y < 0}; analogosly we introce W E (s) W(s) \ {s} an U E (s) U(s) \ {s} for the respective sbsets having their germ at s incle in {(x, y) : y > 0}. This notation is se in Figres 12, 13, 14, 18, 19 an 29, where the ifferent possible relative positions for the stable an nstable manifols are sketche Singlarities on R ±. Recall that R ± are the symmetric rays efine in (15). De to the symmetry of Y ζ we only nee to sty the singlarities on R +. The singlarities on R an their type can be obtaine irectly sing the reversibility property. Moreover, the singlarities occr in pairs: singlarities on R + are reflecte on R with respect to the x-axis, an viceversa. The nmber, location an topological type of the singlarities of Y ζ along R + are escribe in Propositions 9, 10 an 11, where the local bifrcation iagram is fon to be etermine by the srfaces G,G l, G r, H, H l an H r as introce in (4). The following proposition follows from straightforwar calclations.

9 REVERSIBLE CUBIC CENTERS 9 F s 0 + s 0 0 F s 0 0 F 0 Figre 3. F, F, F form a partition of P efine by ifferent configrations of singlarities along R 0, see Proposition 8. F s 0 + F s 0 0 s 0 0 E F 0 E 0 s 0 E 0 s 0 s 0 + Figre 4. F, F, F E, E, E form a partition of P + efine by the ifferent configrations of singlarities along R 0, see Proposition 8. Proposition 9. Let ζ P. (1) If ζ G l, then the vector fiel Y ζ has no singlarities on R +. (2) If ζ G, then the vector fiel Y ζ has one singlarity on R +, that we enote by s 1 = (x 1, y 1 ). In particlar, s 1 = 1/ α ( α, 1), r 1 = s 1 = 1, (16) α = x 1 /y 1 an γ = ( (x 1 ) 2 + (y 1 ) ) /x 1. (17) (3) If ζ G r, then the vector fiel Y ζ has two singlarities on R +, that we enote by s 1 ±. In particlar, s 1 ± = y1 ± ( α, 1), where y1 ± = αγ± α 2 γ 2 4(α 2 +1) = r1 ± 2(α 2, (18) +1) α 2 +1 r± 1 = s 1 ± = αγ± α 2 γ 2 4(α 2 +1), 0 < r 1 < 1 < r+, 1 (19) 2 α 2 +1 α = x 1 + /y1 + an γ = ( (x 1 + )2 + (y 1 + )2 + 1 ) /x 1 +. (20) Analogosly as in Proposition 9 the singlarities on R are enote by s 2 ± or s2, an by symmetry are etermine by s 2 = 1/ α ( α, 1) an s 2 ± = y1 ± ( α, 1).

10 10 M. CAUBERGH AND J. TORREGROSA Proposition 10. Let ζ G r P an let s 1 ± an y± 1 be as efine in (18). Then the singlarities s 1 ± are hyperbolic an are classifie as follows: s 1 s 1 + in G r P + sale stable noe or focs in H l, Hamiltonian center in H an nstable noe or focs in H r in G r P stable noe or focs sale The trace tr an eterminant D of the linearization of (2) at s 1 ± are given by tr(s 1 ± ) = (2α2 λ γ)y 1 ± an D(s1 ± ) = ±2αλ(y1 ± )2 α 2 γ 2 4(α 2 + 1). (21) In particlar sgn(tr(s 1 ±)) = sgn(2α 2 λ γ) an sgn(d(s 1 ±)) = ±sgn(λ). For λ 0 the tangent vectors to the separatrices at the sale point s 1 σ 2 where σ 2 = sgn( λ) are given by v 1,σ 2 ± = (2(yσ 1 2 ) 2, µ 1,σ 2 ± + (2αyσ 1 2 γ)yσ 1 2 ), an the irection of the flow on the corresponing separatrices is etermine by ( µ 1,σ 2 ± = γ (γ + 2α2 λ) ± 1 ) (γ + 2α 2 2 λ) 2 16yσ 1 2 α(α 2 + 1)λ yσ 1 2. Proof. The statement abot the singlarities s 1 ± in case γ > 2 α 2 + 1/α is base on the classification of the trace an eterminant at their linearizations; the formlas in (21) follow from irect calclation. To etermine the asymptotics of the separatrices at the sale point s 1 σ 2, one can calclate the eigenvales µ 1,σ 2 ± an corresponing eigenvectors v 1,σ 2 ± of the linearization J at the singlarity s 1 σ 2 : [ J(s 1 yσ σ 2 ) = 1 2 ( γ + 2αyσ 1 2 ) 2(yσ 1 2 ) 2 αyσ 1 2 (2αyσ λ γ) 2αyσ 1 2 ( yσ αλ) where we se the fact that x 1 σ 2 = αy 1 σ 2 an 1 + γx 1 σ 2 + (x 1 σ 2 ) 2 + (y 1 σ 2 ) 2 = 0. Frthermore to istingish between ifferent separatrices at the sale singlarity or sale-noe along R ±, we introce the open sets I an E sch that I E =, R 2 = I E (R + R {(0, 0)}), I {(x, 0) : x < 0} an E {(x, 0) : x > 0}. In Propositions 10 an 11 the relative position an tangency of the separatrices of the singlarities along R ± is etaile precisely like in Figres 5 an 6; in particlar in which region I or E they start. Moreover these regions are se to escribe possible connections between singlarities later in this paper. If the stable an nstable manifols at a sale or sale-noe s along R ± are enote by W(s) an U(s) respectively, then we enote by W I (s) W(s)\{s} an U I (s) U(s)\{s} the respective sbsets having their germ at s in I. Analogosly we enote by W E (s) W(s)\{s} an U E (s) U(s)\{s} for the respective sbsets having their germ at s in E. This notation is se in Figres 12, 13, 14, 18, 19 an 29. ], (22)

11 REVERSIBLE CUBIC CENTERS 11 γ G s 1 G r s 2 0 s 1 + s 1 s 2 + s G l Figre 5. G l, G, G r form a partition of P efine by ifferent configrations of singlarities along R ±, see Propositions 9, 10 an 11. α γ G s 1 + s 1 H H r G r s 2 + s 2 s s 1 H l G r s 1 + s 2 + s 1 s s 1 s 2 0 s 2 + s 2 0 s 1 s 2 0 s s 2 G l α Figre 6. G l, G H r, G H, G H l, G r H r, G r H, G r H l form a partition of P + efine by ifferent configrations of singlarities along R ±, see Propositions 9, 10 an 11.

12 12 M. CAUBERGH AND J. TORREGROSA Proposition 11. Let ζ G P an let r, s 1 = (x 1, y 1 ), I an E be efine in (16) an (22). Then γ = 2 α 2 + 1/α an the trace tr an eterminant D of the linearization of (2) at the singlarity s 1 are given by tr(s 1 ) = 2α 2 (λ l α )/ α = 2(λ(x 1 ) 3 + (y 1 ) 2 )/(x 1 y 1 ) an D(s 1 ) = 0, where α2 + 1 l α = > 0. (23) α 3 Then, (1) H G P = {(α, 2α 2 l α, l α ) : α > 0} = {(α 0 (λ), 2(α 0 (λ)) 2 λ, λ) : λ > 0}, where the analytic crve α 0 : (0, ) (0, ) is efine by the positive soltion α 0 (λ) of λ 2 α 6 α 2 1 = 0. In particlar, if ζ H G P, then s 1 is a nilpotent csp whose separatrices can asymptotically be parameterize by ( α α , 1 α α 2 + ϕ ± (α) 3 + O( 4 ) ), 0, (24) where ϕ ± (α) = ±(α 2 + 1) 5/4 / 3α 3 = ±1/( 3(x 1 ) 3 y 1 ). (2) If ζ (G P) \ H, then s 1 is a semi-hyperbolic sale-noe, whose separatrices can asymptotically be parameterize by (x 1 + 2, y 1 + φ 1 (α) 2 + φ 2 (α, λ) 4 + O( 5 )), for 0 an (25) (x 1 + 2, y 1 + φ 3 (α, λ) 2 + O( 3 )), for 0, (26) where φ 1 (α) = 1/α = y 1 /x 1, φ 2 (α, λ) = αl 2 α /[2(λ l α)] = (y 1 ) 3 /[2(x 1 ) 2 (λ(x 1 ) 3 + (y 1 ) 2 )] an φ 3 (α, λ) = α(1 αλ α 2 + 1) = x 1 (λx 1 + (y 1 ) 2 )/(y 1 ) 3. The relative position an tangency of the separatrices (24), (25) an (26) with respect to R +, I an E, an the irection of the flow on these separatrices is as in Figre 5 for λ < 0 an Figre 6 for λ > 0. Proof. Since the fnction (0, ) (0, ) : α l α efine by (23) is strictly ecreasing with α, it has an analytic inverse, that we call by α 0. As a conseqence, along the crve γ = 2 α 2 + 1/α, there is a niqe α 0 sch that l α0 = α /α0. 3 For all ζ G H l = G {α < α 0 } it follows that λ > l α0 an for all ζ G H r = G {α > α 0 } it follows that λ < l α0. To analyze the topological type of the singlarity s 1 in P G, we se [9] an Topological Normal Form Theorems for semi-hyperbolic an nilpotent singlarities (see e.g. [1, 10]). Let ζ P G be fixe. A irect calclation shows that therefore J(s 1 ) = 1 α [ 2/α 2 2(λα α ) 2α(λα α ) µ = tr(s 1 ) = 2λα2 α α an D(s1 ) = 0. If λ l α, then µ 0 an hence s 1 is semi-hyperbolic. Notice that when λ < 0 this is always the case an then µ < 0. If λ = l α, then s 1 is nilpotent an nees a separate sty. ] ;

13 REVERSIBLE CUBIC CENTERS 13 Sppose first that λ l α, then we rece (2) to the topological normal form for semi-hyperbolic singlarities, i.e. = f(, v), v = µv + g(, v), where f an g are O ( (, v) 2 ) for (, v) 0, by the transformation ) [ ] (, v) = C (x 1 α + α2 + 1, y 1 α 1, where C = α α(1 λα α ) For any crve C along which v vanishes ientically, its asymptotics for 0 is given by v = 1 2 α(α2 + 1) 3/2 ( α 3 λ + α 2 + 1) O( 3 ). As sch the asymptotics of f restricte to C is given by ρ 2 + O( 3 ), 0, with ρ = α 3 λ(α 2 + 1) α α 2 + 1(α 3 λ α 2 + 1). For λ < 0 it follows that ρ > 0 an µ < 0. For λ > l α one has ρ, µ > 0 an for 0 < λ < l α one has ρ, µ < 0. From the Topological Normal Form Theorem for semihyperbolic singlarities it then follows that s 1 = (x 1, y 1 ) is a sale-noe with the local phase portraits rawn in Figres 5 an 6. Frthermore it can easily be checke that the separatrices at s 1 in the coorinates (x, y) has the asymptotics given in (25) an (26). Next sppose that λ = l α an hence s 1 = (x 1, y 1 ) is a nilpotent singlarity. Then we rece (2) to the stanar normal form for these singlarities, = v + f(, v), v = g(, v), where f(, v) = g(, v) = O ( (, v) 2 ), (, v) 0, by the transformation (x, y) C 1 (x + α α2 + 1, y 1 α2 + 1 ), where C = [ α 1 2 α2 (α 2 + 1) 1 0 For any crve C along which vanishes ientically, its asymptotics for 0 is given by v = α α O( 3 ). As sch the asymptotics of g restricte to C is given by ρ 2 + O( 3 ), 0, with ρ = 2 α α 2 < 0 an f(, v) + g(, v) v = 0. (,v) C Therefore from the Topological Normal Form Theorem for nilpotent singlarities it follows that s 1 is a csp in case λ = l α (see Figre 6). It can easily be checke that the separatrices at s 1 in the coorinates (x, y) have the asymptotics given in (24). To verify the relative position of the separatrices of the sale-noe an its ynamical behavior for λ l α, we notice that (26) can locally be written as y = y 1 + φ 3 (x x 1 ) + O((x x 1 ) 2 ), x x 1, an hence the behavior on it is asymptotically given by ẏ = µ(y y 1 ) + O((y y 1 ) 2 ), y y 1. For λ < l α (resp. λ > l α ) the hyperbolic behavior is attracting (resp. repelling). Frthermore it follows that the non-hyperbolic behavior of the sale-noe happens along the separatrix with parametrization (25). This separatrix is tangent to R + at s 1 an locally near s 1 concave p (resp. concave own) for λ > l α (resp. λ < l α ). To etermine the sense of the flow on the separatrices etermine by (25) an (26) we consier the sign of ṙ for θ near π θ 1 : sgn(ṙ) = sgn(λ)sgn(θ 1 θ) for θ π θ 1, ].

14 14 M. CAUBERGH AND J. TORREGROSA since cot(π θ 1 ) = α. Taking this into accont we fin the sale-noes as rawn in Figres 5 an 6. Similarly the relative position of the separatrices of the csp an its ynamic behavior is fon as rawn in Figre Local bifrcation iagram. In this section we combine the reslts from Sections 3.1 an 3.2 to obtain the local bifrcation iagram in P. It is ths forme by the bifrcation srfaces E, F, G an H restricte to P as efine in (3) an (4), i.e. srfaces passing throgh which the nmber an/or the topological type of singlarities change. All characteristic slices of the local bifrcation iagram for fixe λ < 0 (resp. λ > 0) are rawn in Figre 2 (resp. Figre 7); the relative position of the bifrcation crves is jstifie by Lemma 12 below. Let π 1, π 2 an π 3 enote the canonical projections π 1 (α, γ, λ) = (γ, λ), π 2 (α, γ, λ) = (α, λ) an π 3 (α, γ, λ) = (α, γ). A characteristic slice of the bifrcation iagram P λ is compose by for bifrcation crves E λ, F λ, G λ an H λ. Passing throgh these crves the singlarities on the horizontal axis appear or isappear throgh a csp or split into a center an a sale, epening on the relative orer or mtal intersections of the bifrcation srfaces that are given in Lemma 12. Frthermore the nmber an topological type of the singlarities along R ± are controlle by the bifrcation crves π 3 (H λ ) an π 3 (G λ ). Both can be written as graphs γ = γ(α), α > 0. The latter one contains the parameter vales (α, γ) for which the singlarity of Y ζ along R ± generically is a sale-noe an the other crve contains parameter vales (α, γ) for which Y ζ is Hamiltonian. Lemma 12. Let P, E, F, G, H be as efine in (3) an (4). (1) H λ = if λ < 0. (2) π 3 (G λ ) is ientical for all λ > 0 : π 3 (G λ ) = π 3 (G); as the graph γ = 2 α 2 + 1/α, α > 0, it is strictly ecreasing with respect to α an it has a horizontal asymptote γ = 2 for α an a vertical asymptote at α = 0. (3) H λ0 G λ0 is empty for λ 0 < 0 an a singleton {(α 0, γ 0, λ 0 )} for λ 0 > 0. Frthermore, the crves α 0 : (0, ) (0, ) an γ 0 : (0, ) (2, ) are analytic an strictly ecreasing resp. increasing sch that α 0 = α(λ 0 ) is the positive soltion of α 3 0λ 2 0 α = 0 an γ(λ 0 ) = α 2 0(λ 0 )/α 0 (λ 0 ) with lim γ 0(λ 0 ) an λ 0 0 lim γ 0(λ 0 ) =. λ 0 (4) The intersection E P λ is nonempty if an only if λ 2. In particlar, (a) π 3 (E λ ) < π 3 (G λ ) if 2 λ 4. Eqivalently, for all α > 0 an γ e, γ g 0 with (α, γ e, λ) E λ an (α, γ g, λ) G λ, it hols that γ e 2 < γ g. (b) E λ G λ = {(α + (λ), γ (λ), λ)} for some α + (λ) > 0 an γ (λ) 0 sch that γ (λ) < γ 0 (λ) if 4 < λ < 6. (c) E 6 G 6 = {(1/ 3, 4, 6)}. () E λ G λ = {(α (λ), γ + (λ), λ)} for some α (λ) > 0, γ + (λ) > 0 sch that γ 0 (λ) < γ + (λ) if λ > 6. (5) The intersection F λ is nonempty if an only if λ 2. In particlar, (a) π 3 (F λ ) < π 3 (G λ ), if 2 λ < 0. Or, for all α > 0 an γ f, γ g 0 with (α, γ f, λ) F λ an (α, γ g, λ) G λ, it hols that γ f < 2 < γ g. (b) F λ G λ = {(α 1 (λ), γ 1 (λ), λ)}, for some α 1 (λ) > 0, γ 1 (λ) > 0 sch that γ 0 (λ) < γ 1 (λ) if λ > 0.

15 REVERSIBLE CUBIC CENTERS 15 It trns ot that for fixe λ < 0 the local bifrcation iagram near R 0 an R ± is etermine in a niqe way by the total nmber of singlarities. Moreover if γ 2 α 2 + 1/α, then γ > λ+2; hence, by Propositions 8 an 9, if there appear singlarities along R +, then there are always two singlarities present along R 0. Hence the total nmber of singlarities can be one, two, three, five or seven. Smmarize we have the following reslt. Proposition 13. Let ζ P be fixe. Then the nmber of singlarities of (2) is prime an 1 n 7. The local phase portrait of Y ζ near the rays R 0 an R ± is niqely etermine by the nmber of singlarities n, an therefore enote by n. The slice of the bifrcation iagram of the local phase portraits of Y ζ near R 0 an R ± is shown in Figre 2. In Section 6 we will see that the global bifrcation iagram restricte to P is completely an niqely etermine by the local one. In Section 7 it becomes clear that this is not the case for P +. The following proposition escribes the local bifrcation iagram in this region in terms of the nmber of singlarities. Proposition 14. The slice of the local bifrcation iagram for λ > 0 fixe is shown in Figre 7. Here we se the following notation to istingish between ifferent configrations of singlarities: n h, n G,k h or n k h, 1 n 7, k {l, H, r} an h {, c, }. If Y ζ has n singlarities, the parameter ζ inces locally near the rays R 0 an R ± the phase portrait n h, n G,k h or n k h. In case all singlarities are on the horizontal axis, ζ belongs to n h ; in case that the rays R ± each carry one singlarity (resp. two singlarities), ζ belongs to n G,k h (resp. n k h ). Frthermore the configration of singlarities along R 0 is specifie by the lower inex h accoring to the following scheme: h h R 0 ζ F F one or two c ζ F E zero ζ E E one or two The configration of the singlarities along R ± is specifie by the pper inex, which is of type G,k or k epening whether it has one resp. two singlarities along R ±, accoring to the following scheme: k G,k R ± k k R ± l ζ H l G one l ζ H l G r two H ζ H G one H ζ H G r two r ζ H r G one r ζ H r G r two Proof. By Propositions 8, 9, 10 an 11 an Figres 4 an 6, the topological type of the singlarities of Y ζ along R 0 an R ± an its corresponing nmber are precisely etermine by the notation introce in previos proposition, an so also the local bifrcation iagram of Y ζ near R 0 an R ±.

16 16 M. CAUBERGH AND J. TORREGROSA 6 < λ λ = 6 4 < λ < 6 2 < λ 4 0 < λ 2 Figre 7. Characteristic slices of the local bifrcation iagram for fixe λ > Perioic, homoclinic an heteroclinic orbits For fixe ζ the ivergence is iv(y ζ ) = iv(y ζ )(x, y) = (2α 2 λ γ)y. Hence by Dlac s Theorem (see [10]) we can raw the following conclsion. Proposition 15. If ζ P \ H, then perioic, homoclinic or heteroclinic orbits intersect the horizontal axis. Frthermore all perioic orbits are part of a continm. As a corollary an since H represents the Hamiltonian vector fiels, there are no limit cycles for Y ζ. Frthermore, Corollary 16. None of the ifferential systems (1), (2) nor (5) have limit cycles.

17 REVERSIBLE CUBIC CENTERS Global bifrcation iagram in H P By rather straightforwar algebraic maniplations we obtain from [8] the topological classification of the Hamiltonian reversible vector fiels in (2), i.e. with γ = 2α 2 λ. Then the Hamiltonian is given by H (α,λ) (x, y) = 1 2 (x2 + y 2 ) (2α2 1)λx 3 + α 2 λxy (x2 + y 2 ) 2. In [8] they are represente by (5) with ξ = 0. Here we a to the reslt in [8] the global bifrcation iagram. Theorem 17. Up to topological eqivalence the Hamiltonian reversible vector fiels of (1) or (2) are given by the 2-parameter family Y ζ with γ = 2α 2 λ, α > 0, λ > 0 (i.e ζ H P) an show 14 topologically ifferent phase portraits epening on the parameter (α, λ) as shown in Figre 8. The classification for topological eqivalence is liste accoring to the nmber of singlarities as follows: 1 c ; 2 ; 3 if the singlarities if the singlarities are noncollinear; 4 G,H ; 5 H c an 5 G,H if there that are topologically eqivalent; 7 Hσ if 2α 2 1 < 0 an 7 Hσ if 2α 2 1 > 0, where σ = sgn{(2α 2 1)Ψ(α, λ)} if Ψ(α, λ) 0 an σ = 0 if Ψ(α, λ) = 0 an Ψ reas as are collinear an 3 G,H c are one resp. three singlarities on the line of symmetry; 6 H or 6H Ψ(α, λ) =(2α 2 1)(α 2 + 1) 2 ((2α 2 1) 2 λ 2 4) α 3 (α 6 λ 2 (α 2 + 1)) 3 + 8α 6 λ 2 6α 2 3λ 2 α 2 + λ 2 6. The bifrcation iagram is sketche in Figre 9. (27) 1 c G,H c 4 G,H 5 H c 5 G,H 6 H 6 H 7 H 7 H0 7 H+ 7 H 7 H0 7 H+ Figre 8. Phase portraits in P H.

18 18 M. CAUBERGH AND J. TORREGROSA In Figre 8 it is seen that the phase portraits 6 H an 6H are topologically eqivalent; both representations are incle to visalize better the bifrcation that happens when passing throgh the bifrcation crves in Figre 9. Figre 9. Global bifrcation iagram in P H, projecte in the (α, λ)- plane, see Theorem 17 an Figre 8. Let s recall some notations se in Theorem 17 an Figre 9. The crves labele by E, F, G correspon to the local bifrcations efine in Section 3.3, intersecte with γ = 2α 2 λ an projecte in the (α, λ)-plane. The crves labele by I, J correspon to global bifrcations, more precisely the ones of crossing of separatrices. These only appear when there are seven singlarities. In particlar, for the Hamiltonian case, when the local phase portrait is given by 7 H or 7 H respectively. The bifrcation srfaces I an J restricte to H istingish between 3 possible global phase portraits 7 Hσ an 7 Hσ respectively, where σ {, 0, +}. From [8] it is ece that I restricte to H is the semi-algebraic set Ψ 1 (0) P H F, where Ψ is efine in (27). The crve J restricte to H is the semi-algebraic set {(α, λ) R 2 : α = 1/ 3, λ 6}. To en we raw the global bifrcation iagram of Hamiltonian reversible vector fiels for (5), i.e. with ξ = 0, to complete the reslt in [8], for g > 0, e R. Notice that the family Y (α,2αλ 2,λ), α, λ > 0 is a strict sbfamily of X(g,0,e) R, g < 0, e R, that correspons with e < 2g, g > 0. Althogh for topological eqivalence no other phase portraits are fon than the ones presente in Figre 8, there is a new bifrcation crve that is enote by D. Along D the ray R 0 is triple (i.e. A(0) = A (0) = A (0) = 0, A (0) 0) implying that the singlarities along R 0 are non-elementary. To highlight this fact the corresponing phase portraits are enote by 2 t an 3 t for the case of two an three singlarities respectively, an are topologically eqivalent to 2 an 3 respectively (see Figre 11). In particlar the bifrcation iagram for the Hamiltonian reversible phase portraits of (5) covers the parameter plane π 1 (Q(H P)) {(g, e) : e 2g, g > 0}, as in Figre 10 that ths can be obtaine by transformation of Figre 9 by Q, given in (6), an aing the region e 2g, g > 0 (incling the bifrcation crve D). The transforme bifrcation crves in Figre 10 are calle corresponingly again by E, F, G, I, J. Now the algebraic set I restricte to H is etermine by ψ, with ψ(g, e) =2304g ge+256e g g 3 e 864g 2 e ge 3 96e g g 4 e 2 120g 3 e 3 +81g 2 e 4 54ge 5 +9e 6 32g 6 e 2 24g 4 e 4 +8g 3 e 5, an J restricte to H correspons to the semi-algebraic set {(g, e) : g > 2, e + g = 0}.

19 REVERSIBLE CUBIC CENTERS 19 Figre 10. Global bifrcation iagram of X(g,0,e) R, for g > 0, e R. 2 t 3 t Figre 11. Phase portraits of X R (g,0,e) for e = 2g, g > 0, see Figre Global bifrcation iagram in P In this section we prove Theorem 4 establishing that for λ < 0 the local phase portraits n (where n = 1, 2, 3, 5, 7) near R 0 an R ± in Proposition 13 etermine niqely the global phase portraits. Therefore we se the same notation to refer to the global phase portraits as to refer to the local ones. As a conseqence the global phase portraits can be istingishe by the total nmber of singlarities. To this en we start by a technical lemma. As in Propositions 10 an 11 we se the notations s 1 ± = (x1 ±, y1 ± ) for the sale an s1 = (x 1, y 1 ) for the sale-noe along R +. Lemma 18. Let ζ (G G r ) P. Then the relative position of the singlarities an the irection of the flow of Y ζ in {y 0} is as inicate in Figre 12. More precisely, for ζ G r : (1) The vertical line x = x 1 + intersects the segment bone by the center point (x 0 +, 0) an the sale point (x0, 0), i.e., x0 + < x1 + < x0. (2) The irection of the flow of the vector fiel Y ζ along the vertical x = x 1 + is as follows: ẋ x=x 1 +,0<y<y 1 + ẋ x=x 1 +,y>y 1 + > 0, ẏ x=x 1 +,0<y<y 1 + > 0, < 0 an ẏ x=x 1 +,y>y 1 + < 0. (28)

20 20 M. CAUBERGH AND J. TORREGROSA These properties are sketche in Figre 12(a). By reflection abot the x-axis the behavior in {y < 0} is obtaine. Analogos properties hol for ζ G, only replacing x 1 + an y 1 + by x 1 an y 1 respectively, an are sketche in Figre 12(b). (a) (b) Figre 12. Local phase portrait of Y ζ in {y 0} for (a) G r P an (b) G P, see Lemma 18. Proof. The claims follow by irect algebraic maniplation sing the expression in (20). Notice frthermore that x 0 + > 1 (x1 +, y1 + ) > x1 + an that x0 > x1 + if an only if (x 1 + )2 + (y 1 + ) λx ((x 1 +) 2 + (y 1 +) λx 1 +) 2 4(x 1 +) 2 < 2(x 1 + )2. This last ineqality hols for (y+ 1 )2 +λx 1 + > 0. To obtain the signs in (28), one calclates ẋ x=x 1 + = y(y 1 + y)(y y), ẏ x=x 1 + = x 1 + (y1 + y)(y1 + + y)((y1 + )2 + λx 1 + )/(y1 + )2. Recall that the rays R 0 an R ± are isoclines for ṙ = 0. Proof of Theorem 4. The global phase portraits 1, 2, an 3 are obtaine irectly from the local phase portraits escribe in Propositions 8 an 13 an Figre 3 an clearly are niqely etermine. Using Lemma 18, Figre 12 an the Poincaré-Benixson Theorem we fin that the relative positions of the separatrices in {y 0} in cases 5 an 7 are as pictre in Figres 13 an 14 respectively. Using the reversibility property one obtains that the global phase portraits in cases 5 an 7 are niqely etermine an as rawn in Figre 1. Figre 13. Relative positions of the separatrices in {y 0} in case 5. The proof that the relative positions of the separatrices in case 5 are as rawn in Figre 13, is analogos to the one for Figre 14 in case 7. We only etail this last case. By Lemma 18, in backwar time, the separatrix W I (s 1 + ) has to intersect the horizontal axis in between x 0 + an x 1 +. Next U I (s 1 +) in forwar time has to intersect the horizontal

21 REVERSIBLE CUBIC CENTERS 21 Figre 14. Relative positions of the separatrices in {y 0} in case 7. axis on the left of x 0 +. By the irection of the flow along R + between s 1 an s1 + an the Poincaré-Benixson Theorem, the separatrix U E (s 1 + ) has to connect the singlarities along R + like pictre in Figre 14. Frthermore, in backwar time, W E (s 1 + ) has to intersect the positive horizontal axis. By Poincaré-Benixson Theorem, U E (s 0 ) has to connect the singlarities s 0 an s 1, an the separatrix W E (s 0 ) has to intersect the positive horizontal axis on the left of the intersection W E (s 1 + ) {y = 0}. Therefore the relative positions of the separatrices in {y 0} is niqely etermine as in Figre 14, ening the proof. 7. Global phase portraits in P + The global bifrcation iagram for fixe λ > 0 is obtaine by the local one as escribe in Proposition 14 an shown in Figre 7, aing the global bifrcation crves that etermine the crossing of separatrices. In this case there are a lot more possibilities even for the local phase portrait than in the case treate in Section 6. For one, two an three singlarities the local phase portrait near R 0 an R ± completely etermines the global one, see Proposition 19. The other cases are sbseqently analyze in Sections 7.2 ntil 7.5 starting with the case of the maximal nmber of singlarities. In particlar, in Section 7.2, the classification for this case is proven an smmarize in Proposition 20. Notice that in this case all singlarities are non-egenerate an elementary. This is also the case for five singlarities if all of them ifferent from the origin are along R ±. However most often this is not the case an there are singlarities along R 0 as well as along R ± ; then the phase portrait is obtaine from the case with seven singlarities by coalescence of a sale an a noe to a sale-noe on both rays R ±. Both cases with five singlarities are treate in Section 7.3. Next, in Section 7.4 (resp. 7.5) the case of six (resp. for) singlarities is obtaine as continos bifrcation from the case with seven to five (resp. five to three an six to two) singlarities. In Section 7.5 we also a a etaile nmerical sty that completes the case of for singlarities. Finally in Section 7.6 we raw the global bifrcation iagram with the partial help of nmerical analysis. From the classification reslts in this section, i.e. Propositions 19, 20, 21, 22 an 23, we fin the classification of the phase portraits in P +, that is smmarize in Theorem Less than or eqal to three singlarities. The following proposition escribes the classification of the phase portraits with at most three singlarities an is obtaine by sing the symmetry property an the local classification from Section 3.

22 22 M. CAUBERGH AND J. TORREGROSA Proposition 19. Let ζ P +. The phase portrait of (2) has at most three singlarities if an only if ζ G l (G E F ). Frthermore the global phase portrait is niqely etermine by the local phase portrait an there are exactly 6 topologically ifferent ones, that are rawn in Figre 15 an are liste by increasing nmber of singlarities: 1 c in G l E F ; 2 in G l E or 2 in G l F (these are eqivalent); 3 in G l E, 3 in G l F, 3 G,t c in G H t (where t {l, r}, both etermining topologically the same phase portrait) an 3 G,H c in G H. 1 c G,l c 3 G,H c 3 G,r c 3 Figre 15. Phase portraits of Y ζ with one, two or three singlarities for ζ P +, see Proposition Seven singlarities. Proposition 20. Let ζ P +. The phase portrait of (2) has seven singlarities if an only if ζ (E F ) G r. In this case there are 20 topologically ifferent phase portraits. More precisely, there are 9 (resp. 11) in E G r (resp. F G r ) that are liste in Figre 16 (resp. Figre 17) an all together they are shortly referre to as 7 (resp. 7 ). In particlar, 7 lσ z an 7 rσ z with σ {+, } are eqivalent, for z = (resp. z = ). Notice that the specific composition of the phase portraits in Figres 16 an 17 respects the sbseqent bifrcation phenomena that happen between ajacent phase portraits an is etaile at the en of this section. This also is the reason why some topologically eqivalent phase portraits appear. Proof of Proposition 20. From the local bifrcation iagram in Figre 7 an Proposition 14 we know that the case with seven singlarities istingishes between 7 an 7. To complete the global phase portraits we analyze the separatrices at the sales, an follow the flx forwar an backwar in time. By the reversibility property we can restrict or attention to the half plane {y 0}. We only etail the proof of case 7. The possible intersections of W I (s 1 ) an U E(s 1 ) with the ray R + in backwar resp. forwar time are shown in Figre 18. The case (b) correspons to the Hamiltonian one. The possibilities (a), (b), (c) are referre to as 7 r, 7H, 7l respectively. Next for each of these cases we analyze the possible intersections of W E (s 1 ) an U E (s 0 ) with the vertical axis in backwar resp. forwar time; for frther

23 REVERSIBLE CUBIC CENTERS 23 7 l 7 l0 7 l 7 l1 7 l+ 7 H 7 H0 7 H+ 7 r 7 r0 7 r+ Figre 16. Phase portraits in P + G r E (referre to as 7 ), see Proposition l 7 l0 7 l 7 l1 7 l+ 7 H 7 H0 7 H+ 7 r 7 r1 7 r 7 r0 7 r+ Figre 17. Phase portraits in P + G r F (referre to as 7 ), see Proposition 20.

24 24 M. CAUBERGH AND J. TORREGROSA reference the y-coorinates are enote by ye 1 an y0 E. Aitionally in case (c) we analyze the possible intersections of W I (s 1 ) an U E (s 0 ) with the vertical axis in backwar resp. forwar time; as before the y-coorinates are enote by yi 1 an y0 E. Observe that this leas to 3 possibilities in cases (a) as well as (b), an 5 possibilities in case (c), aing to 11 possibilities in total for case 7. Then clearly always ye 1 < y1 I (if it exists) an so all the possibilities are smmarize by: If y 0 E < y1 E : If y 0 E = y1 E : (a) (b) (c) 7r 7r0 7 H 7 H0 If ye 1 < y0 E : 7r+ 7 H+ 7 l ; 7 l0 ; If y 0 E < y1 I : If y 0 E = y1 I : 7l ; 7l1 ; If y 1 I < y0 E : 7l+. (a) (b) (c) Figre 18. Possibilities for the separatrices in case 7. The cases 7 z0, where z {r, H, l}, correspon to the fact that W E(s 1 ) = U E (s 0 ). The case 7 l1 correspons to the fact that W I(s 1 ) = U E(s 0 ). We now show that all these possibilities occr an correspon to the phase portraits as rawn in Figre 16. The existence of the Hamiltonian phase portraits, i.e. case (b),

25 REVERSIBLE CUBIC CENTERS 25 follows from [8]. In all these cases the separatrices W I (s 1 ) an U E (s 1 ) coincie. Bifrcating from 7 H or 7 H+ the only singlarity that changes its type is s 1 +, an changes from center to stable or nstable focs. By continity, the attracting or repelling natre of s +, Proposition 15 an the reversibility with respect to the horizontal axis, the connection W I (s 1 ) = U E (s 1 ) breaks in only one possible way for each of the cases as rawn in Figre 18(a) an (c). Therefore the global phase portraits enote by 7 r an 7 l+ exist for vales of the parameter close to the Hamiltonian case. Next 7r0 7 l0, 7l, 7l1 an 7 r+, 7r+, 7l (resp. ) exists by continos epenence on the parameter an the existence of 7r (resp. 7 l an 7 l+ ). This ens the proof of this case. The case 7 is obtaine in a similar way as the case 7. Then we istingish the ifferent phase portraits base on the possible intersections of the separatrices at the sales (s 1 an s0 + ) with the straight line throgh the anti-sales (s1 + an s0 ), see Figre 19. It is seen that 7 lσ z an 7 rσ z with σ {+, } are eqivalent in case 7 z, for Figre 19. Possibilities for the separatrices in case 7. z {, }. Comparing the phase portraits in Figre 16 with the ones in Figre 17, it is fon that these phase portraits are mtally ifferent. From the previos proof we notice that the only 2 possible connections in the non- Hamiltonian case 7 are fon an efine by the bifrcation srfaces: J = {ζ : W E (s 1 ) = U E (s 0 )} an K = {ζ : W I (s 1 ) = U E (s 0 )}. (29) In the non-hamiltonian case 7 the 2 connections are fon an efine by the bifrcation srfaces: I = {ζ : W E (s 0 +) = W I (s 1 )} an L = {ζ : U E (s 0 +) = U E (s 1 )}. (30) In Section 7.6 we raw these srfaces in the bifrcation iagram with the ai of nmerical methos. Notice that the specific composition of the ifferent phase portraits in Figres 16 an 17 is accoring to the bifrcation srfaces H, I, J, K an L; the phase portraits on the mile line correspon to H, say the Hamiltonian srface, an the pper resp. lower line correspon to H l resp. H r. In Figre 16 the iagonal connecting the phase portraits referre to as 7 z0 an the branch from 7 H0 with z {l, H, r} correspons to the bifrcation srface J, to 7 l1 correspons to the bifrcation srface K. In Figre 17 with z {l, r} an b = 0 correspons to the bifrcation srface I (resp. L). the iagonal connecting the phase portraits referre to as 7 zb (resp. b = 1) passing throgh 7 H0

26 26 M. CAUBERGH AND J. TORREGROSA 7.3. Five singlarities. In case of five singlarities there are essentially two cases to be istingishe, whether or not all singlarities are non-egenerate an elementary. If it is the case, the singlarities ifferent from the origin are locate along R ±, an the local phase portraits are referre to as 5 c. However, most often this is not the case; then the corresponing parameters belong to G. In Figre 23 the global bifrcation iagram restricte to G is rawn. Notice that only bifrcation crves are rawn that lea to the lower bon in the classification. Frthermore the ashe lines correspon to the bifrcation crves that are nmerically obtaine. Next theorem smmarizes the topological classification that is fon in all cases of five singlarities. Proposition 21. Let ζ P +. The phase portrait of (2) presents five singlarities for ζ (G (E F )) (G r E F ). In this case there are 9 n 12 topologically ifferent phase portraits. (1) There are exactly 3 in G F as shown in Figre 20 an are referre to as 5. (2) There are exactly 2 in G r E F. These are shown in Figre 21 an are referre to as 5 c. In particlar, 5 l c an 5r c are eqivalent. (3) There are 4 n 7 in G E. More precisely, there are exactly 4 in G E (H r H) an 1 n 5 in G E H l. These are shown in Figre 22 an are referre to as 5. In particlar, 5 G,l an 5 G,r are eqivalent. 5 G,l 5 G,l0 5 G,l+ Figre 20. Phase portraits in P + G F (referre to as 5 ). 5 l c 5 H c 5 r c Figre 21. Phase portraits in P + G r E F (referre to as 5 c ). Proof. The proofs of cases 5, 5 c an 5 are similar to the proof of case 7 in Proposition 20. First we se argments of continity to prove the existence of the corresponing global phase portraits. Next, sing the symmetry, we sty the possible intersections of the separatrices at the sale along R 0 an the sale-noe along R + (resp. at the symmetric sales along R + an R ) in cases 5 an 5 (resp. 5 c ) in the pper plane y > 0, sing pictres similar to Figres 18 an 19.

27 REVERSIBLE CUBIC CENTERS 27 5 G,l 5 G,H 5 G,r 5 G,r0 5 G,r+ Figre 22. Phase portraits in P + G E (referre to as 5 ). First we eal with case 5. Bifrcating from 7 l (resp. 7 l ) to 3 there exists 5 G,l+ also 5 G,l0 exists. Base on the (resp. 5 G,l ). Next by continity from 5 G,l+ to 5 G,l relative positions of the separatrices at the sale-noe an the sale, the attracting natre of the noal part of s 1, Proposition 15 an the reversibility property, it trns ot that these are the only possible phase portraits. Now by comparing these phase portraits in G F, it is seen that, p to topological eqivalence, there are exactly 3 that are shortly referre to as 5. Next we eal with case 5 c. Bifrcating from the Hamiltonian case 5 H c there exist 5l c an 5 r c. That these are the only possible phase portraits follows as in case 5, consiering now the relative positions of the separatrices at the sale an the noe an the attracting natre of s 1. Now by comparing these phase portraits in G r E F, it is seen that, p to topological eqivalence, there are exactly 2 that are shortly referre to as 5 c. Next we eal with case 5. Bifrcating from 7 l (resp. 7 r an 7 r+ ) an 3, there exists 5 G,l (resp. 5 G,r an 5 G,r+ ). Next by continity from 5 G,r to 5 G,r+ also 5 G,r0 exists. Finally from the sty of the separatrices at the sale s 0 an the sale-noe s1 in G E there are exactly 4 in G E (H r H) an there are 5 possibilities in G E H l for the intersection of U E (s 0 ) with R + in forwar time, each giving rise to exactly 1 global phase portrait; this can be seen by a pictre similar to Figre 18(c). All these 5 phase portraits are topologically mtally istinct. It is seen that 1 of these phase portraits is 5 G,l, an ths is eqivalent to 5G,r+. Another is eqivalent to 5 G,r, an the other 3 are ifferent from any of the phase portraits in Figres 18, 19 an 20. Therefore there are 4 n 7 in G E an 9 n 12 with five singlarities. However, as can be seen in the bifrcation iagram rawn in Figre 33 in Section 7.6, the region G E H l is far from the region 7 l+ an therefore it is believe that the lower bons in Proposition 21 are the exact nmbers Six singlarities. Proposition 22. Let ζ P +. The phase portrait of (2) presents six singlarities for ζ (E F) G r. In this case there are at least 8 n 9 topologically ifferent phase portraits. (1) There are exactly 4 in F G r as shown in Figre 24 an are referre to as 6. In particlar, 6 l an 6r are eqivalent. (2) There are 6 n 7 in E G r. More precisely, there are exactly 6 in E G r (H l H) an 1 n 3 in E G r H r. These phase portraits are shown in Figre 25 an are referre to as 6. In particlar, 6 r an 6l+ are eqivalent. In particlar, 6 l an 6 r are eqivalent, as also 6H an 6 H are.

28 28 M. CAUBERGH AND J. TORREGROSA Figre 23. Bifrcation iagram restricte to G projecte in the (α, λ)- plane, i.e. π 2 (G P + ). 6 l 6 H 6 r 6 r1 6 r Figre 24. Phase portraits in P + G r F (referre to as 6 ). 6 l 6 l0 6 l 6 l1 6 l+ 6 H 6 r Figre 25. Phase portraits in P + G r E (referre to as 6 ). Proof. The proofs of cases 6 an 6 are similar to the proof of case 7 in Proposition 20. First we se argments of continity to prove the existence of the corresponing global phase portraits. Next, sing the symmetry, we sty the possible intersections of the

29 REVERSIBLE CUBIC CENTERS 29 separatrices at the csp s 0 an the sale s 1 in the pper plane y > 0, sing pictres similar to the ones in Figres 18 an 19. First we eal with case 6. Bifrcating from the Hamiltonian case 6 H there exist 6 l an 6r. Next by continity from 5r c an 7r also 6r exists. As a conseqence, by continity from 6 r an 6r there exists 6r1. Next we eal with case 6. Bifrcating from the Hamiltonian case 6 H, there exist 6 l+ an 6 r. Next by continity from 5l c an 7l also 6 l exists. As a conseqence, by continity from 6 l an 6 l+, there exist 6l0, 6l an 6l1. Base on the relative positions of the separatrices at the csp an the sale, the attracting resp. repelling natre of s 1 +, Proposition 15, the reversibility property an continity, it trns ot that 6 l, 6 H, 6 r, 6 r0 an 6 r+ are the only phase portraits in F G r. Analogosly, it trns ot that 6 l, 6l0, 6l, 6l1, 6l+, 6H, an 6r are the only phase portraits in E G r (H l H). Finally, analyzing the relative positions of the separatrices at the sale an csp as above there are only 2 more phase portraits possible in E G r H r than 6 r, the one that is rawn in Figre 25. These other possibilities are etermine by the relative position of the intersection of the nstable separatrix at the csp with the ray R +. Firstly, this intersection point is at istance smaller than r 1 from the origin, giving rise to 6 l topologically; seconly, the intersection point is at istance larger than r 1 from the origin, giving rise to a phase portrait that is topologically ifferent from 6 r an 6 l an all other phase portraits with six singlarities. Therefore there are at most 3 topologically ifferent phase portraits in E G r H r. Frthermore, since 6 r is topologically eqivalent to 6 l+, there are at least 6 an at most 7 phase portraits in E G r. Comparing the phase portraits of 6 an 6 we fin frthermore that topologically 6 l = 6l+, 6H = 6H an 6r = 6r ; hence there are at least 8 an at most 9 phase portraits with six singlarities. In Figres 26 an 27 the bifrcation iagram is rawn restricte to E an F in the (α, λ)-plane respectively. In particlar the bifrcation srfaces G, H, J, K an L, efine in (4), (29) an (30), also are rawn restricte to E an F; the ashe lines are se to inicate those bifrcation crves that are nmerically obtaine. Figre 26. Bifrcation iagram restricte to E projecte in the (α, λ)- plane, i.e. π 2 (E P + ).

30 30 M. CAUBERGH AND J. TORREGROSA Figre 27. Bifrcation iagram restricte to F projecte in the (α, λ)- plane, i.e. π 2 (F P + ). From this proof it is seen that there cannot exist more bifrcation crves in the region of six singlarities neither in Figre 27 neither in the region G r H l E in Figre 26, that is referre to as 6 l. However from this proof one cannot concle whether there are more bifrcation crves in the region G r H r E in Figre 26, that is referre to as 6 r. Fixing concrete vales of λ, increasing α an stying the relative positions of the separatrices at the csp an the sale, we fin that the hyperbolic separatrix W E (s 1 ) intersects the ray R 0 on the left of the csp. Therefore nmerically no other bifrcations are fon. Taking into accont the nmerically obtaine bifrcation iagram in Figre 28 they cannot appear since 5 r c an 7 r+ are the only that are nearby For singlarities. When Y ζ has exactly for singlarities the local phase portraits can be 4 G,H, 4 G,r, 4G,l or 4 G,l. From Theorem 17 we alreay know that the local phase portrait 4 G,H niqely etermines the global phase portrait an it is rawn in Figre 28. Here we obtain the niqeness of the global phase portrait in case of the local phase portrait 4 G,l, that is proven in Proposition 23 below. In case of the local phase portraits 4 G,r an 4 G,l we have strong nmerical evience that the global phase portraits are niqe as well in these cases an are as rawn in Figre 28; this is explaine after proving Proposition 23. Notice that the vector fiels having exactly for singlarities correspon to G (E F) an etermine the 1-parameter family Y (α,γ(α),λ(α)), where γ(α) = 2 α 2 + 1/α an (λ(α) = γ(α) + 2 or λ(α) = γ(α) 2). (31) Proposition 23. Let ζ P +. The phase portrait of (2) has for singlarities if an only if ζ G (E F). In this case there are 3 n 6 topologically ifferent phase portraits that are rawn in Figre 28. (1) There exists exactly 1 in G F an 1 in G H. These are enote respectively as 4 G,l an 4 G,H. (2) There exist 2 n 5 in (G E) \ H. In particlar, there are 1 n 3 in G E H r of which 4 G,r is one an there are 1 m 5 in G E H l of which

31 4 G,l REVERSIBLE CUBIC CENTERS 31 is one. Frthermore all possible phase portraits in G E H r are possible ones in G E H l. an 4 G,l are eqivalent. In particlar, 4 G,r 4 G,l 4 G,H 4 G,r 4 G,l Figre 28. Phase portraits in G (E H F). Proof. The Hamiltonian case has alreay been treate in Section 5, so we are left with the existence of the phase portraits 4 G,l, 4 G,r an 4 G,l an the nicity of the first one. The existence part follows reasoning in the same way as in the proof of Proposition 22 sing a continity argment from the existence of corresponing phase portraits with three an five singlarities. Therefore we concentrate on the nicity of 4 G,l an to en we briefly escribe the ifferent possibilities in G E to give a fine pper bon for the total nmber of phase portraits with for singlarities. Let ζ sch that the local phase portrait near R 0 an R + is given by 4 G,l. This correspons to γ = 2 α an λ = 2 α α α The niqe singlarity along both R ± is a sale-noe corresponing to G H l an the niqe singlarity along R 0 is a csp corresponing to F. Consier the straight line L throgh the singlarities s 0 an s 1. Let (0, s 0, s 1 ) be the triangle bone by R 0, L an R + as illstrate in Figre 29(a). The nstable separatrix U I (s 1 ) is tangent to R + an lies insie the triangle (0, s 0, s 1 ) for reverse time sfficiently large (t ). The stable separatrix W E (s 0 ) is tangent to R 0 an lies in E bt otsie the triangle (0, s 0, s 1 ) for times sfficiently large (t ). The orbit U I (s 1 ) will leave the triangle (0, s 0, s 1 ) only by crossing R 0 between 0 an s 0. Inee sppose to the contrary that U I (s 1 ) wol leave the triangle by first crossing L between s 0 an s 1. This wol imply the existence of for tangency points along L that are two by two istinct (incling the singlarities s 0 an s 1 ), see Figre 29(b). This is in contraiction with the fact that the system is cbic an irrecible an therefore can have at most three tangency points, see e.g. [10]. Hence by symmetry the orbit U I (s 1 ) connects the sale-noes s 1 an s 2 insie the polygon 0, s 1, s 0, s 2. Now the global phase portrait 4 G,l as presente in Figre 28 clearly is niqely etermine. Now we cont the phase portraits with for singlarities. There is exactly 1 in G F an 1 in G H referre to as 4 G,l an 4 G,H respectively. There are 3 (resp. 5) possible phase portraits in G E H r (resp. G E H l ), an the 3 possible ones in G E H r are topologically eqivalent to 3 of the 5 possible ones in G E H l. One of the phase portraits in G E H r, 4 G,r, is eqivalent to 4G,l. Therefore there are at least 3 an at most 6 phase portraits with for singlarities.