Global Classification of a class of Cubic Vector Fields whose canonical regions are period annuli

We study cubic vector fields with inverse radial symmetry, i.e., of the form ẋ = δx− y + ax + bxy + cy + σ(dx − y)(x + y), ẏ = x + δy + ex + fxy + gy + σ(x + dy)(x + y), having a center at the origin and at infinity; we shortly call them cubic irs-systems. These systems are known to be Hamiltonian or reversible. Here we provide an improvement of the algorithm that characterizes these systems and we give a new normal form. Our main result is the systematic classification of the global phase portraits of the cubic Hamiltonian irs-systems respecting time (i.e. σ = 1) up to topological and diffeomorphic equivalence. In particular there are 22 (resp. 14) topologically different global phase portraits for the Hamiltonian (resp. reversible Hamiltonian) irs-systems on the Poincaré disc. Finally we illustrate how to generalize our results to polynomial irs-systems of arbitrary degree. In particular we study the bifurcation diagram of a 1-parameter subfamily of quintic Hamiltonian irs-systems. Moreover we indicate how to construct a concrete reversible irs-system with a given configuration of singularities respecting their topological type and separatrix connections.


Introduction
Let P and Q be two real polynomials in the variables x and y, then we say that X = (P, Q) : R 2 −→ R 2 is a planar polynomial vector field of degree d if the maximum of the degrees of the polynomials P and Q is d.Such vector fields are called quadratic or cubic if d = 2 or d = 3, respectively.The polynomial differential system associated to the vector field X is ẋ = P (x, y), ẏ = Q(x, y).
Two of the main classical problems in the qualitative theory of real planar polynomial vector fields are the determination of their limit cycles and the center-focus problem; i.e. to distinguish whether a singular point is either a focus or a center.A center is a singular point having a neighborhood fulfilled of periodic orbits with the unique exception of the singular point.
The classification of the centers of the polynomial vector fields is an old problem which started with the quadratic ones by the works of Dulac [1908], Kapteyn [1911Kapteyn [ , 1912]], Bautin [1954], Żo ladek [1994b],. . .; an update on the quadratic centers can be found in [Schlomiuk, 1993a].For the quadratic polynomial vector fields the characterization of centers is completed.There exist many partial results for the centers of polynomial vector fields of degree larger than 2, but we are very far to obtain a complete classification of the centers for the class of all polynomial vector fields of degree 3.In particular the centers of the cubic polynomial vector fields of the form ẋ = −y + P 3 (x, y), ẏ = x + Q 3 (x, y) with P 3 and Q 3 homogeneous polynomials of degree 3, have been classified by Vulpe & Sibirskiȋ [1988], Lloyd & Pearson [1999] and Żo ladek [1994a].In case former cubic systems are reversible through a linear involution and have a fixed set of dimension 1, a classification of their global phase portraits can be found in [Buzzi et al., 2009].
In this paper we study real planar cubic polynomial systems for which the coefficients in the asymptotic expansion of the Poincaré map both near the origin and near infinity are polynomial in the parameter.These systems are of the form ẋ = δx − y + ax2 + bxy + cy 2 + σ(dx − y)(x 2 + y 2 ), ẏ = x + δy + ex 2 + f xy + gy 2 + σ(x + dy)(x 2 + y 2 ), (1) as also was found in [Blows & Rousseau, 1993].Recall that system (1) has a center at infinity if the origin of system (1) is a center after the transformation x = cos θ/r and y = sin θ/r.First we improve the algorithm used in [Blows & Rousseau, 1993] to characterize these systems having simultaneously a center at the origin and at infinity.As a result we find a new normal form classifiying these systems as a Hamiltonian and a reversible class (see Theorem 1).Although the center-focus problem has its own interest, the knowledge of the center conditions near the origin (resp.infinity) also is used to study the maximum number of large amplitude limit cycles (resp.small amplitude limit cycles) by ways of the so-called division-derivation algorithm.In [Blows & Rousseau, 1993] the simultaneous cyclicity problem is also studied and it is proved that the maximum of limit cycles which can appear by simultaneous bifurcation at the origin and at infinity is seven.Some results on the number of large amplitude limit cycles for this class of cubic planar vector fields can be found in [Liu & Chen, 2002], [Liu & Huang, 2006] and [Zhang & Liu, 2006].More precisely they give concrete bifurcations for which seven large amplitude limit cycles appear.This problem of large amplitude limit cycles is also studied in [Huang & Liu, 2004a], [Zhang et al., 2006] and [Zhang & Liu, 2007] for quintic planar vector fields, in [Huang & Liu, 2004b] for septic planar vector fields, in [Caubergh & Dumortier, 2008] for classical Liénard systems of even degree and in [Luca et al., 2009] for certain subclasses of generalized Liénard systems.
After rescaling the parameter σ can be supposed to be ±1.Written in polar coordinates the leading terms of system (1) at infinity are the same as the ones at the origin, up to the sign of σ.Hence, the local phase portrait at infinity is topologically equivalent to the one at the origin (after time reversal for σ = −1.).Therefore a cubic system of type (1) is called a cubic system with inverse radial symmetry or shortly cubic irs-system.If σ > 0 (resp.σ < 0) we say that system (1) is a cubic system with inverse radial symmetry respecting time (resp.reversing time).
Theorem 1.The cubic irs-system (i.e., system (1)) has a center at the origin and at infinity if and only if δ = d = 0 and it is Hamiltonian or reversible, after a rotation, with respect to the change (x, y, t) → (x, −y, −t).That is, after rotation, system (1) satisfies δ = d = 0, b = −2g and f = −2a for the Hamiltonian class or δ = a = c = d = f = 0 for the reversible class, given below in (2) and (8) respectively.
Next we classify the global phase portraits of the Hamiltonian cubic irs-systems respecting time with respect to topological and diffeomorphic equivalence (see Theorems 2, 3, 4 and 5).The classification of global phase portraits of other Hamiltonian systems is done before, see e.g., [Gasull et al., 2000] and [Guillamon & Pantazi, 2008].However their techniques does not apply to the cubic irs-system we consider in this paper.Here we introduce a new systematic technique to systematically study the global phase portraits of Hamiltonian irs-systems.
By Lemma 2 we only need to consider a = 0, g ≥ 0 and c ≥ 0. Next result details the corresponding phase portrait in each region of the parameter space.
Theorem 2. In Table 1 the global phase portraits of the Hamiltonian cubic irs-systems respecting time, i.e.
Note that Table 1 provides all the phase portraits of system X H (g,c,e) according to values of its parameters.Moreover observe that in cases 1, 2 and 3 the classification is done using only the initial parameters of the system.In the study of case 4 there need to be introduced auxiliary parameters in function of the initial ones, which simplifies the classification, see for more details Theorem 7.
Table 1.Classification of the phase portraits of the Hamiltonian cubic irs-systems respecting time, X H (g,c,e) , with respect to diffeomorphic equivalence (Theorem 2).In Theorem 3 we classify all phase portraits for Hamiltonian cubic irs-systems up to topological equivalence.
Theorem 3. The phase portrait of a Hamiltonian cubic irs-system, X H (g,c,e) , is topologically equivalent to one of the 22 phase portraits described in Figure 1.
Theorem 4. The phase portrait of a reversible Hamiltonian cubic irs-system respecting time, X H (g,0,e) , is topologically equivalent to one of the 14 phase portraits presented in Figure 2. Furthermore the phase portrait of these cubic reversible Hamiltonian systems is uniquely determined with respect to topological equivalence in case that the total number of singularities equals 1, 2, 4 or 6.
Throughout this paper we will use the notation |•| for the Euclidean norm on R 2 as well for the absolute value of a real number.
Theorem 5.The global phase portrait of a Hamiltonian cubic irs-system respecting time, X H (g,c,e) has at most 7 singularities of which one is the center at the origin.Furthermore the following statements hold: (1) There are at most 3 distinct straight lines l through the origin that carry 2 or more singularities of (1).Denote these lines by ) is such a straight line through the origin, then l i carries at most 2 singularities different from the origin.( (a) When l i carries 1 singularity different from the origin, then we denote this singularity by s i and we have s i = 1.Furthermore its Hamiltonian value is h i ≡ 1/12.(b) When l i carries 2 singularities different from the origin, then both singularities lie on the same side of the origin and we denote them by , where h i ± denotes the Hamiltonian value at s i ± respectively.(3) Assume that there are k + 1 straight lines (k ≤ 2), say l i , 0 ≤ i ≤ k, carrying at least two singularities, then we order these lines such that where we extend the notation h i − ≡ h i in case l i carries only 1 singularity outside of the origin.(4) A Hamiltonian cubic irs-system has maximally 7 disjoint period annuli.
(5) The global phase portraits can be classified up to topological equivalence in terms of the number of singularities (# s ) and the order of their values of the Hamiltonian.For three configurations of the Hamiltonian values an additional algebraic condition has to be checked to determine the phase portraits uniquely up to topological equivalence.This classification is presented in Table 2.
Table 2. Topological classification of the phase portraits of the cubic Hamiltonian systems based on the number of singularities.The number m (A) is defined as the maximal multiplicity of a zero of A on [0, π) and A and Π are respectively defined in ( 12) and (52).

#s Hamiltonian values
Condition Figure The paper is organized as follows.In section 2 we prove Theorem 1.Using the improved algorithm of Gasull & Torregrosa [2001] to calculate Lyapunov quantities we derive a new normal form for the cubic irs-systems.In section 3 we recall basic notions and results on singular points, on the infinity and equivalent vector fields.Next in section 4 we study the qualitative properties of the 4-parameter Hamiltonian subfamily for σ = 1 in polar coordinates.This leads to the introduction of trigonometric polynomials A and B that play an important role in the description of the bifurcation diagram of the phase portraits (see subsection 4.1).Next in subsection 4.2 we reduce the study of the 4-parameter Hamiltonian family to the one of a 3-parameter Hamiltonian subfamily.Next in subsection 4.3 we analyze the singularities along a 'ray' depending on its multiplicity and in subsection 4.4 we study the properties of the Hamiltonian.The local analysis of the singularities and the properties of the Hamiltonian enable us to classify the global phase portraits of the Hamiltonian class up to diffeomorphic equivalence.For σ = 1 the bifurcation of the global phase portraits are described in section 5 for arbitrary but fixed number of the rays that can carry singularities multiplicity taken into account (sections 5.1, 5.2, 5.3 and 5.4).These characterizations prove Theorems 2 and 5.The proof of Theorem 3 follows by identifying the topologically nonequivalent phase portraits of Figures 10,12,13,17,18,19,20,21,22,23 and 24; the proof of Theorem 4 follows from Theorem 3 by considering the ones appearing for c = 0. Finally, in section 6, using the systematic method previously introduced for these cubic systems, we construct concrete examples of polynomial Hamiltonian and reversible systems of higher degree than 3 with a given configuration of singularities and connections of the separatrices.In a forthcoming paper we treat the reversible class.

Normal Forms for Cubic irs-Systems
In this section we deduce a normal form for the cubic irs-systems.Ending the writing of this paper we found that the centers for these systems also are characterised in [Blows & Rousseau, 1993] by a calculation of Lyapunov quantities.However in [Blows & Rousseau, 1993] calculations are performed at infinity while here we perform calculations near the origin parallel with calculations at infinity.Moreover we include an improvement of the algorithm as introduced in [Gasull & Torregrosa, 2001] what lead to a significant reduction of the calculations.Since our proof is shorter and leads to a different normal form we include it here.
Lyapunov quantities are defined by an algebraic lemma in [Schlomiuk, 1993b] for a weak focus; we recall a slightly generalized version of it which proof can be found in [Caubergh & Dumortier, 2004].
Lemma 1 [Lyapunov quantities].Let (X λ ) , λ ∈ W ⊂ R p be an analytic family of planar vector fields with for some analytic functions c, d : W → R with c (λ) = 0 for all λ.Then there exists a formal power series F λ with and there exist analytic functions V i , i ∈ N such that Furthermore such analytic functions {V i , i ∈ N} are uniquely determined by (3) in the sense that if another set of analytic functions {W i , i ∈ N} satisfies (3), then where (V 0 , V 1 , . . ., V i−1 ) denotes the ideal generated by the analytic functions V j , 0 ≤ j ≤ i − 1.Any set of analytic functions satisfying (3) is called a set of Lyapunov quantities for X λ .
The systems X λ , λ ∈ W having a center at the origin can be characterized by the vanishing of all Lyapunov quantities: if {V i , i ∈ N} is a set of Lyapunov quantities of X λ , then the focus at the origin of X λ 0 is a center if and only if V i (λ 0 ) = 0 for all i ∈ N. In particular the systems X λ , λ ∈ W having a weak focus at the origin can be characterized by the vanishing of the 0th Lyapunov quantity of X λ .
In practice Lyapunov quantities can be calculated by putting the system in polar coordinates (r, θ) where S k (θ) are homogeneous trigonometric polynomials of degree k whose coefficients are polynomials in (a, b, c, d, e, f, g, δ, σ).Denote by r(θ, r 0 ) the solution of (4) such that r = r 0 when θ = 0.In this case and for r small enough we can write for analytic functions u k with u k (0) = 0 for k ≥ 1.The Poincaré return map is defined as In particular if {V i , i ∈ N} is a set of Lyapunov quantities for (1), then for some non-zero constants c i , i ≥ 0 (see [Caubergh & Dumortier, 2004]).Therefore to find the center conditions, it suffices to determine v k = 0, k ≥ 1 recursively where v 1 = u 1 (2π) mod (δ) and v k = u 2k+1 (2π) mod (δ, u 3 (2π) , u 5 (2π) , . . ., u 2k−1 (2π)) for k ≥ 1.By calculating the set v k , k ≥ 1 instead of V k , k ≥ 1 computations and computer time are significantly reduced; if one in addition works in complex notation as does the algorithm introduced in [Gasull & Torregrosa, 2001], the computer time is even more reduced.System (1) can be written in complex coordinates (z, z), as where r l,k−l depend on the parameter (a, b, c, d, e, f, g, δ, σ) .Notice that only in this section z denotes the conjugate of a complex number z and i the number √ −1.
Proof.[Proof of Theorem 1]After the change of variables z = x + iy system (1) writes in the form ( 5) where Then the cubic irs-system (1) has a weak focus at the origin when δ = 0 and after the transformation x = cos θ/r and y = sin θ/r we can say that system (1) has a weak focus at infinity when d = 0.After a rescaling of the variables and time it is not restrictive to assume that σ = ±1.Here we only consider the case σ = 1, the case σ = −1 is analogous.The computation of the Lyapunov quantities according to the algorithm in [Gasull & Torregrosa, 2001] gives are the Lyapunov quantities that control the centers in real coordinates by adding the condition 3 ) and solving system S = {v 1 = 0, v 2 = 0, v 3 = 0, v 4 = 0}.We solve system S for the variables r kl and next we return to the original parameters a, b, c, e, f, g.By taking into account that system (1) is a real differential system we obtain some concrete families that are of Hamiltonian or reversible type.Then we can conclude that system (1) has a center at the origin if and only if the Lyapunov quantities v j , 0 ≤ j ≤ 4 are zero.
Notice that these two classes are invariant after a rotation with respect to the origin.Up to this rotation the Hamiltonian class is the 4-parameter subfamily within system (1) defined by the conditions and the Hamiltonian is given by The reversible class, up to some rotation with respect to the origin, is the 3-parameter subfamily within system (1) defined by the conditions δ = a = c = d = f = 0. Notice that this family is symmetric with respect to the change (x, y, t) → (x, −y, −t).
In subsection 4.2 for σ = 1 we will see that up to linear conjugacy the phase portraits of the Hamiltonian class are represented by the 3-parameter subfamily b = −2g, f = −2a and a = 0. Then the Hamiltonian class can be represented by the 3-parameter family defined by (2), i.e.
with Hamiltonian H = H (g,c,e) given by and the reversible class can be represented by the 3-parameter family Notice that by increasing |ξ| from ξ = 0, the reversible class is born from the Hamiltonian subclass defined by c = 0, i.e., X R (g,0,e) = X H (g,0,e) .These facts are also true for the case σ = −1.

Topological and Diffeomorphic Equivalence
In this section we recall notions and basic theorems that we will use in the classification of the phase portraits of the Hamiltonian cubic irs-systems.We denote by P n (R 2 ) the set of real planar polynomial vector fields of the form X(x, y) = (P (x, y), Q(x, y)) where P and Q are real polynomials in the variables (x, y) of degree n.We denote by p(X) the Poincaré compactified vector field corresponding to X ∈ P n (R 2 ) which is an analytic vector field induced on S 2 , as described in [Gonsales, 1984], or Chapter 5 of [Dumortier et al., 2006].In this way, S 1 is identified to the infinity of R 2 ; in this paper when we speak about infinity, we mean the circle of infinity of X. System (1) doesn't have singularities at infinity, so the invariant circle S 1 at infinity is a periodic orbit.
We say that two polynomial vector fields X and Y on R 2 are topologically equivalent (resp.diffeomorphically equivalent, resp.diffeomorphically linear ) if there exists a homeomorphism (resp.diffeomorphism, resp.isomorphism) on S 2 preserving the infinity S 1 carrying orbits of the flow induced by p(X) into orbits of the flow induced by p(Y ).In particular the phase portraits are drawn in the disk D 2 that is obtained by projecting the northern hemisphere of the Poincaré sphere on the equatorial plane.
We will rely on a result due to Markus [1960], Neumann [1975] and Peixoto [1971] to determine easily whether two diffeomorphically non-equivalent systems are topologically equivalent.To state this result we first recall some definitions from [Dumortier et al., 2006].
We say that a flow R 2 , ϕ is parallel if it is topologically equivalent to the strip flow (i.e., defined by the flow of ẋ = 1, ẏ = 0), the period annulus or annulus flow (i.e., defined by the flow of ṙ = 0, θ = 1) or the spiral or nodal flow (i.e., defined by the flow of ṙ = r, θ = 0).
The boundary of a maximal open region on R 2 on which the flow is parallel is called an extended separatrix skeleton; its structure is precisely described in [Dumortier et al., 2006].The union S of all orbits in the extended separatrix skeleton is a closed set invariant under the flow.A maximal connected component of R 2 \ S, which is necessarily invariant under the flow, is called a canonical region.Given a flow R 2 , ϕ by the completed separatrix skeleton we mean the union of the extended separatrix skeleton of the flow together with one orbit from each of the canonical regions.
Let C 1 and C 2 be the completed separatrix skeletons of the flows R 2 , ϕ 1 and R 2 , ϕ 2 respectively.Then we say that C 1 and C 2 are topologically equivalent if and only if there exists a homeomorphism from R 2 to R 2 that maps the orbits of C 1 to the orbits of C 2 preserving the orientation.
According to the so-called Markus-Neumann-Peixoto Theorem it suffices to describe the completed separatrix skeleton in order to determine the topological equivalence class of a differential system.More precisely it is stated as follows: Theorem 6 [Markus-Neumann-Peixoto].Assume that R 2 , ϕ 1 and R 2 , ϕ 2 are two continuous flows with only isolated singular points.Then these flows are topologically equivalent if and only if their completed separatrix skeletons are equivalent.

Polar coordinates
By (6) the Hamiltonian class is a 4-parameter family formed by the cubic polynomial differential system of the form with Hamiltonian H = H (a,g,c,e) given by For all values of the 4-dimensional parameter (a, g, c, e) the vector field X (a,g,c,e) has a singularity at the origin; it is of center type.To localize the other singularities of X (a,g,c,e) for some arbitrary but fixed value of (a, g, c, e), and to determine their type, we study this system using polar coordinates, x = r cos θ, y = r sin θ.
Thus we obtain: where Although the functions A and B depend on the parameter, we often leave it out of our notation to facilitate the reading.Singularities of (9) correspond to values (r, θ) with To localize the singularities we look for angles θ satisfying A(θ) = 0, i.e., cos θ = 0 or a + (b + e)tanθ + (c + f )tan 2 θ + gtan 3 θ = 0.
In particular since we have

Reduction to 3-parameter subfamily
From the following lemma it follows that we can restrict our study without loss of generality to the 3parameter subfamily X H (g,c,e) defined in (2), i.e., we can assume that a = 0. (17 Moreover we can suppose that g, c ≥ 0 and that there exists 1 ≤ n ≤ 3 such that Lemma 2. For a, g, c, e ∈ R there exist g, c ≥ 0 such that the vector field X (a,g,c,e) is linearly equivalent to X (0,g,c,e) defined in (9).The linear equivalence is given by the composition of a rotation and some of the following symmetry operations: Proof.By a rotation the vector field X (a,g,c,e) is transformed into X (0,g,c,ẽ) for some parameter value (0, g, c, ẽ) .Indeed the rotation θ → θ − θ 0 is in cartesian coordinates defined by the matrix M (θ 0 ) ≡ cos θ 0 − sin θ 0 sin θ 0 cos θ 0 , and transforms the vector field X (a,g,c,e) into the vector field X K(θ 0 )(a,g,c,e) , where the linear map K (θ 0 ) is given by By T 1 (resp.T 2 and T 3 ) the vector field X (a,g,c,e) is transformed into X (−a,−g,−c,−e) (resp.−X (a,−g,c,−e) and −X (−a,g,−c,e) ).
In particular we can fix the sign of the first non-vanishing derivative of A at θ = 0 since As it is convenient in cases that A has at most two zeroes in [0, π) we shall assume that in cases that A has three zeroes in [0, π) we shall assume that October 7, 2010 11:54 CauLliTor2010˙revised Global Classification of a class of Cubic Vector Fields whose canonical regions are period annuli 11

Classification of the singularities
For a given θ * we define the ray θ = θ * by the set {(r, θ * ) : r ≥ 0}.Then for B (θ * ) = −2 system (9) has exactly one singularity on the ray R * different from the center at the origin, and its radius is 1.For B (θ * ) < −2 system (9) has two singularities on the ray R * with radius r * ± .For later use we describe the behavior of r * ± with respect to B = B (θ * ) in Lemma 3, and illustrate its graph in Figure 3. Lemma 3. Let B < −2 and let r ± be the radii defined as The graphs of r ± exhibit the following behavior (see Figure 3): (1) The graph of r + is concave and decreasing to 1 as B increases to −2.
(2) The graph of r − is convex and increasing to 1 as B increases to −2.
(3) r + + r − = −B and r Since A and B are cubic homogeneous polynomials in (cos θ, sin θ) it follows that and the same property holds for their derivatives with respect to θ.As a consequence there are at most three angles 0 ≤ θ 0 , θ 1 , θ 2 ≤ 2π satisfying A (θ i ) = 0 such that the equation r 2 + B (θ i ) r + 1 = 0 has at least one positive solution r, i = 0, 1, 2. As a conclusion, system (9) has at most 7 singularities, one singularity fixed at the origin and the other singularities laying on the rays θ = θ i for i = 0, 1, 2. From the relation (22) all zeroes of A can be derived from the zeroes in [0, π) , by translating these zeroes over integer multiples of π.Furthermore the graph of A on [π, 2π) is found by reflecting the graph of A on [0, π) about the θ-axis and translating it over a distance of π.If A vanishes along the ray θ = 0, then the horizontal axis can carry singularities; singularities are present on the positive horizontal axis, i.e. on the ray θ = 0, only if B(0) ≤ −2.If B(0) > −2, then there are no singularities present on the positive horizontal axis.By (22) if B(0) ≥ 2, then singularities are present on the negative horizontal axis, and if |B(0)| < 2, no singularities occur on the horizontal axis at all.Analogously if A vanishes along the ray θ = θ * , there are singularities present on then there are no singularities neither on θ = θ * nor on θ = θ * + π.
If det J (x * , y * ) = 0, then the singularity (x * , y * ) is degenerate; it is said to be non-elementary if both eigenvalues of J (x * , y * ) vanish.Using a local classification theorem for nilpotent singularities (see e.g., [Dumortier et al., 2006]), the topological type of the degenerate singularities can be determined in cases B (θ * ) < −2 and B (θ * ) = −2 with A ′ (θ * ) = 0. However by a detailed analysis of the asymptotics of the system, we can describe the local behavior of all degenerate singularities more precisely (see Proposition 1).
In Proposition 1 of section 4.3 we summarize all possible types of the Hamiltonian singularities in terms of the multiplicity of A on the ray θ = θ * , and the sign of A(θ) for θ ց θ * .In fact given θ * such that A (θ * ) = 0, i.e., on the ray θ = θ * singularities can be present, the proposition gives the number, the relative position and the type of the singularities on the ray θ = θ * in terms of the first non-vanishing derivative A at θ * .
The proof of this proposition and the principal result strongly relies on the relation between the trigonometric polynomials A and B, as described in the following lemma.
Lemma 4. Let A and B be the cubic trigonometric polynomials defined in (12).Then for all k ≥ 1 and for all θ ∈ R we have that Lemma 5. Let A, B be the cubic trigonometric polynomials as defined in (12) and θ * ∈ R. Then the following statements are equivalent: (2) a = c = e = g = 0.
In particular if one of the above statements is satisfied, then (9) has a global center at the origin linearly equivalent to { ẋ = −y, ẏ = x} .
Case 3 is distinguished from Case 4 by their number of zeroes in (0, π) : in Case 3 A has no zeroes in (0, π) , while in Case 4, A has 2 zeroes in (0, π) .
Due to the number of zeroes of A in [0, 2π] , multiplicity taken into account, we will call Cases 1, 2, 3 and 4 respectively by 'One triple ray', 'One double ray -one simple ray', 'One simple ray -two complex rays', and 'Three simple rays'.
Next proposition describes the local phase portrait of the Hamiltonian system (9) in a neighborhood of the rays θ = θ * along which A vanishes.
Proposition 1.Let A and B be the trigonometric polynomials of degree 3 defined in (12), and suppose that for 1 ≤ n ≤ 3, October 7, 2010 11:54 CauLliTor2010˙revised Global Classification of a class of Cubic Vector Fields whose canonical regions are period annuli 13 Case 1: One triple ray Case 2: One double ray-one simple ray Case 3: One simple ray -two complex rays Case 4: Three simple rays Then the origin is a center.Furthermore (1) If B (θ * ) > −2, then there are no additional singular points on the ray θ = θ * .The local phase portrait of (9) along the ray θ = θ * is given in Figure 5. (2) If B (θ * ) = −2, then there is one additional singular point on the ray θ = θ * for which the radius r = 1.Depending on the first non-vanishing higher order derivative A (n) (θ * ) and its sign, the local phase portrait of (9) along the ray θ = θ * is given in Figure 5. (3) If B (θ * ) < −2, then there are two additional singularities on the ray θ = θ * for which the radii are 0 < r − < 1 < r + with expressions given in (14).Depending on the first non-vanishing higher order derivative A (n) (θ * ) and its sign, the local phase portrait of (9) along the ray θ = θ * is given by Figure 5.
Lemma 6.Let k, n ≥ 1.Consider the planar differential vector field X with a singularity at (x 0 , y 0 ) , such that in polar coordinates (r, θ) the singularity x 0 corresponds to (r 0 , θ 0 ) and X writes as { ṙ = F (r, θ) , θ = G (r, θ)} for some smooth functions F, G with asymptotics where γ = 0, F (r, θ) = O (θ − θ 0 ) n+1 and Ḡ (r, θ) = O (r − r 0 ) k+1 , for θ → θ 0 , r → r 0 .Then, the diffeomorphic type of the singularity (x 0 , y 0 ) depends on the parity of k and n, the ratio k/n and the sign of γ as shown in Figure 6.For k, n both odd the singularity is a topological saddle, for k and n of opposite parity the singularity is a cusp, and for k, n both even the phase portrait near the singularity shows that of a parallel flow.
October 7, 2010 11:54 CauLliTor2010˙revised Global Classification of a class of Cubic Vector Fields whose canonical regions are period annuli 15 where (r, θ) are polar coordinates, see Lemma 6.
(2) From Proposition 1 the Hamiltonian singularities can topologically be summarized to be one of the following three types: center, saddle or cusp.In particular parameters for which B equals −2 for a certain root θ * of A, show locally along the ray R * a topologically equivalent behavior for fixed sign of γ.For n = 1 the singularity different from (0, 0) is a nilpotent cusp; for n = 2 resp.3, these singularities have zero linear part, and therefore we will call them degenerate cusp of order 1 resp.2. Analogously parameters for which B is strictly smaller than −2 for a certain root θ * of A show locally along the ray R * a topologically equivalent behavior for fixed sign of γ, in case n = 1 or 3.The saddles and centers outside the origin are elementary for n = 1 while they are nilpotent for n = 3.For n = 2 the singularities different from (0, 0) are nilpotent cusps.

Properties of the Hamiltonian
In this section we summarize the properties and the relative values of the Hamiltonian H as defined in (10) at the singular points.Observe that the Hamiltonian H depends on the parameter only through B.
Moreover the radii of the singularities are completely determined by the value of B; as a consequence the value of the Hamiltonian and its properties near a ray θ = θ * implicitly depend on the parameter through B = B (θ * ) .
For r, B ∈ R we write Lemma 7. Let H B (r) be the function defined in (30), then its graph is drawn in Figure 7. Furthermore ( i.e., the Hamiltonian value at the singularities (x * , y * ) = (r ± cos θ * , r ± sin θ * ) on the ray θ = θ * for which B = B (θ * ) and r ± is given by ( 21).(4) The graph of the functions h ± are drawn in Figure 8.In particular these functions satisfy the following properties: (a) The function h + is increasing and concave with respect to the variable B < −2, i.e., h ′ + (B) > 0 and The function h − is increasing and convex with respect to the variable B < −2, i.e., h ′ − (B) > 0 and h Proof.The expression in ( 31) is obtained using ( 10) and the fact that r 2 ± = −1−Br ± .Next by substituting the expressions for r ± into h ± (B) , one finds the expression in (31): From this expression the inequalities in statement (4)(f ) are easily derived.In particular statement (4)(e) then follows from For the properties on the derivatives with respect to B in statement (4)(a) and (b) we notice that Lemma 8. Let H B (r) be defined in (30).Then the following statements hold (1) The radial direction of the flow on a fixed ray θ = θ 1 can be read from the graph of H B , where (2) Let (x * , y * ) ∈ R 2 represented in polar coordinates by (r * , θ * ) , r * > 0, where x * = r * cos θ * , y * = r * sin θ * .Then (x * , y * ) is a singularity of (9) if and only if (r * , θ * ) is a solution of as defined in (30).
Proof.Straight-forward derivation of (30) implies the equality in (32).Furthermore it follows that the equations in (33) are equivalent to (13) with From this lemma it follows that the critical points of H B with respect to r correspond exactly to the singularities of the Hamiltonian vector field.Next lemma further analyzes the graph of H B with respect to the critical points of the Hamiltonian along a given ray θ = θ * , for which A (θ * ) = 0 and B = B (θ * ) ; see Figure 7.In this case we can distinguish between the following 3 cases: As a consequence the graphs of H B 1 and H B 2 only coincide at (0, 0) if B 1 = B 2 .In particular the graph of H B 2 lies below the graph of H B 1 .
Proof.Straight-forward calculations show that for all B ∈ R we get ∂ 2 H B ∂r 2 (0) = 1 > 0. For B > −2, there are no other critical points than r = 0, but the graph of H B is convex or convexconcave-convex depending whether B > − √ 3 or −2 ≤ B < − √ 3 (see Figure 7).For B = −2 we have (1) = 0 and Next two lemma's provide a criterion to decide whether a connection between different singularities is possible.

Classification of the Hamiltonian Phase Portraits
Using Proposition 1 we can classify all Hamiltonian singularities and their local behavior in a systematic way.Now to understand the global behavior, like the relative position of the separatrices and the existence of connections between the singularity points, we make use of the global properties of the Hamiltonian H along a ray carrying singularities (see section 4.4) as well as of a continuity argument (Lemma 11).
In first instance the classification is based on the number of complex zeroes of A in [0, π) multiplicity taken into account.For the 3-parameter Hamiltonian family in ( 2) and ( 17) the trigonometric function A in (12) reads as: Next the trigonometric function B for this 3-parameter family reads as B(θ) = e cos 3 θ + 3g cos θ sin 2 θ − c sin 3 θ.
In particular A has always a zero in [0, π) at θ = 0, and Therefore by Lemma 5 we only need to treat four cases for which the graph of A can be found in Figure 4; in terms of the parameter (g, c, e) these cases are characterized as follows: Case 1 or 'One triple ray' e − 2g = c = 0.
We denote by R i the ray Furthermore for i = 0, 1, 2 we denote ± are the values of the Hamiltonian at the singularities s i ± along the ray R i at a distance r ± from the origin.If |B (θ i )| = 2, i.e., when the singularities on R i collapse, then we will write h i : Lemma 12. Let B be the trigonometric function defined in (12).The origin of (2) is a global center if and only if Proof.From Lemma 4 it follows that both maxima in (37) coincide.The assertion follows since a linear center of the Hamiltonian system (2) with θ > 1 for all (r, θ) is a global center.

Case 1: One triple ray
In this case the functions A and B simplify to For g = 0 the functions A and B vanish identically.So the phase portrait consists of a global center.For g > 0 the graph of A is drawn in Figure 4 Case 1.As a consequence one finds A(0) = A ′ (0) = A ′′ (0) = 0, A ′′′ (0) = 6g > 0 and B(0) = e = 2g > 0. ( Clearly all singularities are found on R 0 , which corresponds to the negative horizontal axis.By Proposition 1 the diffeomorphically different phase portraits are determined by the parameter g, see Figure 10.For g < 1 the phase portrait is a global center.When g grows and reaches g = 1, a degenerate singularity s 0 is born on the horizontal axis at (−1, 0) , and the global center is disturbed by a homoclinic loop.Indeed by Lemma 9 (2) , H is strictly increasing with r ≥ 0, and by continuity with respect to the parameter g, the stable and unstable separatrices of the degenerate singularity at (−1, 0) connect, enclosing the center at the origin.Moreover h 0 − = 1/12.When g grows through g = 1, the cusp singularity splits horizontally into two singularities: a saddle s 0 − and a center point s 0 + .By the same arguments as before, Lemma 9 and continuity, we know that a figure eight is born; moreover, h 0 + < h 0 − < 1/12.This ends the proof of Theorems 2, 5 and 4 in Case 1. October 7, 2010 11:54 CauLliTor2010˙revised Global Classification of a class of Cubic Vector Fields whose canonical regions are period annuli 21 5.2.Case 2: One double ray -one simple ray By Lemma 2, we only have to consider parameter values for which e − 2g = 0, c > 0, g ≥ 0.
In this case the graph of A looks like the one in Figure 4 Case 2 and the trigonometric functions A and B can be written as for some smooth function B depending only on θ and α, where Notice that α takes values from 0 to −∞ when g goes from 0 to +∞.Hence the angle θ 1 goes from π/2 to π.By substituting θ = 0 respectively θ = θ 1 in the above expressions one finds and respectively where we used the fact that sin θ 1 = 1 + α 2 −1/2 and θ 1 lies in [π/2, π) .
Clearly A has 1 double zero at θ = 0 and 1 simple zero θ 1 , see Figure 4 Case 2. Moreover we can describe the bifurcation of the phase portraits in terms of α and c.The parameter α determines the ray R 1 ≡ {θ = θ 1 , r ≥ 0} on which singularities can occur besides on the ray R 0 ≡ {θ = π, r ≥ 0}.For fixed value of α the parameter c determines the presence of 1, 2 or 3 singularities on the rays R 0 and R 1 .Indeed for fixed α the amplitude of the function B grows linearly with c, see (40).
For a fixed but arbitrarily value α ≤ 0, we can describe the bifurcation of the phase portraits with respect to c.The values B (0) ≡ B 0 and B (θ 1 ) ≡ B 1 are constants.Furthermore B (0) and B (θ 1 ) linearly depend on c and we can write B 0,c ≡ cB 0 and B 1,c ≡ cB 1 . (41) Additionally it is easy to verify that B 1 < −B 0 < 0, hence for all c > 0, B 1,c < −B 0,c < 0. The bifurcation values c 1 = c 1 (α) and c 2 = c 2 (α) that determine the birth of singularities on the rays R 1 and R 0 are respectively given by and Case 2(a) If 0 ≤ c < c 1 , then B 1,c > −2.The singularity at the origin is a global center of (2).Case 2(b) If c = c 1 , then B 1,c = −2 < −B 0,c .There are 2 singularities, (0, 0) and s 1 = (cos θ 1 , sin θ 1 ) , both lying on R 1 .
Case 2(c) If c 1 < c < c 2 , then B 1,c < −2 < −B 1,c .There are 3 singularities, (0, 0) and s 1 ± , on the ray R 1 and there are no singularities outside this ray: s 1 − is a saddle and s 1 − is a center.The bifurcation of the phase portraits for c passing through c 1 occurs analogously as in section 5.1, but this time the singularities are on ray R 1 instead of the horizontal axis.
Case 2(d) If c = c 2 , then B 1,c < −2 = −B 0,c .There are 4 singularities, (0, 0) and s 1 ± on the ray R 1 , and s 0 = (−1, 0) on the ray R 0 .The type of the singularities s 1 − (saddle) and s 1 − (center) does not change, however their exact position on the ray R 1 does change.The singularity s 0 is a degenerate cusp of order 1 (see Remark 4.1).By Lemma 7 and Figure 11, we know that the Hamiltonian value at the saddle point s 1 − is smaller than at the degenerate cusp singularity s 0 , where h 0 = 1/12.Now because the Hamiltonian along R 0 is strictly increasing with respect to r, there is a homoclinic loop through the degenerate cusp singularity that encloses the non-isolated periodic orbits surrounding the double homoclinic loop through s 1 − .
Case 2(a) Case 2(b) Case 2(c) Case 2(d) If c > c 2 , then there are 5 singularities, (0, 0) and s 1 ± on the ray R 1 , and (0, 0) and s 0 ± on the ray R 0 ; the nilpotent cusps s 0 ± are born from the degenerate cusp s 0 .Clearly the Hamiltonian value h 0 − at the smallest cusp singularity is always bigger than the one h 1 − at the saddle point along the ray R 1 .Therefore only a connection between the saddle singularity s 1 − and the biggest cusp singularity s 0 + is possible.If c now grows to +∞, the corresponding values B 0,c and B 1,c decrease to −∞ respecting their respective order, however as is illustrated in Figure 11, the relative order of the values of the corresponding Hamiltonian values changes at some value c 3 , giving rise to the crossing of homoclinic connections and the appearance of a heteroclinic connection.Lemma 11 ensures the existence of this third bifurcation value and In particular only for c = c 3 there exists a connection between s 1 − and s 0 + .Using the argument of continuity, the increasing of the Hamiltonian with respect to r ≥ 0 and the corresponding graphs in Figure 11, we obtain the phase portraits in the cases c > c 2 as drawn in Figure 12.Using the relation between α and (g, c) we can draw the bifurcation diagram of phase portraits in Case 2 in the (g, c, e)-space; by Lemma 2 it then suffices to present the bifurcation diagram to g, c ≥ 0 in the plane e = 2g as done in Figure 12.In this figure we distinguish between 7 regions which are denoted by R 1 , R 2 , . . ., R 7 respectively for which the typical phase portraits in each region R i , i = 1, . . ., 7, are respectively drawn.
In particular the explicit expression of the bifurcation curves can be computed.Solving (13) for θ = 0 and θ = θ 1 we obtain the value for the radius of the singularities s 0 ± and s 1 ± along the rays R 0 for c ≥ c 2 and R 1 for c ≥ c 1 respectively: and for the corresponding Hamiltonian values at these singularities: Therefore the bifurcation curves defined in ( 42), ( 43) and ( 44) can be presented respectively as the following algebraic sets: Taking into account the number of singularities, the phase portrait also is uniquely determined with respect to diffeomorphic equivalence in case the total number of singularities is less than five.When the maximum number of singularities is reached, five in this case, it is necessary to consider the value of the Hamiltonian at the singularities different from the origin.To be precise it suffices to consider only h 1 − and h 0 + to describe the bifurcation.For completeness in Figure 12 we indicate the relative value of the Hamiltonian values for all singularities.
This ends the proof of Theorems 2 and 5 in Case 2. Notice that c = 0 in Case 2, hence the Hamiltonian systems in Case 2 are not reversible and therefore there is nothing to prove for Theorem 4 in Case 2.

Case 3: One simple ray -two complex rays
By Lemma 2 and ( 19) we can assume without loss of generality that e − 2g > 0. In this case the graphs of A and B look like in Figure 4 Case 3. We have By the assumption g ≥ 0, it follows that e > 0. Therefore all singularities lay on R 0 , which corresponds to the negative horizontal axis {(x, 0) : x ≤ 0}.
Using Proposition 1 and Lemma 9, for a fixed value of g ≥ 0, a bifurcation occurs at e = 2 diffeomorphically equivalent to the one that happens in case e − 2g = c = 0.
For e < 2 the phase portrait is a global center.For e = 2 there is a cuspidal loop with cusp s 0 at (−1, 0) ; by Lemma 9, it follows that h 0 = 1/12.For e > 2 there appears the figure eight corresponding to a double homoclinic saddle loop at s 0 − ; by Lemma 9, it follows that h 0 + < h 0 − < 1/12.The bifurcations of the phase portraits in Case 3 with respect to e are drawn in Figure 13.In particular these three phase portraits correspond with 1, 2 or 3 critical points, respectively.This ends the proof of Case 3 of Theorems 2, 4 and 5. To each parameter value (g, c, e) of Case 4 there corresponds a unique parameter value (α, β, λ) with 0 ≤ −β ≤ α defined by ( 46) and ( 47), and so in Case 4 we can describe the global phase portraits in terms of the new parameter (α, β, λ) .Therefore we fix an arbitrary parameter 0 ≤ −β ≤ α and we study the bifurcation of the global phase portraits in the 1-parameter subfamily ( 2 for some smooth functions B i , i = 0, 1, 2 depending only on (α, β) .
For fixed (α, β) the graph of A looks the same as the graph of A (resp.B and B); indeed it is only a rescaled copy with respect to the vertical coordinate.Denote by R i the ray θ = θ i if B i ≡ B (θ i ) < 0 and θ = θ i + π if Bi ≡ B (θ i ) > 0, i = 0, 1, 2. Increasing λ to +∞ the values B i pass subsequently through 2 and so singularities are born on the corresponding ray R i .The order according to which rays carry singularities for the first time is dictated by the relative order of B i , i = 0, 1, 2. For instance suppose that i j , j ∈ {0, 1, 2} with then singularities occur first on ray R i 0 (as soon as λ = 2/ B i 0 ), next on ray R i 1 (as soon as λ = 2/ B i 1 ) and finally on ray R i 2 (as soon as λ = 2/ B i 2 ).Notice that we do not exclude the simultaneous appearance of singularities on different rays, thus when B i 0 = B i 1 , singularities appear simultaneously on rays R i 0 and R i 1 .Proof.[Proof of Theorem 7 (4) and (5)] The bifurcation of the phase portraits with respect to increasing λ in Cases 4D and 4E are analogous to the one in Case 4A, since the saddle points appearing on all the rays R i correspond to the smallest ones s i − , i = 0, 1, 2. In Case 4D since B i 1 = B i 2 we also have |B i 1 (λ)| = |B i 2 (λ)| for all λ.Therefore in Case 4D first the singularities appear along R i 0 , next the singularities appear simultaneously along R i 1 and R i 2 , and the local phase portrait near R i 2 is an identic copy of the one near R i 1 .The first three bifurcations are completely analogous to the ones in Case 4A, the last two bifurcations follow from an analysis based on the Hamiltonian.
In Case 4E since B i 0 = B i 1 we also have |B i 0 (λ)| = |B i 1 (λ)| for all λ.Therefore in Case 4E first the singularities appear along R i 0 and R 1 simultaneously, where the local phase portraits near R 0 and R 1 are diffeomorphic copies; next the singularities appear along R i 2 .
Proof.[Proof of Theorem 7(6) and 7(7)] The bifurcation of the phase portraits with respect to increasing λ in Case 4F is analogous to the one in Case 4B, since the saddle points appearing on the rays R i j , j = 0, 2 correspond to the smallest ones s i j − , j = 0, 2 and the saddle on R i 1 corresponds to the biggest one s i 1 − .

Fig. 3 .
Fig. 3. Graphs of r + and r − as functions of B, drawn in dashed line and continuous line respectively.
and n odd k and n even k odd and n even k even and n odd

Lemma 9 .
Consider the function H B : R → R defined by (30) for B in R.(1) If B > −2, then H B has 1 critical point over R, r = 0, it is a global minimum.(2) If B = −2,then H B has 2 critical points over R, a global minimum at r = 0 and a point of inflection at r = 1.(3) If B < −2, then H B has 3 critical points over R, H B has a local minimum at r = 0 and at r = r + , and a local maximum at r = r − , where r ± are defined in (21).

Fig. 11 .
Fig. 11.Graphs of h + and h − for fixed value of the parameter (α, c) ; the parameter c increases going from Case 2(a) to Case 2(g).
Fig. 8. Graphs of h + and h − with respect to B; see Lemma 7.