Detecting alien limit cycles near a Hamiltonian 2-saddle cycle

This paper aims at providing an example of a cubic Hamiltonian 2saddle cycle that after bifurcation can give rise to an alien limit cycle; this is a limit cycle that is not controlled by a zero of the related Abelian integral. To guarantee the existence of an alien limit cycle one can verify generic conditions on the Abelian integral and on the transition map associated to the connections of the 2-saddle cycle. In this paper, a general method is developed to compute the first and second derivative of the transition map along a connection between two saddles. Next, a concrete generic Hamiltonian 2-saddle cycle is analyzed using these formula’s to verify the generic relation between the second order derivative of both transition maps, and a calculation of the Abelian integral.


Introduction and settings
We deal with perturbations of Hamiltonian systems: (X (µ,ε) ) : where H(x, y), f (x, y, µ, ε), g(x, y, µ, ε) are C ∞ functions, ε is considered to take small positive values and µ varies in some compact subset K ⊂ R p . Further we abbreviate µ = (µ, ε). We suppose that the flow of X (µ,0) = X H contains a period annulus bounded by a hyperbolic 2-saddle cycle L as in Figure 1. A period annulus is a subset of the plane filled by closed orbits of X H . The hyperbolic 2-saddle cycle consists of two saddle-connections Γ 1 and Γ 2 and two hyperbolic saddles s 1 and s 2 such that s 1 := α(Γ 1 ) = ω(Γ 2 ) and s 2 := α(Γ 2 ) = ω(Γ 1 ). We choose H to be zero on the 2-saddle cycle and strictly positive on the nearby closed orbits. In [9] it is proven that, for µ ∈ K and ε > 0 near zero, L can produce limit cycles that are not controlled by zeros of the related Abelian integral (cfr. (5)); these limit cycles are also called 'alien limit cycles' (cfr. [1]). In [9], one found that exactly one alien limit cycle exists in a 'generic' unfolding (1) of codimension 4, leaving one connection of the 2-saddle cycle unbroken. A precise definition is given in Definition 2.
The principal result in this paper establishes the presence of this bifurcation phenomenon of alien limit cycles in the unfolding (X (µ,ε) ) of the quadratic Hamiltonian system X H with two centers and two heteroclinic loops: 8 xy)], y = 2xy + εy(µ 1 + µ 2 x).
(2) where the Hamiltonian H is given by H(x, y) = y(x 2 + 1 12 Notice that this system was also studied in [6]. Verification of the generic conditions that guarantee the presence of an alien limit cycle that bifurcates from the 2-saddle cycle lying in {y ≤ 0} , is quite involved. For an unfolding to be 'generic', besides a genericity property on the related Abelian integral (cfr. (5) , (6) , (9) and (10)), a genericity property of the second order derivative of the transition map along the saddle connections (cfr (7) , (8) and (11)) have to be satisfied as well. Sections 4, 5 and 6, present useful techniques to check the generic conditions in concrete examples.
In this paper we prove the following results. We obtain general formulas for the second order derivative of the transition map near a saddle connection, that remains unbroken in an smooth unfolding of a Hamiltonian vector field; these formulas are stated in section 6: Corollaries 18 and 19 respectively. Next, in section 7, using the developed machinery, in Section 7 the generic conditions are verified in the concrete system (2) ; we conclude that the generic properties described in Definition 2 all are satisfied; in particular, from the result in [9], we can conclude with: Theorem 1 Let (X (µ,ε) ) be the unfolding of the Hamiltonian vector field X H given in (2) with Hamiltonian H given in (3) ; let L be the 2-saddle cycle with saddle points (−1, 0) and (1, 0) , lying in the half plane {y ≤ 0} . Then, Preprint submitted at DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. The final publication is available at http://www.aimsciences.org/journals/redirecting.jsp?paperID=4488 2. Hence, the bifurcation diagram of limit cycles bifurcating from L with respect to (µ, ε), for (μ, ε) sufficiently small, ε > 0, exhibits a swallow tail catastrophy.
In this paper, we only study the 2-saddle cycle in the half plane {y ≤ 0} ; one can study the 2-saddle cycle in the half plane {y ≥ 0} with the same reasoning.
Notice that the results obtained in sections 4, 5 (and 6) are valid for arbitrary analytic families of vector fields (perturbations of a Hamiltonian vector field); only in section 7, we work with the concrete Hamiltonian unfolding (2) .
The paper is organised as follows. In section 2, the generic conditions are specified; in section 3, appropriate normal forms near the hyperbolic saddles s 1 and s 2 are given, that are used to calculate the second order derivative of the transition maps R 1 µ and R 2 µ in Sections 4, 5 and 6. In section 4, relying on [2], general formulas for the second order derivative of the transition map near a saddle connection, that remains unbroken in an smooth family of vector fields. In section 5 (respectively 6), these formulas are translated for smooth families of vector fields when expressed in normalizing coordinates (respectively for smooth unfoldings of a Hamiltonian vector field). Finally, using the developed machinery, in Section 7 the generic conditions are verified in the concrete Hamiltonian system (2).

Generic conditions
Throughout this article we suppose that (X µ ) is a smooth unfolding of a Hamiltonian vector field X H like in (1) such that X H admits a period annulus bounded by a 2-saddle cycle L like in Figure 1 and where µ varies in some neighbourhood of (µ 0 , 0), µ 0 ∈ K. After a translation in parameter space, one can always suppose that µ 0 = 0. Furthermore, we suppose that the connection Γ 2 remains unbroken by the unfolding.
Then, we consider the regular transition maps R 2 µ from Σ 1 to Σ 3 along Γ 2 defined by the flow of −X (μ,ε) , and R 1 µ from Σ 2 to Σ 4 along Γ 1 , defined by the flow of X (μ,ε) . Let D 1 µ (respectively D 2 µ ) be the corner passages near the saddle s 1 (respectively s 2 ) defined by the flow of X (μ,ε) (respectively −X (μ,ε) ), see Figure 2. We suppose that all these transition maps are expressed in function of the chosen regular parameter on the sections Σ i , i = 1, . . . , 4. They are only locally defined: ε as well as the regular parameter u, v, z, w take on small positive values.  Figure 2: Transition maps near a 2-saddle cycle.
Now the difference map ∆ µ : Σ 1 → Σ 4 is locally defined as: for u > 0, with Clearly, for µ near (µ 0 , 0), µ 0 ∈ K, limit cycles of X µ near L, correspond to positive zeroes u of ∆ µ , for u near 0. In particular, we can write for a C ∞ map∆. As for the traditional displacement map (retour mapping minus identity), the linear part of ∆ with respect to ε is related to the Abelian integral I µ (h) for X (μ,ε) ε (cfr. [9]): where γ h is the non-isolated periodic orbit of X H lying inside of {H = h} and passing through the point u on Σ 1 . Furthermore, it is well-known that I(h, µ) admits an asymptotic expansion in the logarithmic scale 1, h, . . . , h i , h i log h, . . . : there exist smooth functions p, q, r, s inμ such that The coefficients p, q, r, s in this expansion can be calculated using Picard-Fuchs equations (cfr. section 7 for an example). The notion of a generic unfolding of codimension 4 also involves the asymptotics of the regular transition maps R 1 µ and R 2 µ . In Section 6, it will be shown that, up to terms of order O (ε) , ε ↓ 0, the transition maps R 1 µ and R 2 µ are the identity map, when expressed in appropriate normalizing coordinates near the saddles: for some smooth functions β 1 , γ 1 , η 1 in the parameter µ = (µ, ε) and for some smooth functions β 2 , γ 2 , η 2 in the parameter µ = (µ, ε). Furthermore, since Γ 2 remains unbroken, we have that β 2 (µ) = 0, ∀µ, and after performing a parameter dependent coordinate change in u, one can suppose that γ 2 (µ) = 0 yielding to: for some smooth function η 2 in µ = (μ, ε) .
Let R 1 µ (respectively R 2 µ ) be the regular transitions along the connection Γ 1 (respectively Γ 2 ). Then, we say that X (μ,ε) is a generic unfolding of X H of codimension 4, if 1. the Abelian integral is of codimension 3, i.e., and the map is a local submersion at zero.
2. the functions η 1 and η 2 , defined by the asymptotic expansions of R 2 µ and R 1 µ in (7) (respectively (8)), satisfy the following generic condition: Remark 3 The main result in [9] implies that a 2-saddle cycle for such a generic Hamiltonian unfolding X (μ,e) of codimension 4 can produce for limit cycles, while it is clear that the related Abelian integral Iμ can have at most 3 zeroes that bifurcate from h = 0, forμ near 0. This is striking since in case that L is a saddle-loop and the Abelian integral is generic, there is a 1-to-1 correspondence between the bifurcation diagram of the limit cycles perturbing from the Hamiltonian saddle loop and the zeroes of the related Abelian integral (cfr. [14]).

Normal forms at hyperbolic saddles
In this section, we recall some useful normal forms for families of C ∞ vector fields near a hyperbolic saddle, that will simplify the calculations of the quantities η 1 (0) and η 2 (0) ; in particular, we give a specification of the isochore Morse lemma for a Hamiltonian unfolding in Theorem 10. Consider a C ∞ family of planar vector fields (X µ ) with parameter values µ varying in some open set P of R p . Suppose that for some µ 0 ∈ P, X µ0 admits a hyperbolic saddle s, and suppose that the Jordan normal form of DX µ0 (s) is given by with λ 2 < 0 < λ 1 . The ratio of hyperbolicity of X µ0 at s is defined by −λ 2 /λ 1 .
As an easy consequence of the implicit function theorem, one can suppose that the saddle is persistant for all X µ , µ ∈ P. By this we mean that there exists a C ∞ function s : P → R 2 such that each X µ admits a hyperbolic saddle at s µ := s(µ) with s µ0 = s.
The following theorem can be found in [15].
Theorem 5 Let (X µ ) µ∈P be a C ∞ family as above such that X µ0 admits a hyperbolic saddle s. Suppose that the ratio of hyperbolicity of X µ0 at s is rational, given by p/q with p, q ∈ N 1 , (p, q) = 1. Then for each N ∈ N, there exists a neighbourhood P N of µ 0 in parameter space such that the M -jet, M = N (p + q) + 1, at s µ of each X µ , µ ∈ P N , is locally C ∞ -conjugate to: where the coefficients a i (µ) and b i (µ) are smooth in µ. In case the ratio of hyperbolicity of X µ0 at s is irrational, then for every N ∈ N, there exists a neighbourhood P N of µ 0 in parameter space such that the N -jet of X µ , µ ∈ P N , is, locally near s µ , C ∞ linearisable.

Remark 6
1. The above theorem only applies to every finite jet of the family of vector fields, while for an individual vector field we have a normal form for its infinite jet at our disposal.
2. Using the theorem of Sternberg for families ( [13]), it follows immediately from the above theorem that, in case the ratio of hyperbolicity of X µ0 at s is rational, ∀k ∈ N, there exists some N (k) ≥ k such that the family (X µ ), µ varying near µ 0 , is, locally near s, C k -conjugate toX In case of an individual integrable vector field a further simplification near a hyperbolic saddle can be obtained by applying Morse's lemma on the first integral H.
Proposition 7 Let X be an integrable vector field with first integral H and admitting a hyperbolic saddle s. Denote by (u, v) the coordinates near s, given by Morse's lemma, in which H reads uv. Then, near s and expressed in the coordinates (u, v), X reads: up to C ∞ equivalence and a possible coordinate change in (u, v).
Proof. Denote by Y , the vector field, defined locally near the origin, that one obtains after expressing X in the coordinates (u, v). Because uv is a first integral of Y , it is clear that there exists a C ∞ function Y , with Y (0) = 0, such that Y 1 = −uY and Y 2 = vY . After a possible coordinate switch in (u, v), one can always suppose that Y (0) > 0 implying the desired result. We continue by describing another way to obtain the normal form (14), in case X is a Hamiltonian vector field. This method will first reduce X to a formal normal form (13) and can be useful when performing calculations in practice. However, using this method, the normal form (14) is only obtained on each of the half planes {u ≥ 0}, {u ≤ 0}, {v ≥ 0} or {v ≤ 0}, which will be sufficient for our further practical use of it.
We notice that similar reductions of Morse functions are already obtained in [12]. However these results were only valid near critical points that are not saddle points. The method that we propose is based on the following proposition.
Proposition 8 Let X be an integrable vector field with first integral H : V ⊂ R 2 → R that admits a hyperbolic saddle s. Suppose that there exist C ∞ coordinates (u, v), near s, in which X reads: Preprint submitted at DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. The final publication is available at http://www.aimsciences.org/journals/redirecting.jsp?paperID=4488 up to C ∞ equivalence. Then H, expressed in the coordinates (u, v), can be written as a C ∞ function in uv, locally near the origin, on each of the half Proof. We prove the statement for H restricted to H = {u ≥ 0} ; in an analogous way the statement is obtained for the other half planes. The orbits of system (15) are given by the curves {(0, 0)} , Choose c 0 such that (c 0 , 0) ∈ H ∩ V, then we define the map f as follows: for Proposition 9 For a Hamiltonian vector field X H , with Hamiltonian H, given by: and admitting a hyperbolic saddle s, there exist C ∞ coordinates (u, v), near s, in which the ∞-jet of X H reads: for some a i ∈ R, i ∈ N 1 .
Proof. Theorem 5 guarantees the existence of a C ∞ coordinate transformation (x, y) = ϕ 1 (u, v) near s in which the ∞-jet of X H reads: for some λ > 0, a i , b i ∈ R, i ∈ N 1 . We prove that the coefficients a i and b i in (18) coincide. It is easily verified that the coordinate transformation (x, y) = ϕ 1 (u, v) transforms the Hamiltonian vector field into: Preprint submitted at DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. The final publication is available at http://www.aimsciences.org/journals/redirecting.jsp?paperID=4488 On the other hand using Borel's theorem on the realization of formal power series, one finds smooth functions f and g such that In particular ϕ 1 brings X H into with j ∞ R(0) = j ∞ S(0) = 0. After a suitable near-identity transformation, one can suppose that R = S = 0 [7]. Comparing (19) with (20), and abbreviating det It is easily seen that ∀k ∈ N 1 , the 2k-jet at zero of the expression does not contain terms in (uv) i , i ≤ k. Therefore comparing terms in uv of the 2-jet at zero of (21), one sees immediately that a 1 = b 1 . By comparing terms in (uv) k in the 2k-jet at u = v = 0 of (21), one can proceed by induction on k ≥ 1 to prove that a k = b k , ∀k ∈ N 1 . The following theorem is a particular case of the 'isochore Morse lemma' proved by Colin de Verdières (for C ∞ vector fields on R n , n ∈ N) in [3]; however, to keep the paper self-contained without being lengthy, we include here the theorem and its proof for C ∞ vector fields on R 2 .
Theorem 10 Let X H be a Hamiltonian vector field that admits a hyperbolic saddle s at which the eigenvalues of DX H (s) are given by ±λ, λ > 0. Then there exist C ∞ coordinate coordinates ϕ : In particular, if H is one of the half planes {n ≥ 0}, {n ≤ 0}, {m ≥ 0} or {m ≤ 0}, then ϕ can be chosen such that the Hamiltonian H (ϕ (n, m)) = nm on H and the equivalence factor equals 1/ det Dϕ(n, m) = λ + d(nm) on H. Furthermore, we can suppose that {n = 1} is contained inside of C 1 . Proof. From Proposition 9 and using Borel's theorem on the realization of formal power series, one finds a C ∞ coordinate change (x, y) = ϕ 1 (u, v) bringing X H , near s, into: with f being smooth such that j ∞ f (0)(z) = i≥1 a i z i and R and S being flat at zero, i.e. j ∞ R(0) = j ∞ S(0) = 0. Applying a suitable near-identity transformation, one can suppose that the flat terms R and S are zero [7], such that the normal form of X H for C ∞ equivalence reads −u ∂ ∂u + v ∂ ∂v . From Proposition 8, one knows that H, expressed in the new coordinates (u, v), is a function in uv locally near the origin on each of the half planes After a reflection with respect to the u-axis, one can suppose that H 0 (0) > 0. One performs the local transformation (n, m) = ϕ 2 (u, v), with where G 0 (uv) = H 0 (uv). This transformation will leave the linear normal form, up to C ∞ equivalence, invariant but will bring the Hamiltonian into nm on the half plane {n ≥ 0}. In the new coordinates (n, m), X H reads: On the other hand it is a straightforward calculation to verify, using Proposition 8, that f (uv) can be written as a function d in nm locally near the origin on the half plane {n ≥ 0}. Therefore applying the transformation (24) on (23) (with R = S = 0), one finds: for some C ∞ function d with d(0) = 0 implying the result on {n ≥ 0}. The same arguments can be used for obtaining the result on the half planes {n ≤ 0}, {m ≥ 0} or {m ≤ 0}. Furthermore, by performing a dilatation, we can obtain that {n = 1} is contained inside of C 1 .

Transition maps
In this section we give formulas for the first and second derivative of the transition map, based on Diliberto's theorem.
Denote by X a C ∞ planar vector field with flow φ(t, v) := φ t (v), v ∈ R 2 . Take two sections Σ 1 and Σ 2 transverse to some regular orbit of X. Suppose Denote by T (s) the transition map of X from Σ 1 to Σ 2 expressed in the chosen Preprint submitted at DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. The final publication is available at http://www.aimsciences.org/journals/redirecting.jsp?paperID=4488 parameters s and s . In particular the orbit φ t (ψ 1 (s)) crosses the section Σ 2 at the point ψ 2 (T (s)). Let the function τ (s) be the transition time function of X from Σ 1 to Σ 2 expressed in the chosen parameter s; τ (s) is the time needed to go from ψ 1 (s) to ψ 2 (T (s)).
Derivatives of the transition map and the transition time function can be found by means of implicit differentiation of the expression: Denoting for fixed t, D 2 φ(t, v) as the differential of φ(t, v) with respect to v, we find To find the desired derivatives, we use Diliberto's theorem [4] to decompose the vectorial equation (26) with respect to an appropriate orthogonal basis. See also [2], where formulas for T (s) and τ (s) are already obtained using Diliberto's theorem.
The scalar and wedge product between a vector field X with Euclidean coordinates (P, Q) and a vector field X with Euclidean coordinates (P , Q) are denoted as X · X = P P + QQ, and X ∧ X := P Q − QP .

Define the vector field
multiple of the orthogonal vector field: such that X ⊥ · N = 1. The following C ∞ functions are referred to as the curl, the divergence and the curvature of X at p respectively: and: Theorem 11 (Diliberto [4]). Let X be a C ∞ planar vector field with flow has solution D 2 φ(t, p)w = A(t)X(φ t (p)) + B(t)N (φ t (p)), where A(t) := A(t, X, p, w) and B(t) := B(t, X, p, w) are given by: Theorem 12 Let X be a C ∞ vector field. Consider the transition map T (s) between two sections Σ 1 and Σ 2 transverse to the flow of X. Suppose ψ 1 and ψ 2 are regular parametrisations of these sections and Γ s is the orbit starting at ψ 1 (s) and ending in ψ 2 (T (s)). Let the quantities ∆ i (s) , i = 1, 2, be defined as: with i = 1, 2, the derivatives of first and second order of T are given by: where ds represents the arc length element of Γ s and where A(z) := A(z, X) and B(z) := B(z, X), z = (x, y) ∈ R 2 , are given by: with Γ s (z) the orbit starting at ψ 1 (s) and ending in z.

Using (37) this yields
where I is given by with A and B defined as in (31). Substituting (38) and τ (s) = α 2 (s)T (s) − A(τ (s)) (that follows from (34)) into (36) and using: the formula of T (s) follows. The formulas in the following corollary can already be found in [5]. One easily verifies that they are a special case of the formulas stated in Theorem 12.
Corollary 13 Let X = P ∂ ∂x + Q ∂ ∂y be a C ∞ vector field. Suppose Γ is an orbit lying on the x-axis and Σ i = {x = x i } are sections locally transverse to the flow of X at (x i , 0), i = 1, 2 and parametrised by y → (x i , y). Then the first two derivatives of the transition map T along Γ from Σ 1 to Σ 2 read:

Transition along a saddle-connection
In this section, using the general formulas obtained in section 4, we derive formulas for the first and second derivative of the transition map along a 2saddle connection, using normalizing coordinates near the saddles. In particular we don't restrict to individual vector fields but consider the transition near the 2-saddle connection in a family that leaves the saddle-connection unbroken.
We consider a C ∞ family of vector fields (X µ ) µ∈P with parameter values µ varying in some subset P ⊂ R p . We suppose that for µ 0 ∈ P, the vector field X µ0 admits a saddle-connection Γ, with α(Γ) = s 1 and ω(Γ) = s 2 , s 1 and s 2 hyperbolic saddles of X µ0 . Let the vector field (X µ ) be expressed in the coordinates (x, y).
In particular we suppose that for µ near µ 0 , X µ has two hyperbolic saddles s 1 (µ) and s 2 (µ) lying in a neigbourhood of s 1 respectively s 2 such that s i (µ 0 ) = s i , i = 1, 2 and that there exists a saddle-connection Γ µ between them that coincides with Γ for µ = µ 0 .
Furthermore, we suppose that for µ near µ 0 , the connection stays unbroken. This asumption is not restrictive by Remark 4.
Normalizing coordinates near the saddles. We suppose that (X µ ) µ∈P0 can be brought into a normal form at both saddles s 1 and s 2 . The normal form at s i depends on the ratio of hyperbolicity r i (see Theorem 5). From now on, Preprint submitted at DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. The final publication is available at http://www.aimsciences.org/journals/redirecting.jsp?paperID=4488 (n, m) will denote the normalizing coordinates near s 1 or s 2 depending on which saddle, s 1 or s 2 , we are dealing with.
In case r i is given by p i /q i , p i , q i ∈ N 1 , (p i , q i ) = 1, there exist some C k (k ≥ 2) coordinates, near the saddle s i , in which the family (X µ ) µ∈P0 reads: where P i (z, µ) and Q i (z, µ) are polynomials in z = n pi m qi of degree N (k) ≥ k and of order O(z 2 ). The functions λ i , ν i , a i and b i (respectively the polynomials P i and Q i ) in (39) depend in a C ∞ -(respectively C k -) way on the parameter µ.
On the other hand if the ratio of hyperbolicity r i of DX µ0 (s i ) is irrational, then (X µ ) µ∈P0 is C k linearisable near the saddle s i . In particular there exists some C k coordinates near the saddle s i in which (X µ ) µ∈P0 reads: The coordinate transformations expressing the coordinates (x, y) in function of (n, m) are denoted as ϕ 1 µ and ϕ 2 µ near s 1 and s 2 respectively. We choose normalizing coordinates near s 1 (resp. s 2 ) such that points on the positive naxis correspond to points on the unstable (resp. stable) separatrix of s 1 (resp. s 2 ), lying on Γ for µ = µ 0 . This can always be achieved by performing a suitable linear transformation in (n, m).
Let us denote the determinants of the corresponding jacobians of these transformations as Remark that geometrically A i µ (n, m) represents the area of the paralellogram spanned by the vectors ∂ϕ i µ ∂n (n, m) and ∂ϕ i µ ∂m (n, m). The angle between these two vectors is strongly related to the function θ i µ (n, m). Let us state the following lemma, which will be of use later on. It can be applied near both saddles s 1 and s 2 inside the family (X µ ) µ∈P0 .
The transition map from Σ 1 µ to Σ 2 µ expressed in the normalizing coordinate m is denoted as R µ (m). Remark that R µ (m) is only defined for m near zero and µ near µ 0 . Calculating the derivatives of R µ directly using Theorem 12 is not possible. Indeed only a finite jet of ϕ 1 µ and ϕ 2 µ at (0, 0) can be calculated implying that one is not able to calculate the derivatives of the parametrisations of the sections Σ i µ , i = 1, 2. However this can be dealt with by using a limiting process (see also [8]).
Take K 0 > 0 and ε 0 > 0 such that {(x, y) | 0 ≤ n < K 0 , −ε 0 < m < ε 0 } lies in the domains of ϕ 1 µ and ϕ 2 µ . For some 0 < K < K 0 fixed, consider the Consider the part of Γ µ lying between the sections C 1 µ,K and C 2 µ,K , denoted as Γ µ,K , and write Z = T µ,K (Y ) as the transition map along Γ µ,K from C 1 µ,K to C 2 µ,K expressed in the parameter m. Further let F µ,K (respectively G µ,K ) be the transition maps from {n = 1} to {n = K} near s 1 (respectively {n = K} to {n = 1} near s 2 ), expressed using as parameter the normalizing coordinate m. Then the transition map R µ can be seen as the composition R µ = G µ,K • T µ,K • F µ,K .
The first two derivatives of R µ at zero are now given by and (46) Because these equalities hold for every 0 < K < K 0 , one can switch over to the limit for K → 0 causing the chosen sections C 1 µ,K and C 2 µ,K to tend arbitrarily close to the saddles. This process will enable us to calculate the derivatives as stated in the following theorem.
Theorem 15 Let (X µ ) be a C ∞ family admitting for each parameter two hyperbolic saddles s 1 (µ) and s 2 (µ) with a saddle-connection Γ µ between them. Let R µ be the transition map from Σ 1 µ to Σ 2 µ along Γ µ expressed using normalizing coordinates. Consider the normal form at s i , (39) or (40), and the corresponding coordinate transformation ϕ i µ , i = 1, 2. Let Γ µ,K be the part of Γ µ starting Preprint submitted at DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. The final publication is available at http://www.aimsciences.org/journals/redirecting.jsp?paperID=4488 at ϕ 1 µ (K, 0) and ending in ϕ 2 µ (K, 0). Then one has where ds represents the arc length element of Γ µ . Suppose that R µ (0) = 1, then The function I µ (K) disappears for r i / ∈ N. When r i ∈ N it is given by the difference where ω is the traditional compensator defined by (49) Remark 16 In expressions (47) , (48) as well as in (60) , some terms tend to infinity, but the limit value of the expression in the right-hand side exists and is finite, since the limit of the left-hand term is well-defined for K → 0 by definition. In fact, in (47) for instance, the logarithm of the expression in between the brackets [. . .] is given by since its limit for K → 0 is finite, the integral is divergent and its principal part is given by In fact, we are not interested in these principal parts but just in the finite quantities which remain after subtracting these (non-interesting) principal terms. In some sense, it is a question of method: we want to compute quantities which are trivially known to be finite (R µ (0), R µ (0), and next η 1 (0) , η 2 (0)) and the method is to apply expressions which diverge in terms of a parameter K and to retain some finite residue.
Proof. We will successively calculate all derivatives appearing in the right-hand side of formula (45). Because the equality holds for all 0 < K < K 0 , one can take the limit as K → 0 to find the desired derivative R µ (0). For calculating the derivatives F µ,K (0) and G µ,K (0), one can use the formulas of Corollary 13. One computes F µ,K (0) = K −r1(µ) and G µ, The derivative T µ,K (0) can be calculated using formula (30) in Theorem 12. From Lemma 14, it easily follows that One now substitutes (51) into (50) and takes the limit as K → 0. Because the coordinate transformations ϕ 1 µ and ϕ 2 µ are locally diffeomorphisms, A i µ , i = 1, 2 stays away from zero for K near 0 implying (47).

Regular transition maps near a Hamiltonian 2-saddle cycle
In this section, we apply the formulas obtained in the previous section on an unfolding of a Hamiltonian vector field. Consider a family (X µ ) like in (1), µ varying near µ 0 = (µ 0 , 0) with µ 0 ∈ R p , and where X H has a saddle-connection Γ on which the Hamiltonian takes constant value 0. Denote s 1 and s 2 as the hyperbolic saddles that are respectively given by the α-limit and ω-limit of Γ. Assume that Γ persists in the family (X µ ).
Appropriate normalizing coordinates near the saddles. Consider first the Hamiltonian vector field X H near the saddles s 1 and s 2 . Let i = 1 or i = 2. Denote the eigenvalues of DX H (s i ) as ±λ i , λ i > 0. Theorem 10 guarantees the existence of coordinates (n, m) in which X H , near s i , reads: up to a non-zero factor E 0 i (n, m) that equals −λ i for n m = 0. Denote by ψ i 0 the coordinate transformation expressing the old coordinates (x, y) in function of the new ones (n, m). Let H i denote a half plane that contains, in its interior, the separatrices corresponding to the separatrices of s i , that lie on L for ε = 0. One can choose ψ i 0 such that H expressed in the new coordinates reads n m on H i and such that E 0 i (n, m) = −1/ det Dψ i 0 (n, m) on H i . Peforming, if necessary a coordinate switch or a reflection with respect to the origin, one can suppose that points on the positive n-axes correspond to points on Γ 1 lying near s 1 and s 2 respectively; similar, This choice of coordinates implies an orientation on the normalizing coordinate axes such that det Dψ 1 0 (0, 0) = 1/λ 1 and det Dψ 2 0 (0, 0) = −1/λ 2 . The half plane H i can be chosen such that in the new coordinates it will correspond to {n ≥ 0}.
We continue by peforming the transformation (x, y) = ψ i 0 (n, m) on the family (X µ ) yielding: ṅ = n + εf i (n, m, µ), Preprint submitted at DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. The final publication is available at http://www.aimsciences.org/journals/redirecting.jsp?paperID=4488 up to the factor E 0 i (n, m). Applying Theorem 5 yields C k transformations (n, m) = (I + εφ i µ )(n, m) (i = 1, 2), near s i and for µ near µ 0 , transforming (56) into the normal form: where P i (z, µ) and Q i (z, µ) are polynomials in z = nm of finite degree N (k) ≥ k and of order O(z 2 ). Composing all the above performed transformations, one obtains C k transformations (x, y) = ϕ 1 µ (n, m) and (x, y) = ϕ 2 µ (n, m) near the saddles s 1 and s 2 respectively and for µ near µ 0 bringing the family (X µ ) in (57) up to the factor Derivation of the transition along a Hamiltonian saddle-connection, expressed using appropriate normalizing coordinates. As before, choose transverse sections Σ 1 µ and Σ 2 µ corresponding to {n = 1} in normalizing coordinates near s 1 and s 2 respectively. Similar choose transverse sections Σ 2 µ and Σ 4 µ corresponding to {m = 1} in normalizing coordinates near s 1 and s 2 respectively. The sections Σ 1 µ and Σ 3 µ are parametrised using the normalizing coordinate m while the normalizing coordinate n is used in order to parametrize the sections Σ 2 µ and Σ 4 µ . Consider the transition maps R 1 µ (m) and R 2 µ (n) from Σ 1 to Σ 3 and Σ 2 to Σ 4 respectively, see Figure 2. We will derive formulas for the first and second derivative of R µ := R 1 µ at zero. Similar formulas for R 2 µ can be deduced by applying a coordinate switch in normalizing coordinates.
Because we have chosen the normalizing coordinates in such way that for ε = 0 the Hamiltonian H reads nm in the normalizing coordinates near the saddles on {n ≥ 0}, it is easily verified that R µ = I + O(ε).
On the half plane {n ≥ 0}, one can define for i = 1, 2: Further, we letã 0 We can now state the following proposition.
Proposition 17 Suppose (X µ ) is a perturbation of a Hamiltonian vector field as in (1), with µ varying in a neighbourhood of some (µ 0 , 0), µ 0 ∈ R p , such that X H admits a saddle-connection Γ : H = 0, between two hyperbolic saddles s 1 and s 2 , that persists in the family (X µ ). Choose appropriate normalizing coordinates (n, m) near the saddles in which (X µ ) reads as in (57) and consider the functions θ Consider the transition map from Σ 1 µ to Σ 3 µ expressed in the appropriate normalizing coordinates. Then we have Preprint submitted at DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. The final publication is available at http://www.aimsciences.org/journals/redirecting.jsp?paperID=4488 Denoting Γ K as the part of Γ lying between ψ 1 0 (K, 0) and ψ 2 0 (K, 0) and f µ , g µ as the restrictions of f and g to ε = 0, we have with κ 0 (z) the curvature of X H at z, as defined in (27), and where U µ (K) is given by the difference U The coefficient α(µ) is given byã 0 1 (µ) −ã 0 2 (µ) and δ(µ) is given by Proof. We choose appropriate normalizing coordinates (n, m) near the saddles as before and apply Theorem 15. In the normalizing coordinates, all transitions occur in the half plane {n ≥ 0}, even in {n ≥ 0} ∩ {m ≥ 0}. So all calculations in normalizing coordinates can be restricted to the half plane {n ≥ 0}. Formula (59) is just the consequence of the fact that R µ = I +O(ε). However it can also be seen by applying the formula (47). Let us explain how formula (60) follows from Theorem 15.
Noticing that the divergence of X µ reads div X µ = ε div (f, g) + O(ε 2 ) and referring to the normal form in (57), it should be clear for the reader that formula (48) in Theorem 15 reduces to where r 1 (K, µ) is the function given by with all appearing functions defined as above.
Notice that the transformationm = R µ (0)m in normalizing coordinates leaves the equality (64) invariant up to order O(ε 2 ). Therefore, one can always assume that the condition R µ (0) = 1 is satisfied such that it is justified to apply formula (48) for obtaining a formula for R µ (0) up to order O(ε 2 ).
Formulas for calculating η 1 and η 2 . Consider a family (X µ ) like in (1) containing a period annulus bounded by a hyperbolic 2-saddle cycle L, see Figure 1, that leaves the connection Γ 2 unbroken. We choose H to be zero on the 2-saddle cycle and strictly positive on the nearby closed orbits. In the following corollaries, we obtain formulas for η 1 (µ, 0) (resp. η 2 (µ, 0)), defined in (7) (resp. (8)) in the case where one can find a curve in parameter space passing through (µ, 0) along which Γ 1 persists.
Corollary 18 Suppose (X µ ) is a perturbation of a Hamiltonian vector field X H = X (µ 0 ,0) , containing a hyperbolic 2-saddle cycle L, that leaves the connection Γ 2 unbroken. Suppose there exists a curve µ = γ(ε) in parameter space passing through (µ 0 , 0) along which the connection Γ 1 stays unbroken. Choose appropriate normalizing coordinates (n, m) near the saddles in which (X µ ) reads as in (57) and consider the functions θ i 0 and A i µ defined in (58) together with the coefficientsã 0 i (µ) andb 0 i (µ). Let Γ K 1 be the part of Γ 1 lying between ψ 1 0 (K, 0) and ψ 2 0 (K, 0). Denote f µ0 and g µ0 as the restrictions of f and g to µ = (µ 0 , 0) respectively. Then the coefficient η 1 (µ 0 , 0) as defined in (7) reads where A(z) is defined as Both 2-saddle cycles lie inside {H = 0} ; as in the announcement of theorem 1, we will only focus on the 2-saddle cycle that is contained in {H = 0} ∩ {y ≤ 0} , and call it by L. The non-isolated periodic orbits inside L, lie inside {H > 0} . The conditions concerning the Abelian integral. Suppose h ≥ 0 and denote γ h as one of the closed curves inside the annulus of which L is the boundary. The Abelian integral is defined as: with f (x, y,μ, ε) and g(x, y,μ, ε) the functions that appear after the parameter ε in the expression of (X µ ), (2). We now check the conditions (9) and (10).
In what follows, we calculate the coefficients in the expansion of I using Picard-Fuchs equations; write The Abelian integral related to (2) is given by: with I k (h) = γ h y k xdy. Now by direct computation, one easily verifies that: In particular the condition to have a 2-saddle cycle is given by: with X 0 = X H and H defined in (3). It will appear to be sufficient to perform normal form calculations up to order 4.
The linear part at the origin is already in its Jordan form. The transformation (x, y) = (x + 1 4 x 3 + 7 48 xy 2 , y + 1 4 will remove all terms of order less than 4. One concludes that the 3-jet of X is C ω linearisable by the transformation given by the composition of the translation (x, y) = (x + 1, y) with (72). The Hamiltonian expressed in the new coordinates already reads 2xy up to order 5. Therefore the transformation that brings the Hamiltonian in xy is up to order 4 given by a dilatation that one can choose to be (x, y) = (x, y 2 ). After a switch of the normalizing coordinates and a reflection (x, y) → (−x, −y), the positive x and y-axis in normalizing coordinates correspond respectively to the unstable and the stable separatrix of s 2 lying on L. One obtains the following 3-jet of the transformation (x, y) = ψ 2 0 (n, m): up to a factor 2 + O(|n m| 2 ). Near s 1 = (−1, 0), one can make use of the symmetry of X H with respect to the y-axis. The Hamiltonian vector field is invariant under the transformation (x, y, t) → (−x, y, −t), such that the behaviour of X H in the region {(x, y) | −1 < x < −1 + ε 0 } is exactly given by the behaviour of −X H in the region {(x, y) | 1 − ε 0 < x < 1}. Choosing ψ 1 0 := S • ψ 2 0 , where S(x, y) = (−x, y), the 3-jet of (X ε ) is near s 1 transformed into: ṅ = n − 1 2 εn 2 m, m = −m + 1 2 εn m + ε up to a factor −2 + O(|n m| 2 ). Moreover the Hamiltonian expressed in new coordinates reads n m, up to order 5.
Preprint submitted at DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. The final publication is available at http://www.aimsciences.org/journals/redirecting.jsp?paperID=4488 One continues by performing a transformation of the form I + εϕ i ε , i = 1, 2 near s 1 and s 2 respectively, keeping the unperturbed vector field unchanged but removing non-resonant terms of order less than 4, appearing after the parameter ε in expressions (74) and (75).