Simplified normal forms near a degenerate elliptic fixed point in two-parametric families of area-preserving maps

We derive simplified normal forms for an area-preserving map in a neighbourhood of a degenerate resonant elliptic fixed point. Such fixed points appear in generic two-parameter families of area-preserving maps. We also derive a simplified normal form for a generic two-parametric unfolding. The normal forms are used to analyse bifurcations of $n$-periodic orbits. In particular, for $n\ge6$ we find regions of parameters where the normal form has"meandering'' invariant curves.


Introduction
In a Hamiltonian system small oscillations around a periodic orbit are often described using the normal form theory [12,2]. In the case of two degrees of freedom the Poincaré section is used to reduce the problem to studying a family of area-preserving maps in a neighbourhood of a fixed point. The Poincaré map depends on the energy level and possibly on other parameters involved in the problem. A sequence of coordinate changes is used to transform the map to a normal form. In the absence of resonances the normal form is a rotation of the plane, and the angle of the rotation depends on the amplitude. In the presence of resonances the normal form is more complicated.
Let us describe the normal form theory in more details. Let F 0 : R 2 → R 2 be an areapreserving map. We assume that F 0 also preserves orientation. Let the origin be a fixed point: F 0 (0) = 0.
Since F 0 is area-preserving det DF 0 (0) = 1. Therefore the two eigenvalues of the Jacobian matrix DF 0 (0) are λ 0 and λ −1 0 . These eigenvalues are often called multipliers of the fixed point. We will consider an elliptic fixed point when λ 0 is not real. As the map is real-analytic the multipliers are complex conjugate, i.e. λ −1 0 = λ * 0 , and consequently belong to the unit circle, i.e. |λ 0 | = 1. Note that in our case λ 0 = ±1 as it is assumed non-real.
There is a linear area-preserving change of variables such that the Jacobian of F 0 takes the form of a rotation: where the rotation angle α 0 is related to the multiplier: λ 0 = e iα 0 . The classical normal form theory [2] states that there is a formal area-preserving change of coordinates which transforms F 0 into the resonant normal form N 0 such that the formal series N 0 commutes with the rotation: N 0 R α 0 = R α 0 N 0 . Following the method suggested in [15] (see e.g. [3,7]), we consider a formal series H 0 such that where Φ t H 0 is a flow generated by the Hamiltonian H 0 . The Hamiltonian is invariant with respect to the rotation H 0 • R α 0 = H 0 . It follows that the normal form preserves the Hamiltonian: H 0 • N 0 = H 0 since the Hamiltonian flow also preserves H 0 . Returning to the original coordinates we conclude that the original map has a non-trivial formal integral H 0 . This integral provides a powerful tool for analysis of the local dynamics including the stability of the fixed point (see e.g. [1]).
It is natural to describe the normal forms using the symplectic polar coordinates (I, ϕ) defined by the equations A fixed point is called resonant if there exists n ∈ N such that λ n 0 = 1. The least positive n is called the order of the resonance. In the resonant case the formal Hamiltonian takes the form [2]: H 0 (I, ϕ) = I 2 k≥0 a k I k + k≥1 j≥0 b kj I kn/2+j cos(knϕ + β kj ) where a k , b kj and β kn are real coefficients. The normal form is not unique. In the paper [7] we proved that in the non-degenerate case, namely if a 0 b 10 = 0 for n ≥ 4 or b 10 = 0 for n = 3, the Hamiltonian can be simplified substantially: there is a canonical formal change of variables which transforms the Hamiltonian to the form H 0 (I, ϕ) = I 2 A(I) + I n/2 B(I) cos(nϕ) , where A and B are formal series in powers of I. We refer the reader to paper [7] for the detail discussions of possible simplifications for the formal series A and B which depend on the order of the resonance n. In particular, for n ≥ 4 the series B(I) contains even powers only. We note that (5) involves formal series in a single variable I only and therefore is substantially simpler than (4) which involves series in two variables.
In this paper we consider the case when the fixed point of F 0 has a degeneracy of the lowest co-dimension, i.e., in (4), We prove that under these assumptions the normal form Hamiltonian (4) can be transformed to a simplified form which looks similar to (5). The detailed description of the simplified normal form is given in Section 2. We also prove that the simplified normal form is unique and therefore in general there is no room for further simplifications. It is natural to consider a generic unfolding by considering the map F 0 as a member of a generic two-parameter family. In Section 2 we provide simplified normal forms for such families and discuss their uniqueness.
Of course, in general the series of the normal form theory are expected to diverge. Nevertheless they provide rather accurate information about the dynamics of the original map. For example, it follows that for any m ∈ N the partial sumĤ m 0 (a polynomial which includes all terms ofĤ 0 up to the order m) satisfieŝ where O m+1 stands for an analytic function which has a Taylor series without terms of order less than m + 1. In particular, the implicit function theorem can be used to show that the map F 0 has n-periodic points near critical points of the HamiltonianĤ m 0 , and KAM theory can be used to establish existence of invariant curves [1,2].
In the case of a generic one-parameter unfolding of a non-degenerate resonant elliptic fixed point, the normal form provides a description for a chain of islands which is born from the origin when the unfolding parameter changes its sign [2,10,12,14]. A typical picture is shown on Figures 1(a) and (b). This analysis distinguishes the cases of week (n ≥ 5) and strong (n ≤ 4) resonances.
In this paper we use the normal forms to describe bifurcations of n-periodic orbits which appear in a generic two-parameter unfolding of a degenerate map F 0 . In Section 3 we analyse bifurcations of n-periodic points using model normal form Hamiltonians for n ≥ 7, n = 6 and n = 5 separately. It is interesting to note that bifurcation diagrams contain sectors where the leading part of the normal form does not satisfy the twist condition required by a standard KAM theory and invariant curves of "meandering type" are observed similar to ones described in the papers [13,4,9,6].
Finally we note that Theorem 2 of the next section imply that in the case n = 3, 4 the lowest degeneracy has co-dimension three.

Unique normal forms
Suppose that F 0 is a real-analytic area-preserving map with an elliptic fixed point at the origin. We note the theory repeats almost literally in the C ∞ category. Let λ 0 be the multiplier of the fixed point and suppose that λ n 0 = 1 for some n > 2 and λ k 0 = 1 for all integer values of k, 1 ≤ k < n. Then there is an analytic area-preserving change of variables which eliminates all non-resonant terms from the Taylor series of F 0 up to the order n + 1 [12,2]. So we assume that F 0 is already transformed to this form.
It is convenient to rewrite F 0 identifying the plane with coordinates (x, y) and the complex plane of the variable z = x + iy. Equation (3) implies z = √ 2Ie iϕ . In the complex notations the transformed map F 0 takes the form where a, b are polynomial in zz. The coefficients of the polynomials are complex, in particular, a(0) = λ 0 . In [7] we studied the normal forms of the map in the generic case when a (0) = 0 and b(0) = 0. A resonant fixed point with the degeneracy of the twist term corresponds to a(0) n = 1 and a (0) = 0. We note that in the space of real area-preserving maps such maps are of co-dimension two. Indeed, the area-preserving property of the map implies that a(0) is on the unit circle and a(0) −1 a (0) is purely imaginary. Consequently, in a generic situation two real parameters are required to satisfy these two condition simultaneously. Note that another obvious degeneracy of a resonant fixed point corresponds to b(0) = 0. As the coefficient b(0) is complex, this degeneracy has co-dimension three and is not considered in this paper.
The coefficients of the series A 0 and B 0 are defined uniquely by the map F 0 provided the leading order is normalised to ensure b 0 > 0.
This theorem follows from Proposition 5 of Section 4.1. The proofs of this and other theorems from this section are based on a development of the Lee series method [5,11,7,8]. We note that Theorem 1 provides a complete set of formal invariants for the map F 0 .
The following theorem covers the cases of n = 3 and n = 4. • if n = 4, then a k = 0 for k = 1 (mod 4) and b k = 0 for k = 3 (mod 4); The coefficients of the series A 0 and B 0 are defined uniquely by the map F 0 provided the leading order is normalised to ensure b 0 > 0.
The case n = 3 was discussed in [7]. The proof for n = 4 is in Section 4.1. In this case, Theorem 2 has an advantage over the normal form of the paper [7] as the present version eliminates assumptions on the coefficients of the twist part of the map which allows a unified treatment for a wider class of maps without increasing the complexity of the simplified normal form.
Let F p be a two-parametric unfolding of F 0 , where p = (p 1 , p 2 ). We assume that (1) the family F p has an elliptic fixed point at the origin, i.e., F p (0) = 0 for all small p; (2) F p (0) = R αp where the angle α p is an analytic function of p and α 0 = 2π m n , i.e., p = 0 corresponds to a resonance of order n.
Suppose that p is sufficiently small to ensure that |α p − α 0 | < π 2n 2 . Then α p does not satisfy any resonant condition of order less than 2n excepting the original one which corresponds to α p = α 0 . According to the normal form theory, there is an analytic area-preserving change of variables which eliminates all non-resonant terms from the Taylor series of F p up to the order n + 1. We assume that F p is already transformed to this form. Then in the complex coordinates the map F p has the form where a p and b p are polynomial in zz, which depend analytically on p, and a p (0) = λ p . We introduce new parameters δ and ν by The preservation of the area by the map f p implies that δ, ν are real. Moreover, they are defined uniquely, i.e., they are independent of the choice of the coordinate change C p which transforms the original map F p to the form (8).
If the Jacobian of the map (p 1 , p 2 ) → (δ, ν) does not vanish at p = 0, the family F p is a generic unfolding of F 0 . Then the inverse function theorem implies that p can be expressed in terms of (δ, ν). From now on we use p = (δ, ν) instead of the original parameters.
The normal form theory states that there is a formal change of variables which transforms F p to its normal form Hp where the formal sum H p includes only resonant monomials, i.e., monomials of the form h j,k,l,m z jzk δ l ν m with k = j (mod n). The transformation to the normal form and the normal form itself are formal power series in z,z and δ, ν.
We note that the normal form is not unique and can be simplified using a formal tangentto-identity change of variables.
Moreover, the coefficients of the series A(I; δ, ν) and B(I; δ, ν) are defined uniquely.
The proof of this theorem is in Section 4.2.
Theorem 3 implies that for n ≥ 5 the qualitative properties of the normal form can be studied using the following model Hamiltonian: This Hamiltonian keeps only the leading terms of the formal series A and B from the simplified normal form (10). Note that the normalisation of the coefficients in front of I 3 and I n/2 does not restrict generality of the model, as under the assumptions of the theorem the normalisation used in (11) can be achieved by rescaling the variable I, the parameters δ and ν, and the Hamiltonian function h.
We note that in the case n = 3 and 4, the degeneracy in the twist term does not affect the reduction to the normal form. Let α 0 = 2π n and δ = α 0 − α. Then (8) implies that the map can be represented in the form where a ν,δ and b ν,δ are analytic functions of the parameters. where where the coefficients are real-analytic functions of ν and • if n = 3, then a km = b km = 0 for k = 2 (mod 3); • if n = 4, then a km = 0 for k = 1 (mod 4) and b km = 0 for k = 3 (mod 4).
Moreover, the coefficients a km (ν) and b km (ν) are defined uniquely.
We note that the theorem implies that in the cases of n = 3 and n = 4 the bifurcations of n-periodic points in a generic two-parameter family are similar to the generic one-parameter case. This statement can be checked by introducing properly scaled variables: if n = 3, then J = δ −2 I andh = δ −3 H δν . After an additional change of variables, which depends on the value of b 0 (0), the leading part of the Hamiltonian takes the formh 0 = J + J 3/2 cos 3ϕ. If n = 4, then J = δ −1 I andh = δ −2 H δν . After an additional change of variables, which depends on the value of b 0 (0), the leading part of the Hamiltonian takes the formh 0 = J + J 2 (a(ν) + cos 3ϕ). In both cases, the scaling leads to the same Hamiltonian as in the one-parameter case. We note that for n = 4 there is a bifurcation which corresponds to the transition between the stable and unstable sub-cases of the non-degenerate case (i.e. the coefficient a crosses ±1).

Bifurcations of n-periodic points
In order to study bifurcations of critical points for the normal form we use model Hamiltonian functions which involve the least possible number of terms while preserving the qualitative properties of the general case described by the normal form (10). In particular, we provide bifurcation diagrams for saddle critical points located in a small neighbourhood of the origin and illustrate the structure of corresponding critical level sets.

n ≥ 7
For n ≥ 7 the model Hamiltonian function takes the form h = δI + νI 2 + I 3 + I n/2 cos nϕ (13) where I, ϕ are symplectic polar coordinates (3). If δ = ν = 0, the Hamiltonian h has a minimum at the origin. All level lines of h are closed curves around the origin (similar to Fig. 1(a)). Therefore the origin is a stable equilibrium of the normal form. Moreover, since I 3 I n/2 , it remains stable for all sufficiently small values of δ and ν.
In order to analyse the Hamiltonian for small δ and ν, we recall that critical points of h are defined from the equations ∂ I h = ∂ ϕ h = 0. After computing the derivatives, we conclude that all equilibria have either ϕ = 0 or π n (mod 2π n ), and where σ ϕ = cos nϕ ∈ {+1, −1}. The equation (14) can be written in the form where A typical plot of the functions f σ is shown on Fig. 2(a) and (b) for ν < 0 and ν > 0 respectively.
Since the function f σ is independent of δ, the numbers and positions of solutions to (15) can be easily read from the graph. Moreover, as the Hessian of h at a critical point is diagonal, the type of the critical point can be read from the slope of f σ . A straightforward analysis shows that f σ with ν < 0 has a single non-degenerate maximum in the neighbourhood of the origin, and it is monotone for ν > 0. So the equation (14) has two solutions for ν > 0, and from none to four solutions for ν < 0 (depending on the value of δ).
The results of the above analysis are summarised in the bifurcation diagram for critical points of h shown on Figure 3. We note that qualitatively the bifurcation diagram is the same for all n ≥ 7. In a neighbourhood of the origin on the (δ, ν)-plane there are 4 domains which correspond to different numbers of saddle critical points of h. If (δ, ν) ∈ D k , k = 0, 1, 2, there are exactly kn saddle critical points of h. The fourth domain is a narrow sector, which  separates D 0 and D 2 and which we denote by D 1 . In this sector the Hamiltonian has n critical saddle points. Let us describe the level sets of h. When the parameters are in D 0 all level sets of h are closed curves which look like the invariant curves shown on Figure 1(a). When the parameters are in D 1 , the critical level set of h form a chain of n islands similar to one shown on Figure 1(b). When the parameters cross the positive ν-semiaxes moving from D 0 to D 1 , a chain of islands bifurcates from the origin. When the parameters cross the negative ν-semiaxes moving from D 1 to D 2 a second chain of n-islands bifurcates from the origin (a typical picture is shown on Figure 4(a)). In D 2 there is a line on which both chains of islands belong to a single level set of h (see Figure 4(c)). Near this line the level sets of h change their topology without any change in the number of critical points (see Figure 4(d)). We note that in this region the Hamiltonian h has meandering invariant curves similar to ones studied in [13,4]. Finally, the chains of islands disappear through Hamiltonian saddle-node bifurcations when the parameters cross the boundary of D 1 : first the outer one and then the other one (see Figure 4(e) and 4(f)).
In order to derive an approximate expression for the curves bounding the domain D 1 on the bifurcation diagram we note that the corresponding values of δ coincide with the maximum value of the functions f σ (compare with equation (15)). The equation ∂fσ ∂I = 0 takes the form 2ν + 6I + σ n 2 The equations (14) and (16) together define a line on the plane of (δ, ν) along which h has a doubled critical point. In order to solve the equation (16), we rewrite it in the form and apply the method of consecutive approximations starting with the initial approximation I 0 = − ν 3 . A standard estimate from the contraction mapping theorem implies Substituting this approximation into (15) we get This equation defines two lines on the plane (δ, ν) (one for each value of σ). Both lines enter the zero quadratically and differ by O(ν n/2−1 ). In this way we have derived an approximate expressions for the lines which bound the domain D 1 on the bifurcation diagram of Fig. 3. We note that σ = 1 leads to a smaller δ for the same ν compared to σ = −1. Consequently, σ = 1 corresponds to the boundary between D 2 and D 1 , and σ = −1 corresponds to the boundary between D 1 and D 0 .
Assuming 0 < < µ we geth Solving equations (17) with respect to we find that = 1 This expression gives a small value when δ and ν are small. In terms of the original parameters the region 0 < < µ has the form δ < ν 2 4 for ν ≤ 0 and δ < 0 for ν > 0. In this region the chain of islands is approximated by a chain of the pendulum's separatrices. Reversing the scaling (17) we obtain the radius and the width for the chain of islands.
(a) In order to study the changes in the level sets of h near the boundary between D 2 and D 1 we rescale the variables: In the new variables the Hamiltonian takes the form (after skipping a constant term) h = aJ + J 3 + (1 + n/6−1 J) n/2 cos ψ . This explicit expression for J shows thath 0 has exactly two saddle critical points per period for a < − 1 , one critical point for − 1 < a < 1 and no critical points for a > 1 . The critical level sets for a = ± 1 are shown on Figures 5(e) and (f) respectively, which correspond to a Hamiltonian saddle-node bifurcation.
We note that at a = −a c , a c = 3 · 2 −2/3 + O( 2 1 ), the two critical saddle points belong to a single critical level set as shown on Figure 5(b).

n = 6
In the case n = 6 the model Hamiltonian takes the form For δ = ν = 0, the origin is stable if |b 0 | < 1 and unstable if |b 0 | > 1. We assume that b 0 ∈ { −1, 0, 1 } as these three cases correspond to a degeneracy of a higher co-dimension. The critical points of h have either ϕ = 0 or π n (mod 2π n ), and where σ ϕ = cos ϕ = ±1. The equation (22) can be easily solved explicitly but it is more convenient to write it in the form where Since the function f σ is independent of δ, the number and positions of solutions to the equation (23) for given δ and ν can be easily read from the graphs of the functions f + and f − . The analysis leads to different conclusions for the stable and unstable cases. Moreover, the qualitative behaviour of the graphs depend on the sign of ν. In the rest, this analysis is completely straightforward and its results are summarised on the bifurcation diagrams of Figure 6(a) and (b) for the stable and unstable case respectively. In the stable case the bifurcation diagram is similar to the case of a resonance of order n ≥ 7. Although later in this section we will observe some quantitative differences.
In the unstable case the bifurcation diagram consists of four domains: D 1 , D 1 , D 2 and D 2 , which correspond to the existence of n and 2n saddle critical points of h as indicated by the subscript in the domain's name.
In order to derive an equation for the boundaries between the domains on the bifurcation diagram, we note that the corresponding value of δ coincide with a critical value of f σ . The equation ∂fσ ∂I = 0 has a unique solution .
The corresponding critical value is . This equation defines two lines corresponding to double critical points of h (for σ = ±1 respectively). Note that the bifurcation diagrams of Figure 6 include only the parts of the parabolas which correspond to positive critical points. Typical pictures of level sets are illustrated by Fig. 7. The bifurcations of islands are illustrated on Figs. 8 and 9 for the stable and unstable case respectively.
If |δ| ν 2 , this Hamiltonian is approximated bȳ The Hamiltonianh 0 depends on the coefficient b 0 but is independent of both δ and ν. We conclude that outside a narrow sector near the δ-axis, the critical level sets of the model Hamiltonian (21) have the shape which is described by the Hamiltonianh 0 . The level sets for this Hamiltonian are shown on Figure 7. In order to study bifurcations of the critical level sets of the Hamiltonian (21) we use the scaling ν = , δ = 2 a, I = J, h = 3h .
In the new variables the Hamiltonian takes the form We conclude that for ν = 0 the two-parameter family (21) is reduce to the family (26) which depends on one parameter only. We note that the transition from (21) to (26) is just a change of variables and parameters and does not involve any approximations. The bifurcations of the critical level sets ofh is illustrated on Figs. 8 and 9 for the stable and unstable case respectively. Note that the corresponding analysis takes into account thath is to be considered on the domain νJ = 2 I > 0 only.
We note that, in contrast to the cases with n ≥ 6, in the cases of n ≤ 5 the cubic term I 3 can be excluded from the model as it is much smaller than the term proportional to I n/2 and, consequently, does not affect the qualitative properties of the bifurcation diagram. The bifurcation diagram for n = 5 looks similar to the unstable case of n = 6. In particular the equilibrium at the origin is unstable when δ = ν = 0. The corresponding level set {h = 0} is not compact and consists of 5 straight lines defined by the equation cos 5ϕ = 0.
Similar to the previous cases, the critical points of h are determined from the equation ∂h ∂I = 0 and ϕ = 0 or ϕ = π/5 (mod 2π). Therefore the problem is reduced to finding positive solutions for the equations: δ + 2νI + σ 5 2 I 3/2 = 0, σ = ±1. When the parameters (δ, ν) are in D 1 or D 1 , singular level sets of h look similar to the one shown on Figure 10(b).
A chain of n islands is born from the origin and go through a bifurcation shown in Figure 11 as the parameters (δ, ν) move in anticlockwise direction through D 2 or D 2 in the plane of Figure 10(a). We note that the symmetry of the Hamiltonian (27) allows us to consider the case δ ≥ 0 only.
In order to study the shape of the islands we apply the scaling In the new variables the Hamiltonian function (27) takes the form If νδ −1/3 is small, the quadratic term can be ignored and we arrive to the Hamiltonian Its critical points are located at the point J cr = ( 2 5 ) 2/3 , ϕ cr = π/5 (mod 2π) and the corresponding critical value is given byh 0 (J cr , ϕ cr ) = 3 5 ( 2 5 ) 2/3 . So the critical set is defined by the equation This critical set and some non-critical level lines ofh 0 are shown on Figure 10(b). If |δ| |ν| 3 , the critical points ofh are close to the critical points ofh 0 and the shape of the corresponding critical set is approximately defined by the equation (32). Taking into account the scaling, we conclude that the critical counts of the original model (27) are located on the circle I = J cr δ 2/3 (1 + O(νδ −1/3 )).
Note that on a small disk δ 2 + ν 2 ≤ 2 0 1, the assumption |δ| |ν 3 | is violated only in a narrow zone near the δ-axis. The relative area of this zone converges to zero when 0 goes to zero. This zone includes the domains D 2 and D 2 .
In order to study the changes in the critical level sets of h in the region where the above assumption is possibly violated, we introduce another scaling In the new variables the Hamiltonian function (27) takes the form h = aJ + J 2 + J 5/2 cos 5ϕ .
We note that the Hamiltonianh depends on one parameter only. Moreover, the change is defined provided ν = 0. On every cubic curve δ = aν 3 in the plane (δ, ν), the Hamiltonian

Simplification of a degenerate Hamiltonian
It is well known that the complex variables defined by facilitate manipulation with real formal series in variables x, y. It is important to note that z is not necessarily the complex conjugate of z, the latter is denoted by z * in this paper. Of course,z = z * if both x and y are real.
In this variables, a rotation R α 0 takes the form of the multiplication: z → λ 0 z,z → λ * 0z , where λ 0 = e iα 0 . Let λ 0 be resonant of order n, i.e., λ n 0 = 1. Any formal power series h invariant with respect to the rotation R α 0 has the form We say that the series h consists of resonant terms as each term of the series is invariant under the rotation by α 0 . It is easy to come back from the variables (z,z) to the symplectic polar coordinates by substituting z = √ 2Ie iϕ andz = √ 2Ie −iϕ . We note that the formal interpolating Hamiltonian of the resonant normal form theory (described in the introduction) has the form (35) and satisfy two additional properties. First, the series is real-valued, i.e., it has real coefficients when written in terms of x, y variables, which is equivalent to the condition h kl = h * lk for all k, l ≥ 0. Second, the series do not contains terms of order lower than two.
We use tangent-to-identity formal canonical transformations to simplify the series h by eliminating as many resonant terms as possible without braking the symmetry of the series. The simplification procedure depends on the order of the resonance and on some lowest order terms of the series. The following proposition describes one of the possible simplifications of the series which still involve infinitely many coefficients. We note that no further simplification is possible as the coefficients of the simplified series are defined uniquely and consequently can be considered as moduli of the formal canonical classification.

Proposition 5 Let h be a formal power series of the form
If h 22 = 0, h n0 > 0 and h kl = h * lk for all k, l, (and additionally h 33 = 0 for n ≥ 6 only) then there exists a formal tangent-to-identity canonical change of variables which transforms the Hamiltonian h intoh, whereh has the following form.
• If n ≥ 6,h (z,z) = (zz) 3 k≥0 a k (zz) k + (z n +z n ) In all cases the coefficients a k and b k are real. They are defined uniquely provided the leading coefficient is normalised by the condition b 0 > 0.
Proof. The case n = 3 is covered by a theorem of [7].
Case of n = 4. The proposition is proved by induction. Following the classical strategy, we perform a sequence of canonical coordinate changes normalising one order of the formal Hamiltonian at a time.
In the case of n = 4, any resonant monomial has an even order because k = l (mod 4) implies that k + l is even. So any resonant monomial of order 2m has the form where |j| ≤ m 2 . Then any real homogeneous resonant polynomial of order 2m has the form The formal Hamiltonian can be written in the form h = k≥2 h k where h k ∈ H 4 k . In particular, h 2 (z,z) = h 4,0 z 4 + h 0,4z 4 where h 4,0 = h 0,4 as h 4,0 is assumed to be real. We see that h 2 already has the desired form. Setting b 0 = h 40 we write h 2 = a 0 z 2z2 + b 0 (z 4 +z 4 ).
All other orders are transformed to the simplified normal form inductively using the Lie series method (see e.g. [5]).
Let p ≥ 2 and take a polynomial χ p ∈ H 4 p . After the substitution (z,z) → Φ 1 χp (z,z) the Hamiltonian takes the formh We note that the Lie series are not necessarily convergent but the formal sum still has a precise meaning as L χp is a linear operator which increases the order of a monomial and, consequently, each term in the formal sum is represented by a finite sum. In particular, Let Λ : H 4 p → H 4 p+1 be the linear operator defined by It is sometimes called the homological operator. We show thath p+1 takes the form stated in the proposition after an appropriate choice of χ p . Using the definition of Λ we easily check that where Q p,j is defined by (41). Since χ ∈ H 4 p , we can write it in the form χ = where c −j = c * j . So c 0 is real and c j with j ≥ 1 may be complex. Let Then equation (42) is equivalent to the system were we follow the convention that c k = 0 for k > k 0 = p 2 . Because of the real symmetry it is sufficient to consider 0 ≤ j ≤ j 0 = p+1 2 . We note that k 0 = j 0 if p is even, and k 0 = j 0 − 1 if p is odd. Then equation (43) with j = j 0 has the form We also note that the coefficients in the system (43) are purely imaginary, so the equations for real and imaginary parts of c j are disjoint. Let c j = c j + ic j , d j = d j + id j ,d j =d j + id j . First we analyse the imaginary part of the equations: The real symmetry implies that the equation with j = 0 is an identityd 0 = d 0 = 0 as c −1 = c 1 . So we are left with j 0 linear relations which involve k 0 + 1 coefficients c k with 0 ≤ k ≤ k 0 . We immediately note that since b 0 = 0 we can setd j = 0 for 1 ≤ j ≤ j 0 by choosing c j−1 recursively starting from j = j 0 . Thus the imaginary part of d j is eliminated completely.
If p is even, k 0 = j 0 and the coefficient c k 0 remains free. This freedom is related to the obvious fact: the identity Λh The real symmetry implies c 0 = 0 and c −1 = −c 1 , so the first two equations take the form In total we obtain j 0 + 1 linear relations which involve k 0 coefficients c k with 1 ≤ k ≤ k 0 . Then we need to consider the following cases separately. If p = 2k 0 + 1, then j 0 = k 0 + 1. The system (45) with 2 ≤ j ≤ j 0 has a non-degenerate matrix and consequently there are unique c k , 1 ≤ k ≤ k 0 , such thatd j = 0 for 2 ≤ j ≤ j 0 . Then monomials of the form form a subspace complementary to the image of the homological operator.
If p = 2k 0 , then j 0 = k 0 and we have to consider two sub-cases. If k 0 is even (or equivalently p = 0 (mod 4)), then we take equations (45) with 1 ≤ j ≤ j 0 . This system has a non-degenerate matrix 1 and consequently there are unique c k , 1 ≤ k ≤ k 0 , such thatd j = 0 for 1 ≤ j ≤ j 0 . Then monomials of the form d 0 Q p+1,0 withd 0 ∈ R form a subspace complementary to the image of the homological operator. If k 0 is odd (or equivalently p = 2k 0 = 2 (mod 4)), then we take equations (45) with 0 ≤ j ≤ j 0 , j = 1. This system has a non-degenerate matrix and consequently there are unique c k , 1 ≤ k ≤ k 0 , such thatd j = 0 for 0 ≤ j ≤ j 0 , j = 1. Then monomials of the form form a subspace complementary to the image of the homological operator.
Repeating the coordinate changes inductively we conclude that the Hamiltonian can be transformed to the form where c p and d p are real. The last series has the desired form (38).
In order to complete the proof we need to establish uniqueness of the series (38). We note that the transformation constructed in the first part of the proof is not unique because the kernel of Λ is not always empty. Nevertheless the normalised Hamiltonian is unique. Indeed, suppose that h can be transformed to two different simplified normal formsh and h due to non-uniqueness of transformations to the normal form. Then there is a canonical transformation φ such thath =h • φ .
Since the transformation φ is tangent to identity there is a formal real-valued Hamiltonian χ such that φ = Φ 1 χ . Suppose that 2p is the lowest order in the formal series χ. Thenh andh coincide up to the order 2p, i.e. h k = h k for 2 ≤ k ≤ p, and h p+1 =h p+1 + Λ(χ p ) .
Since bothh p+1 andh p+1 are in the subspace complementary to the image of Λ, we conclude that Λ(χ p ) = 0 andh p+1 =h p+1 . Then χ p is in the kernel of Λ, thus either χ p = 0 if p is odd, or χ p = c h p/2 2 for some c ∈ R if p is even. Then the change of variables proved using the following rule for the determinant. Let A = (a ij ) be a square k 0 × k 0 matrix with non-zero elements on the three main diagonals only: a jj = α j , a j,j+1 = β j , a j+1,j = −γ j . Then det A = K k0 where the continuants K j are defined recursively: The conclusion K k0 = 0 is checked with the help of a straightforward induction. also transformsh intoh: It is easy to check that there is a formal Hamiltonian χ such thatφ = Φ 1 χ and the lowest order in χ is at least 2p + 2.
Repeating the arguments inductively we see thath andh coincide at all orders. Hence the simplified normal form is unique.
Case of n = 5. Let H 5 m denote the set of all real-valued polynomials which can be represented as a sum of resonant monomials of the order m. The Hamiltonian h = k≥5 h k , h k ∈ H 5 k . The leading term has the form h 5 = b 0 (z 5 +z 5 ).
The homological operator takes the form It is checked directly that Λ : H 5 p → H 5 p+3 . Then we use induction. Suppose the Hamiltonian h has the desired form for all orders up to p + 2 for some p ≥ 3. Then a change of variables generated by χ ∈ H 5 p gives a new Hamiltonianh = h • Φ 1 χ with the properties We can chose χ to ensure thath p+3 is in a complement to the image of Λ. The results of the study of Λ are summarised in Table 1.
The complement to the image of Λ is empty for p = 0 (mod 5). Otherwise it can be chosen as a real multiple of (zz) (p+3)/2 if p is odd, and of (zz) (p−2)/2 (z 5 +z 5 ) if p is even.
Case of n = 6. In this case, k = l (mod n) implies k + l is even. Let H 6 m denote the set of all real-valued polynomials which can be represented as a sum of resonant monomials of the order 2m.
Thenh m = h m for 3 ≤ m ≤ p + 1 andh It is convenient to denote the resonant monomials of order 2m by for 0 < |j| ≤ m 3 . Then any resonant polynomial which contains only monomials of the order 2m has the form m 3 j=− m 3 c j Q mj . Taking into account that c −j = c * j due to real-valuedness, we conclude that the real dimension of the space dim H 6 m = 1 + 2 m 3 . We compute the action of the homological operator Λ on monomials: The results of the study of Λ are summarised in Table 2.
Let H n m denote the set of all real-valued polynomials which can be represented as a sum of resonant monomials of the δ-order m.
It is convenient to denote resonant monomials by Q m,j = z m+nj−3jzm−3j and Q m,−j = z m−3jzm+nj−3j for 0 ≤ j ≤ m 3 . Then any resonant polynomial which contains only monomials of the δ-order m has the form m 3 j=− m 3 c j Q mj . Taking into account that c −j = c * j due to realvaluedness, we conclude that the real dimension of the space dim H n m = 1 + 2 m 3 . We can write the original Hamiltonian in the form h = p≥3 h p where h p ∈ H n p . The leading term has the form h 3 (z,z) = a 0 z 3z3 + b 0 (z n +z n ).
The conclusions of the study of Λ are the same as in the case of n = 6 and therefore are described by Table 2. It can be checked that, if p = 3k a complement to the image of Λ is generated by (zz) p+2 and, if p = 3k + 1 or p = 3k + 2, it is generated by (zz) p+2 and (zz) p−1 (z n +z n ).
Finally we note that the arguments used to prove the uniqueness on the simplified normal form in the case n = 4 can be easily modified to prove the uniqueness for n ≥ 5.
Remark. In the case of n = 6 it is possible to transform the Hamiltonian to an alternative normal form, namely, h(z,z) = (zz) 3 k≥0 k =5 (mod 6) a k (zz) k + (z n +z n ) k≥0 k =2 (mod 6) b k (zz) k .
The coefficients of this simplified normal form are unique. This normal form has an advantage over the form proposed in Proposition 5 as the condition h 33 = 0 can be dropped from the assumptions. The derivation of this version of the normal form is similar to the case of n = 4 described above. • if n = 5h (z,z; δ, ν) = δzz + ν(zz) 2 + (zz) 3 k,m,j≥0 k =1 (mod 5) a kmj (zz) k δ m ν j +(z 5 +z 5 ) k,m,j≥0 k =4 (mod 5) b kmj (zz) k δ m ν j ,
The coefficients of the series a kmj and b kmj are real and defined uniquely by the series h provided the leading order is normalised to ensure b 000 > 0.
Let χ is a homogenous resonance polynomial of order p − 4M − 2J ≥ 1 for some M and J.
After the change of variables (z,z) → Φ δ M ν J χ (z,z) where Λ is the homological operator from Section 4.1 for n = 3, 4 or 5 respectively. It was been shown that there exists such χ thath p+(n−2),K,M has the desired form.
The uniqueness can be proved using the same type of arguments as we used for individual maps in Section 4.1.