The 1:1 resonance in Hamiltonian systems

TL;DR

This paper investigates the 1:1 resonance bifurcation in two-degree-of-freedom Hamiltonian systems, revealing a co-dimension five unfolding with two moduli parameters, advancing understanding of semisimple resonance cases.

Contribution

It provides a detailed analysis of the 1:1 resonance in Hamiltonian systems, identifying the co-dimension and unfolding parameters, which was previously not well understood.

Findings

Identified the co-dimension of 1:1 resonance as five.

Discovered two moduli parameters in the unfolding.

Connected the normal form symmetry to the frequency ratio.

Abstract

Two-degree-of-freedom Hamiltonian systems with an elliptic equilibrium at the origin are characterised by the frequencies of the linearisation. Considering the frequencies as parameters, the system undergoes a bifurcation when the frequencies pass through a resonance. These bifurcations are well understood for most resonances k:l, but not the semisimple cases 1:1 and 1:-1. A two-degree-of-freedom Hamiltonian system can be approximated to any order by an integrable approximation. The reason is that the normal form of a Hamiltonian system has an additional integral due to the normal form symmetry. The latter is intimately related to the ratio of the frequencies. Thus we study $S^1$--symmetric systems. The question we wish to address is about the co-dimension of such a system in 1:1 resonance with respect to left-right-equivalence, where the right action is $S^1$--equivariant. The result is a co-dimension five unfolding of the central singularity. Two of the unfolding parameters are moduli and the remaining non-modal parameters are the ones found in the linear unfolding of this system.

Tables

Table 1. Table 1: Poisson bracket of the real generators I 𝐼 I of the invariants.

Equations

$$H(q,p)=\frac{1}{2}(p_1^2+p_2^2)+\frac{1}{2}(q_1^2+q_2^2)+\dots$$

$$I_1$$

$$I_2$$

$$I_3$$

$$I_4$$

$$H(I)=I_1+a_1{I}_2^2+a_2{I}_3^2+a_3{I}_4^2+b_1{I}_2^3+b_2{I}_3^3+b_3{I}_4^3$$

$$H(I;\mu )=$$

$$+{\mu }_1{I}_2+{\mu }_2{I}_3+{\mu }_3{I}_4+{\mu }_4{I}_2^3+{\mu }_5{I}_3^3$$

$$(a_1-a_2)(a_2-a_3)(a_3-a_1)b_1{b}_2{b}_3\ne 0.$$

$$\omega (X_H,Y)=dH(Y)$$

$$L_{X_H}(H)=dH(X_H)=\omega (X_H,X_H)=0.$$

$$\{f,g\}=\omega (X_f,X_g).$$

$$\{f,g\}=L_{X_g}(f)=-L_{X_f}(g).$$

$$0=L_{X_H}(f)=\{f,H\}$$

$$[X_f,X_g]=-X_{\{f,g\}}.$$

$$\omega (\xi ,\eta )=⟨\xi \mid \Omega \eta ⟩$$

$$\Omega=\left(\begin{array}[]{rr}0&I\\ -I&0\end{array}\right)$$

$$\{f,g\}=\sum _{i=1}^2\left(\frac{\partial f}{\partial q_i}\frac{\partial g}{\partial p_i}-\frac{\partial f}{\partial p_i}\frac{\partial g}{\partial q_i}\right).$$

$${\dot{q}}_i=\{q_i,H\}=\frac{\partial H}{\partial p_i},{\dot{p}}_i=\{p_i,H\}=-\frac{\partial H}{\partial q_i}$$

$$H=H_2+H_3+\cdots +H_k+\cdots$$

$$\begin{array}[]{cccc}\phi\;:&S^{1}\times{\mathbb{R}}^{4}&\longrightarrow&{\mathbb{R}}^{4}\\ &(\varphi,z)&\mapsto&R_{\varphi}z\end{array}$$

$$R_{\varphi}=\left(\begin{array}[]{cccc}\cos\varphi&-\sin\varphi&0&0\\ \sin\varphi&\hphantom{-}\cos\varphi&0&0\\ 0&0&\cos\varphi&-\sin\varphi\\ 0&0&\sin\varphi&\hphantom{-}\cos\varphi\\ \end{array}\right)$$

$$H_{2}(q_{1},p_{1},q_{2},p_{2})=\mbox{$\tfrac{1}{2}$}(q_{1}^{2}+p_{1}^{2})+\mbox{$\tfrac{1}{2}$}(q_{2}^{2}+p_{2}^{2}).$$

$$I_1$$

$$I_2$$

$$I_3$$

$$I_4$$

$$(q,p)\mapsto I$$

$$\left\{\;(q,p\in{\mathbb{R}}^{4}\mbox{\large$\;\;|\;\;$}h_{2}=I_{1}=\mbox{$\tfrac{1}{2}$}(q_{1}^{2}+p_{1}^{2}+q_{2}^{2}+p_{2}^{2})\;\right\}\enspace.$$

$$(H,H_2)$$

$$I_2^2+I_3^2+I_4^2$$

$$\begin{array}[]{cccc}(I_{2},I_{3},I_{4}):&S_{2I_{1}}^{3}\longrightarrow&S_{I_{1}^{2}}^{2}\end{array}$$

$$S_{2I_{1}}^{3}\;\;=\;\;\left\{\;(q,p)\in T^{*}{\mathbb{R}}^{2}\mbox{\large$\;\;|\;\;$}q_{1}^{2}+q_{2}^{2}+p_{1}^{2}+p_{2}^{2}=2I_{1}\;\right\}$$

$$S_{I_{1}^{2}}^{2}\;\;=\;\;\left\{\;(I_{2},I_{3},I_{4})\in{\mathbb{R}}^{3}\mbox{\large$\;\;|\;\;$}I_{2}^{2}+I_{3}^{2}+I_{4}^{2}=I_{1}^{2}\;\right\}$$

Full text

1. 1 Introduction

2. 1.1 Resonant equilibria

3. 1.2 Informal statement of the main theorem

4. 2 A few facts about Hamiltonian systems

5. 2.1 Symplectic spaces and Hamiltonian systems

6. 2.2 Poisson brackets

7. 2.3 Standard forms

8. 3 Resonant Hamiltonian systems and $S^{1}$–symmetry

9. 3.1 $S^{1}$–symmetry related to the $1{:}1$ resonance

10. 3.2 Reduction of the $S^{1}$–symmetry: Hamiltonian systems on $S^{2}$

11. 4 The universal unfolding

12. 4.1 Equivalence classes for $S^{1}$–invariant Hamiltonian systems

13. 4.1.1 Stable $S^{1}$–invariant mappings, co-dimension and unfolding

14. 4.1.2 The tangent space of $Orb_{\mathcal{F}}$ at $(H,K)$

15. 4.1.3 The restricted tangent space of $Orb_{\mathcal{F}}$ at $(H,K)$

16. 4.2 Statement of main theorem

17. 5 Proof of main theorem

18. 5.1 Preliminary transformations

19. 5.2 $S^{1}$–equivariant vector fields

20. 5.3 The structure of the restricted tangent space

21. 5.4 Splitting into homogeneous parts

22. 6 Discussion

The case $k=l=1$ ($1{:}1$ resonance) turns out to be surprisingly complicated.

J.J. Duistermaat in [10]

1 Introduction

One of the few available methods to study the dynamics of Hamiltonian systems is to concentrate on the equilibria. The motion itself being trivial by definition, one considers the local dynamics and linearises the vector field. A hyperbolic equilibrium, with no eigenvalues on the imaginary axis, is dynamically unstable and on a sufficiently small neighbourhood the motion is completely determined by the linearisation.

In the elliptic case the non-linear terms cannot be disposed of completely, but lead to normal forms of which one hopes that they capture the essence of the dynamics. The reasons for irremovable terms are the resonances between the eigenvalues on the imaginary axis. Excluding zero eigenvalues, the resonances of lowest order, i.e. the $1{:}1$ and $1{:}{-}1$ resonances, relate double pairs of imaginary eigenvalues.

1.1 Resonant equilibria

In the present paper we concentrate on the $1{:}1$ resonance and study an equilibrium around which the Hamiltonian expands as

[TABLE]

where we omit the irrelevant constant term. The Hessian $D^{2}H(0)$ is positive definite and this excludes nilpotent terms. Thus a $1{:}1$ resonance is always semisimple. It occurs persistently in $3$–parameter families, cf. [19, 9, 10, 6, 18]. This is in sharp contrast with the $1{:}{-}1$ resonance where the Hessian is not definite. Then we have to distinguish a semisimple and a non-semisimple case. Unfolding the latter leads to the Hamiltonian Hopf bifurcation, which occurs persistently in $1$–parameter families, cf. [27, 7, 15]. The semisimple $1{:}{-}1$ resonance also occurs persistently in $3$–parameter families, cf. [20, 18, 14]. We expect its unfolding to share features of that of the $1{:}1$ resonance.

A comprehensive study of $k{:}l$ resonances, excluding the $1{:}{\pm}1$ cases, has been made in [10]. It turns out that all higher order cases are very similar to each other. In general the unfolding co-dimension of the unfolding is two, where one parameter can be considered as a detuning of the resonance and the other is a modulus, see [30, 13]. Exceptions are the resonances $1{:}2$ and $1{:}3$ with co-dimensions $1$ and $3$, respectively. Again one of the parameters is a detuning and in the case of $1{:}3$ resonance, two parameters are moduli. In all cases there is a bifurcation associated to the resonance. In general a pair of stable and unstable periodic solutions branches off from the origin. The $1{:}2$ and $1{:}3$ cases have a slightly different unfolding scenario, see [10, 4, 13, 8]. As mentioned before the non-semisimple or nilpotent $1{:}{-}1$ resonance shows a different bifurcation (the Hamiltonian Hopf bifurcation, see [27]) and the bifurcations triggered by the semisimple $1{:}{\pm}1$ resonances are still open.

This paper is organized as follows. In section 1.2 we state an informal version of our main theorem. Although informal it still contains the essential properties of the main theorem. Before proving our main result we review some facts on Hamiltonian systems in section 2. The system we study is in normal form and we discuss the properties we use in section 3, especially the induced $S^{1}$–symmetry. Finally in section 4 we state and in section 5 we prove our main theorem using singularity theory for $S^{1}$–equivariant mappings. The concluding section 6 puts our results in context. Our approach fits in the tradition of [10, 27, 9] and it complements [6].

1.2 Informal statement of the main theorem

In order to state our main result we need a few definitions. Here our aim is not full generality, the main theorem is formulated more precisely in section 4.2.

We study a $C^{\infty}$ Hamiltonian system on ${\mathbb{R}}^{4}$ with standard symplectic form in the neighbourhood of an elliptic equilibrium in $1{:}1$ resonance. We may assume that the equilibrium is at the origin, thus the linear part of the Hamiltonian $H$ at [math] vanishes. The matrix associated to the linearisation of the Hamiltonian vector field has coinciding pairs of eigenvalues with equal symplectic sign, therefore this matrix has no nilpotent part, see [16]. As a consequence the quadratic part of the Hamiltonian in the $1{:}1$ case has Morse index [math]. This contrasts with the $1{:}{-}1$ resonance where the corresponding matrix generically does have a nilpotent part, see [27].

As a first step we apply several (symplectic) co-ordinate transformations. The first of these takes the quadratic part $H_{2}$ of $H$ into the form presented in equation (1). Moreover, after a finite number of normal form transformations (see for example [27]), we may assume that a corresponding part of the Taylor expansion of $H$ Poisson commutes with $H_{2}$. We now make an approximation by restricting to this finite part and call it $H$ again. The flow of $H_{2}$ generates an $S^{1}$ symmetry group and the fact that $H$ and $H_{2}$ Poisson commute implies that $H$ is $S^{1}$–symmetric. The consequences of this approximation are discussed in the remarks following theorem 1.1.

The second step is a reduction with respect to the $S^{1}$ symmetry. Restricted to the $3$–sphere $\{H_{2}=1\}$, the projection mapping involved is a Hopf mapping so the reduced phase space is a $2$–sphere. Then we apply equivariant singularity theory to the map germ $(H,H_{2})$ and find a universal unfolding subject to non-degeneracy conditions on the coefficients in the higher order terms of $H$. By the nature of our method, we can not hope for more than local results and we exploit this fact by switching to germs, see [3, 26, 28]. Very briefly: a map germ is the collection of mappings equal to one another on an arbitrary small neighbourhood of a given point, say [math]. Map germs are essentially determined by their Taylor expansions or even Taylor polynomials in a sense that is made more precise in section 4.1.1. In the sequel we say mapping but tacitly assume map germ.

In order to proceed we need the generators of the $S^{1}$–invariant functions as co-ordinates. These are given by

[TABLE]

see section 3.1 for more details. The generators are not independent but related by the syzygy $I_{1}^{2}=I_{2}^{2}+I_{3}^{2}+I_{4}^{2}$. Nevertheless, $H$ and $H_{2}$ can now be expressed as functions of $I$, that is $H_{2}(I)=I_{1}$ and $H(I)=H_{2}(I)+H_{4}(I)+H_{6}(I)+\cdots+H_{k}(I)$. The final result is given in the next theorem.

Theorem 1.1**.**

A universal unfolding of the $S^{1}$–invariant Hamiltonian

[TABLE]

is given by the five parameter family ($\mu\in{\mathbb{R}}^{5}$)

[TABLE]

provided that the real coefficients $a_{1}$, $a_{2}$, $a_{3}$, $b_{1}$, $b_{2}$ and $b_{3}$ satisfy the non-degeneracy condition

[TABLE]

This theorem holds for $S^{1}$–symmetric Hamiltonian systems in $1{:}1$ resonance. Let us make a few remarks on its scope.

Remark 1.2**.**

The unfolding terms $\mu_{4}I_{2}^{3}$ and $\mu_{5}I_{3}^{3}$ can be replaced by any pair from $I_{2}$, $I_{3}$ and $I_{4}$. 2. 2.

The reduction of the $3$–sphere defined by \mbox{\tfrac{1}{2}}(q_{1}^{2}+p_{1}^{2}+q_{2}^{2}+p_{2}^{2})=h_{2} to the $2$–sphere $I_{2}^{2}+I_{3}^{2}+I_{4}^{2}=h_{2}^{2}$ is regular if $h_{2}\neq 0$, so every point on the reduced phase space corresponds to an $S^{1}$–orbit on the original phase space ${\mathbb{R}}^{4}$. 3. 3.

On the reduced phase space the solution curves are defined by $(H,H_{2})=(h,h_{2})$. Thereby time parametrisation is lost. Solution curves consisting of a single point on the reduced phase space correspond to periodic orbits on ${\mathbb{R}}^{4}$, whereas closed curves on the reduced phase space correspond to $2$–tori on ${\mathbb{R}}^{4}$. The former are generically isolated on $S^{2}$, but the latter come in $1$–parameter families. 4. 4.

Non–$S^{1}$–symmetric perturbations (i.e. including non–$S^{1}$–invariant terms in the Taylor expansion of $H$) do affect our result. However, normal form transformations enable us to make these perturbations as small as we wish. Nevertheless their effect is that families of $2$–tori, on ${\mathbb{R}}^{4}$, do not survive as such. From kam theory one expects that these families are Cantorised, i.e. the $2$–tori persist as a Cantor subfamily of large $2$–dimensional Hausdorff measure, where the dense set of internal resonances leads to gaps in the parametrisation. Periodic orbits, as long as they are elliptic or hyperbolic, do persist, as do their bifurcations. Thus our result gives information on low periodic orbits of general Hamiltonian systems in $1{:}1$ resonance. Homoclinic and heteroclinic connections on the reduced phase space generically do break up under non–$S^{1}$–symmetric perturbations yielding chaotic regions familiar from Poincaré sections of for example the Hénon–Heiles system. 5. 5.

In view of the previous remark, the bifurcation diagram for the equilibrium at [math] on ${\mathbb{R}}^{4}$ with branches of periodic orbits is valid for general Hamiltonian systems in $1{:}1$ resonance.

2 A few facts about Hamiltonian systems

Here we very briefly review some facts from the theory of Hamiltonian systems. We concentrate on ${\mathbb{R}}^{4}$. However everywhere in the following sections ${\mathbb{R}}^{4}$ can be replaced by $M$, a $C^{\infty}$ real symplectic manifold. For a thorough treatment we refer to for example [1, 2].

2.1 Symplectic spaces and Hamiltonian systems

Let $\omega$ be a closed, non-degenerate skew symmetric $2$–form on ${\mathbb{R}}^{4}$, making $({\mathbb{R}}^{4},\omega)$ a symplectic space. Furthermore let $H$ be a function in $C^{\infty}({\mathbb{R}}^{4},{\mathbb{R}})$, then the triple $({\mathbb{R}}^{4},\omega,H)$ is called a smooth real Hamiltonian system. Now let $\mathcal{X}({\mathbb{R}}^{4})$ be the set of smooth vector fields on ${\mathbb{R}}^{4}$. The vector field $X_{\textstyle\!H}\in\mathcal{X}({\mathbb{R}}^{4})$ satisfying

[TABLE]

for all $Y\in\mathcal{X}({\mathbb{R}}^{4})$, is called the Hamiltonian vector field of $H$. The vector field $X_{\textstyle\!H}$ defines the flow of the Hamiltonian system on ${\mathbb{R}}^{4}$, we also call this the flow of $H$. A function $f$ is preserved under the flow of the vector field $X_{\textstyle\!H}$ if and only if the Lie derivative of $f$ is identically zero. Using $L_{X_{H}}(f)=df(X_{\textstyle\!H})$ we find that the Hamiltonian function $H$ is preserved by the flow of $X_{\textstyle\!H}$ because

[TABLE]

The last equality follows from the skew symmetry of $\omega$.

2.2 Poisson brackets

Let $f$ and $g$ be in $C^{\infty}({\mathbb{R}}^{4},{\mathbb{R}})$, then we define the Poisson bracket of $f$ and $g$ as

[TABLE]

It follows from this definition that

[TABLE]

Suppose that the function $f$ is preserved under the flow of $X_{\textstyle\!H}$, then

[TABLE]

and vice verse, so once we have the Poisson bracket we do not need the vector field $X_{\textstyle\!H}$ to determine whether $f$ is preserved under the flow of $H$. Furthermore $\{f,g\}=-\{g,f\}$ so $\{H,H\}=0$ from which again follows that $H$ is preserved under the flow of $X_{\textstyle\!H}$. The Poisson bracket satisfies Jacobi’s identity whence Hamiltonian vector fields form a Lie algebra; in fact we have

[TABLE]

Thus $(C^{\infty}({\mathbb{R}}^{4}),\{\cdot,\cdot\})$ is a Lie algebra of functions.

2.3 Standard forms

Darboux’s theorem now states that there are co-ordinates such that $\omega$ becomes constant. Then by applying linear algebra we can bring $\omega$ into a standard form such that

[TABLE]

for all $\xi,\eta\in{\mathbb{R}}^{4}$. Here $\langle\cdot\mid\cdot\rangle$ is the standard inner product on ${\mathbb{R}}^{4}$ and $\Omega$ is a linear mapping with $\Omega=-\Omega^{t}=-\Omega^{-1}$ which takes the standard form

[TABLE]

on the standard basis $\{e_{1},e_{2},f_{1},f_{2}\}$. Let us take co-ordinates $z=(q_{1},q_{2},p_{1},p_{2})$ with respect to this basis, then the Poisson bracket becomes

[TABLE]

Using the Poisson bracket on these co-ordinates we obtain the canonical equations of motion

[TABLE]

for the Hamiltonian $H$. The Poisson bracket allows us to use functions instead of vector fields, which simplifies many computations.

3 Resonant Hamiltonian systems and $S^{1}$–symmetry

On the symplectic space $({\mathbb{R}}^{4},\omega)$ we consider $C^{\infty}$ Hamiltonian systems with an equilibrium at the origin. Furthermore suppose that the linearisation of the corresponding Hamiltonian vector field has resonant imaginary eigenvalues.

When this system has been transformed into normal form it admits an $S^{1}$–symmetry group. Resonant eigenvalues are not generic, but when they appear in parameter families of Hamiltonian systems they are a source of bifurcations. Therefore it is useful to study unfoldings of resonant systems. Most resonances in $4$–dimensional Hamiltonian systems have been studied before, see [10] and references therein. This approach has to be refined for the $1{:}1$ and $1{:}{-}1$ resonances, where the sign is the symplectic sign. See [27] for an extensive study of the so-called nilpotent $1{:}{-}1$ resonance which in a parameter family gives rise to the Hamiltonian Hopf bifurcation. Our aim here is to study the $1{:}1$ resonance. While this case has already been considered in [6], the arguments presented there are incomplete.

A resonant Hamiltonian system naturally leads to an $S^{1}$–invariant system when passing to a normal form truncation. But we may also consider Hamiltonian systems with an externally given symplectic $S^{1}$–action. Our results hold for such systems as well, provided that the $S^{1}$–action satisfies the conditions in the next section.

3.1 $S^{1}$–symmetry related to the $1{:}1$ resonance

Since we work in the class $C^{\infty}({\mathbb{R}}^{4})$ the Hamiltonian function $H$ has an infinite Taylor series. We now put some more structure on these functions by collecting homogeneous terms, turning $(C^{\infty}({\mathbb{R}}^{4}),\{\cdot,\cdot\})$ into a graded Lie algebra. Then we expand

[TABLE]

with $H_{k}\in{\mathbb{R}}[z]$ homogeneous of degree $k$. The normal form procedure acts in a very nice way on this Lie algebra, for details see [27]. The final result is that for the normal form we have $\{H_{2},H_{k}\}=0$ for all $k$ and therefore $\{H_{2},H\}=0$. This means that the normal form of $H$ is invariant under the flow of $H_{2}$ which is generated by $X_{\textstyle\!H_{2}}$. Now we assume that the linear part $X_{\textstyle\!H_{2}}$ of the vector field $X_{\textstyle\!H}$ is in $1{:}1$ resonance, then (the normal form of) $H$ is $S^{1}$–invariant with respect to the $S^{1}$–action

[TABLE]

where

[TABLE]

and $z=(q_{1},p_{1},q_{2},p_{2})$. The quadratic part of such a Hamiltonian systems reads

[TABLE]

Note that this function has Morse index [math] which is intimately related to the fact that the eigenvalues of the linear part of the corresponding Hamiltonian vector field have equal symplectic sign, see [5].

Every $S^{1}$–invariant $C^{\infty}$–function can be written as a function of so called invariants. This is a consequence of far more general results which we now state. We start with a theorem on invariant polynomials.

Theorem 3.1** (Hilbert, Schwartz).**

Let $\Gamma$ be a compact group which acts linearly on ${\mathbb{R}}^{n}$ and let ${\mathbb{R}}[z]^{\Gamma}$ denote the set of $\Gamma$–invariant polynomials. Then a finite number $r$ of polynomials $\rho_{1},\ldots,\rho_{r}\in{\mathbb{R}}[z]^{\Gamma}$ exist that generate ${\mathbb{R}}[z]^{\Gamma}$. The $\rho_{1},\ldots,\rho_{r}$ form a Hilbert basis and are called generators. Furthermore every $\Gamma$–invariant $C^{\infty}$–function $f\in C^{\infty}({\mathbb{R}}^{n})^{\Gamma}$ can be written as a $C^{\infty}$–function $\hat{f}\in C^{\infty}({\mathbb{R}}^{r})$ of the $r$ generators of ${\mathbb{R}}[z]^{\Gamma}$.

Unfortunately the function $\hat{f}$ need not be unique for there may be syzygies among the $\rho_{j}$.

Let us now determine the invariants of the $S^{1}$–action associated to the $1{:}1$ resonance. These are polynomials on the phase space and they Poisson commute with $H_{2}$.

Lemma 3.2**.**

The generators in ${\mathbb{R}}[q,p]^{S^{1}}$ of the invariants of the $S^{1}$–action associated to the $1{:}1$ resonance are given by

[TABLE]

with syzygy $I_{1}^{2}=I_{2}^{2}+I_{3}^{2}+I_{4}^{2}$.

For a proof we refer to [7].

Thus every $S^{1}$–invariant $C^{\infty}$–function on ${\mathbb{R}}^{4}$ can be written as a $C^{\infty}$–function of the Hilbert basis $\{I_{1},I_{2},I_{3},I_{4}\}$. From now on we restrict to a smaller set of functions, namely the formal series in ${\mathbb{R}}[[I]]$. The reasons we can do this are 1) every polynomial in $I$ is the Taylor series of a $C^{\infty}$-function of $I$; 2) we only allow for a finite number of conditions on the coefficients of a series. The latter means that we do not encounter the subtleties on infinitely flat functions, however see remark 1.2, item 4. Moreover we are only interested in $C^{\infty}$–functions that are zero at the origin. Therefore we only consider formal series without constant terms, denoted by ${\mathbb{R}}[[I]]_{0}$.

Now a function in ${\mathbb{R}}[[I]]_{0}$ is not unique, due to the syzygy among the generators. In this respect it is worth noting that when we consider functions in ${\mathbb{R}}[[I]]_{0}$ modulo the ideal generated by $I_{1}^{2}-(I_{2}^{2}+I_{3}^{2}+I_{4}^{2})$, denoted by ${\mathbb{R}}[[I]]_{0}/{\sim}$, we have the following splitting, see [7]. This splitting is also not unique, but seems natural in view of the syzygy.

Lemma 3.3**.**

${\displaystyle{\mathbb{R}}[[I_{1},I_{2},I_{3},I_{4}]]_{0}/{\sim}\;\;=\;\;{\mathbb{R}}[[I_{2},I_{3},I_{4}]]_{0}\;\oplus\;I_{1}{\mathbb{R}}[[I_{2},I_{3},I_{4}]]_{0}}$.

When chosen in this last space the function $\hat{f}$ in theorem 3.1 is unique. Now that we know the generators of the invariants we can write $H$ and $H_{2}$ as functions of these. In particular, we have $H_{2}(I)=I_{1}$.

3.2 Reduction of the $S^{1}$–symmetry:

Hamiltonian systems on $S^{2}$

We are primarily interested in the flow of $H$. Since the flow of $H$ and the $S^{1}$–action commute ($H$ and $H_{2}$ Poisson commute), the orbits of $H$ through an $S^{1}$–orbit are equivalent. Therefore we wish to reduce to the orbit space ${\mathbb{R}}^{4}/S^{1}$ where points correspond to $S^{1}$–orbits on ${\mathbb{R}}^{4}$. The projection mapping

[TABLE]

defined in lemma 3.2 just does that. It allows us to reduce the dynamics of $H$ on ${\mathbb{R}}^{4}$ to a $2$–dimensional phase space.

The $S^{1}$–action is generated by the vector field $X_{\textstyle\!H_{2}}$. Now $H_{2}$ is preserved by its own flow, therefore the $S^{1}$–action preserves $I_{1}$ which defines a $3$–sphere

[TABLE]

As $H$ and $H_{2}$ Poisson commute, the flow of $H$ also preserves this $3$–sphere. Because of the syzygy $I_{1}^{2}=I_{2}^{2}+I_{3}^{2}+I_{4}^{2}$ the projection mapping takes the flow of $H$ to a $2$–sphere in the reduced phase space; the reduced phase space is determined by $I_{1}=h_{2}$, $I_{1}^{2}=I_{2}^{2}+I_{3}^{2}+I_{4}^{2}$. The reduced dynamics of $H$ can simply be characterised by the level $h$ of $H$. This means that an orbit of the reduced flow of $H$ is determined by the equations

[TABLE]

The reduced dynamics of $H$ consists of curves on a $2$–sphere. Note that in order to know the time parametrisation of these curves we still have to solve a generally difficult differential equation. But we do have a full geometric characterisation.

This leads us to the following. We consider the set of smooth $S^{1}$–invariant mappings $C^{\infty}({\mathbb{R}}^{4},{\mathbb{R}}^{2})^{S^{1}}$ of the form $(H,H_{2})$. The reduced dynamics of $H$ is determined once we specify its value by $(H,H_{2})=(h,h_{2})$. In the next section we address the question whether a polynomial $H$ exists such that this mapping is stable in the sense of singularity theory.

Remark 3.4**.**

For a far more complete account of general regular reduction see for example [1, 2]. More details about the $1{:}1$ resonance can be found in [7] where the projection mapping (3) is shown to be the Hopf mapping from $S^{3}$ to $S^{2}$. 2. 2.

Other resonances like $k{:}l$ give rise to a different reduced phase space, having singularities. These arise from non-trivial isotropy subgroups of the $S^{1}$–symmetry group in these cases. They again turn up in new generators with a higher order syzygy. In $4$–dimensional resonant Hamiltonian systems the situation is relatively simple, there are four generators and one syzygy. In higher dimensions both the number of generators and the number of syzygies depend on the resonance, i.e. on the ratios $k_{1}:k_{2}:\cdots:k_{n}$, making it computationally difficult. Then the Gröbner basis algorithm is indispensable.

Both sides of the syzygy define a Casimir element, i.e. their Poisson brackets with the $I$ vanish. A straightforward calculation yields table 1 of Poisson brackets.

The invariants from lemma 3.2 are sometimes called Hopf variables. Indeed, $I_{1}$ generates the $S^{1}$–symmetry (2) and hence is an integral of motion for every Hamiltonian system with that symmetry. The Hopf mapping

[TABLE]

from the $3$–sphere

[TABLE]

to the $2$–sphere

[TABLE]

performs the reduction to one degree of freedom by identifying points related through (2).

The phase portraits are obtained by intersecting, within ${\mathbb{R}}^{3}$, the level sets of the Hamiltonian $H=H(I_{2},I_{3},I_{4})$ with $S^{2}$. Where $H$ is a Morse function, this yields finitely many centres and saddles, with generically no heteroclinic connections between the latter. Under variation of parameters local and global bifurcations may occur.

4 The universal unfolding

In this section we state our main theorem. First we provide a context for the theorem by introducing the notion of stable mappings under left-right-equivalence.

4.1 Equivalence classes for $S^{1}$–invariant

Hamiltonian systems

The meaning of ‘universal unfolding’ depends on the universe in which we work and the notion of equivalence. As explained in section 3.2 we consider Hamiltonian systems on ${\mathbb{R}}^{4}$ that are $S^{1}$–invariant and can be reduced to $S^{2}$. If we content ourselves with characterising the reduced dynamics of $H$ by the orbits only we just need to specify values of $H$ and $H_{2}$. That is the orbits of the reduced Hamiltonian systems are the fibres of the mapping $(H,H_{2})$. Note however that $H_{2}(I)=I_{1}$ and $H_{2}$ is an integral of the Hamiltonian system. So $I_{1}$ is constant and therefore not to be considered as a variable but rather a parameter. Furthermore note that the fibres of the mappings $(H,H_{2})$ and $(H,H_{2}^{2})$ are identical. Using the relation of the generators of the invariants $I$, we have $H_{2}(I)^{2}=I_{1}^{2}=I_{2}^{2}+I_{3}^{2}+I_{4}^{2}$. This leads us to define $K(I)=H_{2}(I)^{2}$ and consider the mapping $\mathcal{F}(I)=(H(I),K(I))$ on our universe $C^{\infty}({\mathbb{R}}^{4},{\mathbb{R}}^{2})^{S^{1}}_{0}$, the $S^{1}$–invariant $C^{\infty}$–mappings from ${\mathbb{R}}^{4}$ to ${\mathbb{R}}^{2}$ taking $(0,0)$ to $(0,0)$.

A natural notion of equivalence on $C^{\infty}({\mathbb{R}}^{4},{\mathbb{R}}^{2})^{S^{1}}_{0}$ is provided by so called left-right-equivalences, see definition 4.1 below. For if $\mathcal{F}$ and $\mathcal{G}$ are left-right-equivalent then the fibres of $\mathcal{F}$ and $\mathcal{G}$ are diffeomorphic. This in turn implies that the orbits of the $S^{1}$–invariant Hamiltonian systems in $\mathcal{F}=(H,K)$ and $\mathcal{G}=(H^{\prime},K^{\prime})$ can be mapped to each other by a simple diffeomorphism.

Definition 4.1**.**

The mappings $\mathcal{F},\mathcal{G}\in C^{\infty}({\mathbb{R}}^{4},{\mathbb{R}}^{2})^{S^{1}}_{0}$ are called left-right-equivalent if $(\psi,\phi)\in\text{Diff}({\mathbb{R}}^{2})_{0}\times\text{Diff}({\mathbb{R}}^{4})^{S^{1}}_{0}$ exists such that $(\psi,\phi)\cdot\mathcal{F}=\mathcal{G}$, where $(\psi,\phi)\cdot\mathcal{F}=\psi\circ\mathcal{F}\circ\phi$.

4.1.1 Stable $S^{1}$–invariant mappings,

co-dimension and unfolding

The idea of stability of a mapping $\mathcal{F}$ is that every mapping $\mathcal{G}$ nearby $\mathcal{F}$ is equivalent to $\mathcal{F}$, or put differently, that $\mathcal{G}$ is an element of the orbit of $\mathcal{F}$ under left-right-equivalence. Here we give a short overview in a series of definitions and theorems.

Definition 4.2**.**

The orbit of $\mathcal{F}\in C^{\infty}({\mathbb{R}}^{4},{\mathbb{R}}^{2})^{S^{1}}_{0}$ under left-right-equivalences is given by

[TABLE]

To define ‘nearby’ we use the definition of a deformation.

Definition 4.3**.**

A deformation (or unfolding) of a mapping $\mathcal{F}\in C^{\infty}({\mathbb{R}}^{4},{\mathbb{R}}^{2})^{S^{1}}_{0}$ is a $C^{\infty}$–mapping $\mathcal{F}:{\mathbb{R}}^{4}\times{\mathbb{R}}^{p}\longrightarrow{\mathbb{R}}^{2}$ defining a family of $S^{1}$–equivariant $\mathcal{F}_{\nu}$, $\nu\in{\mathbb{R}}^{p}$, such that $\mathcal{F}_{0}=\mathcal{F}$.

This allows to formulate a parametric version of $\mathcal{F}$ being an interior point of the orbit of $\mathcal{F}$.

Definition 4.4**.**

A mapping $\mathcal{F}\in C^{\infty}({\mathbb{R}}^{4},{\mathbb{R}}^{2})^{S^{1}}_{0}$ is called stable if for every deformation $\mathcal{F}_{\nu}$ there is an open neighbourhood $U$ of $0\in{\mathbb{R}}^{p}$ such that for all $\nu\in U$, $\mathcal{F}_{\nu}\in Orb_{\mathcal{F}}$.

The conditions of stability in this sense are hard to check. The conditions of infinitesimal stability are much easier to check and this notion of stability turns out to be equivalent with the previous one.

Definition 4.5**.**

$\mathcal{F}$* is called infinitesimally stable if the tangent space of $Orb_{\mathcal{F}}$ at $\mathcal{F}$ is equal to the tangent space of $C^{\infty}({\mathbb{R}}^{4},{\mathbb{R}}^{2})^{S^{1}}_{0}$ at $\mathcal{F}$.*

A proof of the next theorem can be found in [26].

Theorem 4.6**.**

A mapping is stable if and only if it is infinitesimally stable.

Stable mappings form an open and dense subset of $C^{\infty}({\mathbb{R}}^{4},{\mathbb{R}}^{2})^{S^{1}}_{0}$, see [29]. A mapping that fails to be stable has therefore non-zero co-dimension

Definition 4.7**.**

Two deformations $\mathcal{F}_{\nu}$ and $\mathcal{G}_{\mu}$ are left-right-equivalent if there are $(\psi_{\nu},\phi_{\nu})$ and $\mu(\nu)$ with $\psi_{\nu}\circ\mathcal{F}_{\nu}\circ\phi_{\nu}=\mathcal{G}_{\mu(\nu)}$.

This allows to generalize the previous discussion of mappings to deformations.

Definition 4.8**.**

A versal unfolding is a stable deformation.

The minimal number of parameters of a versal unfolding of a mapping $\mathcal{F}_{0}$ coincides for $C^{\infty}({\mathbb{R}}^{4},{\mathbb{R}}^{2})^{S^{1}}_{0}$ with the co-dimension of $\mathcal{F}_{0}$.

4.1.2 The tangent space of $Orb_{\mathcal{F}}$ at $(H,K)$

Let $\mathcal{X}({\mathbb{R}}^{4})$ be the Lie algebra of $\text{Diff}({\mathbb{R}}^{4})_{0}$ and $\mathcal{X}({\mathbb{R}}^{4})^{S^{1}}$ be the Lie algebra of $\text{Diff}({\mathbb{R}}^{4})^{S^{1}}_{0}$.

Lemma 4.9**.**

The tangent space of $Orb_{\mathcal{F}}$ of $\mathcal{F}\in C^{\infty}({\mathbb{R}}^{4},{\mathbb{R}}^{2})^{S^{1}}_{0}$ at $(H,K)$ is given by

[TABLE]

Proof.

For every near-identity transformation $(\psi,\phi)\in\text{Diff}({\mathbb{R}}^{2})_{0}\times\text{Diff}({\mathbb{R}}^{4})^{S^{1}}_{0}$ there exist $X\in\mathcal{X}({\mathbb{R}}^{2})$ and $Y\in\mathcal{X}({\mathbb{R}}^{4})^{S^{1}}$ such that for some $t\in R$ we have $(\psi,\phi)=(\text{e}^{tX},\text{e}^{tY})$. Then the tangent vectors are $\frac{d}{dt}(\text{e}^{tX}\circ\mathcal{F}\circ\text{e}^{tY})|_{t=0}=X(\mathcal{F})+d\mathcal{F}(Y)$. ∎

Taking a closer look at the tangent space of $Orb_{\mathcal{F}}$ at $\mathcal{F}=(H,K)$ in lemma 4.9; we explicitly have

[TABLE]

In this expression $X$ is any vector field on ${\mathbb{R}}^{2}$, but $Y$ is an $S^{1}$–equivariant vector field on ${\mathbb{R}}^{4}$. Using theorem 4.6 we have to check that every $S^{1}$–equivariant map germ can be written as $(X_{1}(H,K)+Y(H),X_{2}(H,K)+Y(K))$ for a suitable choice of $X$ and $Y$.

4.1.3 The restricted tangent space of $Orb_{\mathcal{F}}$ at $(H,K)$

The $S^{1}$–equivariant vector fields are such that $Y(K)$ can be any function of degree 2 and higher in the set of $S^{1}$–invariant functions on ${\mathbb{R}}^{4}$. This follows from an explicit calculation of these vector fields in section 5.2. Thus the stability of $\mathcal{F}$ is determined by the first component. More precisely we have the following.

Proposition 4.10**.**

The co-dimension of $(H,K)$ in $C^{\infty}({\mathbb{R}}^{4},{\mathbb{R}}^{2})^{S^{1}}_{0}$ with the full group of left-right-equivalences is equal to the co-dimension of $H$ in $C^{\infty}({\mathbb{R}}^{4})^{S^{1}}_{0}$ with the group of left-right-equivalences that fix $K$.

Therefore we restrict to vector fields in $Y\in\mathcal{X}({\mathbb{R}}^{4})^{S^{1}}$ such that $X_{2}(H,K)+Y(K)=0$. Or, from a slightly different point of view, we look for a normal form of the mapping $\mathcal{F}=(H,K)$. But the second component can already be regarded as being in normal form. Therefore we may restrict to transformations that preserve $K$, that is $X_{2}(H,K)+Y(K)=0$.

Lemma 4.11**.**

The set of $S^{1}$–equivariant vector fields $Y$ with $Y(K)\in{\mathbb{R}}[[H,K]]_{0}$ can be decomposed as the direct sum of two modules. The first is a module over ${\mathbb{R}}[[I]]_{0}/{\sim}$ and consists of vector fields $Y\in\mathcal{X}({\mathbb{R}}^{4})^{S^{1}}$ taking $K$ to zero. The second is a module over ${\mathbb{R}}[[H,K]]_{0}$, generated by vector fields $Y\in\mathcal{X}({\mathbb{R}}^{4})^{S^{1}}$ taking $K$ to $K$ or to $H$.

Proof.

$S^{1}$–equivariant vector fields $Y$ such that $Y(K)\in{\mathbb{R}}[[H,K]]_{0}$ are generated by $S^{1}$–equivariant vector fields satisfying one of the three equations $Y(K)=0$, $Y(K)=K$ and $Y(K)=H$. ∎

From now on we consider the restricted tangent space of $Orb_{H}$ under left-right-transformations and we call it $T_{1}$. The restricted tangent space of $Orb_{H}$ is again the sum of two (function) modules $\mathcal{J}\oplus\mathcal{M}$. Suppose $U_{1},\cdots,U_{k}$ generate the solutions of $Y(K)=0$ and $V_{1}$ and $V_{2}$ solve $Y(K)=K$ and $Y(K)=H$, respectively. Furthermore let $F_{i}=U_{i}(H)$ for $i\in\{1,\ldots,k\}$ and $G_{j}=V_{j}(H)$ for $j\in\{1,2\}$. Then we have the following.

Lemma 4.12**.**

The restricted tangent space of $Orb_{H}$ is the sum of two modules $\mathcal{J}\oplus\mathcal{M}$, the first is a module over ${\mathbb{R}}[[I]]_{0}/{\sim}$ and the second is a module over ${\mathbb{R}}[[H,K]]_{0}$. That is, every function $f$ in the tangent space of $Orb_{H}$ is of the form $f=\xi_{1}F_{1}+\cdots+\xi_{k}F_{k}+\eta_{0}+\eta_{1}G_{1}+\eta_{2}G_{2}$, with $\xi_{i}\in{\mathbb{R}}[[I]]_{0}/{\sim}$ and $\eta_{i}\in{\mathbb{R}}[[H,K]]_{0}$.

Thus the question about the co-dimension and universal unfolding of the mapping $\mathcal{F}\in C^{\infty}({\mathbb{R}}^{4},{\mathbb{R}}^{2})^{S^{1}}_{0}$ reduces to finding the co-dimension and a complement of the first component of the tangent space of $\mathcal{F}$ with respect to restricted left-right transformations. This in turn can be reformulated as follows. Let $G$ be the mapping

[TABLE]

Then the questions we want to answer are:

What is the co-dimension of the image of $G$ in ${\mathbb{R}}[[I]]_{0}/{\sim}$ ? 2. 2.

If the latter is nonzero, then what is a complement?

4.2 Statement of main theorem

Our main theorem is about the universal unfolding of the mapping $(H,K):C^{\infty}({\mathbb{R}}^{4})^{S^{1}}_{0}\longrightarrow{\mathbb{R}}^{2}$ with respect to restricted left-right-equivalence from the previous section. That is we consider all left-right transformations that preserve $K$. As explained in section 1.2 we are interested in the fibres of the mapping $(H,K)$. For this mapping we have the following result.

Theorem 4.13**.**

The universal unfolding of the mapping $(H,K)$ with respect to restricted left-right-equivalence is given by

[TABLE]

provided that the real coefficients $a_{1}$, $a_{2}$, $a_{3}$ and $b_{1}$, $b_{2}$ and $b_{3}$ satisfy the non-degeneracy condition

[TABLE]

The parameters $\mu_{4}$ and $\mu_{5}$ are moduli.

5 Proof of main theorem

We now prove our main theorem. Our starting point is the mapping $\mathcal{F}=(H,K)$. We split the higher order terms of $H$ into two parts, $H_{4}$ is of degree $2$ in $I$, $H_{6}$ is of degree $3$.

The proof consists of several steps which we now list.

Apply preliminary transformations to $\mathcal{F}$ to get rid of as many coefficients as possible. 2. 2)

Determine the tangent space of $Orb_{\mathcal{F}}$ at $\mathcal{F}$. 3. 3)

Find the $S^{1}$–equivariant vector fields on ${\mathbb{R}}^{4}$. 4. 4)

Observe that we can restrict to the first component of $\mathcal{F}$ using restricted vector fields. 5. 5)

Observe that we can proceed by degree when we split $C^{\infty}({\mathbb{R}}^{4},{\mathbb{R}})^{S^{1}}_{0}$ as a direct sum of spaces of homogeneous polynomials. The cases of relative large degree turn out to be the easiest. Then we are left with a finite number of low degree cases that have to be treated separately.

5.1 Preliminary transformations

We start with the mapping $\mathcal{F}=(H,K)$, where $H$ is a polynomial of degree $3$ in $I$, that is $H=H_{2}+H_{4}+H_{6}$. We assume that symplectic transformations already have been used exhaustively. But since we consider $\mathcal{F}$ in a more general context, more transformations are allowed.

The first observation is that we can always subtract $H_{2}$ from $H$ because $H_{2}$ is a conserved function in the sense of Hamiltonian systems. Thus we have $H=H_{4}+H_{6}$. Furthermore, since $H_{2}(I)=I_{1}$ we consider $I_{1}$ as a parameter. Therefore $I_{1}$ appears at most in the coefficients of $H$. So in fact $H$ and $K$ only depend on $I_{2}$, $I_{3}$ and $I_{4}$, without further restrictions or relations.

[TABLE]

The second observation is that by a transformation from $id\times SO(3)$ we can always achieve $a_{23}=0$, $a_{24}=0$ and $a_{34}=0$. Note that such a transformation preserves both $K$ and the relation $I_{1}^{2}=I_{2}^{2}+I_{3}^{2}+I_{4}^{2}$.

Remark 5.1**.**

We may include more third degree terms in $H_{6}$, like $I_{2}I_{4}^{2}$. However, they turn out to be unimportant. **

5.2 $S^{1}$–equivariant vector fields

Considering the mapping $(H,K)$ instead of $(H,H_{2})$ where $I_{1}$ is a parameter, we take $I_{2}$, $I_{3}$ and $I_{4}$ as co-ordinates on ${\mathbb{R}}^{3}$ without any restrictions. Now $(H,K)$ is a mapping in $C^{\infty}({\mathbb{R}}^{3},{\mathbb{R}}^{2})_{0}$. Origin preserving transformations on ${\mathbb{R}}^{3}$ are generated by the vector fields

[TABLE]

To define the restricted tangent space of the mapping $\mathcal{F}$ we have to find the vector fields solving $X(K)=0$, $X(K)=K$ and $X(K)=H$.

Lemma 5.2**.**

The vector fields solving $X(K)=0$ are generated by

[TABLE]

The vector fields solving $X(K)=K$ and $X(K)=H$ respectively are generated by

[TABLE]

Proof.

Let X=\mbox{\tfrac{1}{2}}\sum_{i=1}^{9}\xi_{i}X_{i} then

[TABLE]

and after some straightforward calculations the results follow. ∎

5.3 The structure of the restricted tangent space

The restricted tangent space $T_{1}$, see section 4.1.3, is the sum of a module $\mathcal{M}$ and an ideal $\mathcal{J}$ both subsets of ${\mathbb{R}}[[I_{2},I_{3},I_{4}]]_{0}$. $\mathcal{M}$ is a module over ${\mathbb{R}}[[H,K]]_{0}$ and generated by the functions $1$, $G_{1}$ and $G_{2}$. $\mathcal{J}$ is the ideal generated by $F_{1},F_{2},F_{3}$. So if $f\in T_{1}$ then $f=\xi_{1}F_{1}+\xi_{2}F_{2}+\xi_{3}F_{3}+\eta_{0}+\eta_{1}G_{1}+\eta_{2}G_{2}$, with $\xi_{i}\in{\mathbb{R}}[[I_{2},I_{3},I_{4}]]_{0}$ and $\eta_{i}\in{\mathbb{R}}[[H,K]]_{0}$.

In lemma 5.2 we defined the vector fields $U_{1}$, $U_{2}$, $U_{3}$, $V_{1}$ and $V_{2}$. Thus we know the generators of $\mathcal{J}$ and $\mathcal{M}$

[TABLE]

Defining $G_{1}$ as $V_{1}(H)-H$ instead of $V_{1}(H)$ is just convenient but not essential. In the definition above we only show the leading terms of $F_{1},\ldots,G_{2}$.

In principle each term in $f\in{\mathbb{R}}[[I_{2},I_{3},I_{4}]]_{0}$ is an infinite series, but with a term of lowest degree. For our purposes it makes sense to call this the degree of $f$ and the term with lowest degree the leading term. Recall that the degree is at least $1$ as we only consider formal series without constant term. Before using this to define a filtration on $T_{1}$ we formally define the degree of $f$ and the leading term.

Definition 5.3**.**

For $0\neq f\in{\mathbb{R}}[[I_{2},I_{3},I_{4}]]_{0}$ we define the degree of $f$ as $k\in{\mathbb{N}}$ for which $0<\lim_{t\to 0}t^{-k}f(tI)<\infty$. Suppose $k=\text{degree}(f)$ then we call $\mathsf{L}(f)=\lim_{t\to 0}t^{-k}f(tI)$ the leading term of $f$.

The following properties of degree and leading term are almost obvious.

Lemma 5.4**.**

Let $f$ and $g$ be functions (germs) in ${\mathbb{R}}[[I_{2},I_{3},I_{4}]]_{0}$ and let $m$ and $n$ be monomials in ${\mathbb{R}}[[I_{2},I_{3},I_{4}]]_{0}$, then

1. i)

if $m(I)=I^{k}$ then $\text{degree}(m)=|k|$ 2. ii)

if $\text{degree}(m)<\text{degree}(n)=l$ then $\text{degree}(m+n)=\text{degree}(m)$ and $\mathsf{L}(m+n)=m$ 3. iii)

$\text{degree}(f+g)=\min(\text{degree}(f),\text{degree}(g))$* and if $\text{degree}(f)<\text{degree}(g)$ then $\mathsf{L}(f+g)=\mathsf{L}(f)$* 4. iv)

$\text{degree}(f\cdot g)=\text{degree}(f)\cdot\text{degree}(g)$* and $\mathsf{L}(f\cdot g)=\mathsf{L}(f)\mathsf{L}(g)$*

With this notion of degree we define a filtration on ${\mathbb{R}}[[I_{2},I_{3},I_{4}]]_{0}$. Since $\mathcal{J}$ and $\mathcal{M}$ are subsets of ${\mathbb{R}}[[I_{2},I_{3},I_{4}]]_{0}$ they immediately inherit the filtration.

Definition 5.5**.**

For $k\in{\mathbb{N}}_{>0}$ let $\mathcal{R}_{k}$ be the set $\{f\in{\mathbb{R}}[[I_{2},I_{3},I_{4}]]_{0}\;|\;\text{degree}(f)=k\}$. Then we have $\mathcal{R}_{k+1}\subset\mathcal{R}_{k}$ and $\mathcal{R}_{1}={\mathbb{R}}[[I_{2},I_{3},I_{4}]]_{0}$, therefore $\mathcal{R}_{k}$ is a filtration of ${\mathbb{R}}[[I_{2},I_{3},I_{4}]]_{0}$. Similarly $\{\mathcal{J}_{k}\}$ and $\{\mathcal{M}_{k}\}$ are filtrations.

Remark 5.6**.**

As an analogy of a Gröbner basis for polynomial ideals, see [4], we could hope that $T_{1}$ is generated by $\mathsf{L}(F_{1}),\mathsf{L}(F_{2}),\mathsf{L}(F_{3}),1,\mathsf{L}(G_{1}),\mathsf{L}(G_{2})$ in the following sense: every $f\in T_{1}$ can be written as $\xi_{1}\mathsf{L}(F_{1})+\xi_{2}\mathsf{L}(F_{2})+\xi_{3}\mathsf{L}(F_{3})+\eta_{0}+\eta_{1}\mathsf{L}(G_{1})+\eta_{2}\mathsf{L}(G_{2})$, with $\xi_{i}\in{\mathbb{R}}[[I_{2},I_{3},I_{4}]]_{0}$ and $\eta_{i}\in{\mathbb{R}}[[H,K]]_{0}$. **

5.4 Splitting into homogeneous parts

Since the co-dimension of $\mathcal{F}$ as a smooth mapping is the same as the co-dimension of the mapping as a formal power series, we can simplify the problem by looking at homogeneous functions and add the co-dimensions found for each degree starting at degree one. This is carried out in the following chain of assertions.

Let $\mathcal{H}_{k}(I_{2},I_{3},I_{4})$ be the set of all homogeneous functions of degree $k$ in $I_{2}$, $I_{3}$ and $I_{4}$. In fact we have $\mathcal{H}_{k}(I_{2},I_{3},I_{4})=\mathcal{R}_{k}/\mathcal{R}_{k+1}$. Furthermore let $\mathcal{H}_{k}(H,K)$ be the set of all homogeneous functions of degree $k$ in $K$ and $H$, then $\mathcal{H}_{m}(H,K)\subset\mathcal{R}_{2m}$. Since $\mathcal{H}_{m}(H,K)$ is not homogeneous in $I$ we use a projection $\Pi_{k}:\mathcal{R}_{k}\to\mathcal{H}_{k}(I_{2},I_{3},I_{4})$ selecting the homogeneous part of a function $f\in\mathcal{R}_{k}$. The following general result leaves us with a small number of cases.

Proposition 5.7**.**

The co-dimension of $\Pi_{k}(T_{1})$ in $\mathcal{H}_{k}(I_{2},I_{3},I_{4})$ is zero for $k=4$ and $k\geq 6$. Or, put differently, the mapping (odd degree)

[TABLE]

is onto for $m\geq 3$ and also the mapping (even degree)

[TABLE]

is onto for $m\geq 2$.

Thus we have to investigate degrees $1,2,3$ and $5$ separately. First we prove proposition 5.7 in three lemmas. In order to do so it is useful to introduce some notation, which is motivated by the fact that the projection of $\xi_{1}\mathsf{L}(F_{1})+\xi_{2}\mathsf{L}(F_{2})+\xi_{3}\mathsf{L}(F_{3})$ on $\langle I_{2}^{k},I_{3}^{k},I_{4}^{k}\rangle$ is always zero.

Definition 5.8**.**

The space $\mathcal{H}_{k}(I_{2},I_{3},I_{4})$ has a monomial basis denoted by $b_{k}=\{\ldots,I_{2}^{k},I_{3}^{k},I_{4}^{k}\}$. Let $b^{\sharp}$ be the set of monomials $I_{2}^{k}$, $I_{3}^{k}$ and $I_{4}^{k}$. Furthermore let $b_{k}^{\flat}$ be the set of monomials in $b_{k}$ with the monomials in $b^{\sharp}$ excluded. Finally let $B_{k}^{\sharp}$ be the subspace of $\mathcal{H}_{k}(I_{2},I_{3},I_{4})$ spanned by $b_{k}^{\sharp}$, similarly $B_{k}^{\flat}$ is spanned by $b_{k}^{\flat}$.

The next three lemmas treat different parts of proposition 5.7. The following lemma shows that the mapping from $\mathcal{H}_{k-2}(I_{2},I_{3},I_{4})^{3}$ to $B_{k}^{\flat}$ is onto for each $k\geq 2$. Thus we get rid of the first factor of the mapping in proposition 5.7. Later on we use this lemma again for the remaining low degree cases.

Lemma 5.9**.**

The mapping $\mathcal{H}_{k-2}(I_{2},I_{3},I_{4})^{3}\to B_{k}^{\flat}:(\xi_{1},\xi_{2},\xi_{3})\mapsto\xi_{1}\mathsf{L}(F_{1})+\xi_{2}\mathsf{L}(F_{2})+\xi_{3}\mathsf{L}(F_{3})$ is onto provided that $a_{1}-a_{2}\neq 0$, $a_{2}-a_{3}\neq 0$ and $a_{3}-a_{1}\neq 0$ and $k\geq 2$.

Proof.

Every monomial in $B_{k}^{\flat}$ can be written as either $I^{l}I_{2}I_{3}$, $I^{l}I_{3}I_{4}$ or $I^{l}I_{2}I_{4}$ for some multi-index $l$ with $|l|=k-2$. Therefore every $f\in B_{k}^{\flat}$ can be expressed as $\xi_{1}\mathsf{L}(F_{1})+\xi_{2}\mathsf{L}(F_{2})+\xi_{3}\mathsf{L}(F_{3})$ for some $\xi_{i}\in\mathcal{H}_{k-2}(I_{2},I_{3},I_{4})$, but only if $a_{1}-a_{2}\neq 0$, $a_{2}-a_{3}\neq 0$ and $a_{3}-a_{1}\neq 0$. If for example $a_{1}-a_{2}=0$, then $I_{2}I_{3}\not\in T_{1}$. ∎

The following two lemmas show that the second factor of the mapping in proposition 5.7 maps onto $B^{\sharp}$, but we have to distinguish the odd and even degree cases.

Lemma 5.10** (Odd degree).**

The mapping \mathcal{H}_{m-1}(H,K)\to\mathcal{H}_{2m+1}(I_{2},I_{3},I_{4}):\eta_{1}\mapsto\Pi_{2m+1}\big{(}\eta_{1}\mathsf{L}(G_{1})\big{)} followed by projection on $B_{2m+1}^{\sharp}$ is onto provided that $a_{1}-a_{2}\neq 0$, $a_{2}-a_{3}\neq 0$, $a_{3}-a_{1}\neq 0$ and $b_{1}b_{2}b_{3}\neq 0$ and $m\geq 3$.

Proof.

The projection of the functions $K^{m-1}\mathsf{L}(G_{1}),K^{m-2}H\mathsf{L}(G_{1}),\ldots,H^{m-1}\mathsf{L}(G_{1})$ on $B_{2m+1}^{\sharp}$ is given by the vectors in the matrix

[TABLE]

which has rank three as soon as the conditions are met. ∎

Finally we state and prove a lemma for the even degree case.

Lemma 5.11** (Even degree).**

The mapping \mathcal{H}_{m}(H,K)\times\mathcal{H}_{m-1}(H,K)\to\mathcal{H}_{2m}(I_{2},I_{3},I_{4}):(\eta_{0},\eta_{2})\mapsto\Pi_{2m}\big{(}\eta_{0}+\eta_{2}\mathsf{L}(G_{2})\big{)} followed by projection on $B_{2m}^{\sharp}$ is onto provided that $a_{1}-a_{2}\neq 0$, $a_{2}-a_{3}\neq 0$ and $a_{3}-a_{1}\neq 0$ and $m\geq 2$.

Proof.

The projection of the functions

[TABLE]

on $B_{2m}^{\sharp}$ is given by the vectors in the matrix

[TABLE]

which has rank three as soon as the conditions are met. ∎

With these three lemmas we prove proposition 5.7.

Proof of proposition 5.7.

The odd degree part of the proposition is covered by combining lemmas 5.9 and 5.10 showing that the product mapping is onto $\mathcal{H}_{2m+1}(I_{2},I_{3},I_{4})$. Similarly combining lemmas 5.9 and 5.11 shows that in case of even degree the product mapping is onto $\mathcal{H}_{2m}(I_{2},I_{3},I_{4})$. ∎

Finally we consider the remaining cases: degrees $1,2,3$ and $5$. In all cases we follow the same pattern, we determine the co-dimension of \Pi_{k}\big{(}\xi_{1}F_{1}+\xi_{2}F_{2}+\xi_{3}F_{3}+\eta_{0}+\eta_{1}G_{1}+\eta_{2}G_{2}\big{)} in $\mathcal{H}_{k}(I_{2},I_{3},I_{4})$ for $k\in\{1,2,3,5\}$. But in view of lemma 5.9 we only have to consider the projection on $B_{k}^{\sharp}$. The main result of this part is the next proposition.

Proposition 5.12**.**

A complement of $T_{1}$ in $\mathcal{R}$ is spanned by the functions $\langle I_{2},I_{3},I_{4},I_{2}^{3},I_{3}^{3}\rangle$ or $\langle I_{2},I_{3},I_{4},I_{2}^{3},I_{4}^{3}\rangle$ or $\langle I_{2},I_{3},I_{4},I_{3}^{3},I_{4}^{3}\rangle$ as a linear space.

We prove this proposition in several lemmas. The following lemma is immediately clear.

Lemma 5.13** (Degree one).**

A monomial basis of functions of degree one is $\{I_{2},I_{3},I_{4}\}$. Since $T_{1}$ does not contain functions of degree one, the co-dimension in this space is three and a complement is $B_{1}^{\sharp}$ itself.

Thus we get unfolding terms: $\mu_{1}I_{2}$, $\mu_{2}I_{3}$ and $\mu_{3}I_{4}$.

Lemma 5.14** (Degree two).**

Functions of degree two with a nonzero projection on $B_{2}^{\sharp}$ are $K$, $H$ and $G_{2}$. These three functions are independent a soon as $(a_{1}-a_{2})(a_{2}-a_{3})(a_{3}-a_{1})\neq 0$.

Proof.

The projection of $K$, $H$ and $G_{2}$ onto $B_{2}^{\sharp}$ is given by the matrix

[TABLE]

cf. (6). The determinant of $A_{2}^{\sharp}$ is $(a_{1}-a_{2})(a_{2}-a_{3})(a_{3}-a_{1})$. ∎

Lemma 5.15** (Degree three).**

There is only one function in $T_{1}$ with a nonzero projection on $B_{3}^{\sharp}$, namely $G_{1}$. Thus the co-dimension of $T_{1}$ in the space of homogeneous functions of degree three is two. As a complement any pair of $I_{2}^{3}$, $I_{3}^{3}$ and $I_{4}^{3}$ will do. We take for example $\mu_{4}I_{2}^{3}$ and $\mu_{5}I_{3}^{3}$ as unfolding terms, then we must impose the condition $b_{3}\neq 0$.

Proof.

The projection of $G_{1}$, $\mu_{4}I_{2}^{3}$ and $\mu_{5}I_{3}^{3}$ on $B_{3}^{\sharp}$ is given by the matrix

[TABLE]

∎

Lemma 5.16** (Degree five).**

There are only two functions of degree five in $T_{1}$, namely $KG_{1}$ and $HG_{1}$, with a nonzero projection on $B_{5}^{\sharp}$. However, a function $F_{5}\in T_{1}$ exists such that \Pi_{k}\big{(}F_{5}\big{)}=0 for $k\leq 4$ and \Pi_{5}\big{(}F_{5}\big{)}\neq 0. With $F_{5}$ the co-dimension of $T_{1}$ in the space of homogeneous functions of degree five is zero, provided that $(a_{1}-a_{2})(a_{2}-a_{3})(a_{3}-a_{1})b_{1}b_{2}b_{3}\neq 0$.

Proof.

Let

[TABLE]

be a function of degree 4, with $\xi_{1},\ldots,\eta_{22}\in{\mathbb{R}}$. Then a non-trivial solution of \Pi_{4}\big{(}F_{5}\big{)}=0 exists while \Pi_{5}\big{(}F_{5}\big{)}\neq 0. The projection of the functions \Pi_{5}\big{(}KG_{1}\big{)}, \Pi_{5}\big{(}HG_{1}\big{)} and \Pi_{5}\big{(}F_{5}\big{)} onto $B_{5}^{\sharp}$ has the matrix

[TABLE]

and $\det(A_{5}^{\sharp})=(a_{1}-a_{2})(a_{2}-a_{3})(a_{3}-a_{1})b_{1}b_{2}b_{3}$. ∎

The last lemma is about the modal parameters.

Lemma 5.17**.**

Parameters $\mu_{4}$ and $\mu_{5}$ are moduli.

Proof.

Let $H(I;\mu)$ be as in the main theorem 4.13. From the previous proofs it follows almost immediately that the unfoldings of $(H(I;0,0,0,0,0),K)$ and $(H(I;0,0,0,\mu_{4},\mu_{5}),K)$ are equal for small values of $\mu_{4}$ and $\mu_{5}$. Therefore $\mu_{4}$ and $\mu_{5}$ are moduli. ∎

The proof of theorem 4.13 follows from proposition 5.7, proposition 5.12 and lemma 5.17.

Remark 5.18**.**

As a by product we find that $T_{1}$ is not generated by

[TABLE]

See remark 5.6. **

6 Discussion

The dynamics of an $n$–degree-of-freedom Hamiltonian system locally around an elliptic equilibrium at the origin is characterised by an $n$–tuple $\omega\in{\mathbb{R}}^{n}$ of frequencies. When the frequencies satisfy an integer relation $\langle m\mid\omega\rangle\neq 0$ with $m\in{\mathbb{Z}}^{n}$ we say that the frequencies are resonant. For most equilibria the frequencies are non-resonant. However, when the system depends on parameters there are resonances at a dense subset of parameter values. Since low order resonances are accompanied by bifurcations the corresponding points in parameter space are of special interest.

Here we consider two-degree-of-freedom systems. In that case $\omega=(\omega_{1},\omega_{2})$, so $\omega$ is resonant if $\omega_{1}/\omega_{2}$ is an element of ${\mathbb{Q}}$. We may assume without loss of generality that $\omega_{1}$ and $\omega_{2}$ are relative prime integers $k$ and $l$ at resonance. The linear part of the vector field is determined by $\omega=(k,l)$ if $k\neq\pm l$. In linear Hamiltonian systems imaginary eigenvalues, in casu the frequencies $k,l$ have a sign. The sign is related to the Morse index of the Hamiltonian. Therefore a $k{:}l$ resonance is not equivalent to a $k{:}{-}l$ resonance; in particular the $1{:}1$ and $1{:}{-}1$ resonances are not equivalent. Moreover, eigenvalues with equal sign are always semi-simple, whereas the $1{:}{-}1$ resonance can also be nilpotent. Thus there are three resonances with equal frequencies, namely the semi-simple $1{:}{-}1$, the nilpotent $1{:}{-}1$ and the $1{:}1$ resonance. The latter is always semi-simple. The nilpotent $1{:}{-}1$ resonance is what triggers the Hamiltonian Hopf bifurcation.

As indicated in the introduction the $k{:}l$ resonances, with $k,l\in{\mathbb{N}}$, are very similar. In particular, in the sense of section 4.1.1 the co-dimension is $2$, provided that $k{:}l$ is not equal to $1{:}1$, $1{:}2$ or $1{:}3$. The last two exceptional cases have co-dimension $1$ and $3$, respectively. Thus all definite resonances except $1{:}1$ have in common that they occur persistently in $1$–parameter families and if more parameters are present these are moduli, see [10]. In this respect our case of the $1{:}1$ resonance is very exceptional: its co-dimension is $5$, it occurs persistently in $3$–parameter families and two of the unfolding parameters are moduli. When we restrict to the linear unfolding, there is a transformation group acting on the unfolding. This can be used to reduce the number of parameters. Using invariants of this transformation group we find that one of the generators is $\mu_{1}^{2}+\mu_{2}^{2}+\mu_{3}^{2}$. Then in a reduced linear unfolding the $1{:}1$ resonance occurs persistently in a $1$–parameter family, see [17] for more details.

Before applying singularity theory we reduce the $S^{1}$–symmetric system using invariants. Another approach is that in [4] where the system is first reduced to a planar system. Then singularity theory using right equivalence is applied to obtain an unfolding. With a different notion of equivalence one may expect different co-dimensions. In [4], by nature of the method, one finds lower bounds for the co-dimensions. For the resonances $1{:}2$, $1{:}3$ and $1{:}4$ these lower bounds are computed and they coincide with the co-dimensions found in [10], namely $1$, $3$ and $2$, respectively. However, the non-degeneracy conditions of [4] and [10] differ. It would be interesting to compare both methods for the $1{:}1$ resonance.

The results obtained so far are a starting point for extensions and applications. Let us list a few. In general, when a system passes a resonance upon varying one or more parameters, one expects a bifurcation to occur. We see this phenomenon in the resonances mentioned earlier. Therefore we would like to explore the bifurcation scenario of the $1{:}1$ resonance, or more general explore the geometry of level sets of the momentum mapping depending on parameters near $1{:}1$ resonance. A similar program can be carried out for Hamiltonian systems in $1{:}1$ resonance which are also reversible, see [25], or symmetric (other than the $S^{1}$ symmetry induced by the $1{:}1$ resonance). The unfolding of the semisimple $1{:}{-}1$ resonance is similar to the unfolding of the $1{:}{-}1$ resonance, but the bifurcation scenario is most likely very different. A well-known system in $1{:}1$ resonance is the Hénon–Heiles system. Our original plan, to apply the unfolding and bifurcation results, now comes within reach. Furthermore we wish to relate our results to the results in a series of articles by Elipe, Lanchares et al. and Frauendiener [11, 12, 21, 22, 23, 24, 25] for families of $S^{1}$–symmetric Hamiltonian systems. These are the subjects of future publications.

Acknowledgment

It is a pleasure to thank Henk Broer, Richard Cushman, Jaap Top and Gert Vegter for fruitful discussions and suggestions.