Masterclass Finite and Infinite Dimensional Dynamical

Systems

Course: Finite Dimensional Dynamical Systems

Henk Broer and Heinz Hanßmann

Progam information Time and place:

Wednesdays 10:00-13:00starting 14 September 2005Mathematisch InstituutBudapestlaan 6, Utrecht

See http://www.math.uu.nl/mc/index.html andhttp://homepages.cwi.nl/~doelman/course.html

Course material includes

1. H.W.Broer, Notes on perturbation theory 1991, Erasmus ICP Math-ematics and Fundamental Applications, Aristotle University Thessa-loniki, (1993), 44 p.

NB: For main Chapter Structure Preserving Normal Forms, see be-low.

2. M.C. Ciocci, A. Litvak-Hinenzon and H.W. Broer, Survey on dissipa-tive kam theory including quasi-periodic bifurcation theory based onlectures by Henk Broer. In: J. Montaldi and T. Ratiu (eds.): Geo-metric Mechanics and Symmetry: the Peyresq Lectures, LMS LectureNotes Series, 306. Cambridge University Press, 2005, 303-355.

Structure Preserving Normal Forms

Henk Broer

Abstract

We review the formal theory of normal forms of dynamical systems near

equilibrium points. Systems with continuous time, i.e. of vector fields,

or autonomous systems of ODE’s, are considered extensively. This in-

cludes the case where a Lie-algebra structure is preserved, in particular

the Hamiltonian case and circumstances where parameters are present.

Certain related topics are discussed more briefly, such as the cases of a

vector field near a periodic solution or a quasi-periodic invariant torus as

well as the case of a diffeomorphism near a fixed point.

Contents

1 The normal form procedure 2

1.1 Some background, linearization . . . . . . . . . . . . . . . . . . . 21.2 Some preliminaries from differential geometry . . . . . . . . . . . 5

1

1.3 ‘Simple’ in terms of an adjoint action . . . . . . . . . . . . . . . . 71.4 Torus symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.5 On the choices of the complementary space Gm and the normal-

izing transformation . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Preservation of Structure 12

2.1 The Lie-algebra proof . . . . . . . . . . . . . . . . . . . . . . . . 132.2 The Hamiltonian case . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Miscellaneous remarks 18

3.1 A diffeomorphism near a fixed point . . . . . . . . . . . . . . . . 193.2 Near a periodic solution . . . . . . . . . . . . . . . . . . . . . . . 193.3 Near a quasi-periodic torus . . . . . . . . . . . . . . . . . . . . . 213.4 Non-formal aspects . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1 The normal form procedure

The term ‘normal form’ is widely used in mathematics and its meaning is verysensitive for the context. In the case of linear maps from a given vector space toitself, for example, one often considers all possible choices of a basis. Each sucha choice gives a matrix-representation of a given linear map. A suitable choiceof basis now gives the well-known Jordan normal form. This normal form, in asimple way displays certain important properties of the linear map, concerningits spectrum, its eigenspaces, and so on.

Here we are concerned with dynamical systems, such as vector fields (i.e.systems of autonomous ODE’s), or diffeomorphisms. Our main concern willbe to simplify these systems near certain equilibria, by a proper choice of co-ordinates. In particular this simplification concerns the Taylor series at theequilibrium points. Before we explore these ideas further, however, let us firstgive a convenient normal form of a vector field near a non-equilibrium point.

Theorem 1 [30] Let the C∞ vector field X on Rn be given by x = f(x) andassume that f(p) 6= 0. Then there exists a neighbourhood of p with local C∞

coordinates y = (y1, y2, · · · , yn), such that in these coordinates X has the form

y1 = 1

yj = 0,

for 2 ≤ j ≤ n.

Such a chart usually is called a flowbox and the above theorem the FlowboxTheorem. A proof simply can be given using a local C∞ section that cuts theflow of the vector field transversally. For the coordinate y1 then use the time-parametrization of the flow, while the coordinates yj , for 2 ≤ j ≤ n come fromthe section. E.g. compare [30].

2

1.1 Some background, linearization

The idea of simplification near an equilibrium goes back at least to Poincare [25],also compare e.g. Arnold [2]. To fix thoughts, we now let X be a C∞ vectorfield on Rn, with the origin as an equilibrium point. Suppose that X has theform x = Ax + f(x), x ∈ Rn, where A is linear and where f(0) = 0, D0f = 0.The first idea is to apply successive C∞ changes of coordinates of the formId + P, with P a homogeneous polynomial of degree m = 2, 3, · · · , ‘simplifying’the Taylor series step by step.

The most ‘simple’ form that can be obtained in this way, is where all higherorder terms vanish. In that case the normal form is formally linear. Such a casewas treated by Poincare, as we shall investigate now.

To this end may even assume to work on Cn. We assume the eigenvalues ofA to be distinct. A collection λ = (λ1, · · · , λn) is said to be resonant if thereexists a relation of the form

λs =< r, λ >,

for r = (r1, · · · , rn) ∈ Zn, with rk ≥ 0 for all k and with∑

rk ≥ 2. The order ofthe resonance then is the number |r| =

∑

rk. The Poincare Theorem now reads

Theorem 2 [25, 3] If the eigenvalues of A have no resonances, there exists aformal change of variables x = y + O(|y|2), transforming the above vector fieldX, given by

x = Ax + f(x)

to

y = Ay.

We include a proof [3], since this will provide the basis for almost all our furtherconsiderations.

Proof. The formal power series x = y + O(|y|2) is produced in an inductivemanner. Indeed, for m = 2, 3, · · · a polynomial transformation x = y + P (y)is constructed, with P homogeneous of degree m, which removes the terms ofdegree m from the vector field. At the end we have to take the composition ofall these polynomial transformations.1. The basic tool for the m-th step is the following. If the vector fields x = Ax+v(x) + O(|x|m+1) and y = Ay ame related by the transformation x = y + P (y)with P homogeneous of degree m, then

DxPAx − AP (x) = v(x).

This relation usually is called the homological equation, the idea being to de-termine P in terms of v : by this choice of P the term v can be transformedaway.

The proof of this relation is straightforward. In fact,

x = (Id + DyP )Ay =

= (Id + DyP )A(x − P (x) + O(|x|m+1)) =

= Ax + DxPAx − AP (x) + O(|x|m+1),

3

where we used that for the inverse transformation we know y = x − P (x) +O(|x|m+1).

2. For notational convenience we introduce the linear operator adA, the so-called adjoint operator, by

adA(P )(x) := DxPAx − AP (x),

then the homological equation reads adA(P ) = v. So the question is reduced towhether v is in the image of the operator adA. It now appears, that the eigenval-ues of the operator adA can be expressed in those of A. Indeed, if x1, x2, · · · , xn

are the coordinates corresponding to the basis e1, e2, · · · en, again it is a straight-forward computation to show that for P (x) = xres, one has

adA(P ) = (< r, λ > −λs)P.

Here we use the multi-index notation xr = xr1

1 xr2

2 · · ·xrn

n . In fact, for this choiceof h one has AP (x) = λsP (x), while

∂xr

∂xAx =

∑

j

rj

xjxrλjxj =< r, λ > xr .

We conclude that the monomials xres are eigenvectors corresponding to theeigenvalues < r, λ > −λs. So we see that the operator adA is it semisimple,since it has a basis of eigenvectors. Therefore, if the kernel ker adA = 0, itsimage is the whole space of our polynomial vector fields. This, in turn, isexactly what the non-resonance condition on the eigenvalues of A amounts to.More precisely, the homological equation is solvable for each homogeneous partv of degree m, provided there are no resonances up to order m. Also the solutionP then can be chosen from the class of vector fields, which are homogeneouspolynomials of degree m.

3. The induction process now runs as follows. For m ≥ 2, given a form

x = Ax + vm(x) + O(|x|m+1),

we solve the homological equation

adA(Pm) = vm,

then carrying out the transformation x = y + Pm(y). This takes the above formto

y = Ay + wm+1(y) + O(|y|m+2).

As said before, the composition of all the polynomial transformations eventuallygives the desired formal transformation. 2

Remarks.

• Here we do not go into the problems of convergence of the formal series.This question is important, however, when working in the analytic setting.

4

• If resonances are excluded up to a finite order N, we can linearize up tothat order, so obtaining a normal form

y = Ay + O(|y|N+1).

In this case the transformation can be chosen polynomial.

• If the original problem is real, but with the matrix A having non-real eigen-values, we still can keep all transformations real by considering complexconjugate eigenvectors.

We end these preliminaries discussing two other linearization theorems, onedue to Sternberg and the other to Hartman & Grobman. We recall the followingfor a vector field X(x) = Ax+ f(x), x ∈ Rn, with 0 as an equilibrium point, i.e.with f(0) = 0, D0f = 0. The equilibrium 0 is hyperbolic if the matrix A has nopurely imaginary eigenvalues. Sternberg’s Theorem now reads

Theorem 3 [31] Let X and Y be C∞ vector fields on Rn, with 0 as a hyper-

bolic equilibrium point. Also suppose that there exists a formal transformation(Rn, 0) → (Rn, 0) taking the Taylor series of X at 0 to that of Y. Then thereexists a local C∞ diffeomorphism Φ : (Rn, 0) → (Rn, 0), such that Φ∗X = Y.

We recall that Φ∗X(Φ(x)) = DxΦX(x). This means that X and Y are lo-cally conjugate by Φ, the evolution curves of X are mapped to those of Yin a time-preserving manner. In particular Sternberg’s Theorem applies whenthe conclusion of Poincare’s Theorem holds: for Y just take the linear partY (x) = Ax.

Combining these two theorems we find that in the hyperbolic case, underthe exclusion of all resonances, the vector field X is C∞-ly linearizable. TheHartman-Grobman Theorem now says that the non-resonance condition can beomitted, provided we only want for a C0-linearization.

Theorem 4 e.g. [24] Let X be a C∞ vector field on Rn, with 0 as a hyperbolicequilibrium point. Then there exists a local homeomorphism Φ : (Rn, 0) →(Rn, 0), conjugating X locally to its linear part.

1.2 Some preliminaries from differential geometry

Before we develop a more general normal form theory we recall some elementsfrom differential geometry. One central notion used here is that of the Lie-derivative. For simplicity all our objects will be of class C∞. Given a vectorfield X we can take any tensor τ and define its Lie-derivative LXτ with respectto X as the infinitesimal transformation of τ along the flow of X. In this wayLXτ becomes a tensor of the same type as τ. To be more precise, for τ a realfunction f one so defines

LXf(x) = X(f)(x) = df(X)(x) =d

dt

∣

∣

∣

∣

t=0

f(Xt(x)),

i.e. the directional derivative of f with respect to X. Here Xt denotes the flowof X over time t. For τ a vector field Y one similarly defines

5

LXY (x) =d

dt

∣

∣

∣

∣

t=0

(X−t)∗Y (x) =

= limt→0

1

t(X−t)∗Y (x) − Y (x) =

= limh→0

1

hY (x) − (Xh)∗Y (x).

Etc. for differential forms, · · · . Another central notion is the Lie-bracket [X, Y ]defined for any two vector fields X and Y on Rn. This is another vector fielddefined by the commutator

[X, Y ](f) = X(Y (f)) − Y (X(f)).

Here f is any real function on Rn while, as before, X(f) denotes the directionalderivative of f with respect to X, etc.

Let us express the Lie-bracket in coordinates. If X is given by the systemof differential equations xj = Xj(x), with 1 ≤ j ≤ n, then the directionalderivative X(f)is given by X(f) =

∑nj=1 Xj∂f/∂xj. Then, if [X, Y ] is given by

the system xj = Zj(x), for 1 ≤ j ≤ n, of differential equations. one directlyshows that

Zj =

n∑

k=1

(Xk∂Yj

∂xk− Yk

∂Xj

∂xk).

Here Yj relates to Y as Xj does to X. We list some useful properties.

Proposition 5 [30]

• LX(Y1 + Y2) = LXY1 + LXY2 (linearity over R)

• LX(fY ) = X(f) × Y + f × LXY (Leibniz rule)

• [Y, X ] = −[X, Y ] (skew symmetry)

• [[X, Y ], Z] + [[Z, X ], Y ] + [[Y, Z], X ] = 0 (Jacoby identity)

• LXY = [X, Y ]

• [X, Y ] = 0 ⇔ Xt Ys = Ys Xt

Proof. The first four items are left to the reader.The equality LXY = [X, Y ] is easy to prove observing the following. Both

members of the equality are defined intrinsically, so it is enough to check it inany choice of (local) coordinates. Moreover, we can restrict our attention to theset x | X(x) 6= 0. By the Flowbox Theorem 1 we then may assume for thecomponent functions of X that X1(x) = 1, Xj(x) = 0 for 2 ≤ j ≤ n. It is easyto see that both members of the equality now are equal to the vector field Zwith components

Zj =∂Yj

∂x1.

6

Remains the equivalence of the commuting relationships. The commutingof the flows, by a very general argument, implies that the bracket vanishes. Infact, fixing t we see that Xt conjugates the flow of Y to itself, which is equivalentto (Xt)∗Y = Y. This in turn implies that LXY = 0.

Conversely, let c(t) = ((Xt)∗Y )(p). From the fact that LXY = 0 it thenfollows that c(t) ≡ c(0). Before we go into this, observe that this is sufficient forour purposes, since it implies that (Xt)∗Y = Y and therefore Xt Ys = Ys Xt.Finally, the fact that c(t) is constant can be shown as follows.

c′(t) = limh→0

1

hc(t + h) − c(t) =

= limh→0

1

h((Xt+h)∗Y )(p) − ((Xt)∗Y )(p) =

= (Xt)∗ limh→0

1

h((Xh)∗Y )(X−t(p)) − Y (X−t(p)) =

= (Xt)∗LXY (X−t(p)) =

= (Xt)∗(0) =

= 0.

2

1.3 ‘Simple’ in terms of an adjoint action

We now return to the setting of our normal form theory at equilibrium points.So given is the vector field x = X(x), with X(x) = Ax + f(x), x ∈ Rn, whereA is linear and where f(0) = 0, D0f = 0. We recall that it is our general aimto simplify the Taylor series of X at 0.

However, we have not yet said what the word ‘simple’ generally could meanin the present situation. In order to understand what is going on, keeping aneye on Poincare’s Theorem 2, we introduce the adjoint action associated to thelinear part A,, defined on the class of all C∞ vector fields on Rn. To be precise,this adjoint action adA is defined by the Lie-bracket

adA : Y 7→ [A, Y ],

where A is identified with the linear vector field x = Ax. It is easily seen thatthis fits with the notation introduced in Theorem 2.

Let Hm(Rn) denote the space of polynomial vector fields, homogeneous ofdegree m. Then the Taylor series of X can be viewed as an element of the spaceof formal power series

∏∞m=1 Hm(Rn). Also, it directly follows that adA induces

a linear map Hm(Rn) → Hm(Rn), to be denoted by admA. Let

Bm := imadmA,

the image of the map admA in Hm(Rn). Then for any complement Gm of Bm

in Hm(Rn), in the sense that

Bm ⊕ Gm = Hm(Rn),

7

we define the corresponding notion of ‘simpleness’ by requiring the homogeneouspart of degree m to be in Gm. In the case of the Poincare Theorem 2, sinceBm = Hm(Rn), we have Gm = 0.

We now quote a theorem from Dumortier [9], Section 7.6.1. Although itsproof is very similar to the one of Theorem 2, we include it here, also becauseof its format.

Theorem 6 [31, 26] Let X be a C∞ vector field, defined in the neighbourhoodof 0 ∈ Rn, with X(0) = 0 and D0X = A. Also let N ∈ N be given and, form ∈ N, let Bm and Gm be as before. Then there exists, near 0 ∈ Rn, ananalytic change of coordinates Φ : Rn → Rn, with Φ(0) = 0, such that

Φ?X(y) = Ay + g2(y) + · · · + gN (y) + O(|y|N+1),

with gm ∈ Gm, for all m = 2, 3, · · · , N.

Proof. We use induction on N. Let us assume that

X(x) = Ax + g2(x) + · · · + gN−1(x) + fN (x) + O(|x|N+1),

with gm ∈ Gm, for all m = 2, 3, · · · , N − 1 and with fN homogeneous of degreeN.

We consider a coordinate change x = y + P (y), where P is polynomial ofdegree N, see above. For any such P, by substitution we get

(Id + DyP )y = A(y + P (y)) + g2(y) + · · · + gN−1(y)

+ fN (y) + O(|y|N+1),

or

y = (Id + DyP )−1(A(y + P (y)) + g2(y) + · · · + gN−1(y)

+ fN (y) + O(|y|N+1))

= Ay + g2(y) + · · · + gN−1(y) + fN(y) + AP (y) − DyPAy + O(|y|N+1),

using that (Id + DyP )−1 = Id − DyP + O(|y|N ). We conclude that the termsup to order N − 1 are unchanged by this transformation, while the N -th orderterm becomes

fN(y) − adNA(P )(y).

Clearly, a suitable choice of P will put this term in GN . This is the presentanalogue of the homological equation as introduced before.

2

Remark. For simplicity our formulations all are in the C∞-context, but obviouschanges can be made for the case of finite differentiability. The latter case is ofimportance for applications of normal form theory after reduction to a centremanifold.

8

1.4 Torus symmetry

As a special case let the linear part A be semisimple, in the sense that it isdiagonizable over the complex numbers. It then directly follows that also admAis semisimple, which implies that

imadmA ⊕ ker admA = Hm(Rn).

The reader is invited to provide the eigenvalues of admA in terms of those ofA, compare the proof of Theorem 2. In the present case the obvious choice forthe complementary defining ‘simpleness’ is Gm = ker admA. Moreover, the factthat the normalized, viz. simplified, terms gm are in Gm by definition meansthat

[A, gm] = 0.

This, in turn, implies that N -jet of Φ?X, i.e. the normalized part of Φ?X, isinvariant under all linear transformations generated by A :

exp tA, t ∈ R.

Compare the geometric introduction of the previous subsection. For furtherreading also compare Takens [33], Broer [6, 8] or [11].

More generally, let A = As+An be the Jordan canonical splitting in semisim-ple resp. nilpotent part. Then one directly shows that admA = admAs +admAn

is the Jordan canonical splitting, whence, by a general argument it follows that

imadmA + ker admAs = Hm(Rn),

where the sum in general no longer is direct. Now we can choose the comple-mentary spaces Gm such that

Gm ⊂ ker admAs,

ensuring invariance of the normalized part of Φ?X, under all linear transforma-tions

exp tAs, t ∈ R.

The choice of Gm can be further restricted, e.g. such that

Gm ⊆ ker admAs \ imadmAn ⊆ Hm(Rn),

compare Van der Meer [34]. For further discussion on the choice of Gm, seebelow.

Examples Cf. [33]. Consider the case where n = 2 and where

A = As =

(

0 −11 0

)

.

So, the eigenvalues of A are ±i and the transformations exp tAs, t ∈ R with Aas infinitesimal generator, form the rotation group SO(2, R). From this, we canarrive at once at the general shape of the normal form. In fact, the normalized,

9

N -th order part of Φ?X is rotationally symmetric. This impies that, if we passto polar coordinates (r, ϕ),

y1 = r cosϕ, y2 = r sinϕ,

the normalized part of Φ?X obtains the form

ϕ = f(r2)

r = rg(r2),

for certain polynomials f and g, with f(0) = 1 and g(0) = 0. What are thedegrees of f and g? Can all polynomials of this type be obtained as normalforms?

Remarks.

• A more direct, ‘computational’ proof of this result can be given as follows,compare the proof of Theorem 2. Also see e.g. [9, 33, 35]. Indeed,introducing ‘vector field notation’ we can write

A = −x2∂

∂x1+ x1

∂

∂x2=

= i(z∂

∂z− z

∂

∂z),

where we complexified putting z = x1 + ix2 and use the well-knownWirtinger derivatives

∂

∂z=

1

2(

∂

∂x1− i

∂

∂x2)

∂

∂z=

1

2(

∂

∂x1+ i

∂

∂x2).

Now a basis of eigenvectors for admA can be found immediately, com-puting some Lie-brackets, compare the previous subsection. In fact, it isgiven by all monomials

zkzl ∂

∂z, and zkzl ∂

∂z,

with k+l = m. The corresponding eigenvalues are i(k−l−1) viz. i(k−l+1).So again we see, now by a direct inspection, that admA is semisimple andthat we can take Gm = ker admA. This space is spanned by

(zz)r(z∂

∂z± z

∂

∂z),

with 2r + 1 = m − 1, which indeed proves that the normal form is rota-tionally symmetric.

10

• A completely similar case occurs for n = 3, where

A = As =

0 −1 01 0 00 0 0

.

In this case the normalized part in cylindrical coordinates (r, ϕ, z),

y1 = r cosϕ, y2 = r sin ϕ, y3 = z,

in general gets the symmetric form

ϕ = f(r2, z)

r = rg(r2, z)

z = h(r2, z),

for suitable polynomials f, g and h.

As a generalization of this example we state the following proposition, wherethe normalized part exhibits an m-torus symmetry.

Proposition 7 [33] Let X be a C∞ vector field, defined in the neighbourhoodof 0 ∈ Rn, with X(0) = 0 and where A = D0X is semisimple with the eigenval-ues ±iω1,±iω2, · · · ,±iωm and 0. Here 2m ≤ n. Suppose that for given N ∈ N

and all integer vectors (k1, k2, · · · , km),

1 ≤m

∑

j=1

|kj | ≤ N + 1 ⇒m

∑

j=1

kjωj 6= 0.

Then there exists, near 0 ∈ Rn, an analytic change of coordinates Φ : Rn → Rn,with Φ(0) = 0, such that Φ?X, up to terms of order N has the following form.In generalized cylindrical coordinates (ϕ1, · · · , ϕm, r2

1 , · · · , r2m, zn−2m+1, · · · , zn)

it is given by

ϕj = fj(r21 , · · · , r2

m, zn−2m+1, · · · , zn)

rj = rjgj(r21 , · · · , r2

m, zn−2m+1, · · · , zn)

zl = hl(r21 , · · · , r2

m, zn−2m+1, · · · , zn),

where fj(0) = ωj and hl(0) = 0 for 1 ≤ j ≤ m, n − 2m + 1 ≤ l ≤ n.

Proofs can be found e.g. in [9, 33, 35]. In fact, if one introduces a suitablecomplexification, it runs along the same lines as the above remark. For the factthat finitely many non-resonance conditions are needed in order to normalizeup to finite order, also compare a remark following Theorem 2.

Since the system of rj - and zl-equations is independent of the angles ϕj , thiscan be studied separately. A similar remark obviously holds for the earlier ex-amples of this section. This kind of ‘reduction by symmetry’ to lower dimensioncan be of great importance when studying the dynamics of X : it enables us toconsider X, viz. Φ?X, as an N -flat perturbation of the normalized part, whichis largely determined by this lower dimensional reduction.

11

1.5 On the choices of the complementary space Gm and

the normalizing transformation

In the previous subsection we only provided a general (symmetric) shape ofthe normalized part. In concrete examples one has to do more. Indeed, giventhe original Taylor series, one has to compute the coefficients in the normalizedexpansion. This means that many choices have to be made explicit.

To begin with there is the choice of the spaces Gm, which define the notionof ‘simple’. We have seen already that this is not unique. Related to this isthe fact that we have not presented any algorithm producing the vector fieldsP that generate the normalizing transformation.

Moreover observe that, even if the choice of Gm has been fixed, still Pusually is not uniquely determined. In the semisimple case, for example, Pis only determined modulo the kernel Gm = ker admA. The choice becomesunique, however, if we require that e.g. P ∈ Bm, the image of admA.

Remark. To fix thoughts, we consider the first of the above examples, on R2,where the normalized truncation has the rotationally symmetric form

ϕ = f(r2)

r = rg(r2).

Here g(r2) = cr2 + O(|r|4). The coefficient c dynamically is important, justthink of the case where the system is part of a family that might go throughHopf-bifurcation. The computation of (the sign of) c can be quite involved, asit appears from e.g. Marsden & McCracken [21]. Nowadays, however, machine-assisted methods largely have taken over this kind of work.

One general way to choose Gm is the following, e.g. see [9], Section 7.6and [35], Section 2.3:

Gm := ker (admAT).

Here AT denotes the transpose of A, defined by the relation < ATx, y >=<x, Ay >, where < ., . > is an inner product on Rn. A suitable choice for an innerproduct on Hm(Rn) then directly gives that

Gm ⊕ im(admA) = Hm(Rn),

as required. Also here the normal form can be interpreted in terms of symmetry,

namely with respect to the group generated by AT. In the semisimple case, thischoice leads to exactly the same symmetry considerations as before.

For the moment we restrict ourselves to only giving references for furtherreading. Some general references are Bruno [14] Dumortier [9], Section 7.6, orVanderbauwhede [35], Part 2. In the latter reference also a brief description isgiven of the sl(2, R)-theory of Sanders and Cushman [16], which is a powerfultool in the case where the matrix A is nilpotent.

12

2 Preservation of Structure

It goes without saying that normal form theory also is of great interest in specialcases where a given structure has to preserved. To fix thoughts, one may thinkof a symplectic or a volume form that has to be respected. Also a certainsymmetry can have this role, e.g. think of an involution related to reversibility.Another, similar, problem is the presence of external parameters in the system.

A natural language for preservation of such structures is that of Lie-subalgebra’sof general Lie-algebra of vector fields, and the corresponding Lie-subgroup ofthe general Lie-group of diffeomorphisms.

2.1 The Lie-algebra proof

Fortunately the setting of Theorem 6 is almost completely in terms of Lie-brackets. Let us reconsider its proof for a while.

So, given is a C∞ vector field X(x) = Ax + f(x), x ∈ Rn, where A is linearand where f(0) = 0, D0f = 0. We recall that in the inductive procedure atransformation

h = Id + P,

with P a homogeneous polynomial of degree m = 2, 3, · · · , is found, putting thehomogeneous m-th degree part of the Taylor series of X into the ‘good’ spaceGm.

Now, nothing changes in this proof if instead of h = Id + P, we take h = P1,the flow over time 1 of the vector field P : indeed, the effect of this change is notfelt until the order 2m−1. Here we use the following formulae for Xt := (Pt)?X :

[Xt, P ] = admA(P ) + O(|y|m+1) and∂Xt

∂t= [Xt, P ],

compare the geometric introduction to the previous section. In fact, exactly thischoice h = P1 was taken by Roussarie [26], also see the proof of Takens [33].Notice that Φ = hN hN−1 · · · h3 h2. Moreover notice that if the vectorfield P is in a given Lie-algebra of vector fields, its time 1 map P1 is in thecorresponding Lie-group. In particular, if P is Hamiltonian, Pt is canonical orsymplectic, and so on.

For the validity of this set-up for a more general Lie-subalgebra of the Lie-algebra of all C∞ vector fields, one has to study in how far the grading

∞∏

m=1

Hm(Rn)

of the formal power series, as well as the splittings

Bm ⊕ Gm = Hm(Rn),

are compatible with the Lie-algebra at hand. Below we shall explain moreprecisely what this involves. The problem was addressed by Broer [6, 8], interms of graded and filtered Lie-algebra’s. Moreover, the methods concerningthe choice of Gm briefly mentioned at the end of the previous section, all carryover to a more general Lie-algebra set-up. Presently, let us restrict to reviewingsome examples.

13

• The volume preserving and symplectic case. On Rn, resp. R2n, we considera volume form resp. a symplectic form, both of them to be denoted by σ.We assume, that

σ = dx1 ∧ · · · ∧ dxn, resp. σ =

n∑

j=1

dxj ∧ dxj+n.

In both cases, let Xσ denote the Lie-algebra of σ-preserving vector fields,i.e. vector fields X such that LXσ = 0. Here L again denotes the Lie-derivative, compare the geometric terms introduced in section 1. Let ustry to give some impression of how these cases fit in a more general theory.

Indeed, one defines

LXσ(x) =d

dt

∣

∣

∣

∣

t=0

(Xt)∗σ(x) =

= limt→0

1

t(Xt)

∗σ(x) − σ(x).

Properties, similar to Proposition 5, hold here. Since in both cases σ is aclosed form, one shows by ‘The Magic Formula’ [1, 30] that

LXσ(x) = d(ιXσ) + ιXdσ =

= d(ιXσ).

Here ιXσ denotes the flux-operator associated to X and σ : it is the dif-ferential form obtained from σ by substituting X on the first position. Inthe volume-preserving case the latter expression denotes div(X)σ and wesee that preservation of σ exactly means that div(X) = 0 : the divergenceof X vanishes. In the Hamiltonian case we conclude that the 1-form ιXσis closed and hence (locally) of the form dH, for a Hamilton function H. Inboth cases, the fact that for a transformation h the fact that h∗σ = σ im-plies that with X also h∗X is σ-preserving. Moreover, for a σ-preservingvector field P and h = P1 one can show that indeed h∗σ = σ.

One other observation is, that by the homogeneity of the above expressionsfor σ, the homogeneous parts of the Taylor series of σ-preserving vectorfields are again σ-preserving. This exactly means that

Hm(R(2)n) ∩ Xσ,

m = 1, 2, · · · , grades the formal power series corresponding to Xσ. Here,notice that

H1(R(2)n) ∩ Xσ = sl(n, R), resp. sp(2n, R),

the special - resp. the symplectic linear algebra.

14

• External parameters. A C∞ family X = Xλ(x) of vector fields on Rn,with parameter λ ∈ Rp, can be regarded as one C∞ vector field on theproduct space R

n ×Rp. Such a vector field is vertical , in the sense that it

has no components in the λ-direction. In other words, if π : Rn×Rp → Rp

is the natural projection on the second component, X is tangent to thefibres of π. It is easily seen that this property defines a Lie-subalgebra ofthe Lie-algebra of all C∞ vector fields on Rn ×Rp. Again, by the linearityof this projection, the gradings and splittings are compatible to it.

The normal form transformations Φ preserve the parameter λ, i.e. Φ π = Φ. Also, in the N -th order normalization Φ?X, the normalized partconsists of a polynomial in y and λ, while the remainder term is of theform O(|y|N+1 + |λ|N+1).

• The reversible case. In the reversible case a linear involution R is given,while for the vector fields we require R?X = −X. Let XR denote the classof all such reversible vector fields. Also, let C denote the class of all Xsuch that R?X = X. Then, both XR and C are linear spaces of vectorfields. Moreover, C is a Lie-subalgebra. Associated to C is the group ofdiffeomorphisms that commute with R. Also it is easy to see that for eachof these diffeomorphisms Φ one has Φ?(XR) ⊂ XR. The above approachapplies now to this situation in a straightforward manner. The gradingsand splittings fit, while we have to choose the infinitesimal generator Pfrom the set C. Details are left to the reader.

Remark. In the case with parameters, it sometimes is possible to obtain analternative normal form where the normalized part is polynomial in y alone, withcoefficients that depend smoothly on λ. A necessary condition for this to work isthat the origin y = 0 is an equilibrium for all values of λ in some neighbourhoodof 0. To be precise, at the N -th order we van achieve smooth dependence of thecoefficients on λ for λ ∈ ΛN , where ΛN is a neighbourhood of λ = 0, that mayshrink to 0 as N → ∞. So, for N → ∞ only the formal aspect remains, as isthe case in the above approach. This alternative normal form can be obtainedby a proper use of the Implicit Function Theorem in the spaces Hm(Rn); fordetails e.g. see Vanderbauwhede [35], Section 2.2. For another discussion on thistopic, cf. Dumortier [9], section 7.6.2.

In subsection 1.2 the role of symmetry was considered regarding the semisim-ple part of the matrix A. A question is how this discussion generalizes to Lie-subalgebra’s of vector fields.

Example Cf. [6, 8] On R3 let us consider a 1-parameter family Xλ of vector

fields, preserving the standard volume σ = dx1∧dx2∧dx3. Assume that X0(0) =0 while the spectrum of D0X

0 consists of the eigenvalues ±i and 0. For themoment regarding λ as an extra state space coordinate, we obtain a verticalvector field on R4 and we apply a combination of the above considerations. The‘generic’ Jordan normal form then is

A =

0 −1 0 01 0 0 00 0 0 10 0 0 0

,

15

with an obvious canonical splitting in semisimple and nilpotent part. The con-siderations of subsection 1.2 then directly apply to this situation. For any N thisyields a transformation Φ : R

4 → R4, with Φ(0) = 0, preserving both the projec-

tion to the 1-dimensional parameter space and the volume of the 3-dimensionalphase space, such that the normalized, N -th order part of Φ?X(y, λ), in cylin-drical coordinates y1 = r cosϕ, y2 = r sinϕ, y3 = z, has the rotationallysymmetric form

ϕ = f(r2, z, λ)

r = rg(r2, z, λ)

z = h(r2, z, λ),

again, for suitable polynomials f, g and h. Note, that in cylindrical coordinatesthe volume has the form σ = rdr ∧ dϕ ∧ dz. The preservation of this forces acertain relation on the polynomials f, g and h, which one? What can you sayof f(0, 0, 0), h(0, 0, 0), and ∂h/∂λ(0, 0, 0)?

Again the functions f , g and h have to fit with the linear part. In particularwe find that h(r2, z, λ) = λ + az + · · · , observing that for λ 6= 0 the origin is noequilibrium point, compare the above remark.

Remark. If A = As +An is the canonical splitting of A in H1(Rn) = gl(n, R),then automatically both As and An are in the subalgebra under consideration.How? In the volume preserving setting this can be seen directly. In generalthe same holds true as soon as the corresponding linear Lie-group is algebraic,see [6, 8] and the references given there.

2.2 The Hamiltonian case

The normal form theory in the Hamiltonian case goes back at least to Birkhoff [4].Other references are, for instance, Gustavson [18], Arnold [2], Sanders [27],Sanders and Verhulst [28], Van der Meer [34], Broer, Chow and Kim [7]. Asmay be clear from the above, our Lie-algebra approach applies here, especiallysince the symplectic group SP(2n, R) is algebraic. The canonical form hereusually goes with the name Williamson, compare Galin [17] and Kocak [20].

The Lie-algebra of Hamiltonian vector fields can be associated to the Poisson-algebra of Hamilton functions as follows, even in arbitrary symplectic setting. Asbefore, the symplectic form is denoted by σ. We recall, that for any HamiltonianH the corresponding Hamiltonian vector field XH is given by dH = σ(XH , .),Now, let H and K be Hamilton functions with corresponding vector fields XH

resp. XK . Then

XH,K = [XH , XK ],

implying that the map H 7→ XH is a morphism of Lie-algebra’s. By definitionthis map is surjective, what is its kernel?

This implies, that the normal form procedure can be completely (re)phrasedin terms of the Poisson-bracket. We shall now demonstrate this by an example,similar to the previous one.

16

Example Cf. [7] Consider R4 with coordinates (x1, y1, x2, y2) and the stan-dard symplectic form σ = dx1 ∧ dy1 + dx2 ∧ dy2, considering a C∞ familyof Hamiltonian functions Hλ, where λ ∈ R is a parameter. The fact thatdH = σ(XH , .) in coordinates means

xj =∂H

∂yj, yj = − ∂H

∂xj,

for j = 1, 2.We assume that for λ = 0 the origin of R4 is a singularity. Then we expand

as a Taylor series in (x, y, λ)

Hλ(x, y) = H2(x, y, λ) + H3(x, y, λ) + . . . ,

where the Hm is homogeneous of degree m in (x, y, λ). It follows for the corre-sponding Hamiltonian vector fields that

XHm∈ Hm−1(R4),

in particular XH2∈ sp(4, R) ⊂ H1(R4). Let us assume that this linear part

XH2at (x, y, λ) = (0, 0, 0) has eigenvalues ±i and a double eigenvalue 0. One

‘generic’ Willamson’s normal form then is

A =

0 1 0 0−1 0 0 00 0 0 10 0 0 0

,

corresponding to the quadratic Hamilton function

H2(x, y, λ) = I +1

2y2

2,

where I := 12 (x1

2 + y12). It is straightforward to give the generic matrix for the

linear part in the extended state space R5, see above, so we will leave this tothe reader.

The semisimple part As = XI now can be used to obtain a rotationally sym-metric normal form, as before. In fact, for any N ∈ N there exists a canonicaltransformation Φ(x, y, λ), which keeps the parameter fixed, and a polynomialF (I, x2, y2, λ), such that

(H Φ−1)(x, y, λ) = F (I, x2, y2, λ) + O(I + x22 + y2

2 + λ2)(N+1)/2.

Instead of using the adjoint action adA on the spaces Hm−1(R5), we also mayuse the adjoint action

adH2 : f 7→ H2, f,where f is a polynomial function of degree m in (x1, y1, x2, y2, λ). In the vectorfield language, we choose the ‘good’ space Gm−1 ⊂ ker adm−1As, which, in thefunction language, translates to a ‘good’ subset of ker adI.

Whatsoever, the normalized part of the Hamilton function H Φ−1, viz.the vector field Φ?XH = XHΦ−1 , is rotationally symmetric. The fact that the

17

Hamilton function F Poisson-commutes with I exactly amounts to invarianceunder the action generated by the vector field XI , in turn implying that I is anintegral of F. Indeed, if we define a 2π-periodic variable ϕ as follows:

x1 =√

2I sin ϕ, y1 =√

2I cosϕ,

then σ = dI ∧ dϕ + dx2 ∧ dy2, implying that the normalized vector field XF hasthe canonical form

I = 0, ϕ = −∂F

∂I,

x2 =∂F

∂y2, y2 = − ∂F

∂x2.

The pair (I, ϕ) often is referred to as Hamiltonian polar coordinates. Noticethat ϕ is a cyclic variable, making the fact that I is an integral clearly visible.Also observe that ϕ = −1 + · · · . As before, and as in e.g. the central forceproblem, this enables a reduction to lower dimension. Here, the latter twoequations constitute the reduction to 1 degree of freedom: it is a family of planarHamiltonian vector fields, parametrized by I and λ.

Remarks.

• This example has many variations. First of all it can be simplified byomitting parameters and even the zero eigenvalues. The conclusion thenis that a planar Hamilton function with a nondegenerate minimum ormaximum has a formal rotational symmetry, up to canonical coordinatechanges. What happens in the case where the eigenvalues are ±iω forsome ω > 0?

• It also can be made more complicated, compare section 1, Proposition 7.Let us consider the case without parameters, with purely imaginary eigen-values ±iω1,±iω2, . . . ,±iωm and with 2(n−m) zero eigenvalues. Providedthat again a non-resonance condition

1 ≤m

∑

j=1

|kj | ≤ N + 1 ⇒m

∑

j=1

kjωj 6= 0,

is imposed, for integer vectors (k1, k2, · · · , km), we conclude that the N -thorder Hamiltonian by canonical transformations can be given the form

F (I1, · · · , Im, xm+1, ym+1 · · · , xn, yn),

where Ij := 12 (xj

2 + yj2). Writing xj =

√

2Ij sin ϕj , yj =√

2Ij cosϕj ,see above, we obtain the canonical system of equations

Ij = 0, ϕj = − ∂F

∂Ij,

xl =∂F

∂yl, yl = − ∂F

∂xl,

18

1 ≤ j ≤ m, m + 1 ≤ l ≤ n. Notice that now ϕj = −ωj + · · · . The casewhere m = n usually is referred to as the Birkhoff normal form. Then,the variables (Ij , ϕj), 1 ≤ j ≤ m, are a set of action-angle variables forthe normalized part of order N.

• It is to be noted that this kind of Hamiltonian result also can be obtained,where the coordinate changes are constructed using generating functions.For examples e.g. see Siegel and Moser [29].

3 Miscellaneous remarks

This section roughly consists of two parts. To begin with, a number of sub-sections is devoted to related formal normal form results, near fixed points ofdiffeomorphisms and near periodic solutions and invariant tori of vector fields.Finally, a few remarks are made regarding non-formal aspects of the presentapproach.

3.1 A diffeomorphism near a fixed point

First we formulate a result by Takens:

Theorem 8 [32] Let T : Rn → Rn be a C∞ diffeomorphism with T (0) = 0and with a canonical decomposition of the derivative D0T = S+N in semisimpleresp. nilpotent part. Also, let N ∈ N be given, then there exists diffeomorphismΦ and a vector field X, both of class C∞, such that S?X = X and

Φ−1 T Φ = S X1 + O(|y|N+1).

Here, as before, X1 denotes the flow over time 1 of the vector field X. Observethat the vector field X necessarily has the origin as an equilibrium point. More-over, since S?X = X, the vector field X is invariant under the group generatedby S.

The proof is somewhat more involved than section 1, Theorem 6, but it hasthe same spirit. In fact, the Taylor series of T is modified step by step, usingcoordinate changes generated by homogeneous vector fields of the proper degree.

After a reduction to centre manifolds the spectrum of S is on the complexunit circle and Theorem 8 especially is of interest in cases where this spectrumconsists of roots of unity, i.e. in the case of resonance. Compare [32].

Again, the result is completely phrased in terms of Lie- algebra’s and groupsand therefore bears generalization to many contexts with a preserved structure,compare section 2. The normalizing transformations of the induction processthen are generated from the corresponding Lie-algebra. For a symplectic ana-logue see Moser [22]. Also, both in Broer, Chow and Kim [7] and in Broer andVegter [13], symplectic cases with parameters are discussed, where S = Id, theIdentity Map, resp. S = −Id, involving a period-doubling bifurcation.

Remark. Let us consider a symplectic map T of the plane. This means that Tpreserves both the area and the orientation. Assume that T is fixing the origin,while the eigenvalues of S = D0T are on the unit circle, without being rootsof unity. Then S generates the rotation group SO(2, R), so the vector field X,

19

which has divergence zero, in this case is rotationally symmetric. Again thename of Birkhoff often is associated to this result.

3.2 Near a periodic solution

The normal form theory near a periodic solution or closed orbit has a lot ofresemblance to the the local theory we met before.

To fix thoughts, let us consider a C∞ vector field of the form

x = f(x, y)

y = g(x, y),

with (x, y) ∈ S1 × Rn. Here S1 = R/(2πZ). Assuming y = 0 to be a closedorbit, we consider the formal Taylor series with respect to y, with x-periodiccoefficients. By the Floquet theory, we can assume that the coordinates (x, y)are such that

f(x, y) = ω + O(|y|), g(x, y) = Ωy + O(|y|2),where ω ∈ R is the frequency of the closed orbit and Ω ∈ gl(n, R) its Floquetmatrix. Again, the idea is to ‘simplify’ this series further. To this purpose weintroduce a grading as before, letting Hm = Hm(S1×Rn) be the space of vectorfields

Y (x, y) = L(x, y)∂/∂x +

n∑

j=1

Mj(x, y)∂/∂yj,

with L(x, y) and M(x, y) homogeneous in y of degree m−1 resp. m. Notice, thatthis space Hm is infinite-dimensional. However, this is not at all problematicfor the things we are doing here. By this definition, we have that

A := ω∂/∂x + Ωy∂/∂y

is a member of H1 and with this ‘linear’ part we can define an adjoint repre-sentation adA as before, together with linear maps

admA : Hm → Hm.

Again we assume to have a decomposition

Gm ⊕ Im(admA) = Hm,

where the aim is to transform the terms of the series successively into the Gm,for m = 2, 3, 4, · · · .

The story now runs as before. In fact, the proof of section 1, Theorem 6, aswell as its Lie-algebra version of section 2, can be repeated almost verbatim forthis case. Moreover, if Ω = Ωs +Ωn is the canonical splitting in semisimple andnilpotent part, then

ω∂/∂x + Ωsy∂/∂y

20

gives the semisimple part of admA, as can be checked by a direct computation.From this computation one also deduces the non-resonance conditions neededfor the present analogue of section 1, Proposition 7.

Remarks.

• As general references we can mention Bruno [14], Chow and Wang [15],Iooss [19], Sanders and Verhulst [28].

• This approach also is important for non-autonomous systems with periodictime dependence. Then one obtains, as a special case of the above form,

x = ω

y = g(x, y),

so where x ∈ S1 is proportional to the time. For applications to time-

dependent systems on R2, we refer to e.g. Arnold [3] or Broer and Veg-ter [13] and to Broer, Roussarie and Simo [10]. All these applications alsocontain parameters, while the former two, moreover, are Hamiltonian. Itappears that in these cases, as a corollary, a normal form is obtained forthe Poincare- or period map of the system, completely as in the previoussubsection. This is (planar) vector field, the time 1 flow of which gives anormalizing approximation of the given Poincare-map.

• Finally we record that there is a narrow relation between the presentnormalization procedure and averaging. See almost all the references tothis subsection.

3.3 Near a quasi-periodic torus

The approach of the previous subsection also applies near an invariant torus,provided that certain requirements are met. Here we refer to Braaksma, Broerand Huitema [5], Broer and Takens [11] and Bruno [14].

Let us consider a C∞ system

x = f(x, y)

y = g(x, y),

as before, with (x, y) ∈ Tm × Rn. Here Tm = Rm/(2πZ)m. We assume thatf(x, y) = ω + O(|y|), which implies that y = 0 is an invariant m-torus, withon it a constant vector field with frequency-vector ω. We also assume thatg(x, y) = Ωy +O(|y|2), amounting to a kind of Floquet form, with Ω ∈ gl(n, R),independent of x. It has to be noted that unlike for the cases m = 1 and n = 1,in general this ‘Floquet hypothesis’ can not always be satisfied by a suitablechoice of coordinates. Compare [5] and some references therein.

Anyway, granted this hypothesis, we expand in Taylor series as before, nowwith coefficients that are functions on Tm. These coefficients, in turn, are ex-panded in Fourier series. The aim then is, to ‘simplify’ this combined formalseries by successive coordinate changes. To be precise, we wish to normalize this

21

series with respect to y, following the above procedure. As a second require-ment it then is needed that certain diophantine conditions are satisfied on thepair (ω, Ω). Instead of giving general results we again refer to [5, 11, 14] and,moreover, present a simple example with a parameter. Here again a direct linkwith averaging is obtained.

Example Given a family of vector fields X = X(x, λ), with (x, λ) ∈ Tm ×R.We assume that X has the form

x = ω + f(x, λ),

with f(x, 0) ≡ 0. The claim is, that by successive transformations

h : (x, λ) 7→ (x + P (x, λ), λ)

the x-dependence of X can be pushed away to higher and higher order in λ,so ‘simple’ here means x-independent. Therefore, in a proper formalism, thesesystems constitute the spaces Gm.

Indeed, in the induction process we get

X(x) = ω + g2(λ) + · · · + gN−1(λ) + fN (x, λ) + O(|λ|N+1),

compare the proof of section 1, Theorem 6. Writing ξ = x + P (x, λ), withP (x, λ) = P (x, λ)λN , we substitute

ξ = (Id + DξP )x

= ω + g2(λ) + · · · + gN−1(λ)

+ fN (x, λ) + ∂P/∂x(x, λ)ω + O(|λ|N+1),

Where we express everything in x. So we have to satisfy an equation

DxP (x, λ)ω + fN (x, λ) ≡ cλNmodO(|λ|N+1),

for a suitable constant c. Writing fN (x, λ) = fN(x, λ)λN , this amounts to

DxP (x, λ)ω = −fN(x, λ) + c,

the present form of the homological equation. If

fN (x, λ) =∑

k∈Zn

ak(λ)ei<x,k>,

then c = a0, i.e. the m-torus average of fN (x, λ). Moreover,

P (x, λ) =∑

k 6=0

ak(λ)

i < ω, k >ei<x,k>.

This procedure formally only makes sense if k 6= 0 ⇒< ω, k > 6= 0, i.e. if thecomponents of the frequency vecor ω are rationally independent. Even then,the denominator i < ω, k > can become arbitrarily small, so casting doubt onthe convergence. This problem of small divisors is resolved by requiring thediophantine conditions

22

| < ω, k > | ≥ γ|k|−τ ,

for all k 6= 0. Here γ > 0 and τ > n − 1 are prescribed constants. For furtherreference, e.g. see Arnold [3] or Chapter 4 of [9]: for real analytic X, by the Paley-Wiener estimate on the exponential decay of the Fourier coefficients, the solutionP again is real analytic. Also in the C∞-case the situation is quite simple, sincethen the coefficients in both cases decay faster than any polynomial.

3.4 Non-formal aspects

Up to this moment (almost) all considerations have been formal, i.e. in terms ofTaylor series. In general, the Taylor series of a C∞-function, say, Rn → R willbe divergent. On the other hand, any formal power series in n variables occursas the Taylor series of some C∞-function. This is the content of a theorem byE. Borel, cf. Narasimhan [23].

As we have seen, if the normalization procedure is carried out to some finiteorder N, the transformation Φ is a real analytic map. If we take the limit forN → ∞, we only get formal power series Φ, but, by the Borel theorem, a ‘real’C∞-map Φ exists with Φ as its Taylor series.

Let us discuss the consequences of this, say, in the case of section 1, Propo-sition 7. Assuming that there are no resonances at all between the ωj , as acorollary, we find a C∞-map y = Φ(x) and a C∞ vector field p = p(y), suchthat:

• The vector field Φ?X − p, in corresponding generalized cylindrical coordi-nates has the symmetric C∞-form

ϕj = fj(r21 , · · · , r2

m, zn−2m+1, · · · , zn)

rj = rjgj(r21 , · · · , r2

m, zn−2m+1, · · · , zn)

zl = hl(r21 , · · · , r2

m, zn−2m+1, · · · , zn),

where fj(0) = ωj and hl(0) = 0 for 1 ≤ j ≤ m, n − 2m + 1 ≤ l ≤ n.

• The Taylor series of p identically vanishes at y = 0.

Note, that an ∞-ly flat term p has component functions like e−1/y2

1 . We see,that the m-torus symmetry only holds up to such flat terms. Therefore, thissymmetry, if present at all, also can be destroyed again by a generic flat ‘per-turbation’. We refer to Broer and Takens [11], and some quotations from itsbibliography, for further consequences of this idea. The point is, to use a kindof Kupka-Smale argument, which generically prohibits so much symmetry.

Remarks.

• This ‘topological’ idea can also be pursued in some real analytic cases,where it implies generic divergence of the normalizing transformation.Compare Broer and Tangerman [12] and some of its references.

23

• The Borel theorem also can be used in cases like the Hamiltonian or volumepreserving one. Here, we employ the fact that a structure preservingvector field is generated by a function, resp. an (n − 2)-form. Similarlythe structure preserving transformations have some generator. On thesegenerators we then apply the Borel theorem. Many Lie-algebra’s of vectorfields have this ‘Borel property’, saying that a formal power series of atransformation can be represented by a C∞ map in the same structurepreserving setting.

References

[1] R. Abraham and J.E. Marsden. Foundations of Mechanics. Benjamin,1978.

[2] V.I. Arnold. Mathematical Methods of Classical Mechanics. Springer-Verlag, 1980.

[3] V.I Arnold. Geometrical Methods in the Theory of Ordinary DifferentialEquations. Springer-Verlag, 1983.

[4] G.D. Birkhoff. Dynamical Systems. AMS Publications, 1927.

[5] B.L.J. Braaksma, H.W. Broer, and G.B. Huitema. Toward a quasi-periodicbifuration theory. Memoirs AMS 421, 1990.

[6] H. W. Broer. Formal normal form theorems for vector fields and some con-sequences for bifurcations in the volume preserving case. Springer-VerlagLNM 898, 1981.

[7] H. W. Broer, S.N. Chow, and Y. Kim. A normally elliptic hamiltonianbifurcation. Submitted for Publication, 1992.

[8] H.W. Broer. Unfoldings of Singularities in Volume Preserving Vector Fields(thesis). University of Groningen, 1979.

[9] H.W. Broer, F. Dumortier, S.J. van Strien, and F. Takens. Structures inDynamics, Finite Dimensional Deterministic Studies. Studies in Mathe-matical Physics, North Holland, 1991.

[10] H.W. Broer, R. Roussarie, and C. Simo. Invariant circles in the Bogdanov-Takens bifurcation for diffeomorphisms. Ergod. Th. & Dynam. Sys. 16,1996.

[11] H.W. Broer and F. Takens. Formally symmetric normal forms and gener-icity. Dynamics Reported 2, 1989.

[12] H.W. Broer and F. Tangerman. From a differentiable to a real analyticperturbation theory, applications to the Kupka Smale theorems. Ergod.Th. & Dynam. Sys. 6, 1986.

[13] H.W. Broer and G. Vegter. Bifurcational aspects of parametric resonance.Dynamics Reported, New Series 1, 1992.

24

[14] A.D. Bruno. Local Methods in Nonlinear Differential Equations. SpringerSeries in Soviet Mathematics, Springer-Verlag, 1989.

[15] S.N. Chow and D. Wang. Normal forms of codimension 2 bifurcating orbits.AMS, 1985.

[16] R. Cushman and J.A. Sanders. Nilpotent normal forms and representationtheory of sl(2, R). Contemp. Math., 56, 1986.

[17] D.M. Galin. Versal deformations of linear hamiltonian systems. Amer. Soc.Trans. 118, 1982.

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