1. Diss. ETH No. 12744 June 1998
Invariant Manifolds, Passage through Resonance, Stability
and a
Computer Assisted Application to a Synchronous Motor
a Dissertation
submitted to the
Swiss Federal Institute of Technology Zurich
for the degree of
Doctor of Mathematics
presented by
Diego Giuseppe Tognola
Dipl. Math., University of Zurich
born July 3, 1967
citizen of Windisch AG, Switzerland
accepted on the recommendation of
Prof. Dr. U. Kirchgraber, referee
Prof. Dr. E. Zehnder, co–referee
PD Dr. D. Stoffer, co–referee

3. dedicated to all my friends
and
everyone supporting me during this work

5. Contents
Introduction v
1 Reduction to a Planar System 1
2 Averaging and Passage through Resonance in Plane Systems 69
3 The Stability of the Set {h = 0} in Action Angle Coordinates 111
4 Application to a Miniature Synchronous Motor 125
5 Application to Van der Pol’s Equation 229
i

6. Kurzfassung
Ziel dieser Arbeit ist die Untersuchung eines Systems gew¨ohnlicher Differentialgleichungen, welches einen
Miniatur–Synchronmotor modelliert. Dieses System ist ein Spezialfall eines allgemeineren Problems,
welches eigenst¨andig von mathematischem Interesse ist. Aus diesem Grunde wird der erste Teil dieser
Arbeit in einem abstrakter Rahmen durchgef¨uhrt. Der zweite Teil zeigt darauf die Anwendung auf das
physikalische Problem.
Das erste Kapitel behandelt ein hamiltonsches und ein exponentiell stabiles lineares System, welche
durch schwache periodische St¨orungen gekoppelt sind. Das hamiltonsche System mit einem Freiheitsgrad
besitze einen elliptischen Fixpunkt im Ursprung. Im ungest¨orten Fall besitze der Ursprung eine attraktive
Zentrumsmannigfaltigkeit sowie eine stabile Mannigfaltigkeit. Unter Verwendung der Theorie invarianter
Mannigfaltigkeiten weisen wir nach, dass diese Struktur im wesentlichen bestehen bleibt. Die Diskussion
auf die zeitabh¨angige, attraktive invariante Mannigfaltigkeit einschr¨ankend, schliesst das erste Kapitel
mit zwei verschiedenen Darstellungen des resultierenden reduzierten Systems.
Das zweite Kapitel zielt auf eine globale Diskussion des reduzierten Systems ab. Mittelungsmethoden
werden angewendet, um das Problem zu vereinfachen. Wir setzen voraus, dass nur endlich viele Resonan-
zen existieren und teilen den Phasenraum in Nichtresonanz- und Resonanzzonen (sog. ¨aussere und innere
Zonen). Die Nichtresonanzzonen bestehen aus dem ganzen Phasenraum, ausser kleinen Umgebungen
der Resonanzen. In den Resonanzzonen, welche kleine Umgebungen der Resonanzen abdecken, werden
Kriterien f¨ur strikte und fast–strikte Resonanzdurchg¨ange hergeleitet. Fast–strikter Resonanzdurchgang
bedeutet Durchgang aller L¨osungen, mit der m¨oglichen Ausnahme einer asymptotisch kleinen Menge von
L¨osungen, welche in die Resonanz eingefangen werden k¨onnen. Die angewendeten Mittelungsmethoden
in der Nichtresonanzzone sind un¨ublich und erlauben es, die zwei Zonen in einer Weise zu w¨ahlen, sodass
sie ¨uberlappen.
Kapitel drei behandelt die zweite Darstellung des reduzierten Systems und ist f¨ur die Stabilit¨atsdis-
kussion passend. Mit Hilfe der Theorie von Floquet gewinnen wir eine Darstellung, welche es erlaubt,
(In)Stabilit¨at umgehend zu diskutieren. Die Abhandlung deckt auch den degenerierten Fall, in welchem
die (In)Stabilit¨at nicht durch lineare Terme verursacht wird.
Der zweite Teil der Arbeit zeigt die Anwendung der im ersten Teil hergeleiteten Methoden und Resultate.
F¨ur das Problem des Synchronmotors werden explizite N¨aherungen der relevanten Gr¨ossen analytisch
hergeleitet und numerisch ausgewertet. Die theoretischen Schl¨usse auf die Dynamik des Motors werden
durch numerische Simulationen best¨atigt. Es wird gezeigt, dass sich der Motor dem Zustand der stabilen,
synchronen Drehung, moduliert durch die zweite Harmonische, n¨ahert, wenn er gestartet wird. Weitere
Effekte wie der Einfluss mechanischer Reibung and zus¨atzlichem Drehmoment werden diskutiert. Strikter
und fast–strikter Resonanzdurchlauf wird f¨ur gewisse Parameter nachgewiesen.
ii

8. Abstract
The aim of this paper is to study a system of ordinary differential equations, modelling a miniature
synchronous motor. This system is a special case of a more general problem which is of mathematical
interest in itself. Hence an abstract framework is introduced in the first part of this work. The second
part then presents the application to the physical problem.
Chapter one treats a Hamiltonian and an exponentially stable linear system, the two being coupled by
weak periodic perturbations. The Hamiltonian system is of one degree of freedom and admits an elliptic
fixed point at the origin. In the unperturbed case the origin admits an attractive center manifold as well
as a stable manifold. Using invariant manifold theory we establish that this structure essentially persists.
Restricting the discussion to the time–dependant attractive invariant manifold, the first chapter closes
with two different representations of the resulting reduced system.
Chapter two aims at a global discussion of this reduced system. Averaging techniques are applied to
simplify the problem. We assume that there exist finitely many resonances only and split the phase space
into non–resonance and resonance zones (so–called outer and inner zones). The non–resonance zone
consists of the entire phase space except small neighbourhoods of the resonances. In the resonance zone,
which cover small neighbourhoods of the resonances, criteria for strict and almost strict passage through
resonances are deduced. Almost strict passage means passage of all solutions with the possible exception
of an asymptotically small set of solutions which may be captured into the resonance. The averaging
method applied in the non–resonance zone is non–standard and permits to choose the two regions in such
a way that they overlap.
Chapter three deals with the second representation of the reduced system and is suitable for the stability
discussion. Using Floquet’s theory we gain a representation which permits to discuss (in)stability at once.
The treatise covers the degenerate case where (in)stability is not caused by linear terms, as well.
The second part of the paper presents the application of the methods and results derived in part one.
For the problem of the miniature synchronous motor, explicit approximations of the relevant quantities
are deduced analytically and evaluated numerically. The theoretical conclusions on the dynamics of
the motor are confirmed by numerical simulation. The motor is shown to approach the stable state of
synchronous rotation with a small modulation by a second harmonic, when started. Additional effects
such as the influence of mechanical friction and an additional torque are discussed. Strict and almost
strict passage through resonance is established for certain parameters.
iv

9. Introduction
The aim of this work is to study a particular type of miniature synchronous motor. Conventional syn-
chronous motors are characterized by the property that under working conditions the rotor exhibits a
stable rotation, the frequency being that of the power supply (hence the term ”synchronous”). In or-
der to enter the working conditions after switching on the motor, different techniques are suggested in
electrical engineering. Some of these techniques (such as pony motors, induction cages or electronic con-
trols) are rather complicated. Hence in many papers the transient behaviour upon start and the state
of synchronous rotation are treated separately. By contrast, the type of motor considered here features
a simple mechanism which permits a satisfactory physical modelling covering the entire process. This
model has been used by the manufacturer [12] for numerical studies and was presented in a colloquium
talk in the nineteen 80’s. It is represented by the following non–linear time–periodic system of ordinary
differential equations
d2
dτ2
ϑ = −
λ
J
i2
1 + i2
2 sin(ϕ) − ˜̺
d
dτ
ϑ − ˜m
U0 sin(ωτ) = R i1 + L
d
dτ
i1 + λ
d
dτ
sin(ϑ)
U0 sin(ωτ) = R i2 + L
d
dτ
i2 + λ
d
dτ
cos(ϑ) + u
d
dτ
u = i2/C.
(1)
The quantity ϑ is the angle of the rotor with respect to a fixed axis, i1 i2 correspond to the currents in
two parallel circuits and u describes the voltage of a condenser attached to the second power circuit.
Our approach for a mathematical treatise is based on perturbation theory. After some preliminary
transformations and assumptions on the parameters, the system turns out to be a special case of the
following problem
( ˙q, ˙p) = J∇H(q, p) + F(q, p, η, t, ε)
˙η = A η + G(q, p, t, ε).
(1.1)
This generalized problem consists of a one–degree of freedom Hamiltonian system (corresponding to the
mathematical pendulum in the above set of equations) and a linear system, the two being coupled by
small periodic perturbations. As this system is of interest by itself, we introduce a general framework
which might be of use elsewhere, too. The hypotheses we make reflect some of the features of the original
physical problem, however. As to the Hamiltonian system these assumptions include the existence of an
elliptic fixed point at the origin satisfying a non–resonance condition, as well as the existence of domains
foliated by periodic solutions (such as the oscillatory and rotatory solutions of the pendulum equation).
The matrix A is assumed to be exponentially asymptotically stable and the Fourier series with respect
to time t of the 2 π–periodic perturbations F and G are assumed to be finite.
v

10. vi Introduction
The original physical problem suggests two main questions. The discussion of the state of synchronous
rotation which is related to the existence and stability of a periodic solution near (q, p, η) = 0 and hence
is local in nature. On the other hand, solutions describing the transition from start to stationary rotation
are of upmost interest. They require a more global treatment. In a first part of this work a number of
key results for systems of type (1.1) are derived which prove to be a tool kit for concrete applications. In
a second part these results are applied to the miniature synchronous motor.
The first part is split into three self–contained chapters. In chapter 1 it is shown that the fixed point
(q, p, η) = 0 of the unperturbed system generates a unique 2 π–periodic solution. A discussion of its
stability is postponed until chapter 3. A time–dependant shift of the coordinates first yields a problem
which is again of type (1.1), i.e.
( ˙ˇQ, ˙ˇP) = J∇H( ˇQ, ˇP) + ˇF( ˇQ, ˇP, H, t, ε)
˙H = A H + ˇG( ˇQ, ˇP, t, ε),
(1.16)
but satisfies ˇF(0, 0, 0, t, ε) = 0 and ˇG(0, 0, t, ε) = 0. For ε = 0, the ( ˇQ, ˇP)–plane H = 0 corresponds to
the center manifold of the origin, whereas the H–axis ( ˇQ, ˇP) = (0, 0) represents the stable manifold. For
ε = 0 sufficiently small we establish the existence of an integral manifold ( ˇQ, ˇP) = V(t, H, ε), the so–called
strongly stable manifold. This is achieved by adapting a result of Kelley [8] to our situation. Applying
the transformation ( ˇQ, ˇP) = (Q, P) + V(t, H, ε) then yields a system of the form
( ˙Q, ˙P) = J∇H(Q, P) + ˆF(Q, P, H, t, ε)
˙H = A H + ˆG(Q, P, H, t, ε),
(1.87)
where in particular ˆF vanishes on the new H–axis, i.e.
ˆF(0, 0, H, t, ε) = 0 ˆG(0, 0, 0, t, ε) = 0. (1.88)
In a next step we replace (Q, P) by action–angle coordinates (ϕ, h) ∈ R2
. The transformed system is
equivalent to (1.87) if we restrict (Q, P) to regions of periodic solutions of ( ˙Q, ˙P) = J∇H(Q, P). In view
of (1.88) such a region may be a neighbourhood of the fixed point (Q, P, H) = (0, 0, 0) as well. In this
case, the set (h, H) = (0, 0) corresponds to (Q, P, H) = (0, 0, 0) and is invariant. The stability discussion
of (h, H) = (0, 0) therefore yields information on the stability of (Q, P, H) = (0, 0, 0) which eventually
corresponds to synchronous rotation in the case of our model of a synchronous motor. In action–angle
coordinates the system is of the form
˙ϕ = ω(h) + f(t, ϕ, h, H, ε)
˙h = g(t, ϕ, h, H, ε)
˙H = A H + h(t, ϕ, h, H, ε)
(1.110)
where A still denotes the matrix introduced in (1.1) and f, g, h vanish for ε = 0. The unperturbed
problem corresponding to (1.110) suggests the existence of an attractive invariant manifold near H = 0.
The majority of results on the existence of such manifolds (see e.g. Fenichel [4], Hirsch, Pugh, Shub
[6]) are based on a discussion of Lyapunov type numbers of solutions. For the purpose of this work
an approach based on more easily accessible quantities is more convenient, see Kirchgraber [9]. In this
work we apply an adaption by Nipp/Stoffer [13] where the assumptions are expressed in terms of the
vector field. It is here where the introduction of action–angle coordinates turns out to be advantageous.
The attractive invariant manifold we establish admits the representation H = S(t, ϕ, h, ε) with S = 0
for ε = 0. Since all solutions of (1.110) approach the invariant manifold, the discussion then reduces to
the reduced system, i.e. the restriction of eq. (1.110) to the attractive invariant manifold. This reduced
system is two–dimensional but non-autonomous. It is represented in two different forms, either of which
will be used in chapter 2 and chapter 3, respectively.

11. Introduction vii
The first representation of the reduced system given in chapter 1 is used for the global discussion. Taking
into account some additional properties of the original physical problem, chapter 2 deals with a system
of the form
˙ϕ = ω(h) +
3
j=2
εj
k,n∈Z
fj
k,n(h) ei(kϕ+nt)
+ ε4
f4
(t, ϕ, h, ε)
˙h =
3
j=2
εj
k,n∈Z
gj
k,n(h) ei(kϕ+nt)
+ ε4
g4
(t, ϕ, h, ε)
(2.1)
defined for ϕ, h ∈ R. Given km, nm ∈ Z and hm ∈ R such that gj
km,nm
= 0 (j ∈ {2, 3}) and km ω(hm) +
nm = 0 the value hm is called a resonance. We assume that the set of resonances hm is finite. Moreover,
for every resonance hm we require that d
dh ω(hm) = 0 holds. In order to obtain information on the
qualitative behaviour of (2.1) averaging techniques are applied. More precisely we apply time–dependant
near–identity transformations of the form ¯h = h + O(ε2
). This change of coordinates is defined in a
standard way, see Kirchgraber [11] or Sanders/Verhulst [17]. We use it in a somewhat different way,
however. As the transformation is singular in every resonance, it is applied outside a neighbourhood of
the resonances. In order to keep the higher order terms small, the size of the neighbourhood of each
resonance must be chosen appropriately. We show that the neighbourhoods omitted may be chosen to be
O(ε)–small. More precisely, for fixed δ > 0 and choosing |ε| < εO
(δ) the transformation may be applied
outside |ε|
δ –neighbourhoods of the resonances. In this outer region the transformed system then takes the
form
˙ϕ = ω(¯h) + O(ε)
˙¯h = ε2
g2
0,0(¯h) + ε2
δ2
¯g2
(t, ϕ, ¯h, ε, δ) + O(ε3
).
(2.23)
where ¯g2
is still bounded. If on a subset of the outer region the map g2
0,0 is bounded from below, the
parameters δ and |ε| < εO
(δ) may be chosen such that ˙¯h > 0 and thus all solutions leave this subset.
Away from zeroes of g2
0,0 the qualitative behaviour is therefore determined simply by the sign of g2
0,0.
In the inner region, i.e. if h satisfies |h − hm| < 4 |ε|
δ , a different near–identity change of coordinates is
defined. The resulting system then reads as follows
˙ϕ = ω(¯h) + O(ε2
)
˙¯h = ε2
g2
0,0(¯h) + ε2
l∈N∗
g2
lkm,lnm
(¯h) eil(kmϕ+nmt)
+ O(ε3
). (2.25)
Introducing the inner variables
ε ˜h := const ¯h − hm ∀ ¯h − hm < 4
|ε|
δ
ψ := km ϕ + nm t,
(2.28)
and taking into account again some special features which arise in the application of the synchronous
motor, the system takes the form of a km 2 π–periodically perturbed pendulum with external torque,
i.e. it is given by
˙ψ = ε ˜h + ε2 ˜f2
(t, ψ, ˜h, ε)
˙˜h = ε (a0 + ac
1 cos(ψ) + as
1 sin(ψ)) + ε2
˜g2
(t, ψ, ˜h, ε).
(2.29)
The quantities a0, ac
1 and as
1 are determined by the Fourier coefficients g2
0,0 and g2
km,nm
evaluated at
h = hm.

12. viii Introduction
We then treat the following two situations:
1. |a0| > (ac
1)
2
+ (as
1)
2
: For all solutions of the unperturbed system (i.e. (2.29) with the O(ε2
)–
terms dropped) the quantity
˙˜h is bounded from below. For ε sufficiently small, we conclude that
all solutions of (2.29) leave the region ¯h − hm < 4 |ε|
δ . This behaviour is refered to as passage
through resonance.
2. |a0| < (ac
1)2
+ (as
1)2
: The unperturbed system admits a hyperbolic and an elliptic fixed point on
the axis ˜h = 0, generating periodic solutions for (2.29). It then is possible that solutions starting
near the boundary ¯h − hm = 4 |ε|
δ are caught near ˜h = 0 as t → ∞. This effect is called capture
into resonance. Here it is shown, however, that the set of such solutions has size O(ε).
By consequence, the global qualitative behaviour of most solutions is known, once the values of g2
0,0 and
g2
km,nm
at h = hm are known. In chapter 4 the computation of these quantities will be the main point of
interest.
In chapter 3 we consider a system of the form
˙ϕ = Ω0 + f,0
(t, ϕ, ε) + P(h) f,1
(t, ϕ, ε) + P(h)
2
f,2
(t, ϕ, P(h), ε)
˙h = P(h)
d
dh P(h)
g,1
(t, ϕ, ε) + P(h)2
d
dh P(h)
g,2
(t, ϕ, ε) + P(h)3
d
dh P(h)
g,3
(t, ϕ, P(h), ε),
(3.1)
according to the second representation of the reduced system introduced in chapter 1. The use of an
analytical cutting function P in (3.1) is reminiscent of the way in which action–angle coordinates were
introduced. One may set P(h) = h in a neighbourhood of h = 0. We assume Ω0 ∈ 1
2 Z and that f,0
(t, ϕ, ε),
g,1
(t, ϕ, ε) admit the following Fourier representation with respect to ϕ
f,0
(t, ϕ, ε) = f,0
0 (t, ε) + f,0
c (t, ε) cos(2 ϕ) + f,0
s (t, ε) sin(2 ϕ)
g,1
(t, ϕ, ε) = g,1
0 (t, ε) − f,0
s (t, ε) cos(2 ϕ) + f,0
c (t, ε) sin(2 ϕ).
(3.2)
The maps f,1
, g,2
are assumed to be π–antiperiodic (i.e. f,1
(t, ϕ + π, ε) = −f,1
(t, ϕ, ε)). With the help
of Floquet’s theory we derive a near–identity transformation of the form
ϕ = ψ + u(t, ψ, ε) P(h) = r
v(t, ε)
1 + ∂ψu(t, ψ, ε)
(3.4)
transforming (3.1) to the form
˙ψ = ˜Ω(ε) + O(r)
˙r = r g,1
0,0(ε) + r2
ˆg,2
(t, ψ, ε) + r3
ˆg,3
(t, ψ, r, ε)
(3.20)
where ˜Ω(0) = Ω0. Hence the coefficient g,1
0,0(ε) provides a criterion for {r = 0} (and thus {h = 0})
to be asymptotically stable, or unstable, respectively. The quantity g,1
0,0(ε) will be evaluated in chapter
4, in order to prove asymptotic stability of the periodic solution near (q, p, η) = (0, 0, 0) in case of the
synchronous motor problem.

13. Introduction ix
In the second part, chapter 4, we present the application of part one to the model of a miniature
synchronous motor mentioned before. After some preliminary preparations eq. (1) is transformed into
˙q = p
˙p = −
a
2
2
sin(q) + ε (η1 cos(q + t) − η2 sin(q + t)) − ε2
̺ p − ε2
(m + ̺)
˙η1 = −η1 + ε sin(q + t)
˙η2 = −η2 − 2 η3 + ε cos(q + t)
˙η3 = η2 − ε cos(q + t).
(4.14)
The quantity a is rougly equal to λ
R . For fixed a the perturbation parameter ε is given by a λ
U0
.
Here we assume that the voltage U0 of the power supply and the moment of inertia J of the motor are
proportional. Thus, ε tends to 0 provided U0 (and thus J) increases, while the magnetic dipol λ, and the
resistance R are kept fixed. By consequence, the effect of induction generated by the rotating permanent
magnet and exerted on the coils decreases as ε → 0.
In order to obtain preliminary insight into the features of (4.14) we present the results of various nu-
merical simulations carried out with the help of the package dstool [3]. The results found confirm the
analytical discussion given later in this chapter. In addition, they demonstrate that the behaviour in a
neighbourhood of the separatrix of the unperturbed problem of (4.14) is of no particular interest if a is
large. (Since the techniques introduced in part one rely on regions of periodic solutions of the Hamiltonian
system, the neighbourhood of a separatrix is not covered by our analytical approach.)
The main task in chapter 4 is to apply the tools of part one to system (4.14) and to compute the key
quantities g2
0,0(h), g2
km,nm
(h) and g,1
0,0(ε). Among other things this amounts to explicitely construct
suitable approximations of the invariant manifolds introduced in chapter 1. The introduction of action–
angle coordinates associated with the pendulum equation, is based on Fourier series of Jacobian elliptic
functions. Eventually g2
0,0(h) and g2
km,nm
(h) are represented with the help of convolutions of Fourier
series. The complexity of this procedure requires the use of a software package for symbolic and numerical
computations. The author has chosen the Maple [15] software package. Its synthax is simple and legible
for readers with basic knowledge in programming. Hence the source code listed is comprehensible to a
growing community. For various choices of the parameters the dynamics of the model is discussed in terms
of the physical behaviour of the motor. The influence of a mechanical friction (given by the parameter
˜̺) and an external torque ( ˜m) is considered as well. Both situations considered in chapter 2, i.e. the
case of passage of all solutions up to an O(ε)–set as well as the passage of strictly all solutions through
resonances are established. The periodic solution near the origin, corresponding to the synchronous
rotation of the shaft, is shown to be stable for all choices of the parameters. Moreover additional results
are established: the possibility of asynchronous rotations, the modulation of the synchronous rotation
state by a second harmonic as well as a synchronous rotation with large variation of the angular speed
(caused by a capture into resonance). The overall conclusion is that for sufficiently large a the motor
behaves favouritely, i.e. enters the state of stable synchronous rotation when switched on.
Chapter 4 closes with a result on the separatrix region for sufficiently small values of the parameter
a. In this situation, the existence of a global attractive invariant manifold of (4.14) is established. The
corresponding reduced system then is of periodically perturbed pendulum type. Although an approximate
representation of the reduced system is not available, the construction of an approximate Melnikov
function is feasible. The numerical evaluation of the corresponding formula then confirms the results
found by numerical simulation. More precisely, it is established that solutions starting with a frequency
larger than the frequency of the power supply may either enter the state of synchronous rotation or the
frequency may eventually tend to zero.

15. Chapter 1
Reduction to a Planar System
1.1 The System under Consideration
1.1.1 The Differential Equations
In this chapter we consider autonomous ordinary differential equations with a nonautonomous time–
periodic perturbation. For the unperturbed case we assume two independent subsystems, a Hamiltonian
system of one degree of freedom and a stable linear system. More precisely we will discuss equations of
the form
( ˙q, ˙p) = J∇H(q, p) + F(q, p, η, t, ε)
˙η = A η + G(q, p, t, ε),
(1.1)
where (q, p) ∈ R2
, η ∈ Rd
and J := 0
−1
1
0 represents the symplectic normal form.
We assume that A ∈ Rd×d
has only eigenvalues on the left complex halfplane. The Hamiltonian H is
assumed to be of class Cω
(i.e. analytical), the maps F, G are assumed to be Cω
, 2π–periodic with
respect to the time–variable t and vanishing as ε → 0.
1

16. 2 Chapter 1. Reduction to a Planar System
1.1.2 General Assumptions on the System
In this chapter we assume the following statements to be true
GA 1.1. The unperturbed Hamiltonian system
( ˙q, ˙p) = J∇H(q, p) (1.2)
satisfies the following set of assumptions :
(a) System (1.2) admits an elliptic fixed point in the origin. More precisely we assume that
in this situation ∇H(0, 0) = 0, ∂q∂pH(0, 0) = 0 and ∂2
q H(0, 0), ∂2
pH(0, 0) > 0. Moreover
D3
H(0, 0) = 0 holds and Ω0 := ∂2
q H(0, 0) ∂2
pH(0, 0) ∈ N := {0, 1, 2, . . .}.
(b) There exist an interval J = (Jl, Jr) together with a mapping Ω ∈ Cω
(J , R) such that the
solution (q, p)(t; 0, p0) of (1.2) with initial value (0, p0), p0 ∈ J at time t = 0 is periodic in t
with frequency Ω(p0) > 0.
(c) There is an integer r ≥ 0 such that for every 0 ≤ k ≤ r+7 the limit of ∂k
p0
Ω(p0) for p0 → Jl, Jr
exists and does not vanish for k = 0. If 0 ∈ J then lim
p0→0
∂p0 Ω(p0) = 0.
GA 1.2. The real parts of the eigenvalues of A are all negative, bounded by a suitable constant c0 > 0:
ℜ(σ(A)) ≤ −c0.
Here and in what follows, σ(A) denotes the spectrum, i.e. the set of all eigenvalues of the matrix
A. Moreover we assume that A is diagonalizable.
GA 1.3. Consider the Taylor expansion of order 3 in ε = 0 of the maps F and G, i.e. the representation
F(q, p, η, t, ε) =
3
j=1
εj
Fj
(q, p, η, t) + ε4
F4
(q, p, η, t, ε)
G(q, p, t, ε) =
3
j=1
εj
Gj
(q, p, t) + ε4
G4
(q, p, t, ε).
(1.3)
We assume that the maps Fj
, Gj
, j = 1, 2, 3 in (1.3) admit a representation as finite Fourier series1
of degree N ∈ N with respect to t , i.e.
Fj
(q, p, η, t) =
|n|≤N
Fj
n(q, p, η) eint
Gj
(q, p, t) =
|n|≤N
Gj
n(q, p) eint
. (1.4)
GA 1.4. The map F is affine with respect to η, i.e. ∂k
η F(q, p, η, t, ε) = 0 for all k ≥ 2.
1Note that the functions F j
n, Gj
n, n ∈ {−N, . . . , N}, j = 1, 2, 3 are complex valued functions. As system (1.1) is real, it
is easy to see that F j
n = ¯F j
−n, Gj
n = ¯Gj
−n, i.e. the complex conjugate valued functions.

17. 1.1. The System under Consideration 3
1.1.3 A Short Overview on the Strategy Followed
The aim of this first chapter is to derive a plane (non–autonomous) system which asymptotically de-
termines the qualitative behaviour of system (1.1). Considering (1.1) in the unperturbed case (ε = 0)
we see that due to the stability of the matrix A (as assumed in GA 1.2), all solutions tend towards the
(q, p)–plane η = 0. Hence the asymptotic behaviour in the unperturbed case is determined by the plane
Hamiltonian system (1.2).
In the case of a small perturbation (ε = 0 but small) we aim on a reduction to a plane system as
well. However, it will be necessary to consider different regions of the (q, p) phase space separately in
order to derive appropriate coordinates. Using invariant manifold theory we then show the existence
of an attractive two-dimensional (time–dependent) invariant manifold for the corresponding region and
consider the system restricted to this manifold. Following this way we yield a plane system, representing
the asymptotic behaviour in the corresponding domain for the perturbed case as well.
In sections 1.2–1.4 we deal with regions of periodic solutions of the Hamiltonian system (1.2) near an
elliptic fixed point. (Note that the fixed point itself is not included in such a region). As the plane
Hamiltonian system admits an elliptic fixed point at (q, p) = (0, 0), there exists a periodic solution of
the perturbed 2 + d–dimensional system (1.1) near the origin. This is dealed with in section 1.2. Since
the stability of this periodic solution is essential for the asymptotic behaviour of (1.1), we will perform
changes of coordinates in a way such that the region considered may be extended into this periodic
solution. This will be prepared in section 1.4.
Sections 1.5–1.6 deal with any region of periodic solutions of the Hamiltonian system. Introducing action
angle coordinates in section 1.5 it will be possible to establish the existence of an attractive invariant
manifold in section 1.6 and consider the ”restricted” plane system on the region chosen.
The entire process carried out in chapter 1 and chapter 2 will be presented in a form sufficiently explicit
for application on concrete examples. This requires more work in the theoretical part but on the other
hand leads to a form applicable in many situations. Moreover the author has tried to present the steps
carried out in a ”modular” manner, such that the results of certain sections may be applied independently.

18. 4 Chapter 1. Reduction to a Planar System
1.2 The Periodic Solution
1.2.1 The Existence of a Unique Periodic Solution Near the Origin
As mentioned above, in this section we consider the case where the Hamiltonian system (1.2) admits an
elliptic fixed point at the origin. The aim is to establish the existence of a unique 2π–periodic solution
of system (1.1) (for ε sufficiently small), located near the elliptic fixed point at the origin. This will be
carried out by applying the following general result to our situation.
Lemma 1.2.1 Consider an ordinary differential equation of the form
˙x = εp
(f(x) + g(x, t, ε)) , x ∈ Rm
(1.5)
where p ∈ N and f(0) = 0. Let f and g be of class C ˜r
(˜r ≥ 1 or ˜r = ω) and assume that g is T –
periodic with respect to t and vanishes for ε = 0, i.e. g(x, t, 0) = 0 ∀x ∈ Rm
, ∀t ∈ R. Moreover, let
σ (Df(0)) ∩ i 2π
T Z = ∅ if p = 0 and det Df(0) = 0 if p > 0.
Then there exists an ε1 > 0 and a unique map ˇx ∈ C ˜r
(R × (−ε1, ε1), Rm
) such that ˇx(t, 0) = 0 (∀t ∈ R)
and for every |ε| < ε1, the mapping t → ˇx(t, ε) is a T –periodic solution of system (1.5).
PROOF: We prove this lemma in several steps.
1. First, define the map
˜g(x, t, ε) := f(x) − Df(0) x + g(x, t, ε)
for x ∈ Rm
, t ∈ R and ε ∈ R. Then we see that ˜g is T –periodic with respect to t,
˜g(0, t, 0) = 0 and ∂x˜g(0, t, 0) = ∂xg(0, t, 0) = 0. (1.6)
Using this map ˜g we may write (1.5) as follows:
˙x = εp
(Df(0) x + ˜g(x, t, ε)) . (1.7)
Let x(t; t0, x0, ε) denote the solution of (1.5) with initial value x0 at time t0. By the uniqueness of
solutions we have
x(t; t0, x0, ε) = x(t; t1, x(t1; t0, x0, ε), ε). (1.8)
Since ˜g(x, t, ε) is T –periodic with respect to t, it follows that the flow induced by (1.5) is T –periodic
as well, hence
x(t; t0, x0, ε) = x(t + T ; t0 + T, x0, ε). (1.9)
As g vanishes for ε = 0 we finally note that x = 0 is a solution of (1.5) for ε = 0, hence
x(t; t0, 0, 0) = 0 ∀t, t0 ∈ R. (1.10)

19. 1.2. The Periodic Solution 5
2. In a next step we shall establish the existence of a unique initial value ξ(ε) near x = 0 which
corresponds to a T –periodic solution of (1.5). This will be shown by applying the Implicit Function
Theorem to the following map:
R(ε, ξ) :=
1
εp T
eεp
T Df(0)
− IR 2+d ξ +
1
T
T
0
eεp
(T −s) Df(0)
˜g(x(s; 0, ξ, ε), s, ε) ds.
Under the conditions assumed, R ∈ C ˜r
(R × Rm
, Rm
). By (1.6) and (1.10) we find
R(0, ξ) =
1
T eT Df(0)
− IR 2+d ξ p = 0
Df(0) ξ p > 0
such that R(0, 0) = 0. Taking the partial derivative of R(0, ξ) we find
∂ξR(0, 0) =
1
T eT Df(0)
− IR 2+d p = 0
Df(0) p > 0.
Since by assumption σ (Df(0)) ∩ i 2π
T Z = ∅ (if p = 0) and det Df(0) = 0 (if p > 0), we see that
det(∂ξR(0, 0)) =



1
T det eT Df(0)
− IR 2+d = 1
T
λ∈σ(Df(0))
(eT λ
− 1) = 0 p = 0
det Df(0) = 0 p > 0.
Hence it follows by the Implicit Function Theorem that there exists an ε1 > 0 as well as a unique
map ξ ∈ C ˜r
((−ε1, ε1), Rm
) with ξ(0) = 0 such that for every |ε| < ε1, R(ε, ξ(ε)) = 0.
In accordance with the representation (1.7) we write the solution x(t; 0, ξ0, ε) of (1.5) with initial
value ξ0 at time t0 = 0 using the Variation of Constant formula, i.e.
x(t; 0, ξ0, ε) = eεp
t Df(0)
ξ0 + εp
t
0
eεp
(t−s) Df(0)
˜g(x(s; 0, ξ0, ε), s, ε) ds.
By definition of R we thus find
R(ε, ξ) = 0 ⇔ εp
T R(ε, ξ) = 0 ⇔ ξ = x(T ; 0, ξ, ε). (1.11)
Setting ξ = ξ(ε) therefore yields ξ(ε) = x(T ; 0, ξ(ε), ε).
3. It remains to show that for fixed |ε| < ε1 the initial value ξ(ε) generates a periodic solution of (1.5),
indeed. We therefore define ˇx(t, ε) := x(t; 0, ξ(ε), ε). Applying (1.8), (1.9) and (1.11) we find for
any t ∈ R
ˆx(t + T, ε) = x(t + T ; 0, ξ(ε), ε) = x(t + T ; T, x(T ; 0, ξ(ε), ε), ε)
= x(t; 0, x(T ; 0, ξ(ε), ε), ε) = x(t; 0, ξ(ε), ε)
= ˇx(t, ε).
Hence the solution ˇx(t, ε) with initial value ˇx(0, ε) = ξ(ε) is T –periodic .
Moreover, ξ(0) = 0 together with (1.10) imply
ˇx(t, 0) = x(t; 0, ξ(0), 0) = x(t; 0, 0, 0) = 0.

20. 6 Chapter 1. Reduction to a Planar System
4. With the help of the statements given by the Implicit Function Theorem on the uniqueness, range
and domain of the map ξ, it eventually may be shown that the map t → ˇx(t, ε) is the only T –periodic
solution close to the origin satisfying ˇx(t, 0) = 0.
Therefore the statement given in lemma 1.2.1 is proved.
It now is a simple consequence of the preceeding lemma that system (1.1) admits a unique 2π–periodic
solution (ˇq, ˇp, ˇη) close to the origin. This is carried out in the following lemma.
Lemma 1.2.2 There exists ε1 > 0 as well as a unique map (ˇq, ˇp, ˇη) ∈ Cω
(R × (−ε1, ε1), R2+d
) such
that for fixed |ε| < ε1 the map t → (ˇq, ˇp, ˇη)(t, ε) is a 2π–periodic solution of (1.1) and for ε = 0,
(ˇq, ˇp, ˇη)(t, 0) = 0 ∀t ∈ R.
For simplicity we will omit the parameter a in the notation (ˇq, ˇp, ˇη) unless needed explicitely.
PROOF: For x = (q, p, η) ∈ R2+d
we set
f(x) := f(q, p, η) :=
J∇H(q, p)
A η
g(x, t, ε) := g(q, p, η, t, ε) :=
F(q, p, η, t, ε)
G(q, p, t, ε)
.
By assumption GA 1.1a we have f(0) =
J∇H(0, 0)
A 0
= 0 and
σ(Df(0)) = σ
JD2
H(0, 0) 0
0 A
= σ(JD2
H(0, 0)) ∪ σ(A), (1.12)
such that from σ JD2
H(0, 0) = ± i ∂2
q H(0, 0) ∂2
pH(0, 0) and GA 1.1a together with GA 1.2 we
deduce σ (Df(0)) ∩ i Z = ∅.
Taking into account the assumptions made in section 1.1.1 for F, G and H it is readily seen that we are
in the position to apply lemma 1.2.1 (where m = 2 + d, p = 0, ˜r = ω and T = 2π). Hence the proof of
lemma 1.2.2 is a consequence of lemma 1.2.1.

21. 1.2. The Periodic Solution 7
1.2.2 The Transformation into the Periodic Solution
The purpose of this section is to transform the coordinates of system (1.1) in a way, such that the origin
becomes a fixed point. This may be done by performing a (time–dependent) translation into the periodic
solution (ˇq, ˇp, ˇη).
More precisely we will use the Taylor / Fourier expansions (1.3), (1.4) assumed in GA 1.3 to explicitely
calculate a similar representation of the corresponding vector field in the new coordinates. This will be
prepared in the following lemma:
Lemma 1.2.3 Consider a linear inhomogenous differential equation on R2+d
of the following type
˙x = B x +
|n|≤N
bneint
, (1.13)
where bn ∈ C2+d
for every |n| ≤ N, bn = b−n and σ (B)∩i Z = ∅. Then there exists a unique 2π–periodic
solution given by
x(t) =
|n|≤N
[i n IC 2+d − B]−1
bn eint
. (1.14)
PROOF: Note first that since σ (B) ∩i Z = ∅, the inverse of the matrix i n IC 2+d − B exists. It is evident
that the function x presented in (1.14) is 2π–periodic with respect to t. Moreover
˙x(t) − B x(t) =
|n|≤N
i n [i n IC 2+d − B]
−1
bn eint
− B
|n|≤N
[i n IC 2+d − B]
−1
bn eint
=
|n|≤N
[i n IC 2+d − B] [i n IC 2+d − B]
−1
bn eint
=
|n|≤N
bn eint
,
such that x is a solution of (1.13), indeed.
Consider any further 2π–periodic solution y of (1.13). Writing its Fourier expansion y =
n∈N
cn eint
and
calculating ˙y−B y one then compares the result with
|n|≤N
bn eint
which implies cn = [i n IC 2+d − B]
−1
bn
and thus x = y. Hence x is unique as claimed.
We now are in the position to prove the main result of this section.
Proposition 1.2.4 Let (ˇq, ˇp, ˇη) denote the 2π–periodic solution of system (1.1) for |ε| < ε1, asserted in
lemma 1.2.2 and perform the following change of coordinates in the (q, p, η, t, ε)–space:
(q, p, η, t, ε) = ((ˇq, ˇp, ˇη)(t, ε), 0, 0) + ( ˇQ, ˇP, H, t, ε), (1.15)

22. 8 Chapter 1. Reduction to a Planar System
where2
( ˇQ, ˇP) ∈ R2
, H ∈ Rd
, t ∈ R and |ε| < ε1. Then (1.1) transforms into the system
( ˙ˇQ, ˙ˇP) = J∇H( ˇQ, ˇP) + ˇF( ˇQ, ˇP, H, t, ε)
˙H = A H + ˇG( ˇQ, ˇP, t, ε),
(1.16)
where the following statements hold :
• The mappings ˇF and ˇG are of class Cω
, vanish at the origin ( ˇQ, ˇP, H) = 0 and admit the repre-
sentation3
ˇF( ˇQ, ˇP, H, t, ε) =
3
j=1
εj ˇFj
( ˇQ, ˇP, H, t) + ε4 ˇF4
( ˇQ, ˇP, H, t, ε)
ˇG( ˇQ, ˇP, t, ε) =
3
j=1
εj ˇGj
( ˇQ, ˇP, t) + ε4 ˇG4
( ˇQ, ˇP, t, ε)
(1.17)
where ˇFj
and ˇGj
, (j = 1, . . . , 4) are 2π–periodic with respect to t.
• The map H → ˇF( ˇQ, ˇP, H, t, ε) is affine.
• The mappings ˇF1
, ˇF2
, ˇG1
and ˇG2
may be expressed in terms of the original vector field of system
(1.1):
ˇF1
( ˇQ, ˇP, H, t) = F1
( ˇQ, ˇP, H, t) −
|n|≤N
∆(n, ˇQ, ˇP) F1
n(0, 0, 0) eint
ˇF2
( ˇQ, ˇP, H, t) = F2
( ˇQ, ˇP, H, t) −
|n|≤N
∆(n, ˇQ, ˇP) F2
n(0, 0, 0)eint
+
|n|,|¯n|≤N
1
2 JD3
H( ˇQ, ˇP) α1,1
n,1, α1,1
¯n,1
+ ∂(q,p)F1
n( ˇQ, ˇP, H) − ∆(n + ¯n, ˇQ, ˇP) ∂(q,p)F1
n (0, 0, 0) α1,1
¯n,1
+ ∂ηF1
n( ˇQ, ˇP, H) − ∆(n + ¯n, ˇQ, ˇP) ∂ηF1
n(0, 0, 0) α1,1
¯n,2 ei(n+˜n)t
ˇG1
( ˇQ, ˇP, t) = G1
( ˇQ, ˇP, t) − G1
(0, 0, t)
ˇG2
( ˇQ, ˇP, t) = G2
( ˇQ, ˇP, t) − G2
(0, 0, t) +
|n|,|¯n|≤N
∂(q,p)G1
n( ˇQ, ˇP) − ∂(q,p)G1
n(0, 0) α1,1
¯n,1
(1.18)
where
∆(n, ˇQ, ˇP) := [i n IC 2 − JD2
H( ˇQ, ˇP)] i n IC 2 − JD2
H(0, 0)
−1
α1,1
n,1 = i n IC 2 − JD2
H(0, 0)
−1
F1
n(0, 0, 0)
α1,1
n,2 = [i n IC d − A]
−1
G1
n(0, 0).
(1.19)
2The letter H must be read as ”upper eta”
3for the application in chapter 4 it suffices to consider the expansions including terms of order O(ε2) of ˇF and of order
O(ε) of ˇG. The formulae for O(ε3)–terms are provided in order to enable a more detailed discussion on the capture in
resonance, cf. section 2.3.5.

23. 1.2. The Periodic Solution 9
• Moreover, ˇF1
, ˇF2
, ˇG1
and ˇG2
may be represented as Fourier polynomials in t, similar to the
representation (1.4), i.e.
ˇFj
( ˇQ, ˇP, H, t) =
|n|≤jN
ˇFj
n( ˇQ, ˇP, H, t) eint ˇGj
( ˇQ, ˇP, t) =
|n|≤jN
ˇGj
n( ˇQ, ˇP, t) eint
(1.20)
• The values of the map ˇF3
may be expressed in an analogous way:
ˇF3
( ˇQ, ˇP, H, t) =F3
( ˇQ, ˇP, H, t) −
|n|≤N
∆(n, ˇQ, ˇP) F3
n(0, 0, 0)eint
+
|n|,|¯n|≤N
JD3
H( ˇQ, ˇP) α1,1
n,1, α2,1
¯n,1
+ ∂(q,p)F2
n( ˇQ, ˇP, H) − ∆(n + ¯n, ˇQ, ˇP) ∂(q,p)F2
n(0, 0, 0) α1,1
¯n,1
+ ∂ηF2
n( ˇQ, ˇP, H) − ∆(n + ¯n, ˇQ, ˇP) ∂ηF2
n(0, 0, 0) α1,1
¯n,1
+ ∂(q,p)F1
n( ˇQ, ˇP, H) − ∆(n + ¯n, ˇQ, ˇP) ∂(q,p)F1
n(0, 0, 0) α2,1
¯n,1
+ ∂ηF1
n( ˇQ, ˇP, H) − ∆(n + ¯n, ˇQ, ˇP) ∂ηF1
n(0, 0, 0) α2,1
¯n,1 ei(n+¯n)t
+
|n|,|¯n|,|˜n|≤N
JD3
H( ˇQ, ˇP) α1,1
n,1, α2,2
¯n,˜n,1
+ 1
6 JD4
H( ˇQ, ˇP) − ∆(n + ¯n + ˜n, ˇQ, ˇP) JD4
H(0) α1,1
n,1 α1,1
¯n,1 α1,1
˜n,1
+ 1
2 ∂2
(q,p)F1
n( ˇQ, ˇP, H) − ∆(n + ¯n + ˜n, ˇQ, ˇP) ∂2
(q,p)F1
n(0, 0, 0) α1,1
¯n,1, α1,1
˜n,1
+ 1
2 ∂η∂(q,p)F1
n( ˇQ, ˇP, H) − ∆(n + ¯n + ˜n, ˇQ, ˇP) ∂η∂(q,p)F1
n(0, 0, 0) α1,1
˜n,1, α1,1
¯n,2
+ 1
2 ∂(q,p)∂ηF1
n( ˇQ, ˇP, H) − ∆(n + ¯n + ˜n, ˇQ, ˇP) ∂(q,p)∂ηF1
n(0, 0, 0) α1,1
˜n,2, α1,1
¯n,1
+ ∂(q,p)F1
n( ˇQ, ˇP, H) − ∆(n + ¯n, ˇQ, ˇP) ∂(q,p)F1
n (0, 0, 0) α2,2
¯n,˜n,1
+ ∂ηF1
n( ˇQ, ˇP, H) − ∆(n + ¯n, ˇQ, ˇP) ∂ηF1
n(0, 0, 0) α2,2
¯n,˜n,1 ei(n+¯n+˜n)t
.
(1.21)
where in addition
α2,1
n,1 = i n IC 2 − JD2
H(0, 0)
−1
F2
n(0, 0, 0) α2,1
n,2 = [i n IC d − A]
−1
G2
n(0, 0)
α2,2
n,¯n,1 = i (n + ¯n) IC 2 − JD2
H(0, 0)
−1
∂(q,p)F1
n(0, 0, 0) α1,1
n,1 + ∂ηF1
n(0, 0, 0) α1,1
n,2 .
PROOF: In order to simplify the notation, we use the same abbreviations as introduced in the proof of
lemma 1.2.2:
x := (q, p, η) ˇx(t, ε) :=(ˇq, ˇp, ˇη)(t, ε) y := ( ˇQ, ˇP, H)
f(x) :=
J∇H(q, p)
A η
g(x, t, ε) :=
F(q, p, η, t, ε)
G(q, p, t, ε)
(1.22)
such that system (1.1) defined for x ∈ R2+d
reads
˙x = f(x) + g(x, t, ε). (1.23)

24. 10 Chapter 1. Reduction to a Planar System
In view of assumption GA 1.3 we rewrite g(x, t, ε) as follows:
g(x, t, ε) =
3
j=1
εj
gj
(x, t) + ε4
g4
(x, t, ε) =
3
j=1
εj
|n|≤N
gj
n(x) eint
+ ε4
g4
(x, t, ε) (1.24)
where we have set
gj
n(x) =
Fj
n(q, p, η)
Gj
n(q, p)
∈ Cω
(R2+d
, C2+d
), |n| ≤ N, j = 1, 2, 3
and
g4
(x, t, ε) =
F4
(q, p, η, t, ε)
G4
(q, p, t, ε)
∈ Cω
(R2+d
× R × R, R2+d
).
The transformation defined in (1.15) corresponds to the time–dependent, 2π–periodic translation
x = ˇx(t, ε) + y
defined for t ∈ R and |ε| < ε1. Expressing (1.23) in the new coordinates yields
˙y = f(ˇx(t, ε) + y) + g(ˇx(t, ε) + y, t, ε) − ˙ˇx(t, ε) =: ˜f(y, t, ε). (1.25)
Note that ˜f ∈ Cω
(R2+d
× R × (−ε1, ε1), R2+d
) as f, g and ˇx are of class Cω
. As ˇx(t, ε) is a solution of
(1.1) and hence of (1.23) as well, we find
˜f(0, t, ε) = 0. (1.26)
Moreover, as ˇx and g vanish for ε = 0, it follows at once that
˜f(y, t, 0) = f(y). (1.27)
Since the last d components of
˜f(y, t, ε) − ˜f(y, t, 0) =
J ∇H(ˇq + ˇQ, ˇp + ˇP)
A (ˇη + H)
+
F(ˇq + ˇQ, ˇp + ˇP, ˇη + H, t, ε)
G(ˇq + ˇQ, ˇp + ˇP, t, ε)
−
J ∇H( ˇQ, ˇP)
A H
− ˙ˇx(t, ε)
=
J ∇H(ˇq + ˇQ, ˇp + ˇP) − J ∇H( ˇQ, ˇP) − J∇H(ˇq, ˇp)
A ˇη
+
F(ˇq + ˇQ, ˇp + ˇP, ˇη + H, t, ε) − F(ˇq, ˇp, ˇη, t, ε)
G(ˇq + ˇQ, ˇp + ˇP, t, ε) − G(ˇq, ˇp, t, ε)
(1.28)
do not depend on H, we may split this vector ˜f(y, t, ε) − ˜f(y, t, 0) into a two–dimensional component
which depends on H and a second, d–dimensional component, independent of H. More precisely we are
in the position to introduce the following abbreviations :
ˇF( ˇQ, ˇP, H, t, ε)
ˇG( ˇQ, ˇP, t, ε)
:= ˜f(y, t, ε) − ˜f(y, t, 0)
We continue with the following steps:

25. 1.2. The Periodic Solution 11
1. Consider the Taylor expansion of ˜f at ε = 0, i.e. the representation
˜f(y, t, ε) = ˜f(y, t, 0) + ε ∂ε
˜f(y, t, 0) + 1
2 ε2
∂2
ε
˜f(y, t, 0) + 1
6 ε3
∂3
ε
˜f(y, t, 0) + ε4 ˜f4
(y, t, ε) (1.29)
where ˜f4
(y, t, ε) is of class Cω
(R2+d
× R × (−ε1, ε1), R2+d
) and 2π–periodic with respect to t.
Setting
ˇFj
( ˇQ, ˇP, H, t)
ˇGj
( ˇQ, ˇP, t)
:= 1
j! ∂j
ε
˜f(y, t, 0) j = 1, 2, 3
ˇF4
( ˇQ, ˇP, H, t, ε)
ˇG4
( ˇQ, ˇP, t, ε)
:= ˜f4
(y, t, ε),
(1.30)
and taking into account that (1.26), (1.28) imply
ˇF(0, 0, 0, t, ε)
ˇG(0, 0, t, ε)
= −f(0) = 0
we find the first statement claimed to be proved at once.
2. In order to prove the second statement we note that by (1.28), (1.30)
ˇF( ˇQ, ˇP, H, t, ε) = J ∇H(ˇq + ˇQ, ˇp + ˇP) − J ∇H( ˇQ, ˇP) − J∇H(ˇq, ˇp)
+ F(ˇq + ˇQ, ˇp + ˇP, ˇη + H, t, ε) − F(ˇq, ˇp, ˇη, t, ε)
such that the affinity of F assumed in GA 1.4 implies the affinity of ˇF ( with respect to H).
3. We determine the Taylor coefficients in (1.29).
Using (1.26) we have
∂j
ε
˜f(0, t, 0) = 0 j = 1, 2, 3. (1.31)
On the other hand, from definition (1.25) we derive
∂ε
˜f(y, t, ε) = Df(ˇx(t, ε) + y) ∂ε ˇx(t, ε) + ∂xg(ˇx(t, ε) + y, t, ε) ∂εˇx(t, ε)
+ ∂εg(ˇx(t, ε) + y, t, ε) − ∂ε ˙ˇx(t, ε),
(1.32)
∂2
ε
˜f(y, t, ε) = D2
f(ˇx(t, ε) + y) ∂εˇx(t, ε)[2]
+ Df(ˇx(t, ε) + y) ∂2
ε ˇx(t, ε)
+ ∂2
xg(ˇx(t, ε) + y, t, ε) ∂εˇx(t, ε)[2]
+ ∂xg(ˇx(t, ε) + y, t, ε) ∂2
ε ˇx(t, ε)
+ 2 ∂ε∂xg(ˇx(t, ε) + y, t, ε) ∂εˇx(t, ε) + ∂2
ε g(ˇx(t, ε) + y, t, ε)
− ∂2
ε
˙ˇx(t, ε)
(1.33)
∂3
ε
˜f(y, t, ε) = D3
f(ˇx(t, ε) + y) ∂ε ˇx(t, ε)[3]
+ 3 D2
f(ˇx(t, ε) + y) (∂ε ˇx(t, ε), ∂2
ε ˇx(t, ε)) + Df(ˇx(t, ε) + y) ∂3
ε ˇx(t, ε)
+ ∂3
xg(ˇx(t, ε) + y, t, ε) ∂εˇx(t, ε)[3]
+ 3 ∂2
xg(ˇx(t, ε) + y, t, ε) (∂εˇx(t, ε), ∂2
ε ˇx(t, ε)) + ∂xg(ˇx(t, ε) + y, t, ε) ∂3
ε ˇx(t, ε)
+ 3 ∂ε∂2
xg(ˇx(t, ε) + y, t, ε) ∂εˇx(t, ε)[2]
+ 3 ∂2
ε ∂xg(ˇx(t, ε) + y, t, ε) ∂ε ˇx(t, ε)
+ 3 ∂ε∂xg(ˇx(t, ε) + y, t, ε) ∂2
ε ˇx(t, ε) + ∂3
ε g(ˇx(t, ε) + y, t, ε)
− ∂3
ε
˙ˇx(t, ε)
(1.34)

26. 12 Chapter 1. Reduction to a Planar System
where the notation v[j]
must be understood as applying the corresponding multilinear–form on the
j vectors (v, . . . , v). Taking into account that by (1.24)
∂j
xg(y, t, 0) = 0 j = 1, 2, 3 ∂ε∂j
xg(y, t, 0) = ∂j
xg1
(y, t) j = 0, 1, 2
∂2
ε ∂j
xg(y, t, 0) = 2 ∂j
xg2
(y, t) j = 0, 1 ∂3
ε g(y, t, 0) = 6 g3
(y, t),
we therefore see that setting ε = 0, (1.32), (1.33) and (1.34) reduce to
∂ε
˜f(y, t, 0) = Df(y) ∂ε ˇx(t, 0) + g1
(y, t) − ∂t∂ε ˇx(t, 0)
∂2
ε
˜f(y, t, 0) = D2
f(y) ∂ε ˇx(t, 0)[2]
+ Df(y) ∂2
ε ˇx(t, 0)
+ 2 ∂xg1
(y, t) ∂ε ˇx(t, 0) + 2 g2
(y, t) − ∂t∂2
ε ˇx(t, 0)
∂3
ε
˜f(y, t, 0) = D3
f(y) ∂ε ˇx(t, 0)[3]
+ 6 D2
f(y) (∂ε ˇx(t, 0), 1
2 ∂2
ε ˇx(t, 0)) + Df(y) ∂3
ε ˇx(t, 0)
+ 3 ∂2
xg1
(y, t) ∂ε ˇx(t, 0)[2]
+ 6 ∂xg2
(y, t) ∂ε ˇx(t, 0)
+ 6 ∂xg1
(y, t) 1
2 ∂2
ε ˇx(t, 0) + 6 g3
(y, t)
− ∂t∂3
ε ˇx(t, 0).
(1.35)
4. In a next step we compute the functions ∂ε ˇx(t, 0), ∂2
ε ˇx(t, 0) and ∂3
ε ˇx(t, 0) by solving differential
equations :
Recall that by GA 1.1a D3
H(0, 0) = 0 such that by definition of f, D2
f(0) = 0. Therefore (1.24)
together with (1.31), (1.35) yields the following linear inhomogeneous differential equations
∂t∂ε ˇx(t, 0) = Df(0) ∂εˇx(t, 0) + g1
(0, t) = Df(0) ∂ε ˇx(t, 0) +
|n|≤N
g1
n(0) eint
, (1.36)
∂t∂2
ε ˇx(t, 0) = Df(0) ∂2
ε ˇx(t, 0) + 2 ∂xg1
(0, t) ∂εˇx(t, 0) + 2 g2
(0, t) (1.37)
and
∂t∂3
ε ˇx(t, 0) = Df(0) ∂3
ε ˇx(t, 0) + D3
f(0) ∂ε ˇx(t, 0)[3]
+ 3 ∂2
xg1
(0, t) ∂ε ˇx(t, 0)[2]
+ 6 ∂xg2
(0, t) ∂εˇx(t, 0)
+ 6 ∂xg1
(0, t) 1
2 ∂2
ε ˇx(t, 0) + 6 g3
(0, t).
(1.38)
As we have shown in (1.12) in the proof of lemma 1.2.2, σ (Df(0)) ∩ i Z = ∅. Hence lemma 1.2.3
may be applied to equation (1.36). Therefore the unique 2π–periodic solution ∂ε ˇx(t, 0) of (1.36)
is given by
∂ε ˇx(t, 0) =
|n|≤N
α1,1
n eint
, where α1,1
n := [i n IC 2+d − Df(0)]
−1
g1
n(0). (1.39)
Let us rewrite the differential equation (1.37) using (1.24) and (1.39):
∂t∂2
ε ˇx(t, 0) = Df(0) ∂2
ε ˇx(t, 0) + 2


|n|≤N
Dg1
n(0)eint




|¯n|≤N
α1,1
¯n ei¯nt

 + 2
|n|≤N
g2
n(0) eint
= Df(0) ∂2
ε ˇx(t, 0) + 2
|n|≤N
g2
n(0) eint
+ 2
|n|,|¯n|≤N
Dg1
n(0) α1,1
¯n ei(n+¯n)t
.

27. 1.2. The Periodic Solution 13
Solving this equation with the help of lemma 1.2.3 again we obtain
1
2 ∂2
ε ˇx(t, 0) =
|n|≤N
α2,1
n eint
+
|n|,|¯n|≤N
α2,2
n,¯n ei(n+¯n)t
,
with α2,1
n := [i n IC 2+d − Df(0)]
−1
g2
n(0)
α2,2
n,¯n := [i (n + ¯n) IC 2+d − Df(0)]
−1
Dg1
n(0) α1,1
¯n .
(1.40)
Finally we proceed in an analogous way to obtain
1
6 ∂3
ε ˇx(t, 0) =
|n|,|¯n|,|˜n|≤N
α3,3
n,¯n,˜n ei(n+¯n+˜n)t
+
|n|,|¯n|≤N
α3,2
n,¯n ei(n+¯n)t
+
|n|≤N
α3,1
n eint
(1.41)
where
α3,3
n,¯n,˜n = [i (n + ¯n + ˜n) IC 2+d − Df(0)]
−1 1
6 D3
f(0)(α1,1
n , α1,1
¯n , α1,1
˜n )
+ 1
2 D2
g1
n(0)(α1,1
¯n , α1,1
˜n ) + Dg1
n(0) α2,2
¯n,˜n
α3,2
n,¯n = [i (n + ¯n) IC 2+d − Df(0)]−1
Dg1
n(0)α2,1
¯n + Dg2
n(0)α1,1
¯n
α3,1
n = [i n IC 2+d − Df(0)]−1
g3
n(0).
(1.42)
5. In order to gain expressions for the coefficient maps ∂ε
˜f(y, t, 0), 1
2 ∂2
ε
˜f(y, t, 0) and 1
6 ∂3
ε
˜f(y, t, 0) in
terms of known quantities, we combine the results derived in the first two steps. Let us introduce
the notations
∆(n, ˇQ, ˇP) := [i n IC 2 − JD2
H( ˇQ, ˇP)] i n IC 2 − JD2
H(0, 0)
−1
M(n, ˇQ, ˇP) :=
∆(n, ˇQ, ˇP) 0
0 IC d
= [i n IC 2+d − Df(y)] [i n IC 2+d − Df(0)]
−1
.
(1.43)
Note that ∆(n, 0, 0) = IC 2 and M(n, 0, 0) = IC 2+d . Using the identities (1.24) and (1.39) we rewrite
the first equation in (1.35):
∂ε
˜f(y, t, 0) =
|n|≤N
Df(y) α1,1
n + g1
n(y) − i n α1,1
n eint
=
|n|≤N
g1
n(y) − [i n IC 2+d − Df(y)] α1,1
n eint
=
|n|≤N
g1
n(y) − M(n, ˇQ, ˇP) g1
n(0) eint
. (1.44)
The analogous result for 1
2 ∂2
ε
˜f(y, t, 0) is achieved by substituting (1.24), (1.40) into the second

28. 14 Chapter 1. Reduction to a Planar System
equation of (1.35):
1
2 ∂2
ε
˜f(y, t, 0) = 1
2 D2
f(y)


|n|≤N
α1,1
n eint
,
|¯n|≤N
α1,1
¯n ei¯nt


+
|n|≤N
Df(y) α2,1
n eint
+
|n|,|¯n|≤N
Df(y) α2,2
n,¯n ei(n+¯n)t
+


|n|≤N
Dg1
n(y) eint




|¯n|≤N
α1,1
¯n ei¯nt

 +
|n|≤N
g2
n(y)eint
−
|n|≤N
i n α2,1
n eint
−
|n|,|¯n|≤N
i (n + ¯n) α2,2
n,¯n ei(n+¯n)t
=
|n|≤N
Df(y) α2,1
n + g2
n(y) − i n α2,1
n eint
+
|n|,|¯n|≤N
1
2 D2
f(y) α1,1
n , α1,1
¯n + Df(y) α2,2
n,¯n + Dg1
n(y) α1,1
¯n
−i (n + ¯n) α2,2
n,¯n ei(n+¯n)t
.
Using the abbreviations defined in (1.43) together with the definitions of α2,1
n , α2,2
n,¯n given in (1.40)
we find
1
2 ∂2
ε
˜f(y, t, 0) =
|n|≤N
g2
n(y) − [i n IC 2+d − Df(y)] α2,1
n eint
+
|n|,|¯n|≤N
1
2 D2
f(y) α1,1
n , α1,1
¯n + Dg1
n(y) α1,1
¯n
− [i (n + ¯n) IC 2+d − Df(y)] α2,2
n,¯n ei(n+¯n)t
=
|n|≤N
g2
n(y) − [i n IC 2+d − Df(y)] [i n IC 2+d − Df(0)]
−1
g2
n(0) eint
+
|n|,|¯n|≤N
1
2 D2
f(y) α1,1
n , α1,1
¯n + Dg1
n(y) α1,1
¯n
− [i (n + ¯n) IC 2+d − Df(y)] [i (n + ¯n) IC 2+d − Df(0)]
−1
Dg1
n(0) α1,1
¯n ei(n+¯n)t
hence
1
2 ∂2
ε
˜f(y, t, 0) =
|n|≤N
g2
n(y) − M(n, ˇQ, ˇP) g2
n(0) eint
+
|n|,|¯n|≤N
1
2 D2
f(y) α1,1
n , α1,1
¯n + Dg1
n(y) − M(n + ¯n, ˇQ, ˇP)Dg1
n(0) α1,1
¯n ei(n+¯n)t
. (1.45)
In a similar way we deduce the following representation of 1
6 ∂3
ε
˜f(y, t, 0) from (1.24), (1.41) and the

29. 1.2. The Periodic Solution 15
last equation in (1.35)
1
6 ∂3
ε
˜f(y, t, 0) = 1
6 D3
f(y)
|n|≤N
α1,1
n eint
,
|¯n|≤N
α1,1
¯n ei¯nt
,
|˜n|≤N
α1,1
˜n ei˜nt
+D2
f(y)
|n|≤N
α1,1
n eint
,
|¯n|≤N
α2,1
¯n ei¯nt
+D2
f(y)
|n|≤N
α1,1
n eint
,
|¯n|,|˜n|≤N
α2,2
¯n,˜n ei(¯n+˜n)t
+Df(y)
|n|,|¯n|,|˜n|≤N
α3,3
n,¯n,˜n ei(n+¯n+˜n)t
+
|n|,|¯n|≤N
α3,2
n,¯n ei(n+¯n)t
+
|n|≤N
α3,1
n eint
+1
2
|n|≤N
D2
g1
n(y)eint
|¯n|≤N
α1,1
¯n ei¯nt
,
|˜n|≤N
α1,1
˜n ei˜nt
+
|n|≤N
Dg2
n(y)eint
|¯n|≤N
α1,1
¯n ei¯nt
+
|n|≤N
Dg1
n(y)eint
|¯n|≤N
α2,1
¯n ei¯nt
+
|¯n|,|˜n|≤N
α2,2
¯n,˜n ei(¯n+˜n)t
+
|n|≤N
g3
n(y)eint
−
|n|,|¯n|,|˜n|≤N
i (n + ¯n + ˜n) α3,3
n,¯n,˜n ei(n+¯n+˜n)t
−
|n|,|¯n|≤N
i (n + ¯n) α3,2
n,¯n ei(n+¯n)t
−
|n|≤N
i n α3,1
n eint
,
thus
1
6 ∂3
ε
˜f(y, t, 0) =
|n|≤N
Df(y)α3,1
n + g3
n(y) − i n α3,1
n eint
+
|n|,|¯n|≤N
D2
f(y) α1,1
n , α2,1
¯n + Df(y)α3,2
n,¯n + Dg2
n(y)α1,1
¯n
+Dg1
n(y)α2,1
¯n − i (n + ¯n) α3,2
n,¯n ei(n+¯n)t
+
|n|,|¯n|,|˜n|≤N
1
6 D3
f(y) α1,1
n , α1,1
¯n , α1,1
˜n + D2
f(y) α1,1
n , α2,2
¯n,˜n
+Df(y)α3,3
n,¯n,˜n + 1
2 D2
g1
n(y) α1,1
¯n , α1,1
˜n
+Dg1
n(y)α2,2
¯n,˜n − i (n + ¯n + ˜n)α3,3
n,¯n,˜n ei(n+¯n+˜n)t

30. 16 Chapter 1. Reduction to a Planar System
which by (1.40), (1.42) eventually leads to
1
6 ∂3
ε
˜f(y, t, 0) =
|n|≤N
g3
n(y) − M(n, ˇQ, ˇP) g3
n(0) eint
+
|n|,|¯n|≤N
D2
f(y) α1,1
n , α2,1
¯n + Dg2
n(y) − M(n + ¯n, ˇQ, ˇP) Dg2
n(0) α1,1
¯n
+ Dg1
n(y) − M(n + ¯n, ˇQ, ˇP)Dg1
n(0) α2,1
¯n ei(n+¯n)t
+
|n|,|¯n|,|˜n|≤N
D2
f(y) α1,1
n , α2,2
¯n,˜n
+ 1
6 D3
f(y) − M(n + ¯n + ˜n, ˇQ, ˇP) D3
f(0) α1,1
n , α1,1
¯n , α1,1
˜n
+ 1
2 D2
g1
n(y) − M(n + ¯n + ˜n, ˇQ, ˇP) D2
g1
n(0) α1,1
¯n , α1,1
˜n
+ Dg1
n(y) − M(n + ¯n + ˜n, ˇQ, ˇP) Dg1
n(0) α2,2
¯n,˜n) ei(n+¯n+˜n)t
.
(1.46)
6. In a next step, we split the quantities ∂ε
˜f(y, t, 0), 1
2 ∂2
ε
˜f(y, t, 0) and 1
6 ∂3
ε
˜f(y, t, 0) into two compo-
nents, expressed in terms of the maps Fj
n and Gj
n. This will lead us to the formulae claimed in
(1.18) and (1.21).
Using definitions (1.43), (1.24) we rewrite (1.44) as follows :
∂ε
˜f(y, t, 0) =
|n|≤N
F1
n( ˇQ, ˇP, H) − ∆(n, ˇQ, ˇP) F1
n (0, 0, 0)
G1
n( ˇQ, ˇP) − G1
n(0, 0)
eint
. (1.47)
For convenience we split the vectors αj,1
n , α2,2
n,¯n into two components of dimension 2 and d :
αj,1
n =:
αj,1
n,1
αj,1
n,2
α2,2
n,¯n =:
α2,2
n,¯n,1
α2,2
n,¯n,2
By definition (1.22) we find derivatives of f to be diagonal operators in the following sense :
Df(y) ∼=
JD2
H( ˇQ, ˇP) 0
0 A
D2
f(y)
( ˇQ1, ˇP1)
H1
( ˇQ2, ˇP2)
H2
=
JD3
H( ˇQ, ˇP)( ˇQ1, ˇP1)( ˇQ2, ˇP2)
0
(1.48)
D3
f(y)
( ˇQ1, ˇP1)
H1
( ˇQ2, ˇP2)
H2
( ˇQ3, ˇP3)
H3
=
JD4
H( ˇQ, ˇP)( ˇQ1, ˇP1)( ˇQ2, ˇP2)( ˇQ3, ˇP3)
0
.
Note that by simple consequence,
[i n IC 2+d − Df(y)] =
i n IC 2 − JD2
H( ˇQ, ˇP) 0
0 i n IC d − A
.
Together with the representation of α1,1
n introduced above, we obtain
D2
f(y) α1,1
n , αk,j
¯n =
JD3
H( ˇQ, ˇP) α1,1
n,1, αk,j
¯n,1
0
k, j = 1, 2, (1.49)

31. 1.2. The Periodic Solution 17
and as Gj
n does not depend on η, we have
Dgj
n(y) =
∂(q,p)Fj
n( ˇQ, ˇP, H) ∂ηFj
n( ˇQ, ˇP, H)
∂(q,p)Gj
n( ˇQ, ˇP) 0
. (1.50)
Hence equation (1.45) reads
1
2 ∂2
ε
˜f(y, t, 0) =
|n|≤N
F2
n( ˇQ, ˇP, H)
G2
n( ˇQ, ˇP)
−
∆(n, ˇQ, ˇP) F2
n(0, 0, 0)
G2
n(0, 0)
eint
+
|n|,|¯n|≤N
1
2
JD3
H( ˇQ, ˇP) α1,1
n,1, α1,1
¯n,1
0
(1.51)
+
∂(q,p)F1
n( ˇQ, ˇP, H) − ∆(n + ¯n, ˇQ, ˇP) ∂(q,p)F1
n(0, 0, 0) α1,1
¯n,1
∂(q,p)G1
n( ˇQ, ˇP) − ∂(q,p)G1
n(0, 0) α1,1
¯n,1
+
∂ηF1
n( ˇQ, ˇP, H) − ∆(n + ¯n, ˇQ, ˇP) ∂ηF1
n(0, 0, 0) α1,1
¯n,2
0
ei(n+˜n)t
.
We finally calculate the corresponding representation for 1
6 ∂3
ε
˜f(y, t, 0). Since F is affine with respect
to η (GA 1.4) we have ∂2
ηF1
n( ˇQ, ˇP, H) = 0 such that
D2
g1
n(y)(α1,1
¯n , α1,1
˜n ) =


∂2
(q,p)F1
n( ˇQ, ˇP, H) α1,1
¯n,1, α1,1
˜n,1
∂2
(q,p)G1
n( ˇQ, ˇP) α1,1
¯n,1, α1,1
˜n,1

 (1.52)
+
∂η∂(q,p)F1
n( ˇQ, ˇP, H) α1,1
˜n,1, α1,1
¯n,2 + ∂(q,p)∂ηF1
n ( ˇQ, ˇP, H) α1,1
˜n,2, α1,1
¯n,1
0
and considering (6) we find
D3
f(y)(α1,1
n , α1,1
¯n , α1,1
˜n ) =
JD4
H( ˇQ, ˇP) α1,1
n,1 α1,1
¯n,1 α1,1
˜n,1
0
. (1.53)
Applying (1.49)–(1.53) on (1.46) then yields
1
6 ∂3
ε
˜f(y, t, 0) =
|n|≤N
F3
n( ˇQ, ˇP, H) − ∆(n, ˇQ, ˇP) F3
n(0, 0, 0)
G3
n( ˇQ, ˇP) − G3
n(0, 0)
eint
+
|n|,|¯n|≤N
JD3
H( ˇQ, ˇP) α1,1
n,1, α2,1
¯n,1
0
+
∂(q,p)F2
n( ˇQ, ˇP, H) − ∆(n + ¯n, ˇQ, ˇP) ∂(q,p)F2
n (0, 0, 0) α1,1
¯n,1
∂(q,p)G2
n( ˇQ, ˇP) − ∂(q,p)G2
n(0, 0) α1,1
¯n,2
+
∂ηF2
n( ˇQ, ˇP, H) − ∆(n + ¯n, ˇQ, ˇP) ∂ηF2
n(0, 0, 0) α1,1
¯n,1
0
+
∂(q,p)F1
n( ˇQ, ˇP, H) − ∆(n + ¯n, ˇQ, ˇP) ∂(q,p)F1
n (0, 0, 0) α2,1
¯n,1
∂(q,p)G1
n( ˇQ, ˇP) − ∂(q,p)G1
n(0, 0) α2,1
¯n,2
+
∂ηF1
n( ˇQ, ˇP, H) − ∆(n + ¯n, ˇQ, ˇP) ∂ηF1
n(0, 0, 0) α2,1
¯n,1
0
ei(n+¯n)t
(1.54)

32. 18 Chapter 1. Reduction to a Planar System
+
|n|,|¯n|,|˜n|≤N
JD3
H( ˇQ, ˇP) α1,1
n,1, α2,2
¯n,˜n,1
0
+1
6
JD4
H( ˇQ, ˇP) − ∆(n + ¯n + ˜n, ˇQ, ˇP) JD4
H(0) α1,1
n,1 α1,1
¯n,1 α1,1
˜n,1
0
+1
2


∂2
(q,p)F1
n( ˇQ, ˇP, H) − ∆(n + ¯n + ˜n, ˇQ, ˇP) ∂2
(q,p)F1
n(0, 0, 0) α1,1
¯n,1, α1,1
˜n,1
∂2
(q,p)G1
n( ˇQ, ˇP) − ∂2
(q,p)G1
n(0, 0) α1,1
¯n,1, α1,1
˜n,1


+1
2
∂η∂(q,p)F1
n( ˇQ, ˇP, H) − ∆(n + ¯n + ˜n, ˇQ, ˇP) ∂η∂(q,p)F1
n (0, 0, 0) α1,1
˜n,1, α1,1
¯n,2
0
+1
2
∂(q,p)∂ηF1
n( ˇQ, ˇP, H) − ∆(n + ¯n + ˜n, ˇQ, ˇP) ∂(q,p)∂ηF1
n (0, 0, 0) α1,1
˜n,2, α1,1
¯n,1
0
+
∂(q,p)F1
n( ˇQ, ˇP, H) − ∆(n + ¯n, ˇQ, ˇP) ∂(q,p)F1
n(0, 0, 0) α2,2
¯n,˜n,1
∂(q,p)G1
n( ˇQ, ˇP) − ∂(q,p)G1
n(0, 0) α2,2
¯n,˜n,2
+
∂ηF1
n( ˇQ, ˇP, H) − ∆(n + ¯n, ˇQ, ˇP) ∂ηF1
n (0, 0, 0) α2,2
¯n,˜n,1
0
ei(n+¯n+˜n)t
.
7. Summarizing the identities given in (1.25), (1.29) and (1.27) we consider the transformed system
˙y = ˜f(y, t, ε) = f(y) + ε∂ε
˜f(y, t, 0) + 1
2 ε2
∂2
ε
˜f(y, t, 0) + 1
6 ε3
∂3
ε
˜f(y, t, 0) + ε4 ˜f4
(y, t, ε)
which by (1.22), (1.30) may be represented in the form
( ˙ˇQ, ˙ˇP)
˙H
=
J∇H( ˇQ, ˇP)
A H
+
3
j=1
εj
ˇFj
( ˇQ, ˇP, H, t)
ˇGj
( ˇQ, ˇP, t)
+ ε4
ˇF4
( ˇQ, ˇP, H, t, ε)
ˇG4
( ˇQ, ˇP, t, ε)
.
Thus the identities (1.18) hold as it has been established in (1.47), (1.51) respectively.
8. In order to obtain the formula given in (1.21) one has to consider the first two components of
1
6 ∂3
ε
˜f(y, t, 0), which by (1.30) represents the vector–valued map ˇF3
( ˇQ, ˇP, H, t).
9. It remains to prove the formulae given for the quantities α1,1
n,1, α1,1
n,2 etc. :
From the definition of (1.24) of gj
n(y) we have
gj
n(0) =
Fj
n(0, 0, 0)
Gj
n(0, 0)
j = 1, 2, 3
hence, by definitions (1.39), (1.40) of the vectors αj,1
n ,
αj,1
n =
i n IC 2 − JD2
H(0, 0)
−1
Fj
n(0, 0, 0)
[i n IC d − A]
−1
Gj
n(0, 0)
j = 1, 2.
Together with (1.50) this implies
Dg1
n(y) α1,1
¯n =
∂(q,p)F1
n( ˇQ, ˇP, H) α1,1
¯n,1 + ∂ηF1
n( ˇQ, ˇP, H) α1,1
¯n,2
∂(q,p)G1
n( ˇQ, ˇP) α1,1
¯n,1

33. 1.2. The Periodic Solution 19
such that definition (1.40) reads
α2,2
n,¯n =
i (n + ¯n) IC 2 − JD2
H(0, 0)
−1
∂(q,p)F1
n (0, 0, 0) α1,1
n,1 + ∂ηF1
n (0, 0, 0) α1,1
n,2
[i (n + ¯n) IC d − A]
−1
∂(q,p)G1
n(0, 0) α1,1
n,1
.
We therefore have established all assertions made in proposition 1.2.4.

34. 20 Chapter 1. Reduction to a Planar System
1.3 Some Illustrative Examples
As explained in section 1.1.3, the strategy of this chapter consists in proving the existence of a local,
attractive, two–dimensional invariant manifold. Once this step has been accomplished the qualitative
discussion of (1.1) is reduced to the discussion of a plane, non–autonomous system by considering the
system restricted to the attractive invariant manifold.
However there are a few points to be made when entering this line of attack. The majority of the results
on the existence of attractive invariant manifolds are based on the discussion of Lyapunov type numbers
of solutions, hence set in a more abstract framework4
rather than an applicable form. For the purpose
of this work an approach where assumptions are made on known quantities (as the vector field) is more
convenient.
The general setting for this case can be found in a result by Kirchgraber [9]. It supplies the existence
and additional properties of an attractive invariant manifold for mappings without giving smoothness,
however. In this work we will apply an adaption by Nipp / Stoffer [13] which deals with ODE’s and
establishes smoothness as well. The assumptions on the system made by Nipp / Stoffer are expressed
using certain Lipschitz numbers of the vector field and logarithmic norms of derivatives of the vector
field.
However, we must take into account that theLipschitz numbers of the vector field as well as the logarithmic
norms of the derivatives depend on the choice of coordinates. Hence it is of great interest to find
appropriate coordinates in order to obtain satisfactory results.
Thus the difficulties in discussing the assumptions on the Lyapunov type numbers necessary for the more
”abstract approach” are replaced by the problem of defining suitable coordinates, when aiming at the
setup made in [9] and [13]. The following example illustrates how the choice of ”unnatural” coordinates
may restrict the results obtained in an unsatisfactory way.
1.3.1 Example 1 (disadvantegous cartesian coordinates)
Consider the (unperturbed) system (1.16) in the case of H( ˇQ, ˇP) = ˇP2
/2 − cos( ˇQ) of the mathematical
pendulum,
˙ˇQ = ˇP
˙ˇP = − sin( ˇQ)
˙H = A H,
(1.55)
where A < 0. One of the assumptions made in Nipp / Stoffer [13] includes the existence of constants
γ1 ∈ R, γ2 > 0 such that
µ −JD2
H( ˇQ, ˇP) ≤ γ1, µ (A) ≤ −γ2, γ1 < γ2, (1.56)
uniformly, where µ (M) denotes the logarithmic norm of a matrix M (cf. definition 1.4.5). Choosing the
euclidean norm on R2
one has
µ −JD2
H( ˇQ, ˇP) = 1
2 1 − cos( ˇQ) µ (A) = A,
4as, for instance, given in [4], [6]

35. 1.3. Some Illustrative Examples 21
such that if 1 − cos( ˇQ) ≥ 2 |A|, then (1.56) is not satisfied. Thus the existence of an attractive manifold
may not be established but on a subset of ( ˇQ, ˇP) ∈ R2
1 − cos( ˇQ) < 2 |A| depending on A. Since the
hyperplane H = 0 is a global attractive invariant manifold one expects a result independent of the size of
A. Hence the cartesian coordinates ( ˇQ, ˇP) are ”unnatural” even in the unperturbed case ε = 0. We will
see that using certain action angle coordinates, the domain on which an attractive invariant manifold
may be established is equal to the entire region covered by the action angle coordinates, independent of
A.
The next example illustrates a further reason of more practical nature to introduce action angle coordi-
nates.
1.3.2 Example 2 (further reasons to introduce action angle coordinates)
Let us assume for a moment, that the existence of an invariant manifold Mε has been established on a
sufficiently large domain for a perturbed (autonomous) system of the form
˙ˇQ = ˇP + O(ε)
˙ˇP = − sin( ˇQ) + O(ε)
˙H = A H + O(ε),
(1.57)
(where A < 0 again). As we are interested in an explicit representation of the vector field restricted to
the manifold Mε, it will be necessary to calculate Mε. A possible line of attack consist in writing the
so–called equation of invariance: Assuming that Mε is a graph of a map S, i.e.
Mε = ( ˇQ, ˇP, H) ∈ R3
H = S( ˇQ, ˇP, ε)
we find on one hand ˙H = ∂ ˇQS( ˇQ, ˇP, ε) ˇP − ∂ ˇP S( ˇQ, ˇP, ε) sin( ˇQ) + O(ε), while on the other hand (1.57)
implies ˙H = A S( ˇQ, ˇP, ε) + O(ε). In general this yields a partial differential equation impossible to solve
for S explicitely, even if S is expanded with respect to ε.
Considering any region of the ( ˇQ, ˇP)–space excluding the separatrices and fixed points of the unperturbed
system one may define appropriate action angle coordinates such that equation (1.57) transforms into a
system of the form
˙ϕ = ω(h) + O(ε)
˙h = O(ε)
˙H = A H + O(ε).
(1.58)
The equation of invariance then reads
∂ϕS(ϕ, h, ε) ω(h) + O(ε) = A S(ϕ, h, ε) + O(ε).
Proving that the map S is periodic with respect to ϕ and considering its Fourierseries, an ε–expansion
of S may be found explicitely by comparing Fourier coefficients in the equation of invariance. If the
perturbation in (1.57) is non–autonomous, then one may proceed in a similar way, using the Ansatz
S(t, ϕ, h, ε) =
k,n∈Z
Sk,n(h, ε) ei(kϕ+nt)
.

36. 22 Chapter 1. Reduction to a Planar System
We conclude that it is advantageous to use action angle coordinates, if possible. First since the domain
where the existence of an attractive manifold may be established is expected to be maximal in a certain
sense, second because an expansion of the invariant manifold may be found explicitely.
The third example will illustrate briefly how suitable action angle coordinates are defined. Moreover, it
shows that in the case of 0 ∈ J (cf. 1.97 c and the corresponding paragraph in section 1.1.3), one has to
proceed carefully if extending the domain into the periodic solution near the origin.
1.3.3 Example 3 (extending the domain of action angle coordinates)
Let H( ˇQ, ˇP) = 1
2 ( ˇP2
+ ˇQ2
), A = −1 and the perturbation be given as follows:
˙ˇQ = ˇP + ε ˇP ( ˇP − ε H)
˙ˇP = − ˇQ − 2 ε H
˙H = −H.
(1.59)
For p0 ∈ J = R, the corresponding Hamiltonian system admits the periodic solutions
(q, p)(t; 0, p0) = p0
sin(t)
cos(t)
(1.60)
with frequency Ω(p0) = 1. Using the explicit form (1.60) we introduce action angle coordinates by setting
( ˇQ, ˇP) = Φ(ϕ, h) =: P(h)
sin(ϕ)
cos(ϕ)
ϕ ∈ R, h ∈ J (1.61)
where the map P is chosen appropriate and satisfies P(0) = 0. The formal transformation of system
(1.59) into these new coordinates yields
˙ϕ = 1 + ε cos2
(ϕ) (P(h) cos(ϕ) − ε H) + 2 ε
H
P(h)
sin(ϕ)
˙h = ε P(h)
d
dh P(h)
sin(ϕ) cos(ϕ) (P(h) cos(ϕ) − ε H) − 2 ε
H
P(h)
cos(ϕ)
˙H = −H.
(1.62)
As we are in the situation where 0 ∈ J holds, we have P(0) = 0 such that (1.62) is singular in h = 0.
(Note that by definition (1.61), h = 0 corresponds to ( ˇQ, ˇP) = (0, 0) and therefore the periodic solution
(ˇq(t, ε), ˇp(t, ε)) arising near the elliptic fixed point of the unperturbed system).
The extension of the action angle coordinates into h = 0 therefore may not be performed straightforward,
but requires some preliminary preparations.
More precisely one may see that (1.62) is singular due to the fact that the right hand side of the ( ˇQ, ˇP)–
subsystem in (1.59) does not vanish for ( ˇQ, ˇP) = 0. We therefore prepare (1.59) by applying a suitable
transformation:
As the set
( ˇQ, ˇP) ∈ R2
( ˇQ, ˇP) = (−εH, εH) (1.63)

37. 1.3. Some Illustrative Examples 23
is invariant with respect to (1.59), the transformation
( ˇQ, ˇP) = (−εH, εH) + (Q, P) (1.64)
may be performed, yielding the system
˙Q = P + εP (P + ε H)
˙P = −Q
˙H = −H.
(1.65)
Here the right hand side of the ( ˙Q, ˙P)–equation vanishes for (Q, P) = (0, 0), hence the H–axis is invariant
with respect to (1.65). Applying (1.61) on (1.65) then yields
˙ϕ = 1 + ε cos2
(ϕ) (P(h) cos(ϕ) + ε H)
˙h = ε P(h)
d
dh P(h)
sin(ϕ) cos(ϕ) (P(h) cos(ϕ) + ε H)
˙H = −H.
(1.66)
Following the properties of P assumed in 1.97 a, this system admits a Cr+5
–extension into h = 0.
1.3.4 Example 4 (reasons to introduce the map P)
Let us rewrite transformation (1.61) of example 3 for P(h) =
√
2 h :
( ˇQ, ˇP) =
√
2 h
sin(ϕ)
cos(ϕ)
The solution (q, p)(t; 0,
√
2 h) with initial value (0,
√
2 h) at time t = 0 of the corresponding Hamiltonian
system satisfies H((q, p)(t; 0,
√
2 h)) = h for all t ∈ R. Hence for this choice of P, the action variable h
may be viewed as the ”energy” of the solutions. Although these action angle coordinates appear to be
suitable, the corresponding system is not differentiable in h = 0 :
˙ϕ = 1 + ε cos2
(ϕ)
√
2 h cos(ϕ) + ε H
˙h = ε h sin(ϕ) cos(ϕ)
√
2 h cos(ϕ) + ε H
˙H = −H.
In order to extend the corresponding system in action angle coordinates into h = 0 in a sufficient regular
way, assumption 1.97 b on the map P therefore is essential.
Additionally we will see in what follows, that the region of the phase space on which the result given in
[13] may be applied to, must be invariant. Due to this assumption, it may be necessary to introduce a
”cutting function” in order to change the vector field locally, if dealing with regions having non-invariant
boundaries. This may be achieved by choosing P in a suitable way. However, inside the regions (on
any compact subset) P may be taken as the identity P(h) = h. In the case 0 ∈ J we will see that the
set {h = 0} is invariant with respect to the corresponding system. Hence in this situation P(h) = h is
admissible even for small h ≥ 0.

38. 24 Chapter 1. Reduction to a Planar System
1.4 The Strongly Stable Manifold of the Equilibrium Point
Consider system (1.16) for ε = 0. In this unperturbed case the ( ˇQ, ˇP)–hyperplane {H = 0} is a center
manifold of the fixed point ( ˇQ, ˇP, H) = 0 (cf. figure 1.1 where d = 1).
Similarly we find the H–subspace {( ˇQ, ˇP) = 0} to be an
Q
P
H
Figure 1.1: The center manifold and the
stable manifold in the unperturbed case
invariant manifold of (1.16). It contains all solutions lim-
iting in the origin. Hence the H–space corresponds to the
stable manifold of the origin. More generally it may be con-
sidered as an invariant manifold which contains the origin
and may be represented as the graph of the constant map
Rd
∋ H → 0 ∈ R2
.
The aim of this section is to show that in the perturbed case
where ε = 0 (but small) such an invariant graph containing
the origin exists as well. More precisely we will prove the
existence of an invariant manifold of the perturbed system
which contains the origin and may be written as the graph of a (time–dependent) function
V : R × Rd
× R ∋ (t, H, ε) → V(t, H, ε) ∈ R2
where V(t, H, 0) = 0. As demonstrated in section 1.3.3 such an invariant manifold may be used to prepare
the extension of the domain of action angle coordinates if considering regions close to an elliptic fixed
point (i.e. the case 0 ∈ J considered in 1.97 c).
Although the definition of the stable manifold of the origin is unique in the unperturbed situation, the
notion of a stable manifold in the perturbed case may be generalized in different ways. There are basically
two approaches found in literature, based on different aspects of the unperturbed stable manifold:
• As the unperturbed stable manifold consist of all solutions limiting to the fixed point, the perturbed
stable manifold may equally be defined as the set of all orbits appoaching the origin as t → ∞.
However, since the origin is not hyperbolic in our situation various bifurcation scenarios are possible
if ε = 0. As for instance the origin may become globally attractive such that the stable manifold of
the perturbed system would be given by the entire phase space.
• On the other hand, the spectrum of the linearization of the perturbed system may always be divided
into a subset of eigenvalues with real parts of size O(ε) (i.e. the perturbed ”center”– eigenvalues)
and a part of eigenvalues with real parts of size O(1) (the perturbed ”stable”– eigenvalues). From
this point of view, the stable manifold could be defined via the eigenspace corresponding to the
perturbed ”stable”– eigenvalues. This would yield the invariant manifold which consists of the
solutions with the strongest rates of attraction towards the origin.
The definitions for the stable manifold of the perturbed system found in literature are usually based on
either of these two approaches. For our purpose it will be sufficient to content ourselves to establish the
existence of an invariant graph of a map V. Since this approach corresponds to the second approach
listed above, we will refer to this manifold as to the strongly stable manifold.

39. 1.4. The Strongly Stable Manifold of the Equilibrium Point 25
1.4.1 The Existence of the Strongly Stable Manifold
In this first subsection we will state the existence of the strongly stable manifold of system (1.16) for
small parameters ε. The theory found in various contributions (see [8], [10]), which may be applied to
establish the existence of a strongly stable manifold deals with the special case where the linearization of
the perturbation vanishes at the origin.
Thus we are not in the position to apply these results directly5
. However it is possible to modify the
program carried out in [8] in a way such that the statements needed here may be established. We therefore
will not verify all details but confine ourselves with a sketch of the adapted proof strategy.
The main idea to proceed in the more general case where the linearization of the perturbation does not
vanish at the origin consist in writing the map V using a linear map Vλ in the form
V(t, H, ε) := λ Vλ(t, ε/λ2
, H) H (1.67)
where the existence of Vλ is obtained by a contraction mapping argument and λ is a sufficiently small,
fixed parameter. This will be demonstrated in the proof of the following proposition :
Proposition 1.4.1 Given any ̺ > 0 there exists an ε2 = ε2(r, ̺) as well as a map V defined for t ∈ R,
|H| < ̺, |ε| ≤ ε2 with values in R2
and of class Cr+7
(where all derivatives up to order r+7 are uniformly
bounded by 1) such that the graph
Nε := (t, ( ˇQ, ˇP), H) ∈ R × R2
× Rd
( ˇQ, ˇP) = V(t, H, ε), |H| < ̺ (1.68)
is an invariant set of (1.16). Moreover the map V satisfies the following properties :
1. V(t, 0, ε) = 0
2. V(t, H, 0) = 0
3. V is 2π–periodic with respect to t.
The proof of this proposition is carried out in several steps.
• The first step consist in simplifying the notation as follows : Given any fixed 0 < λ < 1 we set
(x, y) := (H, ( ˇQ, ˇP))
ϑ := (t, ǫ) := (t, ε/λ2
).
(1.69)
Using these abbreviations we will rewrite system (1.16) in autonomous form. The independent
variable will be denoted by s and differentiation with respect to s is marked by a dot again (i.e. ˙ϑ).
• Lemma 1.4.2 There exist maps X0, Y0, Y1 and Y2 defined for t ∈ R, |ǫ| < ε1, |x| < ̺ and y ∈ R2
as well as a matrix B ∈ R2×2
such that (1.16) is equivalent to the (autonomous) system
˙ϑ = a
˙x = A x + λ2
X0(ϑ, y; λ) y
˙y = B y + λ2
Y0(ϑ, y; λ) x + λ2
Y1(ϑ, y; λ) y + Y2(y)(y, y)
(1.70)
5The author of this thesis did not find a way to reproduce an estimate analogous to equation (32) in [8] for the situation
discussed there in section 6, i.e. the perturbed case. (For an illustrative example, consider the system ˙x = −x+ε y, ˙y = ε x.)
This eventually gave rise to the modification introduced here.

40. 26 Chapter 1. Reduction to a Planar System
for |x| < ̺, where a =
1
0
. Moreover the following statements are true :
1.71 a. X0, Y0 and Y1 vanish for ϑ = (t, 0), i.e. ǫ = 0.
1.71 b. X0, Y0 and Y1 are ω := (2π, 0)–periodic with respect to ϑ.
1.71 c. X0, Y0 and Y1 are of class Cω
. Hence there exists a b0 < ∞ such that all derivatives up to
order r + 4 are bounded by b0, uniformly with respect to t ∈ R, |x| < ̺, y ∈ R2
and ǫ < ε1.
1.71 d. ℜ(σ(B)) = 0.
Recall that by A we denote the diagonalizable matrix of system (1.1), satisfying ℜ(σ(A)) ≤ −c0
(cf. GA 1.2).
PROOF: For x = H, y = ( ˇQ, ˇP), ϑ = (t, ε/λ2
) we define the quantities X0, Y0, Y1, Y2 and B as
follows:
X0(ϑ, y; λ) :=
1
λ2
1
0
∂( ˇQ, ˇP )
ˇG(σ y, t, ǫ λ2
) dσ Y0(ϑ, y; λ) :=
1
λ2
∂H
ˇF(y, 0, t, ǫ λ2
)
Y1(ϑ, y; λ) :=
1
λ2
1
0
∂( ˇQ, ˇP )
ˇF(σ y, 0, t, ǫ λ2
) dσ Y2(y) :=
1
0
(1 − σ)JD3
H(σ y) dσ
B := JD2
H(0, 0).
As shown in proposition 1.2.4 the map ˇF vanishes for (x, y) = (0, 0) and is affine with respect to
x = H. Hence taking into account that ∂H
ˇF does not depend on x we have
ˇF(y, x, t, ε) = ˇF(y, x, t, ε) − ˇF(y, 0, t, ε) + ˇF(y, 0, t, ε) − ˇF(0, 0, t, ε)
=
1
0
d
dσ
ˇF(y, σ x, t, ε) dσ +
1
0
d
dσ
ˇF(σ y, t, 0, ε) dσ
=
1
0
∂H
ˇF(y, 0, t, ǫ λ2
) x dσ +
1
0
∂(Q,P )
ˇF(σ y, 0, t, ǫ λ2
) y dσ
= λ2
Y0(ϑ, y; λ) x + λ2
Y1(ϑ, y; λ) y.
Using the integral representation of the Taylor remainder term and taking into account ∇H(0, 0) = 0
we find
J∇H( ˇQ, ˇP) = J∇H(0, 0) + JD2
H(0, 0) ˇQ, ˇP +
1
0
(1 − σ)JD3
H(σ y)(y, y) dσ
= B y + Y2(y)(y, y).
Additionally it follows from ˇG(0, 0, t, ε) = 0 that
ˇG(y, t, ε) =
1
0
d
dσ
ˇG(σ y, t, ε) dσ =
1
0
∂( ˇQ, ˇP)
ˇG(σ y, t, ε) y dσ = λ2
X0(ϑ, y; λ) y.

41. 1.4. The Strongly Stable Manifold of the Equilibrium Point 27
• In a next step we define an appropriate space for the maps V used in the ansatz (1.67) :
Definition 1.4.3 Let Xj
denote the following subspace of Cj
–maps taking values in the space
L(Rd
, R2
) of d × 2–matrices :
Xj
:= V ∈ Cj
(R × (−ε1, ε1) × Rd
, L(Rd
, R2
)) V satisfies (1.73 a)–(1.73 c) , (1.72)
where
1.73 a. V is ω–periodic with respect to ϑ
1.73 b. V (ϑ, x) = 0 if ϑ = (t, 0)
1.73 c. V X j < ∞ with V X j := max
α∈N 2+d
0≤|α|≤j
sup
t∈R
|ǫ|≤ε1
sup
|x|<̺
∂ α
(ϑ,x)V (ϑ, x) .
Note that for any multi–index α ∈ N2+d
, |α| := α1 + · · ·+ α2+d and ∂ α
(ϑ,x) := ∂ α1
t ∂ α2
ǫ ∂ α3
x1
. . . ∂
α2+d
xd .
Then (Xj
, . X j ) is a Banach space.
• For any V ∈ Xr+7
we substitute y = λ V (ϑ, x) x into the perturbation terms of (1.70), i.e. consider
the systems
˙ϑ = a
˙x = A x + λ3
X0(ϑ, λ V (ϑ, x) x; λ) V (ϑ, x) x
(1.74)
and
˙y = B y + λ2
Y0(ϑ, λ V (ϑ, x) x; λ) x + λ3
Y1(ϑ, λ V (ϑ, x) x; λ) V (ϑ, x) x
+ λ2
Y2(λ V (ϑ, x) x)(V (ϑ, x) x, V (ϑ, x) x).
(1.75)
Let (ϑ, x)(s) := (ϑ, x)(s; ϑ0, x0; V ) denote the solution of (1.74) with initial value (ϑ0, x0) at time
s = 0 (where ϑ0 := (t0, ε0)) depending on V . We then will show that there exists a Vλ ∈ Xr+7
,
such that
y(s) := λ Vλ((ϑ, x)(s; ϑ0, x0; Vλ))) x(s)
is a solution of (1.75) for V = Vλ. This, however implies immediately that (ϑ, x, y)(s) is a solution
of (1.70). We will establish the existence of such a Vλ in an analogous way to the process given in
[8]. In particular the rescalation parameter λ is necessary to obtain sufficient regularity.
• For any fixed V ∈ BX r+8 (1) where
BX r+8 (1) := V ∈ Xr+8
V X r+8 ≤ 1
the following lemma presents a result on the fundamental solutions associated with (1.74):
Lemma 1.4.4 For any initial value (ϑ0, x0) and any V ∈ BX r+8 (1) let U(s) = U(s; ϑ0, x0; V )
denote the unique solution of
˙U(s) = A + λ3
X0(ϑ(s), λ V (ϑ(s), x(s)) x(s); λ) V (ϑ(s), x(s)) U(s) (1.76)
satisfying U(0) = IRd . Then x(s; ϑ0, x0; V ) = U(s; ϑ0, x0; V ) x0.
Moreover there exists λ1 > 0 and a polynomial π(s) with positive coefficients such that for 0 < λ <
λ1 and |x0| < ̺,
|U(s; ϑ0, x0; V )| ≤ e−
c0
2 s
∂ α
(ϑ0,x0)U(s; ϑ0, x0; V ) ≤ e−
c0
2 s
λ3
π(s) 0 < |α| ≤ r + 8.

42. 28 Chapter 1. Reduction to a Planar System
This lemma 1.4.4 is proved by induction with respect to the length |α| of the multi–index α. The
induction is carried out using the notion of the logarithmic norm (introduced in the following
definition 1.4.5) and the statement given in lemma 1.4.6 :
Definition 1.4.5 Following Stroem [18] we introduce the so–called logarithmic norm of a matrix
M ∈ Rn×n
by
µ (M) := lim
δ→0+
|IRn + δ M| − 1
δ
,
where |.| denotes the matrix norm based on the norm chosen on Rn
.
As a simple consequence of lemma 2 in [18] we find
Lemma 1.4.6 Consider a solution W(s) of the inhomogenous, non–autonomous linear differential
equation
˙W(s) = M(s) W(s) + N(s)
where M(s), N(s) are time–dependent linear operators on Rd
, the logarithmic norm µ(M(s)) is
uniformly bounded by −c0
2 and |N(s)| ≤ λ3
e−
c0
2 s
˜π(s) (˜π is a polynomial with positive coefficients).
Then
|W(s)| ≤ e−
c0
2 s
|W(0)| + λ3
π(s) s ≥ 0.
where π(s) =
s
0
˜π(t) dt has positive coefficients as well.
• As mentioned above, the existence of a map Vλ defining an invariant manifold (see (1.67), (1.68))
is established using the contraction mapping theorem. The definition of the mapping considered
and the proof of its contracting properties are the subject of the next step in this line:
Lemma 1.4.7 There exists a λ2 := λ2(r, ̺) > 0 such that for every V ∈ BX r+8 (1), 0 < λ < λ2,
the image T V of the map T , given by
T V (ϑ0, x0) = − 1
λ
∞
0
e−sB
λ2
Y0(ϑ, λ V (ϑ, x) x; λ) U
+ λ3
Y1(ϑ, λ V (ϑ, x) x; λ) V (ϑ, x) U
+ λ2
Y2(λ V (ϑ, x) x)(V (ϑ, x) x, V (ϑ, x) U) ds
(1.77)
exists. Recall that (ϑ, x)(s) = (ϑ, x)(s; ϑ0, x0; V ), U(s) = U(s; ϑ0, x0; V ) denote solutions of (1.74),
(1.76) respectively.
Moreover, the map T is a contraction from BX r+8 (1) to BX r+8 (1) with respect to the Xr+7
–topology
induced on Xr+8
, i.e.
1.78 a. T V ∈ BX r+8 (1)
1.78 b. T V1 − T V2 X r+7 ≤ 1
2 V1 − V2 X r+7 for all V1, V2 in BX r+8 (1).
The way followed to establish this statement is similar to the one given in [8], p. 558–561. The
estimates found in lemma 1.4.4 are used repeatedly. Furthermore one has to apply lemma 1.4.6 to
derive the scalar bounds for ∂ α
(ϑ,x)T V , ∂ α
(ϑ,x) (T V1 − T V2), respectively.

43. 1.4. The Strongly Stable Manifold of the Equilibrium Point 29
• In order to complete the proof of proposition 1.4.1, let Vλ ∈ Xr+7
denote the unique fixed point
of T , which exists by the contraction mapping theorem. Then the group property of the flow
(ϑ, x)(s; . , . ; Vλ), i.e.
(ϑ, x)(˜s; (ϑ, x)(s; ϑ0, x0; Vλ); Vλ) = (ϑ, x)(s + ˜s; ϑ0, x0; Vλ)
together with Vλ = T Vλ implies that the function
y(s; ϑ0, x0; Vλ) := λ Vλ((ϑ, x)(s; ϑ0, x0; Vλ)) x(s; ϑ0, x0; Vλ)
satisfies(1.75). Hence it eventually follows that fixing any 0 < λ < λ2 and setting ε2 := ε1 λ2
, the
map
V(t, H, ε) := λ Vλ(t, ε/λ2
, H) H t ∈ R, |H| < ̺, |ε| < ε2 (1.79)
defines an invariant manifold with the properties claimed in proposition 1.4.1.
The following remark on the parametrization V of the strongly stable manifold will help us to find an
appropriate representation of the vector field when performing a transformation into the strongly stable
manifold (see next section).
Remark 1.4.8 The map V asserted in proposition 1.4.1 satisfies the following partial differential equation
∂tV(t, H, ε) = J∇H(V(t, H, ε)) + ˇF(V(t, H, ε), H, t, ε) − ∂HV(t, H, ε) A H + ˇG(V(t, H, ε), t, ε) .
PROOF: Since for any solution ( ˇQ, ˇP) = V(t, H, ε) of (1.16) we have
( ˙ˇQ, ˙ˇP) = J∇H(V(t, H, ε)) + ˇF(V(t, H, ε), H, t, ε)
=
d
dt
V(t, H, ε)
= ∂tV(t, H, ε) + ∂HV(t, H, ε) A H + ˇG(V(t, H, ε), t, ε)
independent of the solution ( ˇQ, ˇP) considered, the statement follows at once.

44. 30 Chapter 1. Reduction to a Planar System
1.4.2 The Transformation into the Strongly Stable Manifold
The aim of this section is to transform the ”H–axis” {( ˇQ, ˇP) = 0} of system (1.16) ”into the strongly
stable manifold” Nε (as motivated in (1.64)). We will denote the new coordinates by (Q, P) and calculate
the transformed vector field of (1.16) with respect to these new coordinates. As seen in section 1.3.3
we then expect the H–axis {(Q, P) = (0, 0)} to be invariant with respect to the transformed system. In
order to prepare the discussions to follow, we are interested in deriving representations of the transformed
vector field, similar to (1.17). Hence we will compute the terms of order O(ε) and O(ε2
) in an explicit
form.
The leading ε–terms of V may be calculated in an easy way using the contraction T introduced in (1.77).
More precisely one has to expand the fixed point equation Vλ(t0, ǫ0, x0) = T Vλ(t0, ǫ0, x0) with respect to
ǫ0. Taking into account that D3
H(0, 0) = 0 (GA 1.1a) one then applies (1.69), (1.79) to (1.77), yielding
the identity
V(t0, H0, ε) = ε V1
(t0)H0 + ε2
V2
(t0, H0) H0 + ε3
V3
(t0, H0, ε) H0 (1.80)
where
V1
(t0) =
0
∞
e−sB
∂H
ˇF1
(0, 0, 0, s + t0) esA
ds
V2
(t0, H0) =
0
∞
e−sB
∂H
ˇF2
(0, 0, 0, s + t0) + ∂( ˇQ, ˇP )
ˇF1
(0, 0, 0, s + t0) V1
(s + t0) esA
+ e−sB
∂H∂( ˇQ, ˇP )
ˇF1
(0, 0, 0, s + t0) V1
(s + t0) H0, esA
ds.
(1.81)
As assumed in GA 1.1a, GA 1.2 the matrices A and B are diagonalizable such that the exponentials esA
,
e−sB
admit the representation
esA
=
λ∈σ(A)
es λ
TA,λ TA,λ ∈ Cd×d
e−sB
=
ω∈σ(B)
e−s ω
TB,ω TB,ω ∈ C2×2
(1.82)
and the eigenvalues λ ∈ σ(A) have all negative real part, the eigenvalues ω ∈ σ(B) purely imaginary. In
a straightforward calculation one therefore obtains from (1.20)
V1
(t0) =
|n|≤N
eint0
V1
n
V2
(t0, H0) = V2
0 (t0) + V2
1 (t0, H0) :=
|n|≤2N
V2
n,0 + V2
n,1(H0) eint0
(1.83)
where V2
1 is linear with respect to H0 and we have set
V1
n :=
λ∈σ(A)
ω∈σ(B)
(in − ω + λ)−1
TB,ω ∂H
ˇF1
n(0, 0, 0) TA,λ (1.84)

45. 1.4. The Strongly Stable Manifold of the Equilibrium Point 31
and
V2
n,0 :=
λ∈σ(A)
ω∈σ(B)
(in − ω + λ)−1
TB,ω ∂H
ˇF2
n(0, 0, 0) TA,λ
+
|¯n|,|˜n|≤N
¯n+˜n=n
λ∈σ(A)
ω∈σ(B)
(i(¯n + ˜n) − ω + λ)−1
TB,ω ∂( ˇQ, ˇP )
ˇF1
¯n(0, 0, 0) V1
˜n TA,λ
V2
n,1(H0) :=
|¯n|,|˜n|≤N
¯n+˜n=n
λ∈σ(A)
ω∈σ(B)
(i(¯n + ˜n) − ω + λ)
−1
TB,ω ∂H∂( ˇQ, ˇP )
ˇF1
¯n(0, 0, 0)(V1
˜n H0, TA,λ).
(1.85)
We now are in the position to introduce the transformation announced and to derive an explicit formula
for the ε–expansion of the transformed vector field.
Proposition 1.4.9 For any ̺ > 0, t ∈ R, |H| < ̺ and ε < ε2(r, ̺) we consider the change of coordinates
given by
(( ˇQ, ˇP), H, t, ε) = ((Q, P) + V(t, H, ε), H, t, ε). (1.86)
Then the following statements are true:
• System (1.16) transforms into
( ˙Q, ˙P) = J∇H(Q, P) + ˆF(Q, P, H, t, ε)
˙H = A H + ˆG(Q, P, H, t, ε),
(1.87)
where the maps ˆF, ˆG are of class Cr+7
, 2π–periodic with respect to t and
ˆF(0, 0, H, t, ε) = 0 ˆG(0, 0, 0, t, ε) = 0
ˆF(Q, P, H, t, 0) = 0 ˆG(Q, P, H, t, 0) = 0.
(1.88)
• The mappings ˆF, ˆG admit a representation6
of the form
ˆF(Q, P, H, t, ε) =
3
j=1
εj ˆFj
(Q, P, H, t) + ε4 ˆF4
(Q, P, H, t, ε)
ˆG(Q, P, H, t, ε) =
2
j=1
εj ˆGj
(Q, P, H, t) + ε3 ˆG3
(Q, P, H, t, ε)
(1.89)
6for the application in chapter 4 it suffices to consider the expansions including terms of order O(ε2) of ˆF and of order
O(ε) of ˆG.

46. 32 Chapter 1. Reduction to a Planar System
and more explicitely
ˆF1
(Q, P, H, t) = J D2
H(Q, P) − D2
H(0, 0) V1
(t) H
+ ˇF1
(Q, P, H, t) − ˇF1
(0, 0, H, t)
ˆF2
(Q, P, H, t) = J D2
H(Q, P) − D2
H(0, 0) V2
(t, H) H
+ 1
2 JD3
H(Q, P)(V1
(t)H)[2]
+ ˇF2
(Q, P, H, t) − ˇF2
(0, 0, H, t)
+ ∂( ˇQ, ˇP )
ˇF1
(Q, P, H, t) − ∂( ˇQ, ˇP )
ˇF1
(0, 0, H, t) V1
(t) H
− V1
(t) ˇG1
(Q, P, t)
(1.90)
as well as
ˆG1
(Q, P, H, t) = ˇG1
(Q, P, t)
ˆG2
(Q, P, H, t) = ˇG2
(Q, P, t) + ∂( ˇQ, ˇP )
ˇG1
(Q, P, t) V1
(t) H.
(1.91)
• The map ˆF3
may be written in the form
ˆF3
(Q, P, H, t) = ˇF3
(Q, P, 0, t) − ˇF3
(0, 0, 0, t)
− V1
(t) ˇG2
(Q, P, t) − ˇG2
(0, 0, t) + ˆF3,1
(Q, P, H, t)H
(1.92)
for a suitable map ˆF3,1
: R2
× Rd
× R → L(Rd
, R2
).
• Finally, ˆF1
, ˆF2
, ˆG1
and ˆG2
may be represented as Fourier polynomials in t, i.e.
ˆFj
(Q, P, H, t) =
|n|≤jN
ˆFj
n(Q, P, H, t) eint ˆGj
(Q, P, H, t) =
|n|≤jN
ˆGj
n(Q, P, H, t) eint
.
(1.93)
Note that although we write H in the arguments of ˆG1
in (1.89) for simplicity, this map does not depend
on H.
PROOF: Taking the time derivative of transformation (1.86) and using (1.16) we find
( ˙Q, ˙P) = J∇H((Q, P) + V(t, H, ε)) + ˇF((Q, P) + V(t, H, ε), H, t, ε)
−∂tV(t, H, ε) − ∂HV(t, H, ε) A H + ˇG((Q, P) + V(t, H, ε), t, ε) .
which together with the identity found for ∂tV(t, H, ε) in remark 1.4.8 yields
( ˙Q, ˙P) = J∇H((Q, P) + V(t, H, ε)) − J∇H(V(t, H, ε))
+ ˇF((Q, P) + V(t, H, ε), H, t, ε) − ˇF(V(t, H, ε), H, t, ε)
−∂HV(t, H, ε) ˇG((Q, P) + V(t, H, ε), t, ε) − ˇG(V(t, H, ε), t, ε) .
Setting
ˆF(Q, P, H, t, ε) := J∇H((Q, P) + V(t, H, ε)) − J∇H(V(t, H, ε)) − J∇H(Q, P)
+ ˇF((Q, P) + V(t, H, ε), H, t, ε) − ˇF(V(t, H, ε), H, t, ε)
− ∂HV(t, H, ε) ˇG((Q, P) + V(t, H, ε), t, ε) − ˇG(V(t, H, ε), t, ε)
(1.94)

47. 1.4. The Strongly Stable Manifold of the Equilibrium Point 33
we find ˆF to be of class Cr+7
(since V ∈ Cr+7
) and
( ˙Q, ˙P) = J∇H(Q, P) + ˆF(Q, P, H, t, ε).
Expanding ˆF with respect to V(t, H, ε) yields
ˆF(Q, P, H, t, ε) = JD2
H(Q, P) − JD2
H(0, 0) V(t, H, ε)
+1
2 JD3
H(Q, P) − JD3
H(0, 0) V(t, H, ε)[2]
+O(V(t, H, ε)[3]
)
+ ˇF(Q, P, H, t, ε) − ˇF(0, 0, H, t, ε)
+ ∂( ˇQ, ˇP )
ˇF(Q, P, H, t, ε) − ∂( ˇQ, ˇP )
ˇF(0, 0, H, t, ε) V(t, H, ε)
+1
2 ∂2
( ˇQ, ˇP )
ˇF(Q, P, H, t, ε) − ∂2
( ˇQ, ˇP )
ˇF(0, 0, H, t, ε) V(t, H, ε)[2]
+O(V(t, H, ε)[3]
)
−∂HV(t, H, ε) ˇG(Q, P, t, ε) − ˇG(0, 0, t, ε)
+ ∂( ˇQ, ˇP )
ˇG(Q, P, t, ε) − ∂( ˇQ, ˇP )
ˇG(0, 0, t, ε) V(t, H, ε)
+1
2 ∂2
( ˇQ, ˇP )
ˇG(Q, P, t, ε) − ∂2
( ˇQ, ˇP )
ˇG(0, 0, t, ε) V(t, H, ε)[2]
+O(V(t, H, ε)[3]
) .
Plugging in the expansion of V(t, H, ε) as given in (1.80), i.e.
V(t, H, ε) = ε V1
(t)H + ε2
V2
(t, H) H + ε3
V3
(t, H, ε),
we conclude
ˆF(Q, P, H, t, ε) = ε JD2
H(Q, P) − JD2
H(0, 0) V1
(t) H
+ ˇF1
(Q, P, H, t) − ˇF1
(0, 0, H, t)
+ε2
JD2
H(Q, P) − JD2
H(0, 0) V2
(t, H) H
+1
2 JD3
H(Q, P) − JD3
H(0, 0) V1
(t) H
[2]
+ ˇF2
(Q, P, H, t) − ˇF2
(0, 0, H, t)
+ ∂( ˇQ, ˇP )
ˇF1
(Q, P, H, t) − ∂( ˇQ, ˇP )
ˇF1
(0, 0, H, t) V1
(t) H
−V1
(t) ˇG1
(Q, P, t) − ˇG1
(0, 0, t)
+ε3 ˇF3
(Q, P, 0, t) − ˇF3
(0, 0, 0, t) − V1
(t) ˇG2
(Q, P, t) − ˇG2
(0, 0, t)
+ε3
O(H) + O(ε4
).
(Take into account that the terms included in O(V(t, H, ε)[3]
) are of order ε3
or higher and vanish for
H = 0).
Since D3
H(0, 0) = 0, ˇF3
(0, 0, 0, t) = 0 (cf. GA 1.1b, proposition 1.2.4) the formulae (1.90), (1.92) given
in the claim are established. The representation of ˆG(Q, P, H, t, ε) is found in an easier way :
˙H = A H + ˇG( ˇQ, ˇP, t, ε)
= A H + ˇG((Q, P) + V(t, H, ε), t, ε).

48. 34 Chapter 1. Reduction to a Planar System
Define ˆG(Q, P, H, t, ε) := ˇG((Q, P) + V(t, H, ε), t, ε), then ˆG ∈ Cr+7
, ˙H = A H + ˆG(Q, P, H, t, ε) and
ˆG(Q, P, H, t, ε) = ˇG(Q, P, t, ε) + ∂( ˇQ, ˇP)
ˇG(Q, P, t, ε)V(t, H, ε)
+1
2 ∂2
( ˇQ, ˇP )
ˇG(Q, P, t, ε)V(t, H, ε)[2]
+ O(V(t, H, ε)[3]
)
= ε ˇG1
(Q, P, t) + ε2 ˇG2
(Q, P, t) + ∂( ˇQ, ˇP )
ˇG1
(Q, P, t) V1
(t)H + O(ε3
)
which corresponds to (1.91).
The last statement of proposition 1.4.9 is obtained by plugging (1.93) and (1.83) into the representations
(1.90), (1.91) respectively.
Note that since we have used the non–autonomous representation (1.16), the independent variable cor-
responds to t again. Hence ˙Q etc. denote the derivatives with respect to t.
Remark 1.4.10 It may be readily seen that if substituting F, G by ˆF, ˆG system (1.87) fulfills the
assumptions made in GA 1.1–GA 1.3. By consequence of the transformations carried out the identities
(1.88) hold and the vector fields ˆF, ˆG are of class Cr+7
.
In the next section we will consider systems of this type in general and introduce action angle coordinates.

49. 1.5. The Action Angle Coordinates 35
1.5 The Action Angle Coordinates
In this section we present a possible way to introduce action angle coordinates in regions of periodic
solutions of plane Hamiltonian systems. These action angle coordinates will be helpful to establish the
existence of an attractive invariant manifold and to apply averaging methods on (1.87). However the
steps carried out in this section may be applied on any system of the form
( ˙Q, ˙P) = J∇H(Q, P) + ˆF(Q, P, H, t, ε)
˙H = A H + ˆG(Q, P, H, t, ε),
(1.95)
provided that replacing F, G by ˆF, ˆG, the properties assumed in GA 1.1–GA 1.3 are fulfilled, ˆF, ˆG are
of class Cr+7
and
ˆF(0, 0, H, t, ε) = 0 ˆG(0, 0, 0, t, ε) = 0 (1.96)
holds as well (cf. remark 1.4.10).
In the first section 1.5.1 we define the action angle coordinates and discuss some of their properties.
In section 1.5.2 we then introduce a system in action angle coordinates being equivalent to (1.95) in a
sense. As we are interested in considering regions close to the fixed point (Q, P, H) = (0, 0, 0) as well,
we eventually will show that the system introduced provides sufficient information on the qualitative
behaviour of (1.95) in a neighbourhood of the origin. The purpose of the last section 1.5.3 is to give
an alternative representation of the system in action angle coordinates, aiming at the discussion of the
stability of the origin. Moreover we will prove a result on the regularity of this vector field.
1.5.1 The Definition of the Action Angle Coordinates
Consider an interval J as in GA 1.1b such that the solutions (q, p)(t; 0, p0) of (1.2) with initial value
(0, p0), p0 ∈ J at time t = 0 are periodic in t with frequency Ω(p0) > 0. The initial values of these periodic
solutions give rise to the definition of the action–coordinate. However we admit the action–coordinate h
not necessarily to correspond to p0 directly but to be defined via a further transformation, i.e. p0 = P(h).
For instance, such a change of coordinates may consist in mapping the initial values p0 into the energy
H(0, p0) of the corresponding solutions. As seen in section 1.3.4 this possibly causes regularity problems.
If no transformation is performed at all (i.e. P(h) = h) then the domain of the action–coordinates depends
on J . We prefer the domain of the action–coordinate h to be R, thus independent of J . As we will see
in what follows, it is not necessary to fix the transformation any further at all. Therefore we consider
any mapping P which fulfills the following properties:
1.97 a. P ∈ Cω
(R, R)
1.97 b. P : R → J is bijective and d
dh P(h) = 0 for h = 0.
1.97 c. If 0 ∈ J then P(0) = 0.
1.97 d. All the derivatives dk
dhk P(h), 1 ≤ k ≤ r + 5 are bounded uniformly with respect to h.
The angle–coordinate ϕ basically corresponds to the time variable of the periodic solutions of (1.2)
considered. Although the periods Ω of these solutions generally depend on the initial value P(h), the
angle coordinate ϕ is introduced in a way such that it is 2π–periodic, independent of the particular
solution.

50. 36 Chapter 1. Reduction to a Planar System
Using the solutions (q, p)(t; q0, p0) of the Hamiltonian system (1.2) we introduce a map Φ as follows:
Definition 1.5.1 Consider the maps Ω and P as in GA 1.1b, 1.97 a. We define the following quantities:
1. For any ϕ, h ∈ R let (˜q, ˜p) (ϕ, p0) := (q, p)( ϕ
Ω(p0) ; 0, p0) and set
Φ(ϕ, h) := (˜q, ˜p)(ϕ, P(h)). (1.98)
2. In order to shorten the notation we introduce the map
ω(h) := Ω(P(h)). (1.99)
The first lemma in this section gives a summary of a few properties of the map Φ.
Lemma 1.5.2 The following statements on the maps Ω, Φ are true:
1. The map Φ is of class Cω
(R2
, R2
) and 2π–periodic with respect to ϕ ∈ R.
2. If 0 ∈ J then
Φ(ϕ, 0) = 0. (1.100)
3. Let Ω0 denote the quantity introduced in GA 1.1a. Then
Ω(0) = Ω0. (1.101)
4. For all (ϕ, h) ∈ R2
the Jacobian determinant of Φ satisfies
det D Φ(ϕ, h) = ω(h)−1 d
dh H(0, P(h)). (1.102)
For 0 ∈ J this determinant tends towards zero, i.e. det D Φ(ϕ, h) → 0 as h → 0.
PROOF: The first two statements are simple consequences of GA 1.1 and 1.97 a together with the
definition of Φ. We therefore content ourselves with the proof of assertions 3 and 4.
In order to establish (1.101), let us rescale the (q, p)–coordinates of system (1.2) with a parameter λ > 0:
(q, p) = (λ¯q, λ¯p).
We rewrite the right hand side of system (1.2) in the form of a Taylor polynomial using the integral
formula for the remainder term which in addition with ∇H(0, 0) = 0 yields the expression
˙¯q
˙¯p
= JD2
H(0, 0)
¯q
¯p
+ λ
1
0
(1 − σ) J D3
H(σλ¯q, σλ¯p)(¯q, ¯p)[2]
dσ. (1.103)
Let (¯q, ¯p)(t; 0, ¯p0, λ) denote the solution of (1.103) with initial value (0, ¯p0) at time t = 0, where λ may
take any real value. Consider any p0 ∈ J . Then the function (q, p)(t; 0, λ ¯p0) is a solution of (1.2), with

51. 1.5. The Action Angle Coordinates 37
frequency Ω(λ ¯p0), as it follows from GA 1.1b. Since λ (¯q, ¯p)(t; 0, ¯p0, λ) = (q, p)(t; 0, λ ¯p0), (¯q, ¯p)(t; 0, ¯p0, λ)
has frequency Ω(λ ¯p0), too. For λ = 0 we find by (1.103)
¯q(t; 0, ¯p0, 0)
¯p(t; 0, ¯p0, 0)
= et J D2
H(0,0) 0
¯p0
=




cos(Ω0 t)
∂2
pH(0,0)
∂2
q H(0,0) sin(Ω0 t)
−
∂2
q H(0,0)
∂2
pH(0,0) sin(Ω0 t) cos(Ω0 t)




0
¯p0
(1.104)
Here we have used the assumptions made in GA 1.1a. Thus the frequency Ω(λ ¯p0) of (¯q, ¯p)(t; 0, ¯p0, λ)
tends towards Ω0 as λ → 0, i.e. Ω(0) = Ω0 indeed.
Let us establish the last statement claimed. By definition (1.98) of Φ we have to calculate
det DΦ(ϕ, h) = det
∂ϕ ˜q(ϕ, P(h)) d
dh ˜q(ϕ, P(h))
∂ϕ ˜p(ϕ, P(h)) d
dh ˜p(ϕ, P(h))
where
∂ϕ ˜q(ϕ, P(h)) = ω(h)−1
∂tq( ϕ
ω(h) ; 0, P(h)) = ω(h)−1
∂pH(Φ(ϕ, h))
∂ϕ ˜p(ϕ, P(h)) = ω(h)−1
∂tp( ϕ
ω(h) ; 0, P(h)) = −ω(h)−1
∂qH(Φ(ϕ, h)),
(1.105)
hence
det DΦ(ϕ, h) = ω(h)−1
∂pH(Φ(ϕ, h)) d
dh ˜p(ϕ, P(h)) + ∂qH(Φ(ϕ, h)) d
dh ˜q(ϕ, P(h))
= ω(h)−1 d
dh H(Φ(ϕ, h)).
As H is the Hamiltonian of (1.2),
H(Φ(ϕ, h)) = H((q, p)( ϕ
ω(h) ; 0, P(h))) = H((q, p)(0; 0, P(h))) = H(0, P(h)), (1.106)
thus
d
dh H(Φ(ϕ, h)) = d
dh H(0, P(h)), (1.107)
proving (1.102). For 0 ∈ J we have ω(0) = Ω(0) = Ω0 = 0 and since ∇H(0, 0) = 0
lim
h→0
d
dh H(0, P(h)) = lim
h→0
d
dh O(P(h)
2
) = 0.
Hence the proof of lemma 1.5.2 is complete.

52. 38 Chapter 1. Reduction to a Planar System
By consequence of GA 1.1c the following images of Φ are well defined:
Definition 1.5.3 Let (˜q, ˜p), Φ be the maps introduced in definition 1.5.1. Then we set
LJ := Φ(R, R) LJl
:= (˜q, ˜p) (R, Jl) LJr := (˜q, ˜p) (R, Jr). (1.108)
The indices J , Jl, Jr will remind us on the dependence of these quantities on the corresponding sets.
In figure 1.2 we have illustrated the situation in the case of the mathematical pendulum H(Q, P) =
P2
/2 + a
2
2
(1 − cos(Q)) for two choices of the set J , denoted by Ju, Jc.
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0
0
00
0
00
1
1
11
1
11
0
0
00
0
00
1
1
11
1
11
a
0000000000
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11111
0011
h
0000
00
1111
11
L
h
0011
J
Φ(ϕ, )00000
00000
0000000000
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11111
Φ(ϕ, )
u
c
u
J
J L
c−π π
J
P
Q
LJ
−π π
ϕ
h
= LJ
−π π
ϕ
h
LJ
LJ
r
l
l r
Figure 1.2: Illustration of the map Φ(ϕ, h) in the case of the mathematical pendulum
For the first choice Ju the domain of Φ depicted on the left hand side is mapped into a subset composed
by orbits of rotatory solutions of the pendulum equation, above the separatrix. The lower and upper
boundaries of the range LJ , i.e. LJl
and LJr are distinct. Moreover we see that the two hatched
subregions of the domain are mapped into two different ”strips” contained in LJ .
In the second case Jc we consider a map P satisfying P(h) = −P(−h) such that the images of the two
shaded subregions of the domain coincide. Due to the same reason, the sets LJl
and LJr are identical.
Moreover we emphasize that the origin (Q, P) = (0, 0) is contained in the range LJ .