Duffing oscillator and driven damped pendulum

Comparison between the Duﬃng oscillator and the driven
damped Pendulum
Bachelor Thesis
Yiteng Dang
Student Number: 3688356
Supervisor: Dr. Heinz Hanßmann
June 2013
Abstract
Nonlinear dynamical systems can give rise to chaotic phenomena resulting in complicated dy-
namics. Such systems can be studied both by analytic and numerical methods. Two such analytic
methods are Averaging and Melnikov’s Method, the latter of which predicts intersections between
stable and unstable manifolds which are linked to the presence of a horseshoe similar to Smale’s
horseshoe. The aim of this thesis is to study two chaotic nonlinear systems, the Duﬃng oscillator
and the driven damped pendulum, using these analytic techniques as well as numerical techniques.
Various features of these systems are analyzed and a comparison will be given in the end.
1

Preface
Many physical phenomena are described by nonlinear dynamical systems. Unlike linear systems, non-
linear systems do not always have nice properties such has having well-defined solutions that remain
bounded for finite times. Another feature nonlinear systems often exhibit is known as chaos. Although
this term has become heavily popularized over time, giving rise to titles of films, music albums and video
games, its scientific origin lies in describing dynamical systems that are hard to predict, due to being
extremely volatile with regard to changing conditions. While it is difficult to give a precise definition
of chaos, some main properties includes sensitivity to initial conditions, topological mixing and density
of periodic orbits. Topological mixing is a technical term referring to the property that any region in
phase space will eventually overlap with any other region in phase space given enough time. Density of
periodic orbits can be understood as the property that any point in space is approached arbitrarily close
by some periodic orbit. These concepts should become clearer to the reader after reading the concepts
in the first sections and seeing examples in the later sections.
It should not be surprising that few general properties can be derived about nonlinear systems. Never-
theless, a few statements can be made for a wide class of systems, and this will be dealt with in the first
section of this thesis. The reader is assumed to have basic knowledge of ordinary differential equations,
but no prior knowledge about nonlinear dynamical systems is assumed. Definitions will be given on the
way, and theorems will be mostly stated but not proven while a reference to a proof is usually given.
After establishing basic properties of nonlinear systems, we will discuss two analytic techniques that
allow us to investigate certain types of such systems. These are the Averaging Method and Melnikov’s
Method, both of which are applicable to certain systems that can be treated as perturbations of ’nicer’
systems. The latter will give a useful tool to predict intersections of the stable and unstable manifolds
of a system. The next section on Smale’s horseshoe is entirely meant to illustrate why we should be
interested in these intersections of the stable and unstable manifolds. The short answer is given in the
Smale-Birkhoff Homoclinic Theorem, but in order to have a decent understanding of this theorem it
would be good to have seen Smale’s classical horseshoe. Hence the section will first discuss this classical
horseshoe example and its chaotic dynamics, before going over to an explanation how a horseshoe can
be found in other dynamical systems.
After setting up these preliminary theoretical sections, we arrive at the core of the thesis. Two partic-
ular dynamical systems were chosen to be investigated: the Duffing oscillator and the driven damped
pendulum. The next two sections will explain in detail all kinds of features of these systems, employing
the theorems and tools we introduced in the first three sections. In particular, these sections are written
with a similar structure in mind, so that a fair comparison between the two systems can be given by
comparing subsections with the same or similar names one after each other. The analytic analysis will
be complemented by some numerical work on the Duffing oscillator, which parallels the numerical sim-
ulations of the driven damped pendulum presented elsewhere. Finally, we will lay these sections next to
each other and spell out the similarities and differences by giving a concluding comparison between the
systems.
As a student doing a combined bachelor programme in both mathematics and physics, I had many
options for doing a Bachelor Thesis. Doing lab work for several months did not interest me, neither
could I see the relevance of studying a purely mathematical subject for my future studies. Thus I looked
for a topic that had connections with both mathematics and physics and after some searching arrived
at dynamical systems. I know that differential equations appear everywhere in physics, and perhaps
theories in dynamical systems have been largely inspired by physical examples. Studying such a the-
ory would be different from learning abstract theory and being told that it will be useful in physics at
some later stage, but missing the link with physics entirely, which I experienced somewhat too often in
mathematics courses. In the course of writing this thesis, I have received frequent and close supervision
from Dr. Heinz Hanßmann and would like to thank him sincerely for giving me the idea for this thesis
and giving comments on my progress. Writing this thesis has introduced me to the interesting theory of
dynamical systems which I might not have approach so quickly otherwise.
Yiteng Dang
May 2013
2

Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1 Linear and nonlinear differential equations 4
2 Averaging and Melnikov’s Method 8
2.1 Averaging Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Melnikov’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 The Smale horseshoe 12
3.1 Horseshoes in other systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Hyperbolic structure and a theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 The Smale-Birkhoff Homoclinic Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4 Duffing’s equation 18
4.1 The undamped Hamiltonian system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 Fixed points and local stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.3 Global and structural stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.4 The homoclinic orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.5 The externally forced system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.6 Averaging applied to the Duffing oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.7 Melnikov’s Method applied to the Duffing oscillator . . . . . . . . . . . . . . . . . . . . . . 25
4.8 Numerical solutions of the Duffing oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . 27
5 Driven Damped Pendulum 33
5.1 The undamped Hamiltonian system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
5.2 The damped, unforced system and its fixed points . . . . . . . . . . . . . . . . . . . . . . 34
5.3 Implicit solution to the undamped system . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.4 The homoclinic orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.5 Inside the homoclinic orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.6 Melnikov’s Method applied to the DDP . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.7 Numerical solutions of the DDP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6 Comparison and discussion 39
References 41
3

1 Linear and nonlinear differential equations
In this thesis, we consider a differential equation to be of the form
dx
dt
≡ ˙x = f(x, t), x = x(t) ∈ Rn
, (1.0.1)
where f : U × J ⊆ Rn
× R → Rn
is a smooth map called the vector field describing (1.0.1). When
the vector field is time-independent the system is called autonomous. If the vector field is autonomous,
the local existence and uniqueness theorem (cf. [1], Theorem 1.0.1) states that given initial conditions
x(0) = x0 ∈ U, there is a unique solution φ(x0, t) defined for t on some interval (a, b) which is a solution
to (1.0.1). This solution, sometimes written as x(x0, t) or just x(t), is called a solution curve, orbit or
trajectory of (1.0.1) based at x0.
Definition 1.1. If a local solution exists, the vector field generates a flow φt : U → Rn
defined for
t ∈ I = (a, b) which is a solution of (1.0.1):
d
dt
(φ(x, t))) |t=τ = f(φ(x, τ)), (1.0.2)
where φ(x, t) = φt(x).
A point ¯x ∈ U that satisfies f(¯x) = 0 is called a fixed point of the vector field f. A fixed point ¯x is
stable if for every neighborhood V ⊂ U of ¯x there is a neighborhood V1 ⊂ V such that for every x1 ∈ V1
the solution x(x1, t) is defined and lies in V for all t > 0. If in addition V1 can be chosen such that
x(x1, t) → ¯x as t → ∞ then ¯x is said to be asymptotically stable.
Remark 1.1. Fixed points are often characterized by their stability type. Asymptotically stable fixed
points are called sinks. Unstable fixed points are either sources or saddles. Stable but not asymptotically
stable fixed points are called centers.
One way to establish stability and asymptotic stability of fixed points is by finding a suitable Liapunov
function. This is a positive definite function V : W → R defined on some neighborhood W of a fixed
point ¯x that decreases on solution curves of the differential equation (1.0.1).
Theorem 1.1. Let ¯x be a fixed point for (1.0.1) and V : W → R be a differentiable function defined on
some neighborhood W of ¯x satisfying
(i) V (¯x) ≥ 0 and V (x) = 0 iff x = ¯x.
(ii) d
dt V (x) ≤ 0 in W − {¯x}. Then ¯x is stable. Moreover, if
(iii) d
dt V (x) < 0 in W − {¯x},
then ¯x is asymptotically stable.
A typical example of a Liapunov function is the energy of a mechanical system that can be described
by a Hamiltonian. We will not work will Liapunov functions explicitly, but will refer to such functions
in [2] that show asymptotic stability of fixed points in the Duffing oscillator.
Recall that a linear system is a system of the form
˙x = Ax, x ∈ Rn
, (1.0.3)
where A is an n × n matrix with constant coefficients. For linear systems, we know from ordinary differ-
ential equations that the solutions can be found by finding the eigenvalues and (generalized) eigenvectors
of the matrix A. The flow of the system is simply given by exponentiating the matrix A to give etA
.
A property of the linear system is that under this flow etA
, an eigenvector vj
of A will remain in the
subspace {vj
} the eigenvector spans under the flow. The property that the flow of a set remains within
that set is defined in the notion of an invariant set.
Definition 1.2. A set S ⊂ Rn
is called invariant if for all x ∈ S, φt(x) ∈ S for all t ∈ R.
Hence one can now define invariant subspaces spanned by the eigenvectors of the linear system (1.0.3)
by dividing the eigenvectors into their stability types.
4

Definition 1.3. For a system (1.0.3) we define the following subspaces that remain invariant under the
flow etA
of (1.0.3):
The stable subspace Es
is spanned by the eigenvectors of A with eigenvalues having negative real
parts.
The unstable subspace Eu
is spanned by the eigenvectors of A with eigenvalues having positive real
parts.
The center subspace Ec
is spanned by the eigenvectors of A with eigenvalues having zero real parts.
We will see below how these eigenspace in a sense generalize to manifolds for the nonlinear system.
For a nonlinear system
˙x = f(x), x ∈ Rn
, (1.0.4)
one cannot always find analytic expressions for the flow or solution curves of the system. The existence
and uniqueness theorem still guarantees that solutions are defined locally, but there is no general proce-
dure to find these solutions. One approach to derive local properties of the system (1.0.4) is by finding
its fixed points and linearizing the system at these points.
Definition 1.4. Let ¯x be a fixed point of (1.0.4). The linearization of (1.0.4) at ¯x is the system
˙ξ = Df(¯x)ξ, ξ ∈ Rn
, (1.0.5)
where Df = ∂fi
∂xj
ij
is the Jacobian matrix of f. Locally, then, the linearized system (1.0.5) is a good
approximation of the nonlinear system (1.0.4), and some of the correspondence is given by two key
theorems in dynamical systems theory. To understand these theorems, we need two more definitions.
The first describes a large class of fixed points that does not include center points:
Definition 1.5. A fixed point ¯x of (1.0.4) is called a hyperbolic fixed point if Df(¯x) has no eigenvalues
with zero real part. If all eigenvalues have negative real part, then ¯x is a sink. If all eigenvalues have
positive real part, then ¯x is a source. If there are eigenvalues with positive real part as well as eigenvalues
with negative real part, then ¯x is a saddle.
The second can be seen as a generalization of the invariant subspaces for linear systems described
above:
Definition 1.6. Let ¯x be a fixed point of (1.0.4). The local stable and unstable manifolds of ¯x, defined
on a neighborhood U ⊂ Rn
of ¯x, are the invariant sets
Ws
loc(¯x) = {x ∈ U|φt(x) → ¯x as t → ∞, and φt(x) ∈ U for all t ≥ 0}.
Wu
loc(¯x) = {x ∈ U|φt(x) → ¯x as t → −∞, and φt(x) ∈ U for all t ≤ 0}.
Similarly, the global stable and unstable manifolds are defined by the invariant sets resulting from
letting the local manifolds flow backwards and forward under the vector field:
Definition 1.7. We define the global stable and unstable manifolds of a hyperbolic fixed point ¯x by
Ws
(¯x) =
t≤0
φt (Ws
loc(¯x))
Wu
(¯x) =
t≥0
φt (Wu
loc(¯x)) (1.0.6)
The local existence and uniqueness theorem tells that trajectories cannot intersect themselves, and
this is in particularly true for the stable and unstable manifold of the same point. However, there can
be intersections between the stable and unstable manifolds of the same point.
Definition 1.8. Let p be a hyperbolic fixed point and suppose that Ws
(p) ∩ Wu
(p) =. Let q ∈
Ws
(p) ∩ Wu
(p). Then q is called a homoclinic point and the orbit {φt(q)| − ∞ < t < ∞} is called a
homoclinic orbit. This orbit has the property that φt(q) → p for t → ±∞.
Similarly, if the intersection between the stable and unstable manifolds of two different points inter-
sect, we speak of heteroclinic points and heteroclinic orbits.
5

If ¯x is a hyperbolic fixed points, then the local properties of the system near ¯x are well approximated by
the linearized system. The Hartman-Grobman theorem states that the flow φt of (1.0.4) is homeomorphic
to the flow etDf(¯x)
of the linearized system (1.0.5). The Stable Manifold Theorem for a Fixed Point states
that the local stable and unstable manifolds defined above indeed exist for a hyperbolic fixed point ¯x,
and that they have the same dimensions as and are tangent to the invariant subsets Es
and Eu
of
the linearized system (1.0.5) at ¯x. The proofs of these theorems are found in many standard texts in
dynamical systems theory, and although it is not proven in [1], many references to proofs are given in
the book.
For a discrete system described by a map
xn+1 = Bxn, or x → Bx, (1.0.7)
most definitions and results are very similar to those for the continuous, generally nonlinear system
(1.0.4). The notion of smoothness of the vector field is replaced by infinite differentiability of the map B,
in which case it is called a diffeomorphism. An orbit becomes a set of discrete points {Bn
(x0)|n ∈ Z}. A
careful distinction must be made in the how the eigenvalues of the map determines whether a solution
is attracted to a fixed point, and the stability type of the fixed point. Instead of looking at the real part
of the eigenvalue of the map, one has to determine the modulus of the eigenvalue and compare it to 1
instead of 0. Hence for an eigenvalue λ of an eigenvector v, if λ < 1 then v lies in the stable subspace, if
λ > 1 then v lies in the unstable subspace, and if λ < 1 then v lies in the center subspace. With these
two distinctions made, the analogous versions of the Hartman-Grobman and Stable Manifold Theorems
from above also hold for discrete systems (cf. [1]).
We will encounter discrete maps when we study the Duffing oscillator and driven damped pendulum
(DDP) with periodic forcing. In that case the system takes the general form
˙x = f(x, t), x ∈ Rn
, (1.0.8)
where f is periodic in t, i.e. f(t + T) = f(t) for some T > 0. In this case we have a time-dependent
vector field, and nothing of what we derived above for the autonomous system can be directly applied
to this system. Hence we will need new methods to investigate the behavior of such a system. Since the
vector field is periodic, we might expect some solutions to be periodic as well with the same period T.
For instance, if f describes some periodic forcing of a simple mechanical system we might expect the
system to eventually oscillate in resonance with the forcing and take the same period. We can determine
whether the system is indeed periodic by just looking at the position of the system at discrete points of
time which are distance T apart. In this way we get a series of ’snapshots’ of the system evenly spaced
apart from which we can detect whether the system has a period that is (a fraction of) T or a multiple
of T.
The technical method of making the ’snapshots’ is to construct a Poincaré map. The general definition
of the Poincaré map is rather technical and can be found in §1.5 of [1]. We shall now simply give the
form of this map for our periodically forced system (1.0.8). First note that we can rewrite this system
as the autonomous system
˙x = f(x, θ), ˙θ = 1, (x, θ) ∈ Rn
× S1
, (1.0.9)
so we take into account the periodicity by identifying parts of the real line into S1
= R/TZ. We can
now choose a fixed angle θ0 ∈ S1
and define the cross-section
Σ = {(x, θ) ∈ Rn
× S1
|θ = θ0}. (1.0.10)
The idea is now to find where a point starting on Σ will arrive on Σ the first time it returns. Notice that
it does indeed return, in a fixed time T, due to ˙θ = 1. Hence define the Poincaré map P : Σ → Σ as
P(x0) = π ◦ φT (x0, θ0), (1.0.11)
where x0 ∈ Rn
, φt, if it exists, is the flow of (1.0.9), and π : Rn
× S1
→ Rn
is the projection map. A
periodic solution with period T would hence be seen as a fixed point under P. Notice that harmonics
with period 1
k T with integer k cannot be distinguished from each other, but subharmonics with period
kT would correspond to cycles of k different points on Σ under the Poincaré map.
Notice that by using the Poincaré map, the dimension of the system is reduced by one. Since the system
studied in fact lives one dimension higher, it is now possible to see intersections of orbits in the Poincaré
6

map. These are not real intersections in the original phase space the system is defined on, but are the
result of projecting the system on the hypersurface Σ. A particular case of such intersections are those
between the stable and unstable manifolds, and we shall see later how this is related to chaotic dynamics
in the system.
The two systems studied in this thesis are both described by second order differential equations with
a periodic forcing term. If this periodic forcing is absent, the systems can be cast in the two-dimensional
form
˙x = f(x, y) ˙y = g(x, y), (1.0.12)
where f and g are smooth functions on R2
. For such planar systems, a lot more is known than for the
general nonlinear system. Many theorems that apply specifically to two-dimensional systems but may
not be true in general are treated in §1.8 of [1]. One important result is that two dimensional systems
do not exhibit many of the phenomena associated with chaos. For instance, one can prove that the
only possible nonwandering sets for a planar system are fixed points, closed orbits and unions of fixed
points and the trajectories connecting them. The latter are homoclinic orbits or closed paths formed of
heteroclinic orbits. The notion of a nonwandering set is defined in §1.6 of [1], but can be understood as
a set with the property that there are arbitrary large times when it approaches its initial position. Many
of these properties make it easier to study the planar system, and we shall state and prove one result
which will be needed in our discussion of the Duffing oscillator:
Theorem 1.2 (Bendixson’s Criterion). Let D ⊆ R2
be a simply connected region. If ∂f
∂x + ∂g
∂y < 0 or
> 0 on D, then (1.0.12) has no closed orbits lying entirely in D.
Proof. Suppose γ is a closed orbit lying in D and let S be its interior. Notice that we have
dy
dx
=
dy
dt
dt
dx
=
˙y
˙x
=
g(x, y)
f(x, y)
(1.0.13)
on any solution solution curve of (1.0.12). Hence we have
γ
(f(x, y)dy − g(x, y)dx) = 0 (1.0.14)
which by Green’s Theorem is equivalent to
S
∂f
∂x
+
∂g
∂y
= 0. (1.0.15)
But if ∂f
∂x + ∂g
∂y < 0 or > 0 on D, the integral never vanishes on any subset of D, and hence we conclude
that such a γ does not exist. Thus there are no closed orbits in D.
7

2 Averaging and Melnikov’s Method
In analyzing nonlinear differential equations of the form (1.0.4), one cannot always find analytic solutions
to the system studied. Even if these such solutions exist, the expressions might be too complicated to
be useful, as we will see in our examples of the DDP and the Duffing oscillator. Hence we should look
for methods to treat approximate such systems by simpler systems. Perturbation theory deals with
comparing properties of perturbed systems slightly deviating from a well-known system with those of
the known system. In both case of the Duffing oscillator as well as the DDP, we can treat the systems
as perturbations of autonomous two-dimensional systems, which are easier and better understood in the
light of the previous section. Two such techniques are discussed in [1], and we shall now give the main
results necessary for application to our systems.
2.1 Averaging Theorem
The Averaging Method can be summarized as a technique to study a system described by a periodic
vector field by its time-averaged vector field. More formally, one starts with systems of the form
˙x = f(x, t, ), x ∈ U ⊂ Rn
, t ∈ R, 0 ≤ 1, (2.1.1)
where f is periodic in t with period T. One then studies the system by averaging over a period T in the
time variable:
˙y =
1
T
T
0
f(y, t, 0)dt := ¯f(y) (2.1.2)
It appears that many properties of the averaged system (2.1.2) are then carried over to the original
system when is small ( 1).
A number of useful properties are then perserved in the averaged system. For instance, solutions of the
averaged system and the original system remain close to each other for large timescales (O(1
), hyperbolic
fixed points of the averaged system correspond to fixed points of the original system, and the stable and
unstable manifolds remain close to each other for all times. Also, simple local bifurcations such as the
saddle-node bifurcation and the Hopf bifurcation are perserved under averaging. However, what is not
preserved in general is the global behavior, and in fact the technique would fail dramatically when one
tries to employ it to study nonlinear systems in their chaotic regime. Under suitable conditions, however,
one can establish criteria under which also the global properties are preserved.
Because we shall use the technique in our study of the Duffing oscillator, let us two results on averaging
as presented in [1].
Theorem 2.1 (The Averaging Theorem). Consider a system of the form (2.1.1) and the associated
time-averaged system (2.1.2). Suppose the vector field f(x, t, ) is Ck
for k ≥ 2 and bounded on bounded
sets. Then there exists a Ck
change of coordinates x = y + w(y, t, ) under which (2.1.1) becomes
˙y = ¯f(y) + 2
f1(y, t, ), (2.1.3)
where f1 is periodic in t with period T. Moreover,
(i) If x(t) and y(t) are solutions of (2.1.1) and (2.1.2) based at x0, y0, respectively, at t=0, and |x0 −
y0| = O( ), then |x(t) − y(t)| = O( ) on a time scale t 1
.
(ii) A hyperbolic fixed point p0 of (2.1.2) corresponds uniquely to a hyperbolic periodic orbits γ of
(2.1.1) of the same stability type: there exist 0 > 0 such that γ (t) = p0 + O( ) for all 0 < ≤ 0.
(iii) Let xs
(t) and ys
(t) be solutions lying in the stable manifolds of the hyperbolic fixed point p0 and the
hyperbolic orbit γ . If |xs
(0) − ys
(0)| = O( ), then |xs
(t) − ys
(t)| = O( ) for t ∈ [0, ∞). Similar
results apply to the unstable manifolds on the time interval t ∈ (−∞, 0].
The proof of this theorem can be found in paragraph 4.1 of [1].
The second result concerns the validity of the method in deriving global properties of the system by
means of studying the averaged system. What is meant by global properties in this context is the notion
of topological equivalence between vector fields, or in this case Poincaré maps. Hence one can find criteria
under which the Poincaré maps of the averaged system and the original system are similar, by the notion
of topological equivalence:
8

Definition 2.1. Two maps F, G are topologically equivalent if there exists a homeomorphism h such
that h ◦ F = G ◦ h. Two vector fields f, g are topologically equivalent if there exists a homeomorphism h
which takes orbits of f to orbits of g, preserving senses but not not necessarily parametrization of time.
The criteria are then formulated in the following theorem (Theorem 4.4.1. from [1]):
Theorem 2.2. If the time T flow map P0 of (2.1.2), restricted to a bounded domain D ⊂ Rn
, has a
limit set consisting solely of hyperbolic fixed points and all intersections of stable and unstable manifolds
are transverse, then the corresponding Poincare map P |D of (2.1.1) is topologically equivalent to P0|D
for > 0 sufficiently small.
In particular, if the system contains a homoclinic loop of some saddle point, the manifolds do not
intersect transversally - each point on the homoclinic loop lies in both the stable and unstable manifolds.
Hence in such we cannot expect averaging to produce a system that retains all of the relevant features
of the original system. We shall see in the examples that in such cases we will likely find chaos.
We now want to apply the averaging theorem to systems of the form
¨x + ω2
0x = f(x, ˙x, t), x ∈ R, f(x, ˙x, t + T) = f(x, ˙x, t) (2.1.4)
For = 0 the system is simply the linear oscillator without damping and external forcing, and the
solutions are sinusoidal functions of period 2π
ω0
. Duffing’s equation describes a system that can be recast
in this form, and also the DDP can in principle be recast in this form if we perform a Taylor expansion
for the sinusoidal forcing term.
In order to rewrite (2.1.4) in the form (2.1.1) suitable for averaging, we must apply the invertible Van
der Pol transformation. Introduce the linear transformation
u
v
= A
x
˙x
, A =
cos ωt
k − k
ω sin ωt
k
− sin ωt
k − k
ω cos ωt
k
(2.1.5)
Geometrically, this transformation can be interpreted as the composition of a rotation through angle ωt
k
and a reflection in the x-axis with rescaling factor k
ω . We can see this from the decomposition
A =
cos ωt
k − k
ω sin ωt
k
− sin ωt
k − k
ω cos ωt
k
=
cos ωt
k sin ωt
k
− sin ωt
k cos ωt
k
·
1 0
0 − k
ω
(2.1.6)
To undo the transformation we must first rotate back through the same angle and then undo the rescaling,
so we derive
A−1
=
1 0
0 −ω
k
·
cos ωt
k − sin ωt
k
sin ωt
k cos ωt
k
=
cos ωt
k − sin ωt
k
−ω
k sin ωt
k −ω
k cos ωt
k
. (2.1.7)
The transformation can be understood as going over in coordinates which rotate the plane by the solutions
of the linear oscillator for frequency ω
k . To see this, consider a solution to the linear oscillator with natural
frequency ω0 = ω given by x(t) = cos ωt
k , ˙x(t) = −ω
k sin ωt
k . Under the van der Pol transformation, we
get
u
v
=
cos ωt
k − k
ω sin ωt
k
− sin ωt
k − k
ω cos ωt
k
x(t)
˙x(t)
=
cos2 ωt
k + sin2 ωt
k
− cos ωt
k sin ωt
k + sin ωt
k cos ωt
k
=
1
0
, (2.1.8)
so the solution becomes a fixed point under the rotated coordinates. This is relevant in oscillating systems
such as where we apply a driving force of frequency ω
k ≈ ω0 for integer k, and expect to find solutions
that are close to order k resonance. We then let the coordinates rotate according to this expected solu-
tion and examine how the orbits deviate from this standard orbit. The solutions with k > 1 are called
subharmonics of order k and have a period which is k times the period of the solution with k = 1.
The idea of rotating coordinates is further clarified by rewriting the solution x(t) in terms of polar coordi-
nates in uv plane given by r(t) = u(t)2 + v(t)2, φ = arctan v
u . Starting with the inverse transformation
9

(2.1.7), we get
x(t) = u(t) cos ωt − v(t) sin ωt
= (u(t)
u(t)2 + v(t)2
u(t)2 + v(t)2
cos ωt − v(t)
u(t)2 + v(t)2
u(t)2 + v(t)2
sin ωt)
= u(t)2 + v(t)2(
1
1 + v(t)2
u(t)2
cos ωt −
v
u
1 + v(t)2
u(t)2
sin ωt)
= r(t)(cos φ(t) cos ωt − sin φ(t) sin ωt)
= r(t) cos (ωt + φ(t)), (2.1.9)
where we used cos (arctan x) = 1√
1+x2
, sin (arctan x) = x√
1+x2
with x = v
u . Hence we see that a solution
in the averaged system with uv coordinates describes the varying amplitude and phase by which the
solution deviates from the standard solution of the harmonic oscillator.
Finally, let us derive the ﬁnal result of the Van der Pol transformation on the system (2.1.4) which
casts it in the form (2.1.1) suitable for applying the Averaging Theorem. By writing out the transfor-
mation in detail, we obtain the relations
x(t) = u(t) cos
ωt
k
− v(t) sin
ωt
k
˙x(t) = −u(t)
ω
k
sin
ωt
k
− v(t)
ω
k
cos
ωt
k
(2.1.10)
and
u(t) = x(t) cos
ωt
k
− ˙x(t)
k
ω
sin
ωt
k
v(t) = −x(t) sin
ωt
k
− ˙x(t)
k
ω
cos
ωt
k
(2.1.11)
Diﬀerentiate the expressions for u and v with respect to t and use (2.1.4) to obtain
˙u(t) =
d
dt
x cos
ωt
k
− ˙x
k
ω
sin
ωt
k
= ˙x cos
ωt
k
− x
ω
k
sin
ωt
k
− ¨x
k
ω
sin
ωt
k
− ˙x cos
ωt
k
= − x
ω
k
+ [−ω2
0x + f(x, ˙x, t)]
k
ω
sin
ωt
k
= −
k
ω
ω2
− k2
ω2
0
k2
x + f(x, ˙x, t) sin
ωt
k
(2.1.12)
˙v(t) =
d
dt
−x sin
ωt
k
− ˙x
k
ω
cos
ωt
k
= − ˙x sin
ωt
k
− x
ω
k
cos
ωt
k
− ¨x
k
ω
cos
ωt
k
+ ˙x sin
ωt
k
= − x
ω
k
+ [−ω2
0x + f(x, ˙x, t)]
k
ω
cos
ωt
k
= −
k
ω
ω2
− k2
ω2
0
k2
x + f(x, ˙x, t) cos
ωt
k
(2.1.13)
Therefore, the general transformed system takes the form
˙u(t) = −
k
ω
ω2
− k2
ω2
0
k2
x + f(x, ˙x, t) sin
ωt
k
˙v(t) = −
k
ω
ω2
− k2
ω2
0
k2
x + f(x, ˙x, t) cos
ωt
k
(2.1.14)
Hence, when ω2
−k2
ω2
0 = O( ), i.e. when we are close to order k resonance, the system (2.1.14) becomes
suitable for averaging.
10

2.2 Melnikov’s Method
The conditions under which Theorem 2.2 may be applied suggest that averaging may not be applicable
to all systems with periodic forcing. In particular, consider a system which contains a homoclinic loop
joined to a saddle point. Then all points on this loop are on both the stable and unstable manifolds,
and there is an infinite number of such points. The manifolds do not intersect transversally but coincide,
and hence Theorem 2.2 cannot be applied. From a geometrical perspective, imagine that the manifolds
will separate when we perturb the system slightly and cease to be joined together. There are different
scenarios possible in such a situation: one manifold could pass underneath the other, there could be
transversal intersections of the manifolds, or they may touch in a number of points. Some of these
scenarios are illustrated in Figure .
In order to study the effect of small perturbations on systems which contain such a homoclinic
loop, Melnikov developed a method that comes down to computing a function, now referred to as the
Melnikov function, which gives information on the separation between the manifolds under a perturbation
at different points. The function can be used to predict whether (transversal) intersections between the
manifolds are present. We shall see in the next section that these intersections are linked to the existence
of a horseshoe and how this implies that chaos is present in the system.
Melnikov’s Method is applicable to systems that can be written as a perturbation of a Hamiltonian
system, where the original system contains a homoclinic orbit. To be precise, the system must take the
form
˙x = f(x) + g(x, t), x ∈ R2
, (2.2.1)
where the vector fields
f(x) =
f1(x)
f2(x)
, g(x) =
g1(x)
g2(x)
(2.2.2)
are sufficiently smooth (at least C2
) and bounded on bounded sets, and g is periodic in t with period
T. Also, f should be a Hamiltonian vector field, i.e. there is a Hamiltonian function H(u, v) such that
f1 = ∂H
∂v ,f2 = −∂H
∂u . This system can be recast in the autonomous form
˙x = f(x) + g(x, θ)
˙θ = 1, (x, θ) ∈ R2
× S1
, (2.2.3)
In addition, three more requirements are mentioned in [1], as assumptions (A1)-(A3) on p. 185, but not
all of these are needed to apply the basic results. In fact, for the basic result to hold, the system should
only contain a homoclinic orbit to a saddle point, and an explicit expression q0
(t) for this orbit should
be known. The other two assumptions are only needed when one deals with systems that depend on a
parameter value, where intersections may occur at for some parameter values but not for other. However,
we shall be needing this, since we are studying parameter-dependent systems and want to predict at what
parameter values chaos appears. Hence we summarize these assumptions in words without the technical
details:
1. the unperturbed system contains a homoclinic orbit q(t) to a hyperbolic saddle point p
2. the interior of the closed level curve containing q(t) and p is filled with a continuous family of
periodic orbits
3. the period of these orbits is a differentiable function of the value of the Hamiltonian on the orbit
and the period increases with increasing Hamiltonian.
When is increased from 0, a perturbation lemma (Lemma 4.5.1 on p. [1]) tells that the perturbed
system (2.2.1) has a unique hyperbolic periodic orbit, coinciding with the saddle point for = 0. Further-
more, approximations can be found for perturbations of orbits lying in the stable and unstable manifolds
(Lemma 4.5.2. on the same page). In this way, Melnikov was able to derive an expression for the distance
between the perturbed manifolds in terms of distances between these orbits, which can be approximated
by expressions involving only the known homoclinic orbit q0
(t) and the vector fields f and g. The final
result is the Melnikov function
M(t0) =
∞
−∞
f(q0
(t)) ∧ g(q0
(t), t + t0)dt, (2.2.4)
11

where a ∧ b = a1b2 − a2b1 for a, b ∈ R2
. The Melnikov function should really be understood as a distance
function between the stable and unstable manifolds of the saddle orbit of the perturbed system. The
variable t0 parameterizes different locations on the manifolds, and whenever there is a point where the
Melnikov function vanishes, the manifolds coincide. If this is a simple zero, i.e. if the second derivative
is non-zero, then the manifolds intersect transversally. If there are no zeros at all, the manifolds do not
intersect and we have one of the scenarios where one manifold passes beneath the other (see Figure ).
The theorem and proof concerning this result is formulated in Theorem 4.5.3. of [1].
In the case where the system depends on some parameter µ, we are interested in how the manifolds
change location when µ is changed. In particular, we want to know whether there is some value from
which the manifolds touch and expect transversal intersections beyond that value. Such a phenomenon is
called a homoclinic bifurcation. The occurrence of a homoclinic bifurcation only requires the existence of
a quadratic zero under some conditions. Let M(t0, µ) be the Melnikov function depending on parameter
value µ. If there is a quadratic zero at (τ, µb), i.e. if M(τ, µb) = ∂M
∂t0
(τ, µb) = 0, but ∂2
M
∂t2
0
(τ, µb) = 0 and
∂M
∂µ (τ, µb) = 0, then there is indeed a homoclinic bifurcation at parameter value µb (Theorem 4.5.4., [1]).
3 The Smale horseshoe
The Smale horseshoe map is a classical and fundamental example of a chaotic map. The existence of
maps resembling the horseshoe map in dynamical systems is strongly linked to chaotic dynamics, as shall
be shown in the following sections. I shall first describe the horseshoe following its presentation in [1],
using symbolic dynamics, and prove a number of interesting results.
Let S = [0, 1]×[0, 1] be the unit square in the plane and define a map f : S → R2
that can geometrically
best be described as follows: first perform a linear expansion in the vertical direction and a linear
contraction in the horizontal direction and then fold the result back onto the unit square such that the
intersection S ∩ f(S) consists of two vertical strips V1 and V2. See Figure 3.1.
The stretching in the first step can be given explicitly as the map (x, y) → (λx, µy) with 0 < λ < 1
2 and
Figure 3.1: The Smale horseshoe map. Source: Figure 5.1.1. from [1]
µ > 2. These restrictions on the expansion factors are needed because the resulting strip has still to be
folded onto the unit square, so its vertical length must be greater than 2, while twice its horizontal length
must not exceed 1. Notice that the inverse image of the vertical strips V1, V2 consists of two horizontal
strips H1, H2, both with height µ−1
and width 1. Restricted to such a horizontal band Hi, f can be
described as the performing the map (x, y) → (±λx, ±µy) followed by some translation in space, and for
H2 also a rotation, giving the minus sign. Hence on Hi the map has a uniform Jacobian equal to
Df|(x,y) =
±λ 0
0 ±µ
, (3.0.5)
where the + is taken for H1 and the − for H2.
Define the set of points that remain in S for all time by Λ = {x|fi
(x) ∈ S, −∞ < i < ∞}. This is called
an invariant set. We are interested in the structure of this set, which can be studied by inductively
applying f. Applying f twice, the image of all points still in S is given by S ∩f(S)∩f2
(S), so its inverse
image f−2
(S ∩ f(S) ∩ f2
(S)) describes the set of points that get mapped into S by f2
. The image of
12

this set under f lies in V1 ∪ V2, but under f2
, this set should still remain in S. Therefore, the image
under f must also lie in H1 ∪ H2, and the largest possible set satisfying this is (V1 ∪ V2) ∩ (H1 ∪ H2).
The inverse image of this set consists of four horizontal strips lying inside H1 ∪ H2, each having vertical
length µ−2
, while the image consists of four vertical strips lying inside V1 ∪ V2, see Figure 3.2. We can
repeat this argument inductively to find that the set of points which remain in S after n iterations of f,
given by f−n
(
n
k=0 fk
(S)), consists of 2n
horizontal strips, each with height µ−n
(and width 1). Notice
that from using the chain rule repeatedly we have for x ∈ f−1
(
n
k=0 fk
(S))
Dfn
|x = Df|fn−1(x) ◦ Df|fn−2(x) ◦ . . . ◦ Df|x = (Df|x)
n
=
±λn
0
0 ±µn , (3.0.6)
since each the Jacobian is uniform on H1 ∪ H2 and the each horizontal strip of f−1
(
n
k=0 fk
(S)) lies
inside H1 ∪ H2. The sign is determined by whether the strip gets oriented in the same direction under
the map or becomes reversed. We see that the image of one such horizontal strip is a vertical strip of
width λn
(and height 1), and we conclude that
n
k=0 fk
(S) is the union of 2n
disjoint vertical strips.
The total height of the n-th set obtained in this way f−n
(
n
k=0 fk
(S)) (or vertical strips
n
k=0 fk
(S)
Figure 3.2: The sets f−2
(S ∩ f(S) ∩ f2
(S)) and S ∩ f(S) ∩ f2
(S). Source: Figure 5.1.2. from [1]
is 2n
µ−n
= 2
µ
n
< 1 (or (2λ)
n
< 1). Taking n → ∞, we see that the length of the set goes to 0 in
both cases. Hence we are left with an infinite set of horizontal (or vertical) lines. Moreover, any line is
arbitrarily close to any other line, since the space between the strips contracts uniformly with a factor
depending on µ (or λ). Clearly, the limit set f−∞
(
∞
k=0 fk
(S)) (respectively
∞
k=0 fk
(S)) is closed and
because each point lies on a line that with thickness 0, the interior of the set is empty. We conclude
that the limit set consisting of horizontal (vertical) lines is a Cantor set: a closed set which contains no
interior points or isolated points. The invariant set Λ consists of all points which remain in S in both
the forward and backward time direction, so it consists of the intersection of these two sets. Since each
horizontal line intersects a vertical line in exactly one point, we conclude that the resulting intersecting
set is still Cantor set consisting of discrete points in S.
The question now is how we can describe the (uncountably infinite number of) points lying in Λ, and
describe how they are mapped by f onto other points in Λ. It turns out that the entire set Λ can be
described in terms of bi-infinite sequences of the form a = {ai}∞
i=−∞ with ai ∈ {1, 2}. Namely, we can
describe each point by the sequence in which it passes through the horizontal bands H1 and H2, i.e. ai
takes value 1 if the fi
takes the point to H1 and value 2 for H2. It appears that this sequence of 1s and
2s is unique to every point in Λ, and hence we can establish a bijection φ between our set Λ and set of all
bi-infinite sequences Σ = {{ai}∞
i=−∞|ai ∈ (1, 2)}. Define the shift map σ on Σ such that σ(a) = b with
bi = ai+1, so σ shifts the whole sequence of numbers one place to the left. Then this must correspond
to applying f once to a point in Λ. Hence the bijection φ must satisfy φ(f(x)) = σ(φ(x)) for all x ∈ Λ.
This can be rewritten in the form
f|Λ = φ−1
◦ σ ◦ φ. (3.0.7)
By defining a suitable metric on the set Λ, we can ensure that the map φ is continuous, in which case it
is homeomorphism (because Λ is compact). In this case equation (3.0.7) states that f restricted to its
invariant set Λ is topologically conjugate to the shift map σ (see Definition 1.7.2. in [1]). The proof of
the claims of this paragraph are contained in the proof of Theorem 5.1.1. in [1].
As a consequence, we can study the dynamics of the set Λ by studying sequences of symbols under
shifting operations. This technique is called symbolic dynamics and can be used to derive the following
important results:
13

Theorem 3.1. The invariant set Λ of the horseshoe map f has the following properties:
(i) Λ contains a countable set of periodic orbits of arbitrary long periods.
(ii) Λ contains an uncountable set of bounded nonperiodic motions.
(iii) Λ contains a dense orbit.
(iv) The periodic orbits of Λ are all of saddle type and they are dense in Λ
Furthermore, f|Λ is structurally stable.
Structural stability is defined in section 1.7 of [1]. Briefly summarized, a system is structurally
stable if it is insensitive to small perturbations, in such a way that the map that defines it retains its
topological structure. The fact that Λ is structurally stable is a result by Smale and will not be discussed
here. However, the other claims can be proven using symbolic dynamics.
Proof. Notice that a point x ∈ Λ is periodic with period τ if and only if φ(x) contains repeating sections
of length n. Hence the total number of orbits of period n is simply 2n
and by arranging these orbits
by increasing n, we conclude that the set of periodic orbits is countable. Furthermore, no restriction is
placed on the length of the period, so we have orbits of arbitrary long period. This proves (i).
Every point x ∈ Λ has either a periodic orbit or a nonperiodic orbit. By the bijection φ, the set of points
with a nonperiodic orbit is the complement of the set of sequences that are periodic in Σ. However, Σ
is an uncountable, since each sequence in Σ can be identified with the binary representation of a real
number (in which case we should take 0s and 1s instead of 1s and 2s). Explicitly, the bijection from Σ
to R can be defined as a →
∞
k=−∞ 2kak
, where ak ∈ {0, 1}. The complement of an countable set in
an uncountable set is uncountable. This can be proven by contradiction: the union of countable sets is
countable, so if the complement were countable, then the entire set would also be countable. Therefore
we conclude that the set of nonperiodic orbits in Λ is uncountable. This proves (ii).
In order to discuss denseness of orbits, define a metric on Σ by
d(a, b) =
∞
k=−∞
δk2−|k|
, δi =
0 if ai = bi
1 if ai = bi.
(3.0.8)
Clearly, this definition satisfies d(a, a) = 0 and d(a, b) = d(n, a). For the triangle equality, we can
examine the contribution of the i − th index in d(a, c) compared with d(a, b) + d(b, c). If ai = ci then
either ai = bi = ci or ai = bi, bi = ci. In the first case the contribution is 0 on both sides while in the
second it is 2˙2−|k|
on the right side. If ai = ci then either ai = bi, bi = ci or ai = bi, bi = ci, in which
case the contributions on both sides are equal to 2−|k|
. Hence for each i the contributions to the distance
satisfies the triangle inequality and therefore also d(a, c) ≤ d(a, b) + d(b, c).
This metric tells that two orbits are close together if they agree well on a long central part. To find
a sequence that is dense Σ, we must find a sequence that overlaps with the central part of any other
sequence on an interval of arbitrary long length. Hence this orbit should contain all finite strings of
arbitrary length. Such an orbit can be found by placing all finite strings after one other. The collection
of finite strings is countable, by the argument for part (i) and they all have finite length. Therefore, the
entries of this string can be listed by listing all finite strings one after another. For example, start with
the two strings of length 1, then list the four strings of length 2, and so on:
a0 = 0; a1 = 1;
(a2, a3) = (0, 0); (a4, a5) = (0, 1); (a6, a7) = (1, 1); (a8, a9) = (1, 1)
(a10, a11, a12) = (0, 0, 0); (a13, a14, a15) = (0, 0, 1); etc.
(3.0.9)
We can even take the negative indices arbitrary in this way. Clearly, any finite string can be brought to
the center by applying σ a finite number of times. This proves (iii).
The above argument also shows that the set of periodic orbits is dense in Λ, since for a central part of
arbitrary length, we can find a periodic orbit that contains that central part, because there are periodic
orbits of arbitary period. For the saddle property, notice that the linearisation of f on Λ takes the form
(3.0.5) in each point x ∈ Λ, since Λ ⊂ H1 ∪ H2. The eigenvalues at each point are thus given by ±λ and
±µ, with |λ| < 1 and |µ| > 1. We conclude that every point in Λ is of saddle type. This proves (iv) and
concludes the proof of the theorem.
14

3.1 Horseshoes in other systems
From the preceding section, it should be clear that the horseshoe map is an example of map that exhibits
characteristics that are typical of chaotic systems. For instance, the fact that invariant set Λ is a Cantor
set implies that it becomes very difficult to predict whether a point in unit square S will remain in S for
all time unless one knows the position of the point with extreme accuracy. In any neighbourhood of a
point in Λ there are points not in Λ (this is one of the properties of a Cantor set). Hence the behaviour
of the system is extremely sensitive to initial conditions. The existence of nonperiodic orbits shows that
for certain initial values the orbit continues to be irregular for an indefinite amount of time. Density of
orbits in the set is another property that is typical of chaotic systems.
All these properties makes the horseshoe map an interesting example to study and we want to generalize
this to include examples of similar maps contained within systems such as the Duffing oscillator. This
generalization can be done for the system of a bouncing ball described in section 2.4 of [1], but for the
Duffing oscillator it is not so straightforward. The first problem that arises is that the Duffing oscillator
is a continuous system whereas the bouncing ball and horseshoe systems are both discrete systems.
However, the Duffing oscillator with γ = 0 is reduced to a discrete system by the Poincaré map Pγ. This
Poincaré map is not known explicitly, although approximations can be found, as Holmes discusses in [2].
Using the numerical results obtained from such approximations, we shall try to identify horseshoe-like
phenomena in the Duffing system.
We can proceed by extending the specific example by Smale to more general horseshoes and derive
general results on these extensions. Then we can try to apply these results to the case of the Duffing
oscillator by studying numerical results obtained by approximations of the Poincaré map.
3.2 Hyperbolic structure and a theorem
The following two sections are based on sections 5.2 and 5.3 of [1]. I try to discuss the relevant results
that can be applied to find horseshoes in the Duffing example while avoiding the details of the abundance
of technical definitions and assumptions which the authors use to derive their results. Nevertheless, I
shall start with giving one such technical definition.
Definition 3.1. Let Λ be an invariant set for the discrete dynamical system defined by f : Rn
→ Rn
.
A hyperbolic structure for Λ is a continuous invariant direct sum decomposition TΛRn
= Eu
Λ ⊕ Es
Λ with
the property that there are constants C > 0, 0 < λ < 1 such that:
1. If v ∈ Eu
Λ, then |Df−n
(x)v| ≤ Cλn
|v|.
2. If v ∈ Es
Λ, then |Dfn
(x)v| ≤ Cλn
|v|.
Here TΛRn
= {TxRn
|x ∈ Λ} consists of the tangent spaces at each point in the invariant set and
the decomposition means TxRn
= Eu
x ⊕ Es
x for each x ∈ Λ. Invariance means Df(Eu
x ) = Eu
f(x) and
Df(Es
x) = Es
f(x) and continuity means that the bases in the tangent spaces can be chosen such that they
depend continuously on x.
The set Λ is then referred to as a hyperbolic set. Note that a hyperbolic set is not too different from
a union of hyperbolic points, to which an additional structure is attached. For instance, the hyperbolic
structure requires that the contraction and expansion factors are uniform for all points in the set.
A hyperbolic structure can be put on the Λ from Smale’s horseshoe map. The tangent space decompo-
sition can simply be taken to be the decomposition into the horizontal and vertical directions at each
point. So for each x ∈ Λ, Es
x is a horizontal line and Eu
x a vertical line. Since the Jacobian was given
by (3.0.5) at each point, the contraction and expansion factors are uniform and we can take λ in the
definition to be the max{λ, µ−1
} and C = 1.
Now we want to extend the horseshoe example to more general cases. Firstly, one can imagine that
slightly deforming the horseshoe maps, for instance in a way such that the lines are no longer straight,
might still produce the same dynamics. This is contained in the result on structural stability which was
stated in Theorem 3.1 and apparently has been proven by Smale.
It appears that we can now give criteria under which a two-dimensional map contains an invariant set
that is similar to that of horseshoe map. Loosely summarized, if we can identify disjoint horizontal and
vertical strips (not necessarily straight) in the domain and range of our map, the map sends horizontal
strips to vertical strips and thereby stretches these uniformly in one or the other direction, then the map
is rather similar to Smale’s horseshoe map, apart from the hyperbolic structure. If in addition we can find
15

set (so-called sector bundles) in the horizontal strips that get expanded in one direction and contracted
in the other by Df, then we can also put a hyperbolic structure on the invariant set of this map. These
conditions are formally stated as H1-H3 on p.240 of [1]. Under conditions H1 and H2, Theorem 5.2.4.
on p.241 stated that the map f has an invariant set, and that f is topologically equivalent to a shift on
Σ on this set, i.e. fΛ = h ◦ σ ◦ h−1
. If H3 is also satisfied, this set Λ contains a hyperbolic structure.
Notice that this is means that many properties of the horseshoe are carried over to such a system. The
theorem tells that we can describe the map f in terms of bi-infinite sequences in a similar way as we
could do with the horseshoe map. In particular, the properties we derived in Theorem 3.1 should all
hold, since the proof makes use of the sequence representation of points in Σ only.
In practice, identifying horseshoes by checking H1-H3 is not always a simple task, and even for the
bouncing ball problem a rather complicated discussion is needed. Clearly, there must be a better way
of establishing the existence of chaotic phenomena by the presence of horseshoes. Indeed, recall that
the presence of horseshoes is closely linked with intersections of stable and unstable manifolds, and this
was the motivation behind applying Melnikov’s method to study these intersections. The numerical
data already suggests that there is a close link between intersections between the stable and unstable
manifolds and chaotic dynamics. It appears that such a connection can be proven mathematically.
3.3 The Smale-Birkhoff Homoclinic Theorem
In this section, I will state and discuss consequences of the Smale-Birkhoff Homoclinic Theorem. The
proof can be found on page 252-253 of [1] and makes extensive use of symbolic dynamics. Essentially, this
theorem states that the existence of transversal intersections between the stable and unstable manifold
of some point imply that we can give a symbolic description of the dynamics in a similar way as we did
in the horseshoe example.
Firstly, we need to extend the identifying two horizontal strips that contain the invariant set and keeping
track of in which of the two strips a points gets mapped to, to a more general situation in which we may
have a very different partitioning of the set. This is formally done by constructing Markov partitions of a
set, and identifying orbits with sequences that keep track in which rectangle of the Markov partitioning
a point gets mapped to.
In order to make this more precise, we will need more definitions. Firstly, we want to divide the set
into subsets that can be numbered by some indexing set. We will choose these subsets to be rectangles,
where the formal definition of a rectangle R is a closed subset with the property that for x, y ∈ R, the
stable manifold of one point and the unstable manifold of the other, with the same diameter , intersect
in exactly one point in R. This can be denoted as Ws
(x) ∩ Wu
(y) = p, p ∈ R, and for the Ws
(x) and
Wu
(y) we take the definition as in the stable manifold theorem in [1] (Theorem 5.2.8. on p.246). Hence
in the example of the horseshoe, the rectangles are intersections of rectangles with Λ. Proposition 5.3.1.
of [1] shows that the intersection of the manifolds in this case is also in Λ. In applying this proposition
we must use the notion of an indecomposable set, which is also a requirement for the existence of Markov
partitions as shall be shown later. Consider a dynamical system with flow φt and assume it has a closed
invariant set Λ.
Definition 3.2. A closed invariant set Λ is indecomposable if for every pair of points x, y in Λ and > 0,
there are points x = x0, x1, . . . , xn−1, xn = y and times t1, t2, . . . , tn−1, tn ≤ 1 such that the distance
from φti
to xt is smaller than .
Intuitively, this means that any two points in the set can be connected by a chain of points, each
of which approximates the next arbitrarily closely through the flow map. We can define the analogous
property for a map G by taking ti to be integer and Gti
instead of φti . The attractor of Smale’s horseshoe
map Λ is indecomposable because the map f has an orbit which is dense in Λ. Hence we can take x1 to
be a point in Λ which is close to x and know that there after some number of iterations n, fn
(x1) will
be close to y.
The rectangles defined in this way will be used for the symbolic description of the invariant set Λ. The
idea now is to find a partitioning of Λ into a finite collection of rectangles so that we can identify points
by the numbering of the rectangles it passes through under the concerned mapping. Formally, this is
called a Markov partitioning, and the definition is as follows:
Definition 3.3. A Markov partitioning for Λ is a finite collection of rectangles {R1, . . . , Rm} such that:
1. Λ =
m
i=1 Ri;
2. int Ri ∩ int Rj = ∅ if i = j;
16

3. f(Wu
(x, Ri)) ⊃ Wu
(f(x), Rj) and f(Ws
(x, Ri)) ⊂ Ws
(f(x), Rj) whenever x ∈ int Ri, f(x) ∈ int
Rj.
The interior of a rectangle is defined to be the set of points in R for which there exists a δ > 0 such
that the intersections of the manifolds with diameter δ with Λ are still in R, i.e. Wu
δ (x) ∩ Λ ⊂ R and
Ws
δ (x) ∩ Λ ⊂ R.
We now have a theorem that states that such Markov partitions exist under suitable conditions. To
be precise, the invariant sets must be compact, maximal, indecomposable and hyperbolic, see Theorem
5.3.2.
If we have found such a Markov partitioning, we can describe the dynamics by keeping track of the
rectangles a point passes through and this is formally described in Proposition 5.3.4. This proposition is
very similar to Theorem 5.1.1. for the Smale horseshoe, with a number of important differences. Firstly,
we are considering a partitioning consisting of m rectangles, which means that we should allow the
elements of our sequences to consist of m different integers instead of 2, i.e. ai ∈ {1, . . . , m}. Moreover,
we want to restrict ourselves to sequences which describe orbits that can be reached under the map f,
so we want to exclude cases in which two subsequent elements ai and ai+1 describe a mapping from
one rectangle to the other that is not possible under the map f. This requirement can be stated as int
Rai ∩ f−1
(int Rai+1 ) = ∅ for all i ∈ Z. Let ΣA be the complete set of such bi-infinite sequences that
satisfy this property. Then the proposition says that we can find a map π : ΣA → Λ that is surjective,
so for each point in Λ we can find a sequence in ΣA, but this sequence does not need to be unique. The
reason behind this is that we are dealing with a Markov partition whose rectangles are not necessarily
disjoint, i.e. Ri ∩ Rj = ∅ in general. In the definition we only required that the interiors were disjoint.
Hence if a point has an orbit that passes through a point on the boundary of two rectangles after a
number of iterations n of f, then we can associate two sequences in ΣA whose elements an are different.
However, restricted to the set of points that get mapped from interior to interior only, this map is in fact
bijective. This need to identify sequences with each other can be compared with an identification in the
real numbers, namely that 0.999 . . . = 1. Here, we can also find two different symbolic representations
that eventually describe the same number.
Since the map π is not bijective, we can no longer write f = π ◦ σ ◦ π−1
, where σ is the same shift map
as defined earlier. Nevertheless, the statement f ◦ π = π ◦ σ should still be true. The set ΣA together
with σ is called a subshift of finite type and the statement becomes that f is topologically equivalent to
a subshift of finite type.
We now have for a general system described by a map f the following: if limit set of this map is suitable
(see above), we can find a Markov partition, in which case we can conclude that our map f is topologically
equivalent to a subshift of finite type. Then we can describe the dynamics symbolically. This still seems
removed from clear observations we can make on our system (in general we don’t know what our limit
set is like), but it appears that this is sufficient to make the link with intersections of stable and unstable
manifolds. The details of this link are contained in the proof of the Smale-Birkhoff Homoclinic Theorem,
which is stated as follows:
Theorem 3.2 (The Smale-Birkhoff Homoclinic Theorem). Let f : Rn
→ Rn
be a diffeomorphism such
that p is a hyperbolic fixed point and there exists a point q = p of transversal intersection between Ws
(p)
and Wu
(p). Then f has a hyperbolic invariant set Λ on which f is topologically equivalent to a subshift
of finite type.
For a proof, see p. 252-253 in [1].
17

4 Duffing’s equation
The physical basis for Duffing’s equation is the motion of a buckled beam under forced vibrations.
Duffing’s equation can be used to describe certain driven and damped oscillators. However, unlike the
pendulum, Duffing’s equation is only an approximate model for the physical problem it describes. The
most general form of this equation takes the form
¨x + δ ˙x + βx + αx3
= γ cos ωt. (4.0.1)
There are five parameters in this equation. β measures the strength of the linear force, α is a measure
for the nonlinearity of the force, δ controls the strength of the damping force, γ controls the strength of
the driving force, and ω is the driving frequency. In general, the parameters δ,γ and ω are taken to be
positive real numbers.
We first study the most general form of (4.0.1) with real coefficients α and β. We then motivate why
only certain parameter ranges of (α, β) are interesting to study in depth. We will then proceed to study
these analytically, starting with the simple cases and gradually extending to the general system. We
establish local properties such as fixed points and their stability for the unforced system and analyze
the homoclinic orbit, which forms a rare case in which the orbit can be determined explicitly. We then
can apply the more sophisticated techniques of averaging and Melnikov’s Method, where we will follow
the description given in [1]. Finally, we will study the system by numerically computing its orbits and
Poincaré maps and compare the results with the analytic results.
4.1 The undamped Hamiltonian system
In the case γ = 0, δ = 0, the system is reduced to
¨x + βx + αx3
= 0. (4.1.1)
This system is Hamiltonian with potential
V (x) =
β
2
x2
+
α
4
x4
. (4.1.2)
The associated force as a function of the displacement from equilibrium is
F(x) = −βx − αx3
. (4.1.3)
Indeed, we can write the system as the two-dimensional system
˙u = v =
∂H
∂v
˙v = −βu − αu3
= −
∂H
∂u
, (4.1.4)
where the Hamiltonian is given by
H =
1
2
v2
+
β
2
u2
+
α
4
u4
(4.1.5)
The potential and associated force for different signs of α and β are shown in Figure 4.5. The level
curves of the Hamiltonian are shown in Figure 4.2. Let us analyze the different cases in some more detail.
The case α > 0, β > 0 shown in (a) leads to a potential that in shape resembles that of the harmonic
oscillator, differing only by the addition of a term fourth degree term that will dominate for large values
of x. The dynamics of this potential is qualitatively similar to that of the harmonic oscillator, and we
expect oscillating solutions trapped in the potential well. Adding damping will let the solutions approach
the stable equilibrium x = 0. Adding external forcing will likely produce behavior similar to the forced
linear oscillator, which is studied in depth in classical mechanics (see for instance [4]), although the
addition of the non-linearity could be studied in more detail. Similarly, we expect the case α < 0, β < 0
shown in (b) to be qualitatively similar to the linear system with α = 0, β < 0, which does not seem to
describe any physically interesting situation.
The cases α > 0, β < 0 and α < 0, β > 0 shown in (c) and (d), i.e. when α and β differ in sign, seem
to hold more interesting dynamics. Notice the appearance of two additional fixed points (where F = 0),
18

2 1 1 2
x
10
5
5
10
(a) α = 1, β = 1
2 1 1 2
x
10
5
5
10
(b) α = −1, β = −1
2 1 1 2
x
1.5
1.0
0.5
0.5
1.0
1.5
(c) α = 1, β = −1
2 1 1 2
x
2.0
1.5
1.0
0.5
0.5
1.0
1.5
(d) α = −1, β = 1
Figure 4.1: The potential and the associated force of the undamped, unforced Duffing equation (4.1.1)
for the different sign cases of α and β. The blue line shows the potential (4.1.2), while the red line shows
the force (4.1.3).
2 1 0 1 2
2
1
0
1
2
u
v
Figure 4.2: Level curves of the Hamiltonian of the undamped, unforced Duffing equation (4.1.1) with
α = 1, β = −1. The sinks are indicated in black and the saddle is in red.
which are of a different type than the fixed point x = 0. On a large scale, for large values of x the
potential will be dominated by the x4
term. Hence the situation sketched in (c) shows a system that is
bounded while in (d) there are unbounded solutions that diverge to ±∞. Boundedness is a property that
is generally desirable and allows one to apply more interesting techniques to study the system. Hence
our focus will be on the case α > 0, β < 0 exemplified in (c).
19

Let us therefore concentrate on the case α > 0, β < 0 and rewrite the Duffing equation as
¨x + δ ˙x − βx + αx3
= γ cos ωt, δ, β, α, γ, ω > 0 (4.1.6)
This is also the form studied by Holmes in [2] and description in [1] focusses on the special case where
β = 1, α = 1 in (4.1.6). Physically, we can interpret the choice of signs as follows: the negative linear
term causes an outward force for small values of x until an equilibrium is reached. If x is increased
further, the force will be directed inward towards the equilibrium, hence indicating stability of the fixed
point (we shall show this analytically below). The system with positive cubic term is sometimes referred
to as a hardening spring (for instance in [3]). It appears that the system (4.1.6) is an approximate model
for the dynamical behavior of a buckled beam subject to forced periodic lateral vibrations, subject to
some time-independent forces. For instance this could a beam placed in an inhomogeneous magnetic
field, shaken sinusoidally using an electromagnetic vibration generation, as sketched in Figure 4.3. The
beam is then deflected towards either of the magnets, but also has an unstable equilibrium in the middle,
where the forces of both magnets cancel each other. A slightly more detailed description of the physical
nature of the equation can be found in [2].
0.1 0.2 0.3 0.4
Γ
0.1
0.2
0.3
0.4
∆
Figure 4.3: The physical system modeled by the Duffing equation (4.1.6). Source: Figure 2.2.1 in [1].
4.2 Fixed points and local stability
Let us write the system (4.1.6) without external forcing as the two-dimensional system
˙u = v
˙v = βu − αu3
− δv, β, α, δ > 0 (4.2.1)
The equilibria of the system must satisfy ˙u = ˙v = 0, from which it follows that βu − αu3
= 0. Hence
the three equilibria are at (u, v) = (0, 0) and (u, v) = (± β
α , 0). Linearizing (4.2.1) in these points, we
obtain the Jacobian matrices
0 1
β −δ,
for (u, v) = (0, 0);
0 1
−2β −δ,
for (u, v) = (± β
α , 0). (4.2.2)
From here we read off that the eigenvalue equation at (0, 0) is given by λ2
+ δλ − β = 0, with solution
λ = −δ
2 ± 1
2 δ2 + 4β. For δ = 0 the solutions are simply λ = ±
√
β. Hence we see that (0, 0) has two
real eigenvalues of different sign, thus indicating that (0, 0) is a saddle point. If we introduce friction and
δ > 0, then because δ2 + 4β > δ we still have one positive and one negative eigenvalue, so (0, 0) is again
a saddle. The eigenvalue equation at (±
√
−β, 0) is λ2
+δλ−2β = 0, with solution λ = −δ
2 ± 1
2 δ2 − 8β.
For δ = 0 this becomes simply λ = ±
√
−2β. In this case we find two imaginary eigenvalues, indicating
that the two solutions x = ±
√
β are center points if there are is no friction. If we now let δ > 0,
because of δ2
− 8β < δ2
we have either two negative eigenvalues when or two imaginary eigenvalues with
negative real part. In both cases this indicates that the fixed points are attracting sinks and hence stable.
20

4.3 Global and structural stability
By analyzing the fixed points, we have established local stability at only a few points. This tells us that
solutions near the stable fixed points will be attracted towards these points, but it gives us no information
on what the behavior of solutions far away from these points will look like. Neither do we know whether
the system is structurally stable, so that it is insensitive to small perturbations. However, one can show
both global stability and structural stability by applying standard techniques from dynamical systems
theory.
First, let us look again at the potential sketched in Figure 4.5(c). The shape of the potential with the
two minima suggests that solutions starting on some bounded set should remain in a bounded set. If
friction is added to the system, we expect solutions not only to remain bounded, but also eventually be
attracted to one of the minima. In order to show such global attraction rigorously, one needs to find
suitable Liapunov functions. A first candidate that comes to mind it to simple consider the Hamiltonian
function given in (4.1.5). Let us compute its time-derivative on any solution trajectory of the damped
system (4.2.1):
dH
dt
= v ˙v + βu ˙u + αu3
˙u
= v(−βu − αu3
− δv) + βuv + αu3
v
= −δv2
. (4.3.1)
Hence we see that dH
dt ≤ 0 on the entire plane, again showing the stability of the two sinks, which are
also energy minima. Furthermore, dH
dt = 0 only on the u-axis. Since we have only one other fixed point,
also lying on the u-axis, which is the saddle at (0, 0), any solution not approaching this point should
eventually approach one of the sinks. Hence all solutions not starting on the unstable manifold of (0, 0)
will approach either of the sinks as t → ∞. Nevertheless, because the sinks themselves lie on the u-axis,
it is not possible to find a neighbourhood of these points where dH
dt < 0 and hence one cannot use the
Hamiltonian to prove asymptotic stability. That these neighbourhoods do in fact exist can be shown
by defining a different Liapunov function which can show that all solutions inside the homoclinic loop
will eventually be attracted to the orbit (cf. [2]). Another Liapunov function can be used to show that
solutions starting sufficiently far away from the origin will eventually be attracted to a bounded set (cf.
[2]). Hence there are no solutions diverging from any bounded set and most solutions will be attracted
to either of the sinks.
Furthermore, one can also show that the unforced, damped system (4.2.1) has no closed orbits and
is structurally stable. For this we first apply Bendixson’s Criterion to the vector field of (4.2.1) which
we shall call f = (f1, f2). We find that on the entire plane we have
∂f1
∂u
+
∂f2
∂v
= −δ < 0. (4.3.2)
Hence the system (4.2.1) has no closed orbits lying in any simply connected set lying in R2
, and hence
has no closed orbits at all. Thus we have a planar system with three hyperbolic fixed points and no closed
orbits, which apparently suffices to draw the conclusion that the system is structurally stable. We do not
have the right tools to show this rigorously, but refer to the article by Holmes [2] in which he draws this
conclusion based on these two facts and also to the section on Peixoto’s Theorem for Two-Dimensional
Flows in [1].
4.4 The homoclinic orbit
There are two ways to find the homoclinic orbits: either by identifying the level curve of the Hamiltonian
on which they lie, or by giving a general solution to the Duffing equation without damping and external
forcing. We will use the second approach, which also allows us to show that solutions to the undamped,
unforced Duffing oscillator exist in terms of an elliptic integral. The approach is in essence the same
as for the DDP, as we shall see in the next chapter. We will then find suitable initial values such that
the solution is indeed on one of the homoclinic orbits and show that the elliptic integral simplifies to an
integral that can be evaluated.
We begin with the differential equation
˙u = v
˙v = βu − αu3
, β, α > 0, (4.4.1)
21

which is simply eq. (4.2.1) with δ = 0. Using
˙v =
1
2
d
du
(v2
), (4.4.2)
we now obtain
d
du
(v2
) = 2βu − 2αu3
, (4.4.3)
which can be integrated to give
˙u2
= v2
= βu2
−
1
2
αu4
+ 2C, (4.4.4)
where C is the integration constant. Denote u(t)|t=0 = u0, v(t)|t=0 = ˙u(t)|t=0 = ˙u0. Then
2C = ˙u2
0 − βu2
0 +
1
2
αu4
0. (4.4.5)
Hence we have a single equation involving ˙u and u taking the form
˙u =
du
dt
= ± βu2 −
1
2
αu4 + 2C. (4.4.6)
By seperation of variables and integrating, we then obtain the relation
t = ±
u
u0
du
βu2 − 1
2 αu4 + 2C
. (4.4.7)
The solutions of this equation are again given in terms of Jacobi elliptic integrals, which cannot be
simpliﬁed in general.
For the homoclinic orbit, we know that the solutions lie on the level curve H = 1
2 v2
− 1
2 βu2
+ 1
4 αu4
= 0,
since the energy must be the same as for the saddle point (0, 0) (see also Figure 4.2). The points lying
on the u-axis are therefore (u, v) = (± 2β
α , 0). Choosing these points as initial value (u0, v0), we get
C = 0. Let us now compute the homoclinic orbit with u > 0 explicitly. Using C = 0 and taking a plus
sign in (4.4.8), the integral reduces to
t =
u
u0
du
βu2 − 1
2 αu4
, (4.4.8)
which can be evaluated using the substitution sin φ = α
2β u. Indeed, this gives
t =
u
u0
du
βu2 − 1
2 αu4
= β− 1
2
u
u0
du
u 1 − α
2β u2
= β− 1
2
φ
π
2
cos φ dφ
sin φ 1 − sin2
φ
= β− 1
2
φ
π
2
1
sin φ
= β− 1
2 log | tan
φ
2
|
φ
π
2
= β− 1
2 log tan
φ
2
. (4.4.9)
Reverting this relation, we obtain
φ(t) = 2 arctan e
√
βt
. (4.4.10)
22

By using sin (2 arctan y) = 2y
y2+1 = 2
y+y−1 , we obtain
u(t) =
2β
α
1
2
sin 2 arctan e
√
βt
=
2β
α
1
2
2
e
√
βt + e
√
βt
=
2β
α
1
2
sech βt. (4.4.11)
Thus the right homoclinic orbit based at (u0, v0) = ( 2β
α , 0) are given by
u+
(t) =
2β
α
1
2
sech τ
v+
(t) = β
2
α
1
2
sech τ tanh τ, (4.4.12)
where τ =
√
βt. By symmetry of the system (see for instance [2] par. 2.4), we know that the other
homoclinic orbit to the left of the saddle is given by (u−
(t), v−
(t)) = (−u+
(t), −v+
(t)).
4.5 The externally forced system
If the system is also driven, i.e. if γ > 0, δ > 0 in eq. (4.0.1), the vector field is no longer time-
independent. The usual way of writing the system as an autonomous system is by introducing a new
variable θ ∈ S1
replacing the original time variable t such that ˙θ is constant. We can then write the
general Duffing equation (4.0.1) as
˙u = v
˙v = −βu − αu3
− δv + γ cos ωθ
˙θ = 1 (4.5.1)
The third component shows that (4.5.1) has no fixed points, although we still find periodic solutions. For
γ = 0 the fixed points of the undamped system (4.2.1) become periodic orbits, with period 2π
ω , with the
same stability type. Because the unforced system is structurally stable, we know that for small values of
γ the vector fields in (4.5.1) and (4.2.1) are conjugate, and therefore the periodic orbits will persist for
small values of γ.
Furthermore, if γ is either small or large enough, one can again show global stability for the forced system
by finding suitable Liapunov functions (cf. [2]). However, if γ is neither small nor large enough, these
conclusions cannot be drawn and this is precisely the region in which we will see chaos. Not much else
can be said about the forced system (4.5.1) than these general remarks, and we will need to apply more
sophisticated techniques to derive properties of the system.
4.6 Averaging applied to the Duffing oscillator
One of the techniques discussed that seems suitable for dealing with the time-dependent system (4.5.1)
is of course averaging. In order to apply averaging, consider the Duffing oscillator as a perturbation of
the linear (harmonic) oscillator, and write
¨x + ω2
0x = [−αx3
− δ ˙x + γ cos ωt], (4.6.1)
Notice that we are now taking the Duffing oscillator with positive linear term, while so far we have concen-
trated on the system with negative linear term. Hence the results we obtain cannot be related directly to
the system studied above, but nevertheless serve to illustrate the use of the averaging technique. A sim-
ilar yet in detail somewhat different analysis is performed for the oscillator with negative linear stiffness
in [2]. Notice that the function f(x, t, ) = [−αx3
− δ ˙x + γ cos ωt] is smooth in all variables, bounded on
any bounded set U ⊃ Rn
, and hence the requirements for applying the Averaging Theorem (2.1) are met.
23

Assume that ω2
≈ ω2
0, so we look for solutions near resonance of order one and let ω2
0 − ω2
= Ω.
Then we obtain
˙u(t) = −
1
ω
(ω2
− ω2
0) x + [−αx3
− δ ˙x + γ cos ωt] sin ωt
= −
1
ω
Ω (u cos ωt − v sin ωt) + [−α (u cos ωt − v sin ωt)
3
− δ (−uω sin ωt − vω cos ωt) + γ cos ωt] sin ωt
=
ω
−Ω (u cos ωt − v sin ωt) + α (u cos ωt − v sin ωt)
3
− δ (uω sin ωt + vω cos ωt) − γ cos ωt sin ωt
and similarly we get
˙v(t) =
ω
−Ω (u cos ωt − v sin ωt) + α (u cos ωt − v sin ωt)
3
− δ (uω sin ωt + vω cos ωt) − γ cos ωt cos ωt.(4.6.2)
After expanding brackets and calculating the averages of products of trigonometric functions over one
period 2π
ω we obtain the results
˙u(t) =
2ω
−ωδu − Ωv −
3α
4
(u2
+ v2
)v
˙v(t) =
2ω
Ωu − ωδv +
3α
4
(u2
+ v2
)u − γ
(4.6.3)
We can rewrite the solutions in terms of polar coordinates r =
√
u2 + v2 and φ = arctan v
u . This gives
˙r(t) =
1
r
(u ˙u + v ˙v) =
2ωr
−ωδu2
− Ωuv −
3α
4
(u2
+ v2
)uv + Ωuv − ωδv2
+
3α
4
(u2
+ v2
)uv − γv
=
2ωr
−ωδ(u2
+ v2
) − γv
=
2ω
(−ωδr − γ sin φ) (4.6.4)
and
˙φ(t) =
1
1 + (v
u )2
(−
v ˙u
u2
+
˙v
v
)
=
1
r2
(u˙v − v ˙u)
=
2ωr2
Ωu2
− ωδuv +
3α
4
(u2
+ v2
)u2
− γu + ωδuv + Ωv2
+
3α
4
(u2
+ v2
)v2
=
2ωr2
Ω(u2
+ v2
) +
3α
4
(u2
+ v2
)2
− γu
=
2ω
Ω +
3α
4
r2
−
γ cos φ
r
(4.6.5)
Recall that
x(t) = r(t) cos (ωt + φ(t)), (4.6.6)
so that the solutions r(t) and φ(t) of equations (4.6.4) and (4.6.5) describe a changing amplitude and
phase of the solution. Hence the fixed points of the averaged system correspond to periodic, sinusoidal
orbits that behave just like the solutions of the linear oscillator.
The system (4.6.4)-(4.6.5) can be solved numerically to give the frequency response function for
the Duffing equation, which shows the location of the fixed points against the (normalized) driving
frequency ω
ω0
. An example of such a frequency response function is shown in Figure (4.4). The interesting
phenomenon in these plots is the appearance of multiple equilibria at a certain value of ω
ω0
. Increasing
this ratio results in the sudden appearance of an additional pair of equilibria from the same point different
from the original single equilibrium, of which one is a saddle and the other is a sink. The three solutions
coexist for a range of of parameter values, until the saddle joins with the original sink and both disappear,
leaving only the sink that appeared with the bifurcation. This phenomenon is comparable to a saddle-
node bifurcation, although another stable solution in addition to the saddle and node exists for a range
of parameter values around the bifurcation value. Recall that simple local bifurcations are preserved
24

under averaging, and in particular the saddle-node bifurcation is known to persist (also cf. [1] Theorem
4.3.1.). Hence we can conclude that these bifurcations found in the parameter space of ω are also to be
found for the actual system. These are the jump resonances and flip bifurcations described in §2.2 of [1].
Figure 4.4: Frequency response function for the Duffing equation with α = 0.05, δ = 0.2, γ = 2.5.
Source: Figure 2.2.1 in [1].
We can find that such a bifurcation indeed occurs when we examine the averaged system described
by (4.6.3) by plotting its vector field and finding fixed points and their stability type using a program
such as PPlane.
The question remains whether the averaged system can describe the full system for all ranges of
parameter values we are interested in. The Averaging Theorem guarantees only that local structure is
preserved and even the bifurcation corresponds to a real bifurcation. For sufficiently small, also the
global structure appears to be preserved. Indeed, we can check that the requirements of Theorem 2.2
are met. First, we apply Bendixson’s Criterion to the averaged system (4.6.3) to find that
∂f1
∂u
+
∂f2
∂v
=
1
2
−δ −
3α
4
(2uv) +
1
2
−δ +
3α
4
(2uv) = −δ < 0. (4.6.7)
This allows us to conclude that the planar averaged system (4.6.3) does not contain any closed orbits
or homoclinic loops. As for fixed points and their type, we have to rely on numerical evidence such
as presented in the frequency response curve (4.4) as the fixed points cannot be solved exactly for the
system (4.6.3). If we trust that for a range of parameter values this curve would look similar, then we
would indeed have a finite number of fixed points, each of the hyperbolic. A figure found in [3] suggest
that the fixed points lie on similar curves as (4.4) for a number of different parameter values, differing
only in their shape and the presence of the bifurcation. Hence we expect the requirements of Theorem
2.2 to be met, so that we can conclude that for small , the averaged system and original system have
similar global behavior by the notion of topological equivalence.
4.7 Melnikov’s Method applied to the Duffing oscillator
We now want to apply Melnikov’s Method to the Duffing oscillator. This will enable us to derive an
expression for the region in γδ parameter space in which there are transversal intersections between the
manifolds, and we expect to see chaos only in this region. We will compare the result with numerical
solutions of Duffing’s equation in the next section. First note that the Duffing oscillator can be described
as a perturbation of a Hamiltonian system. The Hamiltonian is given by
H(x, ˙x) =
˙x2
2
− β
x2
2
+ α
x4
4
, (4.7.1)
25

(a) ω = 1.2ω0 (b) ω = 1.5ω0
(c) ω = 1.65ω0 (d) ω = 1.8ω0
Figure 4.5: Vector field of the averaged system for different parameter values of ω, with the same
parameter values as in the frequency response curve and ω0 = 1. The figures show the fixed points and
solutions leading to these fixed points. For the two values for which there is a saddle, the stable and
unstable manifolds of the saddle are shown.
with β, α > 0. Let f = (∂H
∂v , −∂H
∂u ) be the Hamiltonian vector field and let g(u, v, t) be the perturbation
term. Then the correct form of the system is
˙u = f1(u, v) + g1(u, v, t) = v
˙v = f2(u, v) + g2(u, v, t) = βu − αu3
+ (¯γ cos ωt − ¯δv), (4.7.2)
Recall that the original Hamiltonian system has two homoclinic orbits q0
+(t) and q0
−(t) where q0
+(t) =
−q0
−(t) and q0
+(t) = 2β
α
1
2
sech τ as given in eq. (4.4.12). Furthermore, it should be clear from Figure
4.2 that the interiors of {q0
+(t)}∪{(0, 0)} and {q0
−(t)}∪{(0, 0)} are filled with periodic orbits of increasing
period. We shall not explicitly show that the period is a differentiable function of the energy, but assume
that also this condition is satisfied and we may apply Melnikov’s Method. We insert our expressions for
f1, f2, g1, g2 with the homoclinic orbit q0
+(t) and its derivate v0
+(t) as given in eq. (4.4.12) into eq. (2.2.4)
26

to obtain
M(t0) =
∞
−∞
(f1(q0
+(t))g2(q0
+(t), t + t0) − f2(q0
+(t))g1(q0
+(t), t + t0) dt
=
∞
−∞
v0
+(t) ( ¯f cos ω (t + t0) − ¯δv) dt
= ¯γ
∞
−∞
v0
+(t) cos ω (t + t0) − ¯δ
∞
−∞
v0
+(t)2
dt
= ¯γ
∞
−∞
v0
+(t) cos (ωt) cos (ωt0) dt − ¯γ
∞
−∞
v0
+(t) sin (ωt) sin (ωt0) − ¯δ
∞
−∞
v0
+(t)2
dt
= ¯γ
2β
α
1
2
cos (ωt0)
∞
−∞
sech τ tanh τ cos (ωβ− 1
2 τ) dτ − sin (ωt0)
∞
−∞
sech τ tanh τ sin (ωβ− 1
2 τ) dτ
− ¯δβ
3
2
2
α
∞
−∞
sech2
τ tanh2
τ dτ (4.7.3)
The first integral is over an odd function and hence vanishes. The second integral gives
πωsech πω
2
√
β
√
β
and
the third integral gives 2
3 . The final expression we obtain for the Melnikov function is given by
M(t0) =
4
3
¯δ
β
3
2
α
+ (
2
α
)
1
2 ¯γπω sech
πω
2β
1
2
sin ωt0. (4.7.4)
The only dependence on t0 is in the last term sin ωt0, so we see that the ratio ¯γ
¯δ
determines whether
the function becomes zero. Notice that ¯γ
¯δ
= γ
δ , so we will replace the variables with bar with variables
without bar. If the ratio γ
δ is very small, then clearly the function will remain positive for all values of t0
and the manifolds do not intersect. If it is large enough, the amplitude of the sine will be large enough
for zeros to appear. The critical value at which the function touches zero is when
4
3
δ
β
3
2
α
= (
2
α
)
1
2 γπω sech
πω
2β
1
2
(4.7.5)
This gives the following critical ratio between the parameters:
R(ω, α, β) =
γc
δ
=
4
3
β
3
2
(2α)
1
2 πω
cosh
πω
2β
1
2
. (4.7.6)
If γ > R(ω, α, β)δ then the manifolds will intersect transversally, while at the critical value γ =
R(ω, α, β)δ they touch.
4.8 Numerical solutions of the Duffing oscillator
The analysis of the Duffing oscillator is inspired by the numerical solutions of the DDP presented in [4]
by Taylor and by the numerical results in [2] and [1], although an exact investigation of this kind has
most likely not been done before. We chose to examine the qualitative behaviour for different values in
the two parameters γ and δ, while keeping the other constants fixed. We wanted to know whether how
chaos sets in the Duffing oscillator by increasing the driving amplitude γ at different values of δ. By
varying both γ and δ, we could also look for links between chaotic solutions and intersections of stable
and unstable manifolds as predicted by Melnikov’s Method in the previous part.
We first examine the behaviour of (4.1.6) in different points in γδ space, with 0 ≤ γ, δ ≤ 0.4 while
fixing ω = 1, α = 1, β = 1. We also choose the same initial conditions x(0) = 1, x (0) = 0 for all our
points. We then examined the time-displacement graph, phase space solution and Poincare sections of
the system, such as shown in Figure 4.6. We initially chose evenly spaced points in γδ space and then
took samples from more points where the behaviour was unpredicted, such as in the case of a sudden
transition to chaos or a periodic solution appearing within the chaotic regime. In the following, we refer
to a period k solution when the solution has a period which is k times the base period (which in this
case turns out to be 2π. The results of these numerical simulations are shown in Figure 4.8, and we shall
discuss the results below.
27

1.5 1.0 0.5 0.5 1.0 1.5
x t
1.5
1.0
0.5
0.5
1.0
1.5
x t
(a) γ = 0.1
3160 3180 3200 3220 3240 3260
t
0.95
1.00
1.05
1.10
x t
(b) γ = 0.1
1.5 1.0 0.5 0.5 1.0 1.5
x t
1.5
1.0
0.5
0.5
1.0
1.5
x t
(c) γ = 0.1
1.5 1.0 0.5 0.5 1.0 1.5
x t
1.5
1.0
0.5
0.5
1.0
1.5
x t
(d) γ = 0.2
3160 3180 3200 3220 3240 3260
t
0.8
0.9
1.0
1.1
1.2
x t
(e) γ = 0.2
1.5 1.0 0.5 0.5 1.0 1.5
x t
1.5
1.0
0.5
0.5
1.0
1.5
x t
(f) γ = 0.2
1.5 1.0 0.5 0.5 1.0 1.5
x t
1.5
1.0
0.5
0.5
1.0
1.5
x t
(g) γ = 0.3
3160 3180 3200 3220 3240 3260
t
1.0
0.5
0.5
1.0
1.5
x t
(h) γ = 0.3
1.5 1.0 0.5 0.5 1.0 1.5
x t
1.5
1.0
0.5
0.5
1.0
1.5
x t
(i) γ = 0.3
Figure 4.6: Numerical solutions of the Duffing oscillator (4.1.6) with ω = 1, α = 1, β = 1, δ = 0.25 and
different values for γ, up to time 2π × 1000. The left column shows the displacement x(t) against time,
the middle column the orbit in phase space (x(t), x (t)), and the right column shows Poincaré sections.
We chose the interval of time for the left and middle column to be [2π × 500, 2π × 520], allowing for 20
periods of the natural motion, while for the Poincare map we chose 100 points starting at t = 2π × 500
in order to allow more accurate determination of the period of periodic solutions.
Figure 4.7: The presence of a saddle orbit is presented in (a) of this figure from [1]. This orbit could not
be found in our numerical solutions.
28

0.1 0.2 0.3 0.4
Γ
0.1
0.2
0.3
0.4
∆
Figure 4.8: A sketch of the qualitative behavior of the Duffing oscillator for different values of δ and γ.
The black line indicates a (sudden) transition between a period one orbit and chaos. The black points
are values of γ ± 0.02 at which chaos was first observed for fixed δ. The grey line results from a closer
look where the intervals in γ are chosen to be 0.005. The blue line indicates the transition from a small
period one orbit left or right of the origin to a large period one orbit. The region below or right to
the dotted blue line indicates where Melnikov’s Method predicts intersections between the stable and
unstable manifolds. The red, orange and yellow points indicate values where a periodic orbits of period
3, 5 and 11 were detected.
For low values of γ and δ, the system attracts to a period 1 orbit either left or right of the origin.
The size of the orbit in phase space appears to increase continuously as γ increases. For our choice of
initial values, most of the time the system is attracted to the right orbit. However, increasing γ further
shows that solutions can cross the origin and attract to the other orbit if γ is large enough. If γ is the
increased further, suddenly a large period 1 orbit appears, which entirely encloses the region where the
two small orbits were located. Further increase in γ shows no qualitative changes and suggests that this
large orbit persist for a large range of γ. Whether these are the only period 1 solutions of the system
is unclear, since one would have to examine a whole range of initial values in order to determine this
from the graphs. In particular, if there are orbits of saddle type, then they will normally not be detected
unless the initial values are chosen to lie precisely on such an orbit. A figure from [1] suggests that there
is also an orbit of saddle type present (see Figure 4.9).
We increased γ gradually for different values of fixed δ and were unable to see period doubling for all
values of δ between 0 and 0.4 we chose. Taking smaller step size (up to 0.01) near the region where this
transition into chaos appeared also did not show any period doubling. The data, however, suggested a
relation between δ and the value of γ for which chaos sets up. It seems to be that chaos sets in at greater
values of γ when the δ is greater, as indicated by the black and grey lines in Figure 4.8. The relation is
not exactly linear, but seems close to linear. An explanation might be the physical argument that one
expects that a stronger damping requires stronger driving force to produce the same effect.
For low damping coefficients (δ ≤ 0.12), the chaotic solutions could not be observed in the chosen
parameter range of γ. There is, however, a bifurcation from a small period one orbit into a large period
one orbit discussed above, and the transition value of γ in this case seems have a similar dependence on
δ as the transition to chaos. The large period one orbit sometimes coexists with the strange attractor
as an attracting set, similar to what is presented by Guckenheimer and Holmes in Figure 4.9(b). Notice
the overlap at δ=0.1 between the grey and blue lines. For this value a chaos was only observed at the
specified grey point, while increasing γ up till γ=0.5 showed no transition to chaos.
Furthermore, the presence of several periodic orbits of uneven period have been detected for certain
parameter values. These have been detected for values beyond or very close to the value at which chaos
29

first sets in. Solutions of peroid 3, 5 and 11 have been identified by looking at the Poincaré map of the
system. The location of these periodic orbits can be understood by Melnikov’s Method for subharmonic
orbits, which is beyond the scope of this thesis.
One can ask whether these periodic orbits are only present at very small regions in the γδ parameter
space, and whether they are stable. Stability was examined by doing another set of measurements at
the parameter values where these periodic orbits appear, in which the initial value for x(0) was adjusted
in an interval [0, 1.2]. For each choice of x(0) we examined solution for large values of t for which initial
transients have died out. These show no difference at all and this suggests that the periodic orbits of
period 3, 5 and 11 have relatively large attracting sets and are stable. For fixed values of δ, we will
compute bifurcation diagrams below that show that indeed the region of γ for which periodic orbits can
be observed is often quite narrow. The dotted line shows the line calculated with Melnikov’s Method with
slope R(ω, α, β) as in (4.7.6) and divides the parameter space into regions where the manifolds intersect
and where they do not intersect. If γ > R(ω, α, β)δ, the manifolds intersect and we expect to see chaotic
behavior. This corresponds with the region right to or below the dotted line in Figure 4.8. Indeed, we see
that all chaotic behavior is found beyond this line, although there is a region beyond this line where chaos
does not seem to be present. However, the Smale-Birkhoff Homoclinic Theorem does clearly state that
in the presence of transversal intersections between the manifolds, there should be a horseshoe present
in the system, and one of the features of the horseshoe is the presence of chaotic, non-periodic solutions.
Whether these have simply not been detected by our choice of initial conditions, or that this is due to a
certain inaccuracy in Melnikov’s Method or our application of it, or that we should be more careful in
interpreting the Smale-Birkhoff Homoclinic Theorem, is not clear. We do mention that there are similar
findings of Holmes in [2], where he mentions that the solutions take longer to stabilize in this regime but
are not yet chaotic. Moreover, we mention that the parameters at which chaos sets in (indicated by the
black and grey lines) are close to linear, but that extending these lines to higher parameter ranges would
result in an intersection with the dotted line. Such a potential intersection will likely not take place,
mostly because Melnikov’s Method applies only for small parameter values of γ and δ and the dotted
line cannot be expected to be extended linearly to larger values of γ and δ.
Now that we have a rough overview of the qualitative behavior of the Duffing oscillator in γδ parameter
space, we wish to examine the transition to chaos more closely. For this we fix one parameter, which we
chose to be δ, slowly vary the other parameter, and make a bifurcation diagram. Specifically, we chose
δ = 0.3, and initially varied γ between 0.25 and 0.42, in steps of 0.001, and used the Poincaré map to
determine the position at roughly 200 subsequent times. The result is the diagram in Figure 4.9 which
we shall analyze now.
Starting from γ = 0.25, we see the presence of a period one orbit, which suddenly jumps to another
period one orbit at γ = 0.261, followed by a strange irregularity. This appears to be a periodic orbit and
closer examination shows that this appears to be a period thirteen orbit. Then at γ = 0.273 suddenly
chaos sets in, without periodic doubling taking place in advance, and the chaos persists for entire range
of γ studied except for a number of windows where the system returns to periodic. The first notable
appearance is at γ = 0.3, where a period give orbit persist for a small range of values. Another large
window appears at γ = 0.363, where a period three orbit can be seen, followed by what might look
like a period doubling. Also smaller intervals of periodic behavior are also present, with a width of a
single point, but these are barely visible in this diagram. One such occasion is at γ = 0.40, where we
already observed a period three orbit as indicated in Figure 4.8, and this orbit can indeed be seen in the
diagram when we zoom in enough, as we shall see below. Hence it is interesting to note that the three
periodic orbits we found when constructing Figure 4.8 are all present in the bifurcation diagram, but
they represent intervals of periodic motion of very different size.
The period thirteen orbit mentioned above has been studied in somewhat more detail and is found to
appear for a very narrow range of parameter values (smaller than 0.0001) around γ = 0.267. Furthermore,
by changing the initial value x(0) it is also apparent that it can only be found near x(0) = 1.0 with a
difference of less than 0.01. Hence it is likely that this is an unstable saddle orbit coexisting with the two
stable period one orbits which exist for a range of parameter values, or simply an nonexisting solution
resulting from inaccuracies in the numerical method, but this seems less likely.
Let us now take a closer look at what looks like a period doubling by zooming in in the region [0.37, 0.39].
We now take stepsize 0.0002 and construct a similar diagram, shown in Figure 4.10. We see that indeed
a bifurcation seems to take place around γ = 0.3726, but this is not a period doubling. Instead, it
appears that the period three orbit splits into at least two different period three orbits. The two upper
branches indicating two of the stable positions of the period three orbit split into two branches, but our
solution lies on either of the two branches, suggesting that there is another solution with different initial
30

values lying on the other branch. The branches then thicken in size, showing that the inaccuracy of
the position increases and hence the periodicity is being disturbed, to finally end up in chaos around
γ = 0.380, although an exact point where chaos sets in is difficult to be mentioned.
Finally, let us zoom in around γ = 0.40 to see the presence of the period three orbit in more detail.
We choose γ in the interval [0.39, 0.41], again with stepsize 0.0002. Now we can clearly distinguish
a small interval containing γ = 0.40 where the motion is periodic, although it is interesting to note
that within this interval there is appears to be a period four motion present one value before 0.400.
Such irregularities can be investigated in depth, but will likely raise more questions to which we do not
have a definite answer with our current knowledge. Possibly such irregularities are related to coexisting
solutions, sometimes unstable ones, which are hard to detect but which we stumbled upon by chance.
Notice also the second window around γ = 0.408, which appears to show a period nine orbit which we
had not encountered before. There are even narrower intervals of periodicity to be spotted, for instance
at γ = 0.406. This suggests that the presence of periodic solutions within the chaotic regime persists for
smaller scales and is possibly found at all length scales.
0.30 0.35 0.40
Γ
1.0
0.5
0.0
0.5
1.0
x t
Figure 4.9: A bifurcation diagram of the Duffing oscillator for γ ∈ [0.25, 0.42], in steps of 0.001.
31

0.375 0.380 0.385 0.390
Γ
1.0
0.5
0.5
1.0
x t
0.395 0.400 0.405 0.410
Γ
1.0
0.5
0.5
1.0
x t
32

Duffing oscillator and driven damped pendulum

Duffing oscillator and driven damped pendulum

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