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
Transcript
Comparison between the Duffing oscillator and the driven
damped Pendulum
Bachelor Thesis
Yiteng DangStudent 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 analyticmethods are Averaging and Melnikov’s Method, the latter of which predicts intersections betweenstable and unstable manifolds which are linked to the presence of a horseshoe similar to Smale’shorseshoe. The aim of this thesis is to study two chaotic nonlinear systems, the Duffing oscillatorand 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.
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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 remainbounded for finite times. Another feature nonlinear systems often exhibit is known as chaos. Althoughthis term has become heavily popularized over time, giving rise to titles of films, music albums and videogames, its scientific origin lies in describing dynamical systems that are hard to predict, due to beingextremely volatile with regard to changing conditions. While it is difficult to give a precise definitionof chaos, some main properties includes sensitivity to initial conditions, topological mixing and densityof periodic orbits. Topological mixing is a technical term referring to the property that any region inphase space will eventually overlap with any other region in phase space given enough time. Density ofperiodic orbits can be understood as the property that any point in space is approached arbitrarily closeby some periodic orbit. These concepts should become clearer to the reader after reading the conceptsin 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 firstsection 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 theway, 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 thatallow us to investigate certain types of such systems. These are the Averaging Method and Melnikov’sMethod, 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 manifoldsof a system. The next section on Smale’s horseshoe is entirely meant to illustrate why we should beinterested in these intersections of the stable and unstable manifolds. The short answer is given in theSmale-Birkhoff Homoclinic Theorem, but in order to have a decent understanding of this theorem itwould be good to have seen Smale’s classical horseshoe. Hence the section will first discuss this classicalhorseshoe example and its chaotic dynamics, before going over to an explanation how a horseshoe canbe 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 dampedpendulum. The next two sections will explain in detail all kinds of features of these systems, employingthe theorems and tools we introduced in the first three sections. In particular, these sections are writtenwith a similar structure in mind, so that a fair comparison between the two systems can be given bycomparing subsections with the same or similar names one after each other. The analytic analysis willbe 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 toeach other and spell out the similarities and differences by giving a concluding comparison between thesystems.As a student doing a combined bachelor programme in both mathematics and physics, I had manyoptions for doing a Bachelor Thesis. Doing lab work for several months did not interest me, neithercould I see the relevance of studying a purely mathematical subject for my future studies. Thus I lookedfor a topic that had connections with both mathematics and physics and after some searching arrivedat dynamical systems. I know that differential equations appear everywhere in physics, and perhapstheories 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 atsome later stage, but missing the link with physics entirely, which I experienced somewhat too often inmathematics courses. In the course of writing this thesis, I have received frequent and close supervisionfrom Dr. Heinz Hanßmann and would like to thank him sincerely for giving me the idea for this thesisand giving comments on my progress. Writing this thesis has introduced me to the interesting theory ofdynamical systems which I might not have approach so quickly otherwise.
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). Whenthe 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 conditionsx(0) = x0 ∈ U , there is a unique solution φ(x0, t) defined for t on some interval (a, b) which is a solutionto (1.0.1). This solution, sometimes written as x(x0, t) or just x(t), is called a solution curve, orbit ortrajectory 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 fort ∈ 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 isstable 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 thatx(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 fixedpoints are called sinks. Unstable fixed points are either sources or saddles. Stable but not asymptoticallystable fixed points are called centers.
One way to establish stability and asymptotic stability of fixed points is by finding a suitable Liapunovfunction. This is a positive definite function V : W → R defined on some neighborhood W of a fixedpoint 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 onsome neighborhood W of x satisfying
(i) V (x) ≥ 0 and V (x) = 0 iff x = x.
(ii) ddtV (x) ≤ 0 in W − x. Then x is stable. Moreover, if
(iii) ddtV (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 describedby a Hamiltonian. We will not work will Liapunov functions explicitly, but will refer to such functionsin [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) eigenvectorsof 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 thesubspace vj the eigenvector spans under the flow. The property that the flow of a set remains withinthat 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.
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Definition 1.3. For a system (1.0.3) we define the following subspaces that remain invariant under theflow etA of (1.0.3):
The stable subspace Es is spanned by the eigenvectors of A with eigenvalues having negative realparts.
The unstable subspace Eu is spanned by the eigenvectors of A with eigenvalues having positive realparts.
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 existenceand 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 findingits 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 keytheorems 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 eigenvalueswith zero real part. If all eigenvalues have negative real part, then x is a sink. If all eigenvalues havepositive real part, then x is a source. If there are eigenvalues with positive real part as well as eigenvalueswith negative real part, then x is a saddle.
The second can be seen as a generalization of the invariant subspaces for linear systems describedabove:
Definition 1.6. Let x be a fixed point of (1.0.4). The local stable and unstable manifolds of x, definedon a neighborhood U ⊂ Rn of x, are the invariant sets
W sloc(x) = x ∈ U |φt(x)→ x as t→∞, and φt(x) ∈ U for all t ≥ 0.
Wuloc(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 fromletting 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
W s(x) =⋃t≤0
φt (W sloc(x))
Wu(x) =⋃t≥0
φt (Wuloc(x)) (1.0.6)
The local existence and uniqueness theorem tells that trajectories cannot intersect themselves, andthis is in particularly true for the stable and unstable manifold of the same point. However, there canbe intersections between the stable and unstable manifolds of the same point.
Definition 1.8. Let p be a hyperbolic fixed point and suppose that W s(p) ∩ Wu(p) 6=. Let q ∈W s(p) ∩Wu(p). Then q is called a homoclinic point and the orbit φt(q)| − ∞ < t < ∞ is called ahomoclinic 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.
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If x is a hyperbolic fixed points, then the local properties of the system near x are well approximated bythe linearized system. The Hartman-Grobman theorem states that the flow φt of (1.0.4) is homeomorphicto the flow etDf(x) of the linearized system (1.0.5). The Stable Manifold Theorem for a Fixed Point statesthat 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 ofthe linearized system (1.0.5) at x. The proofs of these theorems are found in many standard texts indynamical systems theory, and although it is not proven in [1], many references to proofs are given inthe book.
For a discrete system described by a map
xn+1 = Bxn, or x 7→ 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. Acareful distinction must be made in the how the eigenvalues of the map determines whether a solutionis attracted to a fixed point, and the stability type of the fixed point. Instead of looking at the real partof the eigenvalue of the map, one has to determine the modulus of the eigenvalue and compare it to 1instead of 0. Hence for an eigenvalue λ of an eigenvector v, if λ < 1 then v lies in the stable subspace, ifλ > 1 then v lies in the unstable subspace, and if λ < 1 then v lies in the center subspace. With thesetwo distinctions made, the analogous versions of the Hartman-Grobman and Stable Manifold Theoremsfrom 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-dependentvector field, and nothing of what we derived above for the autonomous system can be directly appliedto this system. Hence we will need new methods to investigate the behavior of such a system. Since thevector 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 thesystem to eventually oscillate in resonance with the forcing and take the same period. We can determinewhether the system is indeed periodic by just looking at the position of the system at discrete points oftime which are distance T apart. In this way we get a series of ’snapshots’ of the system evenly spacedapart from which we can detect whether the system has a period that is (a fraction of) T or a multipleof T .
The technical method of making the ’snapshots’ is to construct a Poincare map. The general definitionof the Poincare map is rather technical and can be found in §1.5 of [1]. We shall now simply give theform of this map for our periodically forced system (1.0.8). First note that we can rewrite this systemas 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 cannow 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 thatit does indeed return, in a fixed time T , due to θ = 1. Hence define the Poincare 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. Aperiodic solution with period T would hence be seen as a fixed point under P . Notice that harmonicswith period 1
kT with integer k cannot be distinguished from each other, but subharmonics with periodkT would correspond to cycles of k different points on Σ under the Poincare map.Notice that by using the Poincare map, the dimension of the system is reduced by one. Since the systemstudied in fact lives one dimension higher, it is now possible to see intersections of orbits in the Poincare
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map. These are not real intersections in the original phase space the system is defined on, but are theresult of projecting the system on the hypersurface Σ. A particular case of such intersections are thosebetween the stable and unstable manifolds, and we shall see later how this is related to chaotic dynamicsin the system.
The two systems studied in this thesis are both described by second order differential equations witha periodic forcing term. If this periodic forcing is absent, the systems can be cast in the two-dimensionalform
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 thegeneral nonlinear system. Many theorems that apply specifically to two-dimensional systems but maynot be true in general are treated in §1.8 of [1]. One important result is that two dimensional systemsdo not exhibit many of the phenomena associated with chaos. For instance, one can prove that theonly possible nonwandering sets for a planar system are fixed points, closed orbits and unions of fixedpoints and the trajectories connecting them. The latter are homoclinic orbits or closed paths formed ofheteroclinic orbits. The notion of a nonwandering set is defined in §1.6 of [1], but can be understood asa set with the property that there are arbitrary large times when it approaches its initial position. Manyof these properties make it easier to study the planar system, and we shall state and prove one resultwhich 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 concludethat such a γ does not exist. Thus there are no closed orbits in D.
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2 Averaging and Melnikov’s Method
In analyzing nonlinear differential equations of the form (1.0.4), one cannot always find analytic solutionsto the system studied. Even if these such solutions exist, the expressions might be too complicated tobe useful, as we will see in our examples of the DDP and the Duffing oscillator. Hence we should lookfor methods to treat approximate such systems by simpler systems. Perturbation theory deals withcomparing properties of perturbed systems slightly deviating from a well-known system with those ofthe known system. In both case of the Duffing oscillator as well as the DDP, we can treat the systemsas perturbations of autonomous two-dimensional systems, which are easier and better understood in thelight of the previous section. Two such techniques are discussed in [1], and we shall now give the mainresults 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 periodicvector field by its time-averaged vector field. More formally, one starts with systems of the form
x = εf(x, t, ε), x ∈ U ⊂ Rn, t ∈ R, 0 ≤ ε 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 thetime variable:
y = ε1
T
∫ T
0
f(y, t, 0)dt := εf(y) (2.1.2)
It appears that many properties of the averaged system (2.1.2) are then carried over to the originalsystem when ε is small (ε 1).A number of useful properties are then perserved in the averaged system. For instance, solutions of theaveraged system and the original system remain close to each other for large timescales (O( 1
ε ), hyperbolicfixed points of the averaged system correspond to fixed points of the original system, and the stable andunstable manifolds remain close to each other for all times. Also, simple local bifurcations such as thesaddle-node bifurcation and the Hopf bifurcation are perserved under averaging. However, what is notpreserved in general is the global behavior, and in fact the technique would fail dramatically when onetries 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 averagingas presented in [1].
Theorem 2.1 (The Averaging Theorem). Consider a system of the form (2.1.1) and the associatedtime-averaged system (2.1.2). Suppose the vector field f(x, t, ε) is Ck for k ≥ 2 and bounded on boundedsets. Then there exists a Ck change of coordinates x = y + εw(y, t, ε) under which (2.1.1) becomes
y = εf(y) + ε2f1(y, t, ε), (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(ε), then |x(t)− y(t)| = O(ε) on a time scale t 1
ε .
(ii) A hyperbolic fixed point p0 of (2.1.2) corresponds uniquely to a hyperbolic periodic orbits γε of(2.1.1) of the same stability type: there exist ε0 > 0 such that γε(t) = p0 +O(ε) for all 0 < ε ≤ ε0.
(iii) Let xs(t) and ys(t) be solutions lying in the stable manifolds of the hyperbolic fixed point p0 and thehyperbolic orbit γε. If |xs(0) − ys(0)| = O(ε), then |xs(t) − ys(t)| = O(ε) for t ∈ [0,∞). Similarresults 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 bymeans of studying the averaged system. What is meant by global properties in this context is the notionof topological equivalence between vector fields, or in this case Poincare maps. Hence one can find criteriaunder which the Poincare maps of the averaged system and the original system are similar, by the notionof topological equivalence:
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Definition 2.1. Two maps F,G are topologically equivalent if there exists a homeomorphism h suchthat h F = G h. Two vector fields f, g are topologically equivalent if there exists a homeomorphism hwhich 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 alimit set consisting solely of hyperbolic fixed points and all intersections of stable and unstable manifoldsare transverse, then the corresponding Poincare map Pε|D of (2.1.1) is topologically equivalent to P0|Dfor ε > 0 sufficiently small.
In particular, if the system contains a homoclinic loop of some saddle point, the manifolds do notintersect 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 featuresof 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+ ω20x = εf(x, x, t), x ∈ R, f(x, x, t+ T ) = f(x, x, t) (2.1.4)
For ε = 0 the system is simply the linear oscillator without damping and external forcing, and thesolutions 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 expansionfor 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 Vander Pol transformation. Introduce the linear transformation(
uv
)= A
(xx
), A =
(cos ωtk − k
ω sin ωtk
− sin ωtk − k
ω cos ωtk
)(2.1.5)
Geometrically, this transformation can be interpreted as the composition of a rotation through angle ωtk
and a reflection in the x-axis with rescaling factor kω . We can see this from the decomposition
A =
(cos ωtk − k
ω sin ωtk
− sin ωtk − k
ω cos ωtk
)=
(cos ωtk sin ωt
k− sin ωt
k cos ωtk
)·(
1 00 − 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 00 −ωk
)·(
cos ωtk − sin ωtk
sin ωtk cos ωtk
)=
(cos ωtk − sin ωt
k−ωk sin ωt
k −ωk cos ωtk
). (2.1.7)
The transformation can be understood as going over in coordinates which rotate the plane by the solutionsof the linear oscillator for frequency ω
k . To see this, consider a solution to the linear oscillator with naturalfrequency ω0 = ω given by x(t) = cos ωtk , x(t) = −ωk sin ωt
k . Under the van der Pol transformation, weget (
uv
)=
(cos ωtk − k
ω sin ωtk
− sin ωtk − k
ω cos ωtk
)(x(t)x(t)
)=
(cos2 ωt
k + sin2 ωtk
− cos ωtk sin ωtk + sin ωt
k cos ωtk
)=
(10
), (2.1.8)
so the solution becomes a fixed point under the rotated coordinates. This is relevant in oscillating systemssuch as where we apply a driving force of frequency ω
k ≈ ω0 for integer k, and expect to find solutionsthat 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 calledsubharmonics 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
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(2.1.7), we get
x(t) = u(t) cosωt− v(t) sinωt
= (u(t)
√u(t)2 + v(t)2√u(t)2 + v(t)2
cosωt− v(t)
√u(t)2 + v(t)2√u(t)2 + v(t)2
sinωt)
=√u(t)2 + v(t)2(
1√1 + v(t)2
u(t)2
cosωt−vu√
1 + v(t)2
u(t)2
sinωt)
= r(t)(cosφ(t) cosωt− sinφ(t) sinωt)
= r(t) cos (ωt+ φ(t)), (2.1.9)
where we used cos (arctanx) = 1√1+x2
, sin (arctanx) = x√1+x2
with x = vu . Hence we see that a solution
in the averaged system with uv coordinates describes the varying amplitude and phase by which thesolution 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) whichcasts 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) cosωt
k− v(t) sin
ωt
k
x(t) = −u(t)ω
ksin
ωt
k− v(t)
ω
kcos
ωt
k(2.1.10)
and
u(t) = x(t) cosωt
k− x(t)
k
ωsin
ωt
k
v(t) = −x(t) sinωt
k− x(t)
k
ωcos
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
ωsin
ωt
k
)= x cos
ωt
k− xω
ksin
ωt
k− x k
ωsin
ωt
k− x cos
ωt
k
= −(xω
k+ [−ω2
0x+ εf(x, x, t)]k
ω
)sin
ωt
k
= − kω
([ω2 − k2ω2
0
k2
]x+ εf(x, x, t)
)sin
ωt
k(2.1.12)
v(t) =d
dt
(−x sin
ωt
k− x k
ωcos
ωt
k
)= −x sin
ωt
k− xω
kcos
ωt
k− x k
ωcos
ωt
k+ x sin
ωt
k
= −(xω
k+ [−ω2
0x+ εf(x, x, t)]k
ω
)cos
ωt
k
= − kω
([ω2 − k2ω2
0
k2
]x+ εf(x, x, t)
)cos
ωt
k(2.1.13)
Therefore, the general transformed system takes the form
u(t) = − kω
([ω2 − k2ω2
0
k2
]x+ εf(x, x, t)
)sin
ωt
k
v(t) = − kω
([ω2 − k2ω2
0
k2
]x+ εf(x, x, t)
)cos
ωt
k(2.1.14)
Hence, when ω2−k2ω20 = O(ε), 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 applicableto all systems with periodic forcing. In particular, consider a system which contains a homoclinic loopjoined 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 manifoldswill separate when we perturb the system slightly and cease to be joined together. There are differentscenarios possible in such a situation: one manifold could pass underneath the other, there could betransversal intersections of the manifolds, or they may touch in a number of points. Some of thesescenarios are illustrated in Figure .
In order to study the effect of small perturbations on systems which contain such a homoclinicloop, Melnikov developed a method that comes down to computing a function, now referred to as theMelnikov function, which gives information on the separation between the manifolds under a perturbationat different points. The function can be used to predict whether (transversal) intersections between themanifolds are present. We shall see in the next section that these intersections are linked to the existenceof 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 Hamiltoniansystem, where the original system contains a homoclinic orbit. To be precise, the system must take theform
x = f(x) + εg(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 periodT . Also, f should be a Hamiltonian vector field, i.e. there is a Hamiltonian function H(u, v) such thatf1 = ∂H
∂v ,f2 = −∂H∂u . This system can be recast in the autonomous form
x = f(x) + εg(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 notall of these are needed to apply the basic results. In fact, for the basic result to hold, the system shouldonly contain a homoclinic orbit to a saddle point, and an explicit expression q0(t) for this orbit shouldbe known. The other two assumptions are only needed when one deals with systems that depend on aparameter 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 whatparameter values chaos appears. Hence we summarize these assumptions in words without the technicaldetails:
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 ofperiodic orbits
3. the period of these orbits is a differentiable function of the value of the Hamiltonian on the orbitand the period increases with increasing Hamiltonian.
When ε is increased from 0, a perturbation lemma (Lemma 4.5.1 on p. [1]) tells that the perturbedsystem (2.2.1) has a unique hyperbolic periodic orbit, coinciding with the saddle point for ε = 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 distancebetween the perturbed manifolds in terms of distances between these orbits, which can be approximatedby expressions involving only the known homoclinic orbit q0(t) and the vector fields f and g. The finalresult 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 distancefunction between the stable and unstable manifolds of the saddle orbit of the perturbed system. Thevariable t0 parameterizes different locations on the manifolds, and whenever there is a point where theMelnikov function vanishes, the manifolds coincide. If this is a simple zero, i.e. if the second derivativeis non-zero, then the manifolds intersect transversally. If there are no zeros at all, the manifolds do notintersect 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 manifoldschange location when µ is changed. In particular, we want to know whether there is some value fromwhich the manifolds touch and expect transversal intersections beyond that value. Such a phenomenon iscalled a homoclinic bifurcation. The occurrence of a homoclinic bifurcation only requires the existence ofa 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 ∂2M∂t20
(τ, µb) 6= 0 and∂M∂µ (τ, µb) 6= 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 ofmaps resembling the horseshoe map in dynamical systems is strongly linked to chaotic dynamics, as shallbe 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 geometricallybest be described as follows: first perform a linear expansion in the vertical direction and a linearcontraction in the horizontal direction and then fold the result back onto the unit square such that theintersection 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 befolded onto the unit square, so its vertical length must be greater than 2, while twice its horizontal lengthmust not exceed 1. Notice that the inverse image of the vertical strips V1, V2 consists of two horizontalstrips H1, H2, both with height µ−1 and width 1. Restricted to such a horizontal band Hi, f can bedescribed as the performing the map (x, y) 7→ (±λx,±µy) followed by some translation in space, and forH2 also a rotation, giving the minus sign. Hence on Hi the map has a uniform Jacobian equal to
Df |(x,y) =
(±λ 00 ±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|f i(x) ∈ S,−∞ < i <∞. This is calledan invariant set. We are interested in the structure of this set, which can be studied by inductivelyapplying f . Applying f twice, the image of all points still in S is given by S∩f(S)∩f2(S), so its inverseimage 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 imageunder 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 verticallength µ−2, while the image consists of four vertical strips lying inside V1 ∪ V2, see Figure 3.2. We canrepeat this argument inductively to find that the set of points which remain in S after n iterations of f ,given by f−n(
⋃nk=0 f
k(S)), consists of 2n horizontal strips, each with height µ−n (and width 1). Noticethat from using the chain rule repeatedly we have for x ∈ f−1(
since each the Jacobian is uniform on H1 ∪ H2 and the each horizontal strip of f−1(⋃nk=0 f
k(S)) liesinside H1 ∪H2. The sign is determined by whether the strip gets oriented in the same direction underthe map or becomes reversed. We see that the image of one such horizontal strip is a vertical strip ofwidth λn (and height 1), and we conclude that
⋃nk=0 f
k(S) is the union of 2n disjoint vertical strips.The total height of the n-th set obtained in this way f−n(
⋃nk=0 f
k(S)) (or vertical strips⋃nk=0 f
k(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 2nµ−n =(
2µ
)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 isarbitrarily close to any other line, since the space between the strips contracts uniformly with a factordepending on µ (or λ). Clearly, the limit set f−∞(
⋃∞k=0 f
k(S)) (respectively⋃∞k=0 f
k(S)) is closed andbecause each point lies on a line that with thickness 0, the interior of the set is empty. We concludethat the limit set consisting of horizontal (vertical) lines is a Cantor set: a closed set which contains nointerior points or isolated points. The invariant set Λ consists of all points which remain in S in boththe forward and backward time direction, so it consists of the intersection of these two sets. Since eachhorizontal line intersects a vertical line in exactly one point, we conclude that the resulting intersectingset 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, anddescribe how they are mapped by f onto other points in Λ. It turns out that the entire set Λ can bedescribed in terms of bi-infinite sequences of the form a = ai∞i=−∞ with ai ∈ 1, 2. Namely, we candescribe each point by the sequence in which it passes through the horizontal bands H1 and H2, i.e. aitakes value 1 if the f i takes the point to H1 and value 2 for H2. It appears that this sequence of 1s and2s is unique to every point in Λ, and hence we can establish a bijection φ between our set Λ and set of allbi-infinite sequences Σ = ai∞i=−∞|ai ∈ (1, 2). Define the shift map σ on Σ such that σ(a) = b withbi = ai+1, so σ shifts the whole sequence of numbers one place to the left. Then this must correspondto 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 itis homeomorphism (because Λ is compact). In this case equation (3.0.7) states that f restricted to itsinvariant set Λ is topologically conjugate to the shift map σ (see Definition 1.7.2. in [1]). The proof ofthe 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 undershifting operations. This technique is called symbolic dynamics and can be used to derive the followingimportant 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 structurallystable if it is insensitive to small perturbations, in such a way that the map that defines it retains itstopological structure. The fact that Λ is structurally stable is a result by Smale and will not be discussedhere. 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 sectionsof length n. Hence the total number of orbits of period n is simply 2n and by arranging these orbitsby increasing n, we conclude that the set of periodic orbits is countable. Furthermore, no restriction isplaced 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 pointswith a nonperiodic orbit is the complement of the set of sequences that are periodic in Σ. However, Σis an uncountable, since each sequence in Σ can be identified with the binary representation of a realnumber (in which case we should take 0s and 1s instead of 1s and 2s). Explicitly, the bijection from Σto R can be defined as a 7→
∑∞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 iscountable, so if the complement were countable, then the entire set would also be countable. Thereforewe 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 6= bi.(3.0.8)
Clearly, this definition satisfies d(a,a) = 0 and d(a,b) = d(n,a). For the triangle equality, we canexamine the contribution of the i − th index in d(a, c) compared with d(a,b) + d(b, c). If ai = ci theneither ai = bi = ci or ai 6= bi, bi 6= ci. In the first case the contribution is 0 on both sides while in thesecond it is 22−|k| on the right side. If ai 6= ci then either ai 6= bi, bi = ci or ai = bi, bi 6= ci, in whichcase the contributions on both sides are equal to 2−|k|. Hence for each i the contributions to the distancesatisfies 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 finda sequence that is dense Σ, we must find a sequence that overlaps with the central part of any othersequence on an interval of arbitrary long length. Hence this orbit should contain all finite strings ofarbitrary length. Such an orbit can be found by placing all finite strings after one other. The collectionof finite strings is countable, by the argument for part (i) and they all have finite length. Therefore, theentries of this string can be listed by listing all finite strings one after another. For example, start withthe two strings of length 1, then list the four strings of length 2, and so on:
We can even take the negative indices arbitrary in this way. Clearly, any finite string can be brought tothe 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 ofarbitrary length, we can find a periodic orbit that contains that central part, because there are periodicorbits 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) andconcludes 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 exhibitscharacteristics that are typical of chaotic systems. For instance, the fact that invariant set Λ is a Cantorset implies that it becomes very difficult to predict whether a point in unit square S will remain in S forall time unless one knows the position of the point with extreme accuracy. In any neighbourhood of apoint in Λ there are points not in Λ (this is one of the properties of a Cantor set). Hence the behaviourof the system is extremely sensitive to initial conditions. The existence of nonperiodic orbits shows thatfor certain initial values the orbit continues to be irregular for an indefinite amount of time. Density oforbits 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 generalizethis to include examples of similar maps contained within systems such as the Duffing oscillator. Thisgeneralization can be done for the system of a bouncing ball described in section 2.4 of [1], but for theDuffing oscillator it is not so straightforward. The first problem that arises is that the Duffing oscillatoris a continuous system whereas the bouncing ball and horseshoe systems are both discrete systems.However, the Duffing oscillator with γ 6= 0 is reduced to a discrete system by the Poincare map Pγ . ThisPoincare 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-likephenomena in the Duffing system.We can proceed by extending the specific example by Smale to more general horseshoes and derivegeneral results on these extensions. Then we can try to apply these results to the case of the Duffingoscillator by studying numerical results obtained by approximations of the Poincare 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 resultsthat can be applied to find horseshoes in the Duffing example while avoiding the details of the abundanceof technical definitions and assumptions which the authors use to derive their results. Nevertheless, Ishall 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 = EuΛ ⊕ EsΛ withthe property that there are constants C > 0, 0 < λ < 1 such that:
1. If v ∈ EuΛ, then |Df−n(x)v| ≤ Cλn|v|.
2. If v ∈ EsΛ, then |Dfn(x)v| ≤ Cλn|v|.
Here TΛRn = TxRn|x ∈ Λ consists of the tangent spaces at each point in the invariant set andthe decomposition means TxRn = Eux ⊕ Esx for each x ∈ Λ. Invariance means Df(Eux ) = Euf(x) and
Df(Esx) = Esf(x) and continuity means that the bases in the tangent spaces can be chosen such that theydepend continuously on x.The set Λ is then referred to as a hyperbolic set. Note that a hyperbolic set is not too different froma union of hyperbolic points, to which an additional structure is attached. For instance, the hyperbolicstructure 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 eachpoint. So for each x ∈ Λ, Esx is a horizontal line and Eux a vertical line. Since the Jacobian was givenby (3.0.5) at each point, the contraction and expansion factors are uniform and we can take λ in thedefinition to be the maxλ, µ−1 and C = 1.
Now we want to extend the horseshoe example to more general cases. Firstly, one can imagine thatslightly 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 wasstated 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 setthat is similar to that of horseshoe map. Loosely summarized, if we can identify disjoint horizontal andvertical strips (not necessarily straight) in the domain and range of our map, the map sends horizontalstrips to vertical strips and thereby stretches these uniformly in one or the other direction, then the mapis 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 contractedin the other by Df , then we can also put a hyperbolic structure on the invariant set of this map. Theseconditions 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 onΣ 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. Thetheorem tells that we can describe the map f in terms of bi-infinite sequences in a similar way as wecould do with the horseshoe map. In particular, the properties we derived in Theorem 3.1 should allhold, 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 thebouncing ball problem a rather complicated discussion is needed. Clearly, there must be a better wayof establishing the existence of chaotic phenomena by the presence of horseshoes. Indeed, recall thatthe presence of horseshoes is closely linked with intersections of stable and unstable manifolds, and thiswas the motivation behind applying Melnikov’s method to study these intersections. The numericaldata already suggests that there is a close link between intersections between the stable and unstablemanifolds 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. Theproof can be found on page 252-253 of [1] and makes extensive use of symbolic dynamics. Essentially, thistheorem states that the existence of transversal intersections between the stable and unstable manifoldof some point imply that we can give a symbolic description of the dynamics in a similar way as we didin the horseshoe example.Firstly, we need to extend the identifying two horizontal strips that contain the invariant set and keepingtrack of in which of the two strips a points gets mapped to, to a more general situation in which we mayhave a very different partitioning of the set. This is formally done by constructing Markov partitions of aset, and identifying orbits with sequences that keep track in which rectangle of the Markov partitioninga point gets mapped to.In order to make this more precise, we will need more definitions. Firstly, we want to divide the setinto 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, thestable manifold of one point and the unstable manifold of the other, with the same diameter ε, intersectin exactly one point in R. This can be denoted as W s
ε (x) ∩Wuε (y) = p, p ∈ R, and for the W s
ε (x) andWuε (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 propositionwe must use the notion of an indecomposable set, which is also a requirement for the existence of Markovpartitions as shall be shown later. Consider a dynamical system with flow φt and assume it has a closedinvariant set Λ.
Definition 3.2. A closed invariant set Λ is indecomposable if for every pair of points x, y in Λ and ε > 0,there are points x = x0, x1, . . . , xn−1, xn = y and times t1, t2, . . . , tn−1, tn ≤ 1 such that the distancefrom φti to xt is smaller than ε.
Intuitively, this means that any two points in the set can be connected by a chain of points, eachof which approximates the next arbitrarily closely through the flow map. We can define the analogousproperty for a map G by taking ti to be integer and Gti instead of φti . The attractor of Smale’s horseshoemap Λ is indecomposable because the map f has an orbit which is dense in Λ. Hence we can take x1 tobe a point in Λ which is ε close to x and know that there after some number of iterations n, fn(x1) willbe ε close to y.The rectangles defined in this way will be used for the symbolic description of the invariant set Λ. Theidea now is to find a partitioning of Λ into a finite collection of rectangles so that we can identify pointsby the numbering of the rectangles it passes through under the concerned mapping. Formally, this iscalled 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 =⋃mi=1Ri;
2. int Ri ∩ int Rj = ∅ if i 6= j;
16
3. f(Wu(x,Ri)) ⊃ Wu(f(x), Rj) and f(W s(x,Ri)) ⊂ W s(f(x), Rj) whenever x ∈ int Ri, f(x) ∈ intRj .
The interior of a rectangle is defined to be the set of points in R for which there exists a δ > 0 suchthat the intersections of the manifolds with diameter δ with Λ are still in R, i.e. Wu
δ (x) ∩ Λ ⊂ R andW sδ (x) ∩ Λ ⊂ R.
We now have a theorem that states that such Markov partitions exist under suitable conditions. Tobe precise, the invariant sets must be compact, maximal, indecomposable and hyperbolic, see Theorem5.3.2.If we have found such a Markov partitioning, we can describe the dynamics by keeping track of therectangles a point passes through and this is formally described in Proposition 5.3.4. This proposition isvery 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 theelements 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 fromone rectangle to the other that is not possible under the map f . This requirement can be stated as intRai ∩ f−1(intRai+1) 6= ∅ for all i ∈ Z. Let ΣA be the complete set of such bi-infinite sequences thatsatisfy 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. Thereason behind this is that we are dealing with a Markov partition whose rectangles are not necessarilydisjoint, i.e. Ri ∩ Rj 6= ∅ 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 anumber 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 factbijective. This need to identify sequences with each other can be compared with an identification in thereal numbers, namely that 0.999 . . . = 1. Here, we can also find two different symbolic representationsthat 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 mapas defined earlier. Nevertheless, the statement f π = π σ should still be true. The set ΣA togetherwith σ is called a subshift of finite type and the statement becomes that f is topologically equivalent toa 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 topologicallyequivalent to a subshift of finite type. Then we can describe the dynamics symbolically. This still seemsremoved from clear observations we can make on our system (in general we don’t know what our limitset is like), but it appears that this is sufficient to make the link with intersections of stable and unstablemanifolds. 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 suchthat p is a hyperbolic fixed point and there exists a point q 6= p of transversal intersection between W s(p)and Wu(p). Then f has a hyperbolic invariant set Λ on which f is topologically equivalent to a subshiftof 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 thependulum, Duffing’s equation is only an approximate model for the physical problem it describes. Themost general form of this equation takes the form
x+ δx+ βx+ αx3 = γ cosωt. (4.0.1)
There are five parameters in this equation. β measures the strength of the linear force, α is a measurefor the nonlinearity of the force, δ controls the strength of the damping force, γ controls the strength ofthe driving force, and ω is the driving frequency. In general, the parameters δ,γ and ω are taken to bepositive real numbers.We first study the most general form of (4.0.1) with real coefficients α and β. We then motivate whyonly certain parameter ranges of (α, β) are interesting to study in depth. We will then proceed to studythese analytically, starting with the simple cases and gradually extending to the general system. Weestablish local properties such as fixed points and their stability for the unforced system and analyzethe homoclinic orbit, which forms a rare case in which the orbit can be determined explicitly. We thencan apply the more sophisticated techniques of averaging and Melnikov’s Method, where we will followthe description given in [1]. Finally, we will study the system by numerically computing its orbits andPoincare 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
2x2 +
α
4x4. (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
2v2 +
β
2u2 +
α
4u4 (4.1.5)
The potential and associated force for different signs of α and β are shown in Figure 4.5. The levelcurves 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 harmonicoscillator, differing only by the addition of a term fourth degree term that will dominate for large valuesof x. The dynamics of this potential is qualitatively similar to that of the harmonic oscillator, and weexpect oscillating solutions trapped in the potential well. Adding damping will let the solutions approachthe stable equilibrium x = 0. Adding external forcing will likely produce behavior similar to the forcedlinear oscillator, which is studied in depth in classical mechanics (see for instance [4]), although theaddition of the non-linearity could be studied in more detail. Similarly, we expect the case α < 0, β < 0shown in (b) to be qualitatively similar to the linear system with α = 0, β < 0, which does not seem todescribe 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, seemto hold more interesting dynamics. Notice the appearance of two additional fixed points (where F = 0),
18
-2 -1 1 2x
-10
-5
5
10
(a) α = 1, β = 1
-2 -1 1 2x
-10
-5
5
10
(b) α = −1, β = −1
-2 -1 1 2x
-1.5
-1.0
-0.5
0.5
1.0
1.5
(c) α = 1, β = −1
-2 -1 1 2x
-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 showsthe 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) withα = 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 thepotential will be dominated by the x4 term. Hence the situation sketched in (c) shows a system that isbounded while in (d) there are unbounded solutions that diverge to ±∞. Boundedness is a property thatis generally desirable and allows one to apply more interesting techniques to study the system. Henceour 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
This is also the form studied by Holmes in [2] and description in [1] focusses on the special case whereβ = 1, α = 1 in (4.1.6). Physically, we can interpret the choice of signs as follows: the negative linearterm causes an outward force for small values of x until an equilibrium is reached. If x is increasedfurther, the force will be directed inward towards the equilibrium, hence indicating stability of the fixedpoint (we shall show this analytically below). The system with positive cubic term is sometimes referredto as a hardening spring (for instance in [3]). It appears that the system (4.1.6) is an approximate modelfor the dynamical behavior of a buckled beam subject to forced periodic lateral vibrations, subject tosome time-independent forces. For instance this could a beam placed in an inhomogeneous magneticfield, shaken sinusoidally using an electromagnetic vibration generation, as sketched in Figure 4.3. Thebeam 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 physicalnature of the equation can be found in [2].
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
ææ
æ
æ
æ
æ
æ
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) = (±√
βα , 0). Linearizing (4.2.1) in these points, we
obtain the Jacobian matrices(0 1β −δ,
)for (u, v) = (0, 0);
(0 1−2β −δ,
)for (u, v) = (±
√βα , 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 = − δ2 ±12
√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 = − δ2 ±
12
√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 withnegative 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 thatsolutions near the stable fixed points will be attracted towards these points, but it gives us no informationon what the behavior of solutions far away from these points will look like. Neither do we know whetherthe system is structurally stable, so that it is insensitive to small perturbations. However, one can showboth global stability and structural stability by applying standard techniques from dynamical systemstheory.First, let us look again at the potential sketched in Figure 4.5(c). The shape of the potential with thetwo minima suggests that solutions starting on some bounded set should remain in a bounded set. Iffriction is added to the system, we expect solutions not only to remain bounded, but also eventually beattracted to one of the minima. In order to show such global attraction rigorously, one needs to findsuitable Liapunov functions. A first candidate that comes to mind it to simple consider the Hamiltonianfunction given in (4.1.5). Let us compute its time-derivative on any solution trajectory of the dampedsystem (4.2.1):
dH
dt= vv + βuu+ αu3u
= v(−βu− αu3 − δv) + βuv + αu3v
= −δv2. (4.3.1)
Hence we see that dHdt ≤ 0 on the entire plane, again showing the stability of the two sinks, which are
also energy minima. Furthermore, dHdt = 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 shouldeventually 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 theHamiltonian to prove asymptotic stability. That these neighbourhoods do in fact exist can be shownby defining a different Liapunov function which can show that all solutions inside the homoclinic loopwill eventually be attracted to the orbit (cf. [2]). Another Liapunov function can be used to show thatsolutions 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 attractedto either of the sinks.
Furthermore, one can also show that the unforced, damped system (4.2.1) has no closed orbits andis structurally stable. For this we first apply Bendixson’s Criterion to the vector field of (4.2.1) whichwe 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 hencehas no closed orbits at all. Thus we have a planar system with three hyperbolic fixed points and no closedorbits, which apparently suffices to draw the conclusion that the system is structurally stable. We do nothave the right tools to show this rigorously, but refer to the article by Holmes [2] in which he draws thisconclusion based on these two facts and also to the section on Peixoto’s Theorem for Two-DimensionalFlows in [1].
4.4 The homoclinic orbit
There are two ways to find the homoclinic orbits: either by identifying the level curve of the Hamiltonianon which they lie, or by giving a general solution to the Duffing equation without damping and externalforcing. 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 sameas for the DDP, as we shall see in the next chapter. We will then find suitable initial values such thatthe solution is indeed on one of the homoclinic orbits and show that the elliptic integral simplifies to anintegral 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
2αu4 + 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 = u20 − βu2
0 +1
2αu4
0. (4.4.5)
Hence we have a single equation involving u and u taking the form
u =du
dt= ±
√βu2 − 1
2αu4 + 2C. (4.4.6)
By seperation of variables and integrating, we then obtain the relation
t = ±∫ u
u0
du√βu2 − 1
2αu4 + 2C
. (4.4.7)
The solutions of this equation are again given in terms of Jacobi elliptic integrals, which cannot besimplified in general.For the homoclinic orbit, we know that the solutions lie on the level curve H = 1
2v2− 1
2βu2 + 1
4αu4 = 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) = (±√
2βα , 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 plussign in (4.4.8), the integral reduces to
t =
∫ u
u0
du√βu2 − 1
2αu4, (4.4.8)
which can be evaluated using the substitution sinφ =√
α2βu. Indeed, this gives
t =
∫ u
u0
du√βu2 − 1
2αu4
= β−12
∫ u
u0
du
u√
1− α2βu
2
= β−12
∫ φ
π2
cosφdφ
sinφ√
1− sin2 φ
= β−12
∫ φ
π2
1
sinφ
= β−12
[log | tan
φ
2|]φπ2
= β−12 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) = 2yy2+1 = 2
y+y−1 , we obtain
u(t) =
(2β
α
) 12
sin[2 arctan
(e√βt)]
=
(2β
α
) 12 2
e√βt + e
√βt
=
(2β
α
) 12
sech√βt. (4.4.11)
Thus the right homoclinic orbit based at (u0, v0) = (√
2βα , 0) are given by
u+(t) =
(2β
α
) 12
sech τ
v+(t) = β
(2
α
) 12
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 newvariable θ ∈ S1 replacing the original time variable t such that θ is constant. We can then write thegeneral Duffing equation (4.0.1) as
u = v
v = −βu− αu3 − δv + γ cosωθ
θ = 1 (4.5.1)
The third component shows that (4.5.1) has no fixed points, although we still find periodic solutions. Forγ = 0 the fixed points of the undamped system (4.2.1) become periodic orbits, with period 2π
ω , with thesame stability type. Because the unforced system is structurally stable, we know that for small values ofγ the vector fields in (4.5.1) and (4.2.1) are conjugate, and therefore the periodic orbits will persist forsmall values of γ.Furthermore, if γ is either small or large enough, one can again show global stability for the forced systemby finding suitable Liapunov functions (cf. [2]). However, if γ is neither small nor large enough, theseconclusions cannot be drawn and this is precisely the region in which we will see chaos. Not much elsecan be said about the forced system (4.5.1) than these general remarks, and we will need to apply moresophisticated 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 ofthe linear (harmonic) oscillator, and write
x+ ω20x = ε[−αx3 − δx+ γ cosωt], (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 tothe 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 stiffnessin [2]. Notice that the function f(x, t, ε) = ε[−αx3− δx+γ cosωt] is smooth in all variables, bounded onany bounded set U ⊃ Rn, and hence the requirements for applying the Averaging Theorem (2.1) are met.
23
Assume that ω2 ≈ ω20 , so we look for solutions near resonance of order one and let ω2
0 − ω2 = εΩ.Then we obtain
u(t) = − 1
ω
([(ω2 − ω2
0)]x+ ε[−αx3 − δx+ γ cosωt]
)sinωt
= − 1
ω
(εΩ (u cosωt− v sinωt) + ε[−α (u cosωt− v sinωt)
3 − δ (−uω sinωt− vω cosωt) + γ cosωt])
sinωt
=ε
ω
(−Ω (u cosωt− v sinωt) + α (u cosωt− v sinωt)
3 − δ (uω sinωt+ vω cosωt)− γ cosωt)
sinωt
and similarly we get
v(t) =ε
ω
(−Ω (u cosωt− v sinωt) + α (u cosωt− v sinωt)
3 − δ (uω sinωt+ vω cosωt)− γ cosωt)
cosωt.(4.6.2)
After expanding brackets and calculating the averages of products of trigonometric functions over oneperiod 2π
ω we obtain the results
u(t) =ε
2ω
(−ωδu− Ωv − 3α
4(u2 + v2)v
)v(t) =
ε
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
(vu
). This gives
r(t) =1
r(uu+ vv) =
ε
2ωr
(−ωδu2 − Ωuv − 3α
4(u2 + v2)uv + Ωuv − ωδv2 +
3α
4(u2 + v2)uv − γv
)=
ε
2ωr
(−ωδ(u2 + v2)− γv
)=
ε
2ω(−ωδr − γ sinφ) (4.6.4)
and
φ(t) =1
1 + ( vu )2(−vu
u2+v
v)
=1
r2(uv − vu)
=ε
2ωr2
(Ωu2 − ωδuv +
3α
4(u2 + v2)u2 − γu+ ωδuv + Ωv2 +
3α
4(u2 + v2)v2
)=
ε
2ωr2
(Ω(u2 + v2) +
3α
4(u2 + v2)2 − γu
)=
ε
2ω
(Ω +
3α
4r2 − γ cosφ
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 andphase of the solution. Hence the fixed points of the averaged system correspond to periodic, sinusoidalorbits 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 forthe Duffing equation, which shows the location of the fixed points against the (normalized) drivingfrequency 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 ωω0
. Increasingthis ratio results in the sudden appearance of an additional pair of equilibria from the same point differentfrom the original single equilibrium, of which one is a saddle and the other is a sink. The three solutionscoexist 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 rangeof 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] Theorem4.3.1.). Hence we can conclude that these bifurcations found in the parameter space of ω are also to befound 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 εα = 0.05,εδ = 0.2,εγ = 2.5.Source: Figure 2.2.1 in [1].
We can find that such a bifurcation indeed occurs when we examine the averaged system describedby (4.6.3) by plotting its vector field and finding fixed points and their stability type using a programsuch as PPlane.
The question remains whether the averaged system can describe the full system for all ranges ofparameter values we are interested in. The Averaging Theorem guarantees only that local structure ispreserved and even the bifurcation corresponds to a real bifurcation. For ε sufficiently small, also theglobal structure appears to be preserved. Indeed, we can check that the requirements of Theorem 2.2are 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 orbitsor homoclinic loops. As for fixed points and their type, we have to rely on numerical evidence suchas presented in the frequency response curve (4.4) as the fixed points cannot be solved exactly for thesystem (4.6.3). If we trust that for a range of parameter values this curve would look similar, then wewould indeed have a finite number of fixed points, each of the hyperbolic. A figure found in [3] suggestthat the fixed points lie on similar curves as (4.4) for a number of different parameter values, differingonly in their shape and the presence of the bifurcation. Hence we expect the requirements of Theorem2.2 to be met, so that we can conclude that for small ε, the averaged system and original system havesimilar 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 anexpression for the region in γδ parameter space in which there are transversal intersections between themanifolds, and we expect to see chaos only in this region. We will compare the result with numericalsolutions of Duffing’s equation in the next section. First note that the Duffing oscillator can be describedas a perturbation of a Hamiltonian system. The Hamiltonian is given by
H(x, x) =x2
2− β x
2
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 sameparameter values as in the frequency response curve and ω0 = 1. The figures show the fixed points andsolutions leading to these fixed points. For the two values for which there is a saddle, the stable andunstable manifolds of the saddle are shown.
with β, α > 0. Let f = (∂H∂v ,−∂H∂u ) be the Hamiltonian vector field and let εg(u, v, t) be the perturbation
Recall that the original Hamiltonian system has two homoclinic orbits q0+(t) and q0
−(t) where q0+(t) =
−q0−(t) and q0
+(t) =(
2βα
) 12
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 increasingperiod. We shall not explicitly show that the period is a differentiable function of the energy, but assumethat also this condition is satisfied and we may apply Melnikov’s Method. We insert our expressions forf1, 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 cosω (t+ t0)− δv)
]dt
= γ
∫ ∞−∞
v0+(t) cosω (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
) 12[cos (ωt0)
∫ ∞−∞
sech τ tanh τ cos (ωβ−12 τ) dτ − sin (ωt0)
∫ ∞−∞
sech τ tanh τ sin (ωβ−12 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 givesπωsech
(πω2√β
)√β
and
the third integral gives 23 . The final expression we obtain for the Melnikov function is given by
M(t0) =4
3δβ
32
α+ (
2
α)
12 γπω sech
(πω
2β12
)sinωt0. (4.7.4)
The only dependence on t0 is in the last term sinωt0, so we see that the ratio γδ
determines whether
the function becomes zero. Notice that γδ
= γδ , so we will replace the variables with bar with variables
without bar. If the ratio γδ 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 enoughfor zeros to appear. The critical value at which the function touches zero is when
4
3δβ
32
α= (
2
α)
12 γπω sech
(πω
2β12
)(4.7.5)
This gives the following critical ratio between the parameters:
R(ω, α, β) =γcδ
=4
3
β32
(2α)12πω
cosh
(πω
2β12
). (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 hasmost likely not been done before. We chose to examine the qualitative behaviour for different values inthe two parameters γ and δ, while keeping the other constants fixed. We wanted to know whether howchaos sets in the Duffing oscillator by increasing the driving amplitude γ at different values of δ. Byvarying both γ and δ, we could also look for links between chaotic solutions and intersections of stableand 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 whilefixing ω = 1, α = 1, β = 1. We also choose the same initial conditions x(0) = 1, x′(0) = 0 for all ourpoints. We then examined the time-displacement graph, phase space solution and Poincare sections ofthe system, such as shown in Figure 4.6. We initially chose evenly spaced points in γδ space and thentook samples from more points where the behaviour was unpredicted, such as in the case of a suddentransition to chaos or a periodic solution appearing within the chaotic regime. In the following, we referto a period k solution when the solution has a period which is k times the base period (which in thiscase turns out to be 2π. The results of these numerical simulations are shown in Figure 4.8, and we shalldiscuss the results below.
27
-1.5 -1.0 -0.5 0.5 1.0 1.5xHtL
-1.5
-1.0
-0.5
0.5
1.0
1.5
x HtL
(a) γ = 0.1
3160 3180 3200 3220 3240 3260t
0.95
1.00
1.05
1.10
xHtL
(b) γ = 0.1
-1.5 -1.0 -0.5 0.5 1.0 1.5xHtL
-1.5
-1.0
-0.5
0.5
1.0
1.5
x HtL
(c) γ = 0.1
-1.5 -1.0 -0.5 0.5 1.0 1.5xHtL
-1.5
-1.0
-0.5
0.5
1.0
1.5
x HtL
(d) γ = 0.2
3160 3180 3200 3220 3240 3260t
0.8
0.9
1.0
1.1
1.2
xHtL
(e) γ = 0.2
-1.5 -1.0 -0.5 0.5 1.0 1.5xHtL
-1.5
-1.0
-0.5
0.5
1.0
1.5
x HtL
(f) γ = 0.2
-1.5 -1.0 -0.5 0.5 1.0 1.5xHtL
-1.5
-1.0
-0.5
0.5
1.0
1.5
x HtL
(g) γ = 0.3
3160 3180 3200 3220 3240 3260t
-1.0
-0.5
0.5
1.0
1.5
xHtL
(h) γ = 0.3
-1.5 -1.0 -0.5 0.5 1.0 1.5xHtL
-1.5
-1.0
-0.5
0.5
1.0
1.5
x HtL
(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 anddifferent 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 Poincare sections.We chose the interval of time for the left and middle column to be [2π × 500, 2π × 520], allowing for 20periods of the natural motion, while for the Poincare map we chose 100 points starting at t = 2π × 500in 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 notbe found in our numerical solutions.
28
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
ææ
æ
æ
æ
æ
æ
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 pointsare values of γ ± 0.02 at which chaos was first observed for fixed δ. The grey line results from a closerlook where the intervals in γ are chosen to be 0.005. The blue line indicates the transition from a smallperiod one orbit left or right of the origin to a large period one orbit. The region below or right tothe dotted blue line indicates where Melnikov’s Method predicts intersections between the stable andunstable manifolds. The red, orange and yellow points indicate values where a periodic orbits of period3, 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 ofinitial values, most of the time the system is attracted to the right orbit. However, increasing γ furthershows that solutions can cross the origin and attract to the other orbit if γ is large enough. If γ is theincreased further, suddenly a large period 1 orbit appears, which entirely encloses the region where thetwo small orbits were located. Further increase in γ shows no qualitative changes and suggests that thislarge orbit persist for a large range of γ. Whether these are the only period 1 solutions of the systemis unclear, since one would have to examine a whole range of initial values in order to determine thisfrom the graphs. In particular, if there are orbits of saddle type, then they will normally not be detectedunless the initial values are chosen to lie precisely on such an orbit. A figure from [1] suggests that thereis 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 allvalues of δ between 0 and 0.4 we chose. Taking smaller step size (up to 0.01) near the region where thistransition into chaos appeared also did not show any period doubling. The data, however, suggested arelation between δ and the value of γ for which chaos sets up. It seems to be that chaos sets in at greatervalues of γ when the δ is greater, as indicated by the black and grey lines in Figure 4.8. The relation isnot exactly linear, but seems close to linear. An explanation might be the physical argument that oneexpects 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 chosenparameter range of γ. There is, however, a bifurcation from a small period one orbit into a large periodone orbit discussed above, and the transition value of γ in this case seems have a similar dependence onδ as the transition to chaos. The large period one orbit sometimes coexists with the strange attractoras an attracting set, similar to what is presented by Guckenheimer and Holmes in Figure 4.9(b). Noticethe overlap at δ=0.1 between the grey and blue lines. For this value a chaos was only observed at thespecified 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 certainparameter 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 Poincare map of thesystem. The location of these periodic orbits can be understood by Melnikov’s Method for subharmonicorbits, 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 parameterspace, and whether they are stable. Stability was examined by doing another set of measurements atthe parameter values where these periodic orbits appear, in which the initial value for x(0) was adjustedin an interval [0, 1.2]. For each choice of x(0) we examined solution for large values of t for which initialtransients have died out. These show no difference at all and this suggests that the periodic orbits ofperiod 3, 5 and 11 have relatively large attracting sets and are stable. For fixed values of δ, we willcompute bifurcation diagrams below that show that indeed the region of γ for which periodic orbits canbe observed is often quite narrow. The dotted line shows the line calculated with Melnikov’s Method withslope R(ω, α, β) as in (4.7.6) and divides the parameter space into regions where the manifolds intersectand where they do not intersect. If γ > R(ω, α, β)δ, the manifolds intersect and we expect to see chaoticbehavior. This corresponds with the region right to or below the dotted line in Figure 4.8. Indeed, we seethat all chaotic behavior is found beyond this line, although there is a region beyond this line where chaosdoes not seem to be present. However, the Smale-Birkhoff Homoclinic Theorem does clearly state thatin the presence of transversal intersections between the manifolds, there should be a horseshoe presentin 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 acertain inaccuracy in Melnikov’s Method or our application of it, or that we should be more careful ininterpreting the Smale-Birkhoff Homoclinic Theorem, is not clear. We do mention that there are similarfindings of Holmes in [2], where he mentions that the solutions take longer to stabilize in this regime butare not yet chaotic. Moreover, we mention that the parameters at which chaos sets in (indicated by theblack and grey lines) are close to linear, but that extending these lines to higher parameter ranges wouldresult 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 dottedline 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 parameterspace, we wish to examine the transition to chaos more closely. For this we fix one parameter, which wechose to be δ, slowly vary the other parameter, and make a bifurcation diagram. Specifically, we choseδ = 0.3, and initially varied γ between 0.25 and 0.42, in steps of 0.001, and used the Poincare map todetermine the position at roughly 200 subsequent times. The result is the diagram in Figure 4.9 whichwe shall analyze now.Starting from γ = 0.25, we see the presence of a period one orbit, which suddenly jumps to anotherperiod one orbit at γ = 0.261, followed by a strange irregularity. This appears to be a periodic orbit andcloser examination shows that this appears to be a period thirteen orbit. Then at γ = 0.273 suddenlychaos sets in, without periodic doubling taking place in advance, and the chaos persists for entire rangeof γ studied except for a number of windows where the system returns to periodic. The first notableappearance is at γ = 0.3, where a period give orbit persist for a small range of values. Another largewindow appears at γ = 0.363, where a period three orbit can be seen, followed by what might looklike a period doubling. Also smaller intervals of periodic behavior are also present, with a width of asingle point, but these are barely visible in this diagram. One such occasion is at γ = 0.40, where wealready observed a period three orbit as indicated in Figure 4.8, and this orbit can indeed be seen in thediagram when we zoom in enough, as we shall see below. Hence it is interesting to note that the threeperiodic orbits we found when constructing Figure 4.8 are all present in the bifurcation diagram, butthey 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 toappear 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 adifference of less than 0.01. Hence it is likely that this is an unstable saddle orbit coexisting with the twostable period one orbits which exist for a range of parameter values, or simply an nonexisting solutionresulting 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 indeeda bifurcation seems to take place around γ = 0.3726, but this is not a period doubling. Instead, itappears that the period three orbit splits into at least two different period three orbits. The two upperbranches indicating two of the stable positions of the period three orbit split into two branches, but oursolution 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 ofthe position increases and hence the periodicity is being disturbed, to finally end up in chaos aroundγ = 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 distinguisha small interval containing γ = 0.40 where the motion is periodic, although it is interesting to notethat 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 nothave a definite answer with our current knowledge. Possibly such irregularities are related to coexistingsolutions, 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 wehad not encountered before. There are even narrower intervals of periodicity to be spotted, for instanceat γ = 0.406. This suggests that the presence of periodic solutions within the chaotic regime persists forsmaller scales and is possibly found at all length scales.
0.30 0.35 0.40Γ
-1.0
-0.5
0.0
0.5
1.0
xHtL
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 0.390Γ
-1.0
-0.5
0.5
1.0
xHtL
Figure 4.10: A bifurcation diagram of the Duffing oscillator for γ ∈ [0.37, 0.39], in steps of 0.0002.
0.395 0.400 0.405 0.410Γ
-1.0
-0.5
0.5
1.0
xHtL
Figure 4.11: A bifurcation diagram of the Duffing oscillator for γ ∈ [0.39, 0.41], in steps of 0.0002.
32
5 Driven Damped Pendulum
The next system we will study is the driven damped dendulum (DDP). Physically, the DDP can serveas a model for both the mathematical pendulum as well as the physical pendulum. In the first case, thependulum is considered to consist of a point particle attached to a massless thread, while the seconddescribes a continuous body rotating around a fixed point. We consider systems with damping thatdepends linearly on the velocity and periodic external forcing. It can be shown using elementary lawsfrom classical mechanics that both systems can be described using a differential equation of the sameform (cf. for instance Taylor, [4]):
φ+ 2δφ+ ω20 sinφ = γω2
0 cosωt. (5.0.1)
Here φ describes the angular position of the pendulum. The constant ω0 is related to fixed physicalproperties of the pendulum and equals
√gl for the mathematical pendulum. The damping force is linear
with damping coefficient δ. γ relates to the maximum amplitude of the periodic force.
5.1 The undamped Hamiltonian system
The pendulum without damping and external forcing is a Hamiltonian system with Hamiltonian
H =1
2v2 − ω2
0 cosu, (5.1.1)
where u = φ, v = φ. Indeed, the system can be written in the form
u =∂H
∂v= v
v = −∂H∂u
= −ω20 sinu (5.1.2)
The level curves are shown in Figure 5.1. The closed curves correspond to periodic, sinusoidal motion
-6 -4 -2 0 2 4 6
-2
-1
0
1
2
Φ
v
Figure 5.1: Level curves of the pendulum without damping and external forcing.
of the pendulum. These exist for a range of energies until the closed curve is replaced by a heteroclinicorbit between two fixed points whose angles differ by 2π and hence describe the same position. This isthe case when the pendulum has precisely the amount of energy to reach the upward position but willtake an infinite amount of time to do so. If we further increase the energy, the level set contains twodisjoint curves that are periodic in φ and extend infinitely in the φ direction. This behavior correspondsto the pendulum going over its top in either direction and making full swings one after another.The undamped system can in fact be solved analytically, although its solutions cannot be expressed interms of elementary functions. Its solutions are given in terms of Jacobi elliptic functions, which will notbe discussed any further in this thesis. It is noteworthy to mention that the solutions can be handled bymathematical software such as Mathematica to generate plots such as Figure 3.2. Notice its resemblanceto normal sinusoidal functions.
33
5 10 15 20 25 30t
-1.5
-1.0
-0.5
0.5
1.0
1.5
Φ
Figure 5.2: An exact solution to φ+ ω20 sinφ = 0.
5.2 The damped, unforced system and its fixed points
Let us introduce friction but still leave out external forcing. In this case the system is described by thedifferential equation
φ+ 2δφ+ ω20 sinφ = 0. (5.2.1)
Notice that the infinite series for the sine results in an infinite number of stable points at (2k+1)π, k ∈ Z.However, most of these points are identical since the physical position of the pendulum at any instant oftime can be described by an angle between 0 and 2π (where 0 and 2π) describe the same position. Onecan also say that the system is periodic in φ with period 2π since replacing φ by φ+ 2π in (5.0.1) doesnot change the system. Hence the local properties such as fixed points can be studied on R/Z, althoughthe global behavior can only be taken into account if one does not mod out by 2π, so that informationabout the number of times the pendulum has made a full cycle is preserved.Thus is suffices to study the fixed points at φ = 0 and φ = π. The first corresponds to the situation whenthe pendulum hangs vertically downwards, while the second describes the pendulum hanging verticallyupside down. From experience we know the first position is stable and the second is unstable, but thiscan also be shown by linearizing the system in these points, as will be shown below.
Let us now examine the stability of the fixed points of the unforced system 5.2.1. First we will rewritethe system as
φ = v
v = −ω20 sinφ− 2δv (5.2.2)
These are given by (φ, v) = (0, 0) and (φ, v) = (π, 0). Linearization of the system at a point (φ0, v0) givesthe Jacobian
Df(φ0, v0) =
(0 1
−ω20 cosφ0 −2δ
)(5.2.3)
Hence, we find the following linearisations around (0, 0) and (π, 0):
Df(0, 0) =
(0 1−ω2
0 −2δ
)Df(π, 0) =
(0 1ω2
0 −2δ
)(5.2.4)
It follows that the eigenvalues of the linearized systems are given by
λ1,2 = −δ ±√δ2 − ω2
0 if φ0 = 0 λ1,2 = −δ ±√δ2 + ω2
0 if φ0 = π. (5.2.5)
For (0, 0), we can distinguish between the cases δ > ω0 and δ < ω0. If δ > ω0, then the square root termis positive but smaller than δ, so both eigenvalues are negative and real. If δ < ω0, then the square rootterm is imaginary and there are have two imaginary eigenvalues whose real parts are negative. In bothcases, the fixed point (0, 0) is stable. On the other hand, the square root term for (π, 0) is always biggerthan the first term. There are always get two real eigenvalues, of which one is positive and the othernegative. Therefore, (π, 0) is an instable saddle point.
34
5.3 Implicit solution to the undamped system
We will now return to the undamped pendulum, for which we already remarked that analytic solutionsexist, and derive an expression for the period of the motion. We will show that the period increases withthe energy of the system in a continuous, even differentiable way, until the homoclinic orbit is reached.We will then derive an analytic expression for this orbit, as well as derive properties of the system forhigher energies. All this will turn out to be useful in applying Melnikov’s Method to the DDP in thenext section.We start by writing the system in its Hamiltonian form
φ = v
v = −ω20 sinφ. (5.3.1)
By applying the chain rule, we obtain the relation
v =dv
dt=dv
dφ
dφ
dt=dv
dφv =
1
2
d
dφ(v2). (5.3.2)
By comparing with equation 5.3.1, we obtain
d
dφ(v2) = −2ω2
0 sinφ, (5.3.3)
and by writing out the left side,
2v dv = −2ω20 sinφdφ, (5.3.4)
which can be integrated to give
φ2 = v2 = 2ω20 cosφ+ 2C, (5.3.5)
where 2C is the integration constant. Denote φ(t)|t=0 = φ0, v(t)|t=0 = φ(t)|t=0 = φ0. Then
2C = φ20 − 2ω2
0 cosφ0. (5.3.6)
Hence we have reduced our system to a single differential equation involving only φ and φ. In terms ofφ, this equation takes the form
φ =dφ
dt= ±
√2(C + ω2
0 cosφ). (5.3.7)
By seperation of variables and integrating, we then obtain the relation
t = ±∫ φ
φ0
dφ√2(C + ω2
0 cosφ). (5.3.8)
This is an elliptic integral of which the solutions are given by the Jacobi elliptic functions mentionedearlier. We will not go into depth discussing these functions, but focus on some special cases that allowus to derive meaningful results.
5.4 The homoclinic orbit
We will now derive an expression for the homoclinic orbit of the unstable saddle point (π, 0). Firstremark that the integration constant C of equation (5.3.6) is identical to the Hamiltonian of the system(eq. 5.1.1). From Figure 5.1, we see the presence of a level curve passing through the unstable saddlepoint that contains the homoclinic orbit we are looking for. The Hamiltonian of the homoclinic orbitshould coincide with the energy of the saddle point, which is given by H = ω2
0 . Hence we can set C = ω20 ,
so that the implicit expression for φ becomes
t =1
ω0
∫ φ
φ0
dφ√2(1 + cosφ)
. (5.4.1)
35
This integral can be evaluated exactly to give
t =1
ω0log
[tan(π4 −
φ0
4 )
tan(π4 −φ4 )
]. (5.4.2)
Indeed, one can check that
d
dφlog
[tan(
π
4− φ
4)
]= −
sec2(π4 −φ4 )
4 tan(π4 −φ4 )
= − 1
4 cos (π4 −φ4 ) sin(π4 −
φ4 )
= − 1
2(cos π2 + cos φ2 )
= − 1
2 cos φ2= − 1√
2(1 + cosφ). (5.4.3)
We can revert this relation to obtain
φ(t) = π − 4 arctan
[e−ω0t tan(
π
4− φ0
4)
](5.4.4)
We can check that this is indeed the homoclinic orbit. Indeed, when t → ∞, the exponential andtherefore the argument of the arctangent goes to 0, hence φ(t) → π. When t → −∞, this term goes to+∞, so we get φ(t)→ π−4(π2 ) = −π. This is indeed the behavior of the homoclinic orbit we are lookingfor. We can also give an expression for the velocity of the orbit as a function of time:
v(t) =d
dtφ(t) =
4ω0 tan(π4 −φ0
4 )e−ω0t
1 + e−2ω0t tan(π4 −φ0
4 ). (5.4.5)
5.5 Inside the homoclinic orbits
Let the initial conditions satisfy 0 < φ0 < π, φ0 = 0. Then the constant C is given by C = −ω20 cosφ.
Thus the integral (5.3.8) takes the form
t =1
ω0
∫ φ
φ0
dφ√2(cosφ− cosφ0)
. (5.5.1)
By using the correct substitutions and taking φ = 0 we obtain an expression that is equal to T4 , where
T is the period of the motion. The details of the derivation can be found in Awrejcewicz (ref), but wewill use the end result
T =4
ω0
∫ π2
0
dξ√1− sin2 φ0
2 sin2 ξ. (5.5.2)
Notice that the integral is well-defined, because | sin2 φ0
2 sin2 ξ| < 1 on the interval [0, π2 ]. We can alsoexpand the integrand using the binomial theorem,
1√1− sin2 φ0
2 sin2 ξ= 1− 1
2sin2 φ0
2sin2 ξ + ... (5.5.3)
Substituting into the integral expression for T , and using that ω0 =√
gl , we get that the first terms are
given by
T ≈ 2π
√g
l(1 +
1
4sin2 φ02) (5.5.4)
From this expression we see that the period increases as φ0 increases. The standard small angle approx-imation T ≈ 2π
√gl is obtained by taking only the constant term.
We now want to know how the period depends on the energy. Recall that the energy is given byc0φ
2 − ω20 cosφ., which in our case reduces to
E = −ω20 cosφ0. (5.5.5)
36
We can rewrite this relation as
sin2 φ0
2=
1
2(E
ω20
+ 1) (5.5.6)
Hence we can express T in terms of E as
T =4
ω0
∫ π2
0
dξ√1− 1
2 ( Eω2
0+ 1) sin2 ξ
. (5.5.7)
Notice that the integrand is a continuous, differentiable function of E and t, with derivative given by
∂
∂E
1√1− 1
2 ( Eω2
0+ 1) sin2 ξ
=sin2 ξ
4ω20(1− 1
2 ( Eω2
0+ 1) sin2 ξ)3/2
≥ 0 for 0 ≤ ξ ≤ π2 (5.5.8)
Hence also the derivative is a continuous function of E. By Theorems 2.5 and 2.11 from “Diktaat Functiesen Reeksen” ([7], we conclude that the period T is a continuous, differentiable function of the energy E,with derivative
∂T
∂E=
4
ω0
∫ π2
0
sin2 ξ
4ω20(1− 1
2 ( Eω2
0+ 1) sin2 ξ)3/2
≥ 0 for 0 ≤ ξ ≤ π2 (5.5.9)
5.6 Melnikov’s Method applied to the DDP
We are now in the right position to apply Melnikov’s Method to the DDP. First we check that the systemhas indeed the required properties, mentioned on p. 184-185 of [1], that makes it possible to apply theMelnikov analysis. Hence we treat our system (5.0.1) as a perturbation of the Hamiltonian system withHamiltonian given by (5.2.2). Also, smoothness is not an issue since both the original Hamiltonian vector
field
(v
−ω20 sinu
)as well as the perturbation
(0
−2δv + γω20 cosωt
)are smooth vector fields. Also, the
perturbed system is periodic in t with period 2πω . Specifically, we can check that the assumptions (A1)-
(A3) are met.In the previous section, we showed the presence of a homoclinic orbit connecting to the saddle point
p0 := (π, 0), explicitly given by eq. 5.4.4. Hence let q0(t) := π − 4 arctan[e−ω0t tan(π4 −
φ0
4 )].
The set Γ0 = q0(t)|t ∈ R ∪ p0 forms part of the level curve of the Hamiltonian given by H = ω20 . It
is clear from figure 5.1 that the interior of Γ0 is filled with a closed level curves on which a continuousfamily of periodic solutions can be found. We would need to work with the Jacobi elliptic functions thatare solutions to (5.3.8) in order to show the continuous dependence on a parameter which we can chooseas φ0 or the energy. However, for φ0 < π, we can already see the continuous dependence on φ0 by thecontinuity of the integrand in φ0, and under the assumption that the integral in (5.3.8) converges we canestablish continuous dependence on φ0 of the right-hand side of (5.3.8). If φ0 → π, the solution thengoes to the homoclinic orbit (5.4.4) continuously.Finally, we have shown in section 5.5 that the period T of the solutions inside the homoclinic orbit is adifferentiable function of the energy E and ∂T
∂E < 0 inside Γ0.
Thus the conditions (A1)-(A3) are met and we can apply Melnikov’s Method. Let v0(t) = ddtq
0(t), β =
tan(π4 −φ0
4 ). We can then write
v(t) =4ω0βe
−ω0t
1 + e−2ω0tβ=
4ω0β
eω0t + e−ω0tβ(5.6.1)
The Melnikov function for q0(t) is given by
M(t0) =
∫ ∞−∞
v0(t)[γω2
0 cos(t+ t0)− 2δv0(t)]dt
= γω20
(∫ ∞−∞
v0(t) cos(t)dt
)cos(t0)− γω2
0
(∫ ∞−∞
v0(t) sin(t)dt
)sin(t0)− 2δ
∫ ∞−∞
v0(t)2
= 4βγω30
(∫ ∞−∞
cos(t)
eω0t + e−ω0tβdt
)cos(t0)− 4βγω3
0
(∫ ∞−∞
sin(t)
eω0t + e−ω0tβdt
)sin(t0)
−32ω20β
2δ
∫ ∞−∞
1
(eω0t + e−ω0tβ)2dt (5.6.2)
37
Unfortunately, the general form of these integrals cannot be expressed in terms of elementary functions.The solutions can be given in terms of hypergeometric functions, which also fall beyond the scope of thisthesis. However, if we choose a value for φ0, specifically the value φ0 = 0 corresponding to β = 1, thenwe can give an explicit formula. In that case the denominators reduce to expressions in cosh t0, which isan even function. We then see that the second integral vanishes, since the integrand is a product of aneven and odd function. The remaining two integrals can be evaluated to give the result:
M(t0) = 4γω30(πsech
(π2ω
)2ω
) cos(t0)− 32ω20δ(
1
2ω) =
2πω30γ
ωsech
( π2ω
)cos(t0)− 16ω2
0δ
ω(5.6.3)
Let us define
R0(ω) =8
πω0cosh
( π2ω
). (5.6.4)
If γδ > R0(ω), then the Melnikov function will have zeroes and the manifolds will intersect. The number
R0(ω) can be interpreted as the slope of a line in γδ plane which separates the parameter values for δand γ for which the behavior can be thought of as non-chaotic and those for which it can be thoughtof as chaotic. We should compare the solutions to those of the Duffing oscillator, given on p. 192-193of [1]. The resemblance is clear, although with the current knowledge we cannot attach any furtherinterpretation to this.
5.7 Numerical solutions of the DDP
For the DDP, the qualitative behaviour for different regions in parameter space has been investigatedrather extensively by Taylor using numerical calculations (cf [4]). We will not go into the details of this,but give a short summary of the main findings. Taylor chose to examine the DDP for different values ofthe driving coefficient while holding all other parameters fixed. By gradually increasing γ in eq. (5.0.1),a period doubling cascade was observed. The intervals between each period doubling were found to berelated to a universal constant named the Feigenbaum constant, which could be used to predict the limitof a geometrical series which indicates the onset of chaos. Beyond this value of γ, chaos was indeedobserved in the system, characterized by amongst others the lack of apparent periodicity of the solutionseven for long times and sensitivity of initial conditions which results solutions starting with close initialconditions diverging exponentially from each other - the rate of this exponential divergence is called theLiapunov coefficient. Within this chaotic regime there were also regions in which non-chaotic solutionswere present, often with a period which is an uneven number times the base period. One of the toolsused to examine this ’route to chaos’ was the bifurcation diagram in which the location of the DDP at anumber of fixed points with evenly spaced intervals - what we now recognize as resulting from a Poincaremap - is plotted against the variable parameter, which is the driving coefficient γ in this case. This isclearly visualized in Figure 5.3, for which we took the same parameter values and initial values as Taylor([4]).
1.065 1.070 1.075 1.080 1.085 1.090Γ
-0.6
-0.4
-0.2
0.0
ΦHtL
Figure 5.3: A bifurcation diagram of the DDP with γ ∈ [1.06, 1.09], in steps of 0.0005. The parametervalues are ω = 2π, ω0 = 3
2ω, β = 14ω0 and initial conditions are x(0) = −π2 , x
′(0) = 0.
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6 Comparison and discussion
In this section the research question proposed at the beginning of this thesis will be addressed: “Whatare similarities and differences between the Duffing oscillator and the driven damped pendulum?”. Withour outline of the main properties of both systems, it is now possible to give a comparison between thesystems and mention some key differences and similarities between the systems.To begin with, let us note that both systems can to some degree be approximated by the damped drivenharmonic oscillator, which can be described by
x+ δx+ ω 20 x = F (t), (6.0.1)
where ω0 is the natural frequency of the undamped and unforced system, δ is the damping coefficientand F (t) is some general forcing which we can assume to be periodic. For the Duffing oscillator, this iscertainly true if the contribution from the nonlinear term, measured by α, is small. For the DDP, wecan expand sinφ into a Taylor series and by neglecting all terms of order higher than one we obtain theharmonic oscillator. For the harmonic oscillator, explicit solutions are known for all parameter ranges(cf. Chapter 5 in [4]) and the solutions are non-chaotic. The essential difference with the harmonicoscillator is that the systems we studied are not linear, and this results in chaotic solutions in certainparameter ranges. The relative proximity of both systems to the harmonic oscillator is also apparentfrom the solutions of the undamped unforced Hamiltonian system for both systems, which clearly showedperiodic solutions that resemble sinusoidal ones such as shown in Figure 5.2. However, the differencewith the harmonic oscillator is that both systems contain a homoclinic orbit attached to a saddle, as aresult of the nonlinear terms, and the breaking of this orbit under perturbations can give rise to chaoticphenomena as predicted by the Smale-Birkhoff Homoclinic Theorem and Melnikov’s Method.Moreover, it appeared that in both cases even the Hamiltonian system was too complex to yield a solutionthat could be expressed in terms of elementary functions, and we had to be satisfied with elliptic integralsthat could not be simplified further in general. However, for the special case of the homoclinic orbit,we were able to compute this integral and obtain analytic solutions of the homoclinic orbit expressed interms of mostly trigonometric and exponential function. It is unclear where there are underlying reasonsbehind the fact that the calculation for the homoclinic orbit rather miraculously simplifies to yield aresult in terms of elementary functions.Due to the this knowledge of the homoclinic orbits, and other nice properties of the systems, we couldapply Melnikov’s Method to both systems. In both cases we could find a line in γδ space passing throughthe origin which separates the regions in which transversal intersections occur and those in which themanifolds do not intersect. For the Duffing oscillator, we have seen that chaotic solutions are indeedfound only in region in which transversal intersections are predicted and hence does not conflict thepredictions from Melnikov’s Method. It would be an interesting to also study the DDP in γδ space tosee how the numerical solutions agree with the predicted behavior.Another result of both systems’ closeness to the harmonic oscillator is that they can in principle beapproximated using averaging techniques. We have not applied averaging to the DDP, because it wouldfirst require a Taylor expansion of the term containing sinφ and subsequently a treatment of higher orderterms as a perturbation, which would most likely only be justified for small angles. Nevertheless, in thenon-chaotic regime we would expect averaging applied to the DDP to give fairly reliable results, as wehave seen for the Duffing oscillator, where we found the presence of a bifurcation which would probablynot have been detected using other analytical methods. It should be stated, however, that averaginglikely fails in the chaotic regime. One of the main premises of Holmes’ article [2] is that averagingapplied to the Duffing oscillator does not give reliable results in the chaotic regime. Hence it is arguablehow much valuable information one can extract from studying these systems using techniques such asaveraging.There is also an important physical difference between the systems. The position of the pendulumis described by an angle φ which in principle is taken as an element of S1, but can also be taken asan element of R especially when one is interested in the dynamical behavior over a long time and thependulum makes at least one full turn within this time. This ambiguity is absent in the Duffing oscillator,where the position simply represents a one-dimensional deviation from the origin and hence can be takenin R. As a result, we should in principle have checked that the theorems we employed also apply tothe systems defined on manifolds, such as the DDP. Since it is mentioned in the beginning of [1] that“almost all the methods [. . .] also generalize to dynamical systems whose phase spaces are differentiablemanifolds” (page xii), we expect this not be a problem. Therefore, we could generally take φ ∈ S1.Furthermore, the Duffing oscillator has one saddle point and two sinks, while the DDP has one saddle
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and one sink. Consequently, the Duffing oscillator has two homoclinic orbits, one to the left and theother to the right, while the pendulum has just one homoclinic orbit.Finally, the transition to chaos we studied for both systems appeared to be different. We were unableto find period doubling bifurcations for the Duffing oscillator for the parameter ranges studied. It isknown that period doubling does also occur for the Duffing oscillator (cf. §2.2 of [1]). For the DDP, wewere able to visualize the period doubling cascade by computing a bifurcation diagram. However, it isnot know whether such period doubling bifurcations occur in all parameter ranges. It could be possiblethat there are also parameter ranges where the transition to chaos for the DDP is sudden, instead offollowing a period doubling cascade, but this is not known. One possible explanation of the absence ofperiod doubling is that we chose unfortunate initial conditions next to the parameter values, and thateven for these parameter values there are initial values for which such period doubling does take place.This does not seem very likely, however, since we investigated the transition to chaos in a rather detailedway for different initial conditions for certain parameter ranges.
Finally, let us make a remark that there are still many aspects of these systems which are notyet understood after writing this thesis. Many of the numerical results of section 4.8 are not entirelyunderstood and hard to relate to theories in the references we used. Another interesting phenomenonis the presence of symmetries in the Duffing oscillator and possibly also the DDP. These are shortlydiscussed by Holmes §2.4 of [2]. Many solutions come in pairs, such as the stable period one solutions weobserved in the non-chaotic regime, while others are self-similar, such as the homoclinic orbit. However,in the chaotic regime these symmetries apparently disappear. This is seen by the examples of chaoticattracting sets and Poincare maps in the chaotic regime as shown in [1], which clearly show structuresthat do not have left-right symmetry. In general, the presence of symmetries and whether they persistin the presence of chaos is beyond the scope of this thesis, but would allow for interesting further study.
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References
[1] Guckenheimer, J. and Holmes, P., 1983, Nonlinear Oscillations, Dynamical Systems, and Bifurca-tions of Vector Fields, Springer-Verlag, New York.
[2] Holmes, P., 1979, A nonlinear oscillator with a strange attractor, Philosophical Transactions of TheRoyal Society of London, Vol.292 No.1394, p.419-448.
[4] Taylor, J., 2005, Classical Mechanics, University Science Books, United States.
[5] Awrejcewicz, J., 2012, Classical Mechanics: Dynamics, Advances in Mechanics and Mathematics29, Springer, New York.
[6] Kuznetsov, Y.A., 2011, Basis Differentiaalvergelijkingen, lecture notes for the course “Differenti-aalvergelijkingen”, Mathematisch Instituut, Utrecht.
[7] Duistermaat, J.J., 2011, Functies en Reeksen, lecture notes for the course “Functies en Reeksen”,Mathematisch Instituut, Utrecht.
[8] Kalmar-Nagy, T. and Balachandran, B., 2011, Forced harmonic vibration of a Duffing oscillatorwith linear viscous damping, in: The Duffing Equation: Nonlinear Oscillators and their Behaviour(Edited by I. Kovacic and M. J. Brennan), John Wiley & Sons.
