rotation intervals for quasi-periodically forced …

ROTATION INTERVALS FOR

QUASI-PERIODICALLY FORCED

CIRCLE MAPS

A thesis submitted to the University of Manchester

for the degree of Doctor of Philosophy

in the Faculty of Engineering and Physical Sciences

2012

Silvia Pina-Romero

School of Mathematics

Contents

Abstract 7

Declaration 8

Copyright Statement 9

Acknowledgements 10

1 Introduction 12

1.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Background and context 16

2.1 General theory of circle maps . . . . . . . . . . . . . . . . . . . . . . 16

2.2 Literature review of circle map dynamics . . . . . . . . . . . . . . . . 19

2.2.1 Invertible circle maps . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.2 Non-invertible circle maps . . . . . . . . . . . . . . . . . . . . 21

2.2.3 Invertible quasi-periodically forced circle maps . . . . . . . . . 27

2.2.4 Rational case . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2.5 Irrational case . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.2.6 Non-invertible quasi-periodically forced circle maps . . . . . . 30

3 Piecewise example 32

3.1 Construction of piecewise maps . . . . . . . . . . . . . . . . . . . . . 33

3.2 Dynamics of the piecewise family of maps. . . . . . . . . . . . . . . . 33

3.3 Theoretical prediction . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2

3.3.1 Comparison of dynamics . . . . . . . . . . . . . . . . . . . . . 39

4 Observed rotation intervals 42

5 A plausible explanation; our hypothesis 46

5.1 Uniform hypothesis formulation . . . . . . . . . . . . . . . . . . . . . 48

5.2 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

6 Statistical test 53

6.1 Chi-Square goodness of fit test . . . . . . . . . . . . . . . . . . . . . . 55

6.2 Kolmogorov-Smirnov test . . . . . . . . . . . . . . . . . . . . . . . . . 57

7 Borders of rotation intervals 62

8 Length of rotation interval 67

9 Conclusions 71

Bibliography 73

A Using different initial θs 79

Word count 9671

3

List of Figures

2.1 Devil’s staircase for b = .99 . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2 Diagram of tongues from [41] . . . . . . . . . . . . . . . . . . . . . . 21

2.3 Extended diagram of tongues from [7] . . . . . . . . . . . . . . . . . 22

2.4 To the right F (m2) ≤ F (1) and F (m1) ≥ F (0), as in [7]. To the left

F (m2) > F (1) and F (m1) < F (0). . . . . . . . . . . . . . . . . . . . . 23

2.5 Construction of maps H−, H+ and some other Hµ. . . . . . . . . . . 24

2.6 A and Z regions, with stripes and dots respectively. The observable

paths are labelled (a), (b) and (c). . . . . . . . . . . . . . . . . . . . . 25

2.7 To the left the graph of F1.1938,1.1938(x). To the right a close up of

F1.185,1.1938(x), F2,1.1938(x). . . . . . . . . . . . . . . . . . . . . . . . . 25

2.8 Case 1, parameter values b = 1.2, a ∈ [.19, .21], and 50000 iterates . . 26

2.9 Case 2, parameter values: b = 2.8, a = .3, and 50000 iterates . . . . . 26

2.10 Case 3, non-trivial case with b = 5 . . . . . . . . . . . . . . . . . . . . 27

2.11 ρ = 1/2 from [18] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.12 ρ = 12− 1

2ω from [18] . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1 Representation of the bifurcations suspected to be similar; to the left

H2(x) and to the right a piecewise linear map. . . . . . . . . . . . . . 32

3.2 Simplest case a < d . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3 Schematic view of the dynamics when a > d . . . . . . . . . . . . . . 35

3.4 Slight increase on a might not modify ρ. . . . . . . . . . . . . . . . . 36

3.5 Change of coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.6 The computed rotation number with 10000 iterations . . . . . . . . . 39

4

3.7 Proposed fittings for the right and left ends near the bifurcation to-

gether with the computed rotation number. . . . . . . . . . . . . . . 40

3.8 d=0.5, the dashed line corresponds to the proposed fitting on [40] for

an arbitrary value of C1 = .75, the other two lines correspond to the

computed borders of phase-locked regions. . . . . . . . . . . . . . . . 41

3.9 d=0.7, the dashed line corresponds to the proposed fitting on [40] for

an arbitrary value of C1 = 1.1, the other two lines correspond to the

computed borders of phase-locked regions. . . . . . . . . . . . . . . . 41

4.1 Monotone maps defined above . . . . . . . . . . . . . . . . . . . . . . 43

4.2 Small c value in both pictures (0.01). To the left b = 1.2 and to the

right b = 2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.3 Increasing c value. To the left b = 1.2 and c = 1.85, to the right b = 2.8

and c = .33 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.4 Large c value equal to a hundred in both cases. To the left b = 1.2

and to the right b = 2.8 . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.2 Histograms showing the change on distribution for a small (0.1) and a

large (50) value of c of the first 10000 iteration . . . . . . . . . . . . . 51

5.3 Normalized histograms for values of c equal to 1.4 and 278. . . . . . . 52

5.4 Difference between the expected and the observed occurrences for

many increasing values of c. The number of bins considered was 12 to

the left, and 18 to the right. . . . . . . . . . . . . . . . . . . . . . . . 52

6.2 The χ2 statistic for fixed b = 2, a = 0.2, a choice of 30 bins and 100000

iterations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

6.3 Computed χ2 statistic for different initial conditions . . . . . . . . . . 57

6.4 Computed Kolmogorov-Smirnov statistic . . . . . . . . . . . . . . . . 59

6.5 Computed Kolmogorov-Smirnov statistic for a set of eight random ini-

tial conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5

6.6 Mean of DN for a set of random initial conditions and a together with

the suggested function to approximate it. . . . . . . . . . . . . . . . . 60

7.1 Calculation the integral of G+(x). . . . . . . . . . . . . . . . . . . . . 63

7.2 Comparison between the numerical and theoretical borders of rotation

intervals, with c = 200, b = 1.2 and 100000 iterations. . . . . . . . . . 66

8.1 The cubic unfolding, the rotation interval is represented by the area

between the graphs G+ and G− . . . . . . . . . . . . . . . . . . . . . 69

A.1 Rotation numbers for different initial θ, 10000 iterates and c = 100. . 81

6

The University of Manchester

Silvia Pina-RomeroDoctor of PhilosophyRotation intervals for quasi-periodically forced circle mapsNovember 27, 2012

This work investigates some aspects of the dynamics of non-invertible quasi-periodic circle maps, from the point of view of rotation numbers and their structurein parameter space.

Circle maps and quasi-periodically forced circle maps have been widely used asa model for a broad range of physical phenomena. From the mathematical point ofview they have also received considerable attention because of the many interestingfeatures they exhibit.

The system used is given by the maps

xn =

[xn−1 + a+

b

2πsin(2πxn−1) + c sin(2πθn−1)

]mod 1

θn = θn−1 + ω mod1 1

where a, b, c, ω are real constants. In addition, b and ω are restricted, respectively,to values larger than one and irrational.

A fundamental part of this thesis consists of numerical approximations of rotationintervals using and adapting of the work of Boyland (1986) to the quasi-periodic case.Particular emphasis was given to the case of large coupling strength in quasi-periodicforcing.

Examination of the computed rotation numbers for the large coupling case, to-gether with previous claims suggesting that for large coupling strength the b-termcould be neglected (see Ding (1989)), led to the formulation of an ergodic argumentwhich is statistically supported. This argument indicates that, for this case, thequalitative behavior of rotation number depends linearly on a. It is also shown thatthe length of the rotation interval, when the transition from a trivial rotation inter-val (invertible case) to a non-trivial rotation interval occurs, it develops locally as auniversal unfolding.

A different map, piecewise monotone, and structurally similar to the maps definedto calculate the edges of rotation intervals in Boyland (1986), is studied to illustratehow the rotation number grows. The edges of rotation intervals are analyticallycalculated and matched with numerical observations.

7

Declaration

No portion of the work referred to in this thesis has been

submitted in support of an application for another degree

or qualification of this or any other university or other

institute of learning.

8

Copyright Statement

i. The author of this thesis (including any appendices and/or schedules to this

thesis) owns any copyright in it (the “Copyright”) and s/he has given The

University of Manchester the right to use such Copyright for any administrative,

promotional, educational and/or teaching purposes.

ii. Copies of this thesis, either in full or in extracts, may be made only in accor-

dance with the regulations of the John Rylands University Library of Manch-

ester. Details of these regulations may be obtained from the Librarian. This

page must form part of any such copies made.

iii. The ownership of any patents, designs, trade marks and any and all other

intellectual property rights except for the Copyright (the “Intellectual Property

Rights”) and any reproductions of copyright works, for example graphs and

tables (“Reproductions”), which may be described in this thesis, may not be

owned by the author and may be owned by third parties. Such Intellectual

Property Rights and Reproductions cannot and must not be made available

for use without the prior written permission of the owner(s) of the relevant

Intellectual Property Rights and/or Reproductions.

iv. Further information on the conditions under which disclosure, publication and

exploitation of this thesis, the Copyright and any Intellectual Property Rights

and/or Reproductions described in it may take place is available from the Head

of the School of Mathematics.

9

Acknowledgements

I would like to thank my supervisor Prof. Paul Glendinning for his valuable guidance

and patience and my examiners, Mark Muldoon and Rob Sturman, for their time

and helpful advise.

Besides, I would like to thank the University of Manchester for providing a good

environment and facilities to complete this project.

This study would not have been possible without the funding given by CONA-

CYT. I extend my gratitude to Fundacion ESRU, Fundacion Sofia Kovalevskaia and

The Mexican Mathematical Society for their financial support on key stages of the de-

gree. Also, I thank the Instituto Tecnologico Autonomo de Mexico for their patience

and support.

Many thanks are owed to the wonderful collegues who helped me with techni-

cal and mathematical issues, most specially, Citlalitl Nava, Rosa Duran and Jim

Arnold. Also, I thank Sergio Hernandez for his help and suggestions regarding the

computational part.

I gratefully acknowledge the assistance by my good friend Pedro Olivares with

the submission process.

In addition, I take this opportunity to express my deep gratitude for the friendship

and company of the many friends that have joined me in this journey. I thank them

for those endless chats in Big Hands and Sandbar and for making me laugh so much.

I sincerely thank my father for so many lessons, not only on maths, but on life.

I thank my mother and her partner for showing unconditional support throughout

this project.

10

I am truly grateful for my sister’s visits and every adventure together. I also

thank her for always being there even when there was an ocean in between.

Finally, I want to say a special thank you to Gabriel Pareyon, notonalmi, for

sharing his time with me, for listening and encouraging me. Thank you for making

everyday special.

11

Chapter 1

Introduction

Circle map dynamics have been formally studied since the nineteenth century—both

for their mathematical intrinsic interest—particularly as model for the transition to

chaos, and for the broad range of applications that they have.

In physics, there are many examples of nonlinear systems which are periodically or

quasi-periodically forced; both theoretical and experimental, for example, the forced

Van der Pol equation and Duffing’s equation [37].

Experimentally, in the work by He R. et al. [28] an electronic Josephson-junction

simulator, driven by two independent ac sources, is used to observe the transition

from quasiperiodicity to chaos.

Moreover, quasi-periodic systems are used in the work by Coullet, P. et al. [11]

to model doubly periodic flows and, years later, Held G.A . et al [29] report quasi-

periodic, mode-locked, and chaotic states in their experiments with an electron-hole

plasma excited with an external perturbation.

In additon, Ditto W.L. et al. in [15] present a two-frequency quasi-periodically

driven, buckled, magnetoelastic ribbon experiment to explore the existence of a

strange non-chaotic attractor, whereas Bak [4], Janner and Janssen [35] and Aubry

[3], use quasi-periodically forced systems to deal with modulated structures with in-

commensurable periods with the basic lattice, as it could be for example, interacting

gas atoms adsorbed on a crystalline substrate.

In biology a well know example is the work by Glass et al. [20] in which they

12

CHAPTER 1. INTRODUCTION 13

study a periodically stimulated cardiacosillator and also the work by Hoppenstead

et al. [33] on which radial isochron clocks and their response to external signals and

coupling with other radial isochron clocks are studied as a quasi-periodically forced

system. For more applications see [21].

By ‘dynamics’ we mean the description and analysis of the behaviour of iterates

of points.

Very often the Arnold map has been considered as a paradigmatic example of

circle map dynamics. The Arnold map can be expressed by xn = xn−1 + a +

b2π

sin(2πxn−1) mod 1 , where a and b are parameters in R and xn ∈ S1.

Coupling a circle map with another periodic map, results in a forced circle map.

In particular, coupling the Arnold map with the circle map given by θn = θn−1 + ω,

where ω 6∈ Q, and represents a rotation by an irrational, yields a quasi-periodically

forced circle map:

xn = xn−1 + a+b

2πsin(2πxn−1) + c sin(2πθn−1) mod 1

θn = θn−1 + ω mod 1

Opposite to the case of circle maps (for which the dynamics are fully understood

for all parameter values), in the quasi-periodic case there are regions in parameter

space that still require further study as we do not yet understand its dynamics.

A very helpful tool used to describe circle dynamics is the rotation number. Intu-

itively a rotation number is a measure of the average rotation that a point is translated

with each iteration of the map.

They can also be thought of as the rate of rotation.

It is know that in the case of homeomorphisms, all points rotate at the same

asymptotic rate, whereas in the case of non-invertible maps, different rotation num-

bers may occur depending on the initial conditions, however the set of rotation num-

bers form a closed interval [34].

For many authors the quasi-periodically forced map is a natural generalization of

the Arnold map [53]. Therefore, a very common approach to its study consists of


taking the results on circle map dynamics and, whenever possible, extending them

to the quasi-periodic forced case. This is the line we follow in this piece of work.

After [31] we know that rotation numbers are well defined on fibres, i.e. for a

fixed rotation of the coupled map. Therefore, the description of its dynamics is also

found in terms of rotation numbers or intervals.

In this thesis we investigated the dynamics of quasi-periodically forced circle maps

on a set of parameters that have not yet been understood, in particular we focused

on the case where the strength of the coupling, given by parameter c, is large.

We did it by computing the borders of rotation intervals using the algorithm

on [53] and adapting the results of [7]. The contribution made with this research,

was to provide strong evidence of how rotation numbers behave (under previous

assumptions), as well as showing how rotation intervals grow near the border of

invertibility.

We found that when parameter c is large the upper and lower boundaries of the

rotation number could be approximated by a + K1 and a − K2 respectively, where

K1 and K2 are constants which depend on the value of the parameter b. Also that

for b just above one, the rotation interval grows as 94π2 (b− 1)2.

1.1 Outline

The first section of following chapter (i.e. Chapter 2), introduces the basic theoretical

knowledge needed to understand the problem that concerns this thesis. Special atten-

tion is given to the concept of rotation numbers as they constitute a very important

tool for the description and analysis of the dynamics of circle and quasi-periodically

forced circle maps. No proofs are made since they can be found in most of the

standard literature, see for example [42], [38] and [12].

The second part of Chapter 2, establishes the background of the problem and

summarizes the specialized literature in the particular case of the Arnold map:

xn = xn−1 + a+b

2πsin(2πxn−1) mod 1. (1.1)


This chapter also sets the questions and provides cues towards possible solutions.

On Chapter 3, we make a deeper analysis of some aspects of non-invertible circle

maps (i.e. when the value of b exceeds one in equation 1.1. Specifically we look at

the transition to chaos marked by the creation of a rotation interval. In order to do

so, we construct a parametric family of maps which are algebraically simple enough

to formulate analytical results, yet they show equivalent dynamics to certain cases of

Arnold circle maps.

The approach we use to investigate the dynamics of non-invertible quasi-periodically

forced circle maps consist of extending the results of [7] to the quasi-periodic case.

The results are shown in Chapter 4.

Chapter 5 states a likely explanation for the observations in Chapter 4, and briefly

introduces some basic concepts of ergodic theory.

The following two chapters (6 and 7) are taken from a recently published paper

[26].

On Chapter 6 we formalize the numerical evidence that supports the hypothesis

exposed on Chapter 5. For this purpose we provide key concepts on probability which

lead to the Chi-square and Kolmogorov-Smirnov goodness of fit tests.

For Chapter 7 we assume the hypothesis on Chapter 5 to be correct. This allows

us to explicitly compute the borders of rotation intervals in the case of large coupling

by integrating some associated monotone maps.

Chapter 8 shows that the growth of a rotation interval corresponds to a cubic

unfolding from catastrophe theory near the border of invertibility. We develop it by

noticing that our map locally behaves as a cubic map.

The last chapter summarizes what the contributions are and explains how they

fit in the general panorama of dynamical systems.

In Appendix A, we look at the rotation number for different initial values of θ when

the b term is zero. We obtain an explicit expression for the n-th approximation of ρ

depending on the initial θ plus a constant ψ determined by n and the quasi-periodic

forcing ω (ω 6∈ Q).

Chapter 2

Background and context of

rotation intervals

2.1 General theory of circle maps

This chapter gives a brief summary on the general theory of circle maps and quasi-

periodically forced circle maps which are necessary to describe and understand their

dynamics.

The approach adopted throughout this thesis is that of discrete dynamical sys-

tems, where the state of a element at a time n, xn, is determined by its previous state

xn−1. The generation of a discrete dynamical system is an iterative process requiring

a open or compact set X [32], and a function f from such a set into itself.

In the theory of discrete dynamical systems functions are often called ‘maps’. A

map f yields the dynamical system xn = f(xn−1).

For now, let X be the circle S1, and let f : S1 → S1.

The notation to indicate the nth-composition of f with itself is

fn(x) = f ◦ f · · · f(x)︸︷︷︸n

For x0 ∈ S1 consider the sequence {x0, f(x0), f2(x0), ..., f

n(x0), ...}. This sequence

is called the ‘orbit’ (or ‘forward orbit’) of x under f , and is commonly denoted by

o(x0, f). Alternatively, it is written as the sequence {xn}∞n=0.

16

CHAPTER 2. BACKGROUND AND CONTEXT 17

For invertible maps, the inverse of f , f−1, defines the ‘backward’ orbit of x0 in an

analogous way, as the sequence of pre-images of x0: o−1(x0, f) = {x0, f−1(x0), . . . f−n(x0), . . .}.

A point x such that f(x) = x is called a ‘fixed point’. When f q(x) = x, x is called

a ‘periodic point’ of prime period q, assuming that for all i = 0, ..., q − 1, f i(x) 6= x.

Dynamics in the circle are usually studied using the concept of rotation numbers.

To define and compute rotation numbers, first we need to define the lift of a circle

map.

Make π a covering map π : R → S1 (i.e. a map that wraps R around S1) then

F : R→ R is a ‘lift’ of f if π ◦ F = f ◦ π. As the covering map is not unique, there

could be an infinite number of lifts. However, the difference between any two lifts of

the same map f is always an integer.

The ‘degree’ of f is the difference F (x + 1) − F (x). For invertible circle maps

this number is 1 or -1 depending on whether the map is orientation preserving or

orientation reversing, respectively. Consequently, in the orientation preserving case

F (x)− x is a periodic function.

It is known since Poincare that if f is an invertible circle map, then limn→∞Fn(x)−x

n

exists for every x ∈ R and it is independent of x. This limit is the rotation number

of f .

ρ = limn→∞

F n(x)− xn

(2.1)

(1).

Intuitively, rotation numbers can be thought as a measure of the average rotation

of points that f induces as n → ∞. For non invertible maps, it is know that the

rotation number may not be unique and depends on the chosen x.

Henceforth, ρ denotes the rotation number.

Rotation numbers using different lifts may differ but only by an integer.

1For rotation numbers in the non-invertible case see subchapter (2.2.2)


The conditions for the existence of rotation numbers have been redefined to their

simplest and most essential form. Initially, it was thought that the existence of ρ

followed directly from having the lift of an orientation preserving homeomorphism

F : R→ R which was also continuous, strictly increasing and degree one.

Later, in [47] it was proved that the lift does not necessarily needs to be the strictly

increasing and that this condition could be substituted with a more relaxed one: a

non-decreasing condition. In [50] the authors prove that the only two conditions for

the lift of a circle map to ensure the existence of rotation numbers are that it is both,

non-decreasing and degree one.

Allowing the map not to fulfil the non decreasing assumption (or strictly monotone

one) in [50] led to the theory of rotation intervals.

As mentioned in the Introduction, a paradigmatic example is the Arnold map

given by F (x) = x+ a+ b2π

sin(2πx).

In general terms, when restricted to invertible maps, we can see two kinds of

dynamics corresponding to cases when ρ is rational or irrational.

In the particular case of Arnold maps, ρ is determined by the choice of parameters

a and b.

Some relevant features of rotation numbers concerning this piece of work are the

following:

The rotation number is rational (i.e. there exist p, q ∈ N such that ρ = p/q) if and

only if F q(x) = x + p for some x ∈ R. If f is a homeomorphism that has a periodic

point then any orbit is asymptotic to a periodic orbit. If f has a periodic point x0 of

period k, then x0 is a fixed point of fk.

On the other hand, f does not have any periodic points if and only if ρ(f) is

irrational [12]. In the particular case of C2 maps, a irrational rotation number implies

that all orbits are dense. A homeomorphism with an irrational rotation number is

equivalent to a rotation with the same rotation number [42] in the sense that orbits

from both maps lie in the circle in the same order.


2.2 Literature review of circle map dynamics

In this section we consider the Arnold map to explain the dynamics of circle maps.

The invertibility plays a key role in the description and study of the dynamics, which

is why we show the invertible and non-invertible case separately.

2.2.1 Invertible circle maps

For values of b ≤ 1 the map is invertible and there is a unique rotation number for

each choice of parameters [31]. In this section in order to emphasize this dependence,

whenever relevant, we will use the notation ρa,b instead of just ρ.

An important part of understanding the dynamics is related to how rotation

numbers are organized in the parameter space (a, b). This issue has been thoroughly

studied [40],[7]. A common approach consists of fixing one of the parameters, usually

b, and analyzing the behaviour of ρ as the other parameter varies.

In the simplest case, when b is fixed at the value 0 the map is a rotation by a, so

ρa,0 = a.

For values 0 < b < 1 the map is a diffeomorphism. Consider a fixed b∗ ∈ (0, 1]

then it is easy to see that:

• ρa,b∗ is a continuous function of a. [1]

• ρa,b∗ is non decreasing. This is easily corroborated by noting that if a1 ≤ a2 then

Fa1(x) ≤ Fa2(x) for all x ∈ R and so F na1

(x) ≤ F na2

(x), which by the definition

implies ρa1,b∗ ≤ ρa2,b∗ .

• For every rational ρa,b∗ there exist a non trivial interval A ⊂ [0, 1] such that for

every a ∈ A, ρa,b∗ is precisely that rational. On the contrary, for every irrational

ρa,b∗ there is a single value of a that yields such ρa,b∗ .

The behaviour of the graph of ρa,b∗ follows from the observations above. It is

commonly known as devil’s staircase. It is the kind of graph known as a Cantor func-

tion (i.e. it’s constant on intervals corresponding to rational numbers and everywhere

continuous) [12].


Figure 2.1: Devil’s staircase for b = .99

The structure of rotation numbers in the parameter space (initially with b < 1),

is presented in [2],[30]. Regions with the same rational rotation number appear to

form cones with the tip at the line b = 0 and opening up as b approaches 1, while

irrational rotation numbers make lines connecting b = 0 with b = 1. Each region

with a given rotation number is usually called a tongue or a phase locked region.

The width of rational tongues is inversely proportional to the denominator, in the

sense that the width decreases as the denominator increases.

In term of dynamics of the Arnold map, inside each tongue there are exactly two

fixed points (or periodic solutions); one is attracting and the other is repelling. At

the boundary, the two fixed points are annihilated in a saddle-node bifurcation.

The observable dynamics for different choices in parameter space can be grouped

into two [18]: inside rational tongues there’s a unique attractor which is a set of

points in an invariant circle. The other kind is quasi-periodic motion which is dense

in the circle [46].

The numerical study of tongues is possible after the work of [50], in which it is

proved that for all x,

ρ− 1

n≤ F n(x)− x

n≤ ρ+

1

n. (2.2)


Figure 2.2: Diagram of tongues from [41]

This bound allows us to compute ρ, to the desired order by taking any initial

condition and iterating it in a computer. In the literature, the tongue with rotation

number zero is an archetype example for rational tongues because if F has rational

rotation number p/q then F q−p has rotation number 0. And a map has zero rotation

number if and only if it has a fixed point.

2.2.2 Non-invertible circle maps

For non-invertible maps the whole behaviour changes drastically as not every x ∈ R

has necessarily the same rotation number. However, the set of rotation numbers for a

given choice of parameters is a closed interval, which we denote as [ρ−, ρ+] (possibly

trivial), [45], [34].

The detailed structure of tongues in the non-invertible region is presented in [40]

and in [7] through very different approaches. They show that, as the value of the

parameter b grows beyond 1, rational tongues continue to spread out, while irrational

regions begin the process of opening out in a cone shape manner, overlapping in both

cases.


The former paper aims to describe this structure in terms of the occurring bifur-

cations and focuses on the transition to chaos which happens when a single rotation

number turns into a rotation interval [ρ−, ρ+]; the latter deals with the numerical

computation of ρ.

Figure 2.3: Extended diagram of tongues from [7]

In order to understand the structure of tongues in the non-invertible region it is

useful to make a difference between regions where the rotation number is a rational

pq

for some p and q, i.e. ρ = pq, and regions where a rational p

qbelongs to its rotation

interval pq∈ [ρ−, ρ+]. Following the approach of [41] we denote the first as A p

qand

the latter as Z pq

(see figure 2.6).

Because the rotation number is not necessarily unique and the movement is not

necessarily orientation preserving, taking one single initial condition and iterating it

does not yield significant information in terms of average displacements, as it did in

the invertible case. In [7] P. Boyland tackles this problem by developing a family of

invertible maps whose rotation numbers coincide with the rotation numbers of the

original non-invertible map.

The main idea is as follows.


For a fixed map f with lift F = x + a + b2πsin(2πx), and b > 1, the map has

two critical points m1 and m2, with F (m1) and F (m2) minimum and maximum

respectively, [7] restricts the parameter values so F (m2) ≤ F (1), F (m1) ≥ F (0). This

last restriction limits the parameter space in the b direction to those regions where a

unique rotation number may exist. However, in our work, in order to investigate also

the regions where the rotation interval is always non-trivial, we also allow F (m2) >

F (1) and F (m1) < F (0).

Figure 2.4: To the right F (m2) ≤ F (1) and F (m1) ≥ F (0), as in [7]. To the leftF (m2) > F (1) and F (m1) < F (0).

He introduces a parameter µ ∈ [1, 2], which determines a family of invertible

functions. This family is then used to calculate the rotation numbers of F .

For each µ, there are unique points z1,µ ∈ [m1, 1] and z2,µ ∈ [0,m2] with

F (z1,µ) = F (z2,µ) = F (m1) + (µ− 1)(F (m2)− F (m1)).

Note that z1,1 = m1 and z2,2 = m2.

For each µ ∈ [1, 2] define Hµ : [0, 1]→ R by,

Hµ =

F (m1) + (µ− 1)(F (m2)− F (m1)) for x ∈ [z2,µ, z1,µ]

F (x) otherwise

Extending Hµ to the real line using Hµ(x + 1) = Hµ(x) + 1, we can say H

corresponds to the lift of an invertible circle map hµ and hence we may calculate the

rotation number for these maps.


Figure 2.5: Construction of maps H−, H+ and some other Hµ.

The following two results link the rotation numbers of Fa,b and Hµ and allow us

to use Hµ calculate the rotation set of F .

1. ρ(f) = [ρ−, ρ+] = [ρ(h1), ρ(h2)]

2. For each r ∈ ρ(f) there exist µ ∈ [1, 2] with r = ρ(hµ).

In short, the ideas supporting the above statements are that for any µ ∈ [1, 2] and

the corresponding maps Hµ and F there is always an orbit of x under F that never

falls into the interval where Hµ and F disagree. Then, it suffices to recall that Hµ is

non-decreasing and has a unique ρ [7] that depends continuously on µ.

As stated before, the borders of tongues ρ−, ρ+ are due to saddle node bifurcations.

Those saddle nodes correspond to saddle nodes on the graphs of H1 and H2 which

may be classified in two different categories in terms of the dynamics around them,

as we will shortly see.

Because of the structure of tongues in the non-invertible region, three different

types of evolution of rotation numbers are observed when fixing a value of the pa-

rameter b, and moving along a path on a (see figure 2.6).

Following the approach of [54], and for algebraic simplicity, let Fa,b denote the

map Fa,b(x) = x+a+ b sin(2πx). Keep in mind that the previous parameter b relates


Figure 2.6: A and Z regions, with stripes and dots respectively. The observable pathsare labelled (a), (b) and (c).

to b as b = b(2π).

When b > b0 = 1/(2π) the map is non invertible and has two critical points.

For a = b, Fb,b the map has a saddle node bifurcation at x = 3/4 (see figure 2.2.2).

Figure 2.7: To the left the graph of F1.1938,1.1938(x). To the right a close up ofF1.185,1.1938(x), F2,1.1938(x).

However, for b just slightly larger than b0, the map is locally invertible so all orbits

are asymptotic towards the fixed point 34. This causes ρ− and ρ+ to agree around b0

(see figure 2.2.2).

In parameter space, these values correspond to where both A pq

and Z pq

share


Figure 2.8: Case 1, parameter values b = 1.2, a ∈ [.19, .21], and 50000 iterates

boundaries (see path (a) in figure 2.6). As a result, if we look at paths with fixed

b and increasing a, when the tongue ends both ρ− and ρ+ grow as Ka12 , for some

constant K [40].

Greater increasing of b results in larger F (m1), until its value exceeds that of

F (34) jumping over the saddle node [54]. A nontrivial rotation interval appears at

this point, although some of orbits might still be ’trapped’ at x = 34. In parameter

space this case corresponds to the region where the boundaries A pq

and Z pq

differ (see

path (b) in figure 2.6). H2 stays inside the tongue (i.e. keeps the rotation number pq)

while H1 faces a heteroclinic loop and grows as Klog(1/a)

for some constant K [40] (see

figure 2.8).

Figure 2.9: Case 2, parameter values: b = 2.8, a = .3, and 50000 iterates

A third kind of behaviour occurs for even larger values of b, when F (m2) > F (1)


and F (m1) < F (0). This case, consist of paths along a where A pq

does not longer

exist, so that ρ is always non-trivial (see path (c) in figure 2.6).

Figure 2.10: Case 3, non-trivial case with b = 5

This last case is not dealt with in [7] or [40] as it has no sense in the context of

transition to chaos, as it may be always chaotic.

2.2.3 Invertible quasi-periodically forced circle maps

As previously mentioned in the introduction, the quasi-periodically forced case may

be thought of as an extension of circle maps. It is, however, much richer in terms of

dynamics and also slightly more complicated.

Let (x, θ) ∈ R2, the system has the form:

xn+1 = F (xn, θn)

θn+1 = θn + ω

where ω is an irrational number and it happens that F (x + 1, θ) = F (x, θ) + 1

and F (x, θ+ 1) = F (x, θ), so it is the lift of a map of the torus. Then, it makes sense

to take both x and θ mod 1 and to think of them as representing angles.

F (x, θ) = x+ a+b

2πsin(2πx) + c sin(2πθ) (2.3)

G(θ) = θ + ω (2.4)


Once again, the invertibility of the map f determines essential properties of the

dynamics of the maps, so, again, we separate both cases for their study.

The first observation is that the system is in S1 x S1, in contrast to circle maps,

which are in S1. There is also one more dimension in parameter space added by

parameter c. It exhibits new types of dynamics such as quasi-periodic motion with

two and three incommensurate frequencies, chaotic attractor and strange non-chaotic

attractors [46]

Literature has focused on extending or adapting the existing results of circle maps

to quasi-periodically forced circle maps, however, in many cases the generalizations

have not been straight forward and other difficulties have emerged.

Firstly, the existence of rotation numbers is not immediate because ordering is no

longer present for arbitrary pairs of points (x, θ) and such feature plays a key role in

the proof of the existence of rotation numbers.

Nevertheless, [30] proved that, restricted to b < 1, the rotation number exists and

that it is independent of the chosen (x, θ). The proof is based on the monotonicity

for each fixed θ (i.e. restricted to invertible maps) and assumes that ω 6∈ Q [30]. In

this section we restrict ourselves to values of b to be less than one.

It is important to note that, in quasi-periodically forced maps, the rotation num-

ber following the limit definition [ see 2.5] may be referring to different motions; the

θ-direction or the x-direction. The first case, ρθ, is always trivial because it is simply

a rotation by ω, so the representative and interesting one is the latter

ρx = limn→∞

F n(x, θ)− xn

(2.5)

In a way, we can too -in this context- make a distinction between ‘rational’ and

‘irrational’ cases, nevertheless we must make some observations; to start with ‘ratio-

nal’ rotation numbers will now correspond to those numbers of the form ρ = pq

+ ω rs

for some rational pq

and rs. Dynamically, it means that the former two periodic or-

bits inside the rational tongues in circle maps become two invariant curves in the


quasi-periodic case. The periodic motion with period q turns into two-frequency

quasi-periodic behavior with q ‘branches’. The attractor may be a finite union of

closed invariant curves on S1 x S1.

In contrast, when such rational numbers pq

and rs

do not exist, we say that ρ is

‘irrational’ in an analogous way to irrational rotation number for circle maps. The

motion is said to be incommensurate (or three frequency quasi-periodic) and the

attractor may be the whole S1 x S1.

In both cases, strange non-chaotic attractors may be present [14].

In order to understand the effect of the quasi-periodic forcing, in particular related

to its strength, a similar approach to that of Chapter 4 can be found in the literature.

It consists on fixing one or more parameters and numerically observing the changes

that different values of c induce on the parameter space structure.

2.2.4 Rational case

As long as c is small the structure of tongues remains similar to that of the unforced

case. The bounds of tongues are characterized by a saddle node bifurcation of the

two invariant curves inside the tongue[18]. At larger values of c, the boundaries of

tongues might be characterized not only by a saddle node, but by the formation of

a strange non-chaotic attractor, which is the product of the collision, on a countable

set of points in θ, of the two invariant curves inside the tongue.

This change of dynamics leads to a change in the characteristic cone shape to that

of a leaf [18] (see figure 2.11). For higher period q, the decrease of width occurs for

smaller values of c.

2.2.5 Irrational case

For c = 0 the width of the ‘irrational tongue’ is zero. A slight increase on the

parameter c leads to an increase width of the tongue, monotonically to the value


Figure 2.11: ρ = 1/2 from [18]

of b. Further values of c cause the shape to become that of a leaf and deviate its

orientation (see figure 2.12).

A second problem arises when numerically calculating the rotation number be-

cause the bounds (2.2) do not longer hold. However, a new characterization is pro-

posed in [53] which has explicit error bounds in an analogous manner to those in [50]

which we used for all our computations. The method is briefly described in chapter

5.

2.2.6 Non-invertible quasi-periodically forced circle maps

Regarding quasi-periodically forced maps, a relevant fact for this piece of work follows

from a result in [25]. An interpretation of that result states that “moving along a path

in parameter space from a non-chaotic state of an invertible quasi-periodically forced

map to a chaotic non-invertible map [...] it is necessary to create strange non-chaotic

attractors”. Consequently, the transition to chaos sometimes is studied thorough the

appearance of SNAs. Along the way, the authors generalize some results for non-

invertible circle maps to non-invertible quasi-periodically forced circle maps. Among

them, and particularly relevant to us, they prove that the set of rotation numbers is


Figure 2.12: ρ = 12− 1

2ω from [18]

an interval, including the possibility of a point (for further information see Chapter

4).

Chapter 3

Piecewise example

In chapter 2, we described how in [40] they detail the two possible ways that ρ±

may develop right after crossing the borders of the mode locked region (i.e. tongue).

Graphically, as explained in the previous section, these scalings are related to the

way in which ρ± loses the tangency with the identity.

In this section we detail the breaking of a mode locked region for a function which

has a similar bifurcation to that of figure 2.9 (chapter 2). To do so we use a family of

piecewise monotone functions whose rotation numbers go through such a bifurcation

as the parameter values change.

Figure 3.1: Representation of the bifurcations suspected to be similar; to the leftH2(x) and to the right a piecewise linear map.

32

CHAPTER 3. PIECEWISE EXAMPLE 33

3.1 Construction of piecewise maps

Let x ∈ [0, 1], for a given pair of parameters a and d both in [0, 1], the function

will have a plateau of length d and height a, starting at 0; and then a straight

line connecting the end of the plateau with (1, 1 + a). Its dynamics depend on the

parameters a and d.

Formally,

f(x) =

a if x ∈ [0, d)

kx+m if x ∈ [d, 1)

where k = 11−d and m = a− d

1−d

Make F : R → R, F (x + 1) = F (x) + 1, such that F is the lift of the circle map

f above. Because is non-decreasing it has a well defined rotation number for each

choice of parameters.

The approach we followed was to fix d and describe what happens when we move

along a. We chose this approach for the sake of congruence with the Arnold circle

map, as the bifurcation that we want to represent is the one that occurs in the corner

case along a range of values of a (see figure 3.1).

3.2 Dynamics of the piecewise family of maps.

Essentially, there are the two types of dynamics depending whether a is larger or

smaller the value of d, which graphically correspond to whether the plateau intersects

the identity line or not.

Whenever a < d the situation is very simple, all points become fixed points after

a small number of iterations, which means that its average translation is zero (see

figure 3.2).

It is straightforward to check that if x ∈ [0, d), then F n(x) = a for all n > 0.

For x ∈ [d, 1], fn(x) becomes fixed when F n(x) > 1 and F n(x) < 1 + a. The

iteration in which this situation occurs depends on whether they are before or after


Figure 3.2: Simplest case a < d

the point where f(x) = a + x1−d intersects the identity. Either way, after a small

number of iterations Fm(x) > 1 so fm+1(x) = a with the exception of x = −a(1−d)1−(1−d)

which is fixed itself. For this reason, we can intuitively state that F (x), where a < d,

also has an average zero translation.

Observe that because F is non decreasing and degree one, the rotation number

exist and is independent of the initial x. In particular we may assume the initial x

to be d. Therefore for a < d, ρ = 0.

The bifurcation that we are interested occurs at a = d. As soon as d < a, F no

longer has any fixed points. We take δ to be a measure of how much the plateau

is lifted after loosing the tangency, that is δ = a − d and we describe the rotation

numbers for small δs.

Take x = d, we may do so for our purposes because of the independence of initial

conditions. On the first iteration F (d) will become a, then because the slope is larger

than one and the line is above the identity, F 2(d) > F (d), this process will continue

and eventually for some n, F n(d) > 1 and F n−1(d) < 1. This iteration is particularly


important because jumping over 1 means, in terms of circle dynamics, returning to the

initial point and starting the cycle all over again. The rotation number is, therefore,

for this given choice of parameters, the reciprocal of the number of iterations necessary

to achieve so.

For x ∈ [d, 1) it’s easy to see that a initial number of iterations will inevitably

take its image to [1, 1 + a] and the cycle described above will start from there (see

figure 3.3). This first few iterations will not be reflected in the rotation number, as

the rotation number is defined by a limit.

Figure 3.3: Schematic view of the dynamics when a > d

Let m be the number of iterations necessary to have that Fm(d) falls into the

interval [1, 1 + a], then ρ = 1m

. Notice that for some slightly larger values, that is

d+ δ with a small δ, Fm(d+ δ) may still remain inside the inteval [1, 1 + a], and thus

d and d+ δ have the same rotation number.

The set of values for which d+ δ share a rotation number is called a phase locked

regions.


Figure 3.4: Slight increase on a might not modify ρ.

3.3 Theoretical prediction

Our next task is to calculate explicitly the starting and end points of phase locked

regions for a given a fixed d, with respect to the difference δ. To achieve so, we need

to find the regions where Fm(d) ∈ [1, 1 + a].

Looking closely at a small δ, which sets the problem just after the bifurcation, we

can make a coordinate change by setting the ‘new’ origin at (d, d) and may think of

the map as F = kx + δ, where x in the new system of coordinates corresponds to

x+ d in the previous coordinates (see figure 3.5).

Then, as long as F n(x) < 1, the n-th iterate would be given by:


Figure 3.5: Change of coordinates

F (x) = kx+ δ (3.1)

...

F n(x) = knx+ δn−1∑r=0

kr (3.2)

= knx+δ(1− kn)

1− k(3.3)

We may take the initial x to be 0, which for the same reasons than the ones above,

is just as good as any other choice in terms of the rotation number but simplifies the

algebraic calculations so,

F n(0) =δ(1− kn)

1− k

A phase locked region, with a rotation number 1n, will prevail while F n(0) ∈

[1−a, 1). This reduces the question of finding phase locked regions to that of finding

the different values of δ under which,

1− a ≤ δ(1− kn)

1− k≤ 1 (3.4)


It is also clear from this expression that we can have n (and therefore ρ) as a

function of all the fixed constants and δ.

Notice that because k > 1, k − 1 > 0. Also, as 0 < a < 1, 1 − a > 0. Algebraic

manipulation of equation 3.4 yields:

(1− a)(1− k) ≥ δ(1− kn) ≥ (1− k)

(1−a)(1−k)δ

≥ 1− kn ≥ 1−kδ

(1−a)(1−k)δ

− 1 ≥ −kn ≥ 1−kδ− 1

−(

(1−a)(1−k)δ

− 1)≤ kn ≤ −

(1−kδ− 1)

(1−a)(k−1)δ

+ 1 ≤ kn ≤ k−1δ

+ 1

δ+(1−a)(k−1)δ

≤ kn ≤ δ+(k−1)δ

log( δ+(1−a)(k−1)δ

) ≤ n log(k) ≤ log( δ+(k−1)δ

)

log(δ+(1−a)(k−1)

δ)

log(k)≤ n ≤ log(

δ+(k−1)δ

)

log(k)

log(δ+(1−a)(k−1))−log(δ)log(k)

≤ n ≤ log(δ+(k−1))−log(δ)log(k)

log(k)log(δ+(1−a)(k−1))−log(δ) ≥

1n≥ log(k)

log(δ+(k−1))−log(δ)log(k)

log(δ+(1−a)(k−1))+log( 1δ)≥ 1

n≥ log(k)

log(δ+(k−1))+log( 1δ)

(3.5)

And because ρ is actually the inverse of n, the phase locked regions are trapped

between the last two numbers, i.e.,

log(k)

log(δ + (1− a)(k − 1)) + log(1δ)≥ 1

n(3.6)

and

1

n≥ log(k)

log(δ + (k − 1)) + log(1δ)

(3.7)

To verify the prediction, we calculated the rotation number iterating x = 0 ten

thousand times (so the result was within ± 110000

). See figure 3.6.

As we can see in figure 3.7 the computed borders contain the phase locked regions.

Note that the rotation number as a function of a does not need to be continuous

as the piecewise maps are not differomorphisms.


Figure 3.6: The computed rotation number with 10000 iterations

3.3.1 Comparison of dynamics

As mentioned at the beginning, the aim of this chapter was to construct a family of

functions with a similar qualitative behaviour, i.e. a similar bifurcation, to the one

occurring in the Arnold map. The advantage presented by these new piecewise maps

is that due to their simpler algebraic expression, we can easily make the numerical

calculations of phase locked regions and then, compare them with the theoretical

behaviour of Arnold maps.

From [18] we know that a similar picture remains for a small quasi-periodic forcing.

The behaviour of Arnold maps was detailed in [40] (see chapter 2). They found

that the rotation number within some parameter ranges, and a variable δ correspond-

ing to a measure of the distance from the tangency evolves as

ρ+ =p

q+

C1

log(1δ)

(3.8)

for some p, q and some constant C1, of which no information is given.

For a small δ, the calculated borders of phase-locked regions behave in a similar


Figure 3.7: Proposed fittings for the right and left ends near the bifurcation togetherwith the computed rotation number.

way to proposed scaling in [40] (above). This can be seen from the fact that log(k) is

a constant and the expression in the denominator is dominated by the term log(1δ)

Furthermore, we realized (experimentally) that we could find a value for C1 such

that ρ can be located between the borders

log(k)

log(δ + (1− a)(k − 1)) + log(1δ)

(3.9)

and

log(k)

log(δ + (k − 1)) + log(1δ). (3.10)

As it can be seen in figures 3.8 and 3.9.

Although a formal proof of the equivalence between the scalings has yet to be

done, the rough qualitative analysis presented here suggest that they do.


Figure 3.8: d=0.5, the dashed line corresponds to the proposed fitting on [40] for anarbitrary value of C1 = .75, the other two lines correspond to the computed bordersof phase-locked regions.

Figure 3.9: d=0.7, the dashed line corresponds to the proposed fitting on [40] for anarbitrary value of C1 = 1.1, the other two lines correspond to the computed bordersof phase-locked regions.

Chapter 4

Observed rotation intervals

The aim of this piece of work was to numerically investigate the boundaries of tongues

in the quasi-periodically forced non-invertible case. Its contribution can be split in

two parts, first, in this chapter, we show what the dynamics are under certain param-

eter ranges. Secondly, in the next chapter, we give strong numerical and statistical

evidence to support our observations.

In congruence with most of the literature, we used rotation numbers to analyze

the dynamics. Assuming that a rotation interval may exist for some parameter values

(i.e. the rotation number is not unique), we chose to observe the behaviour of the

borders ρ− and ρ+. We were able to calculate them accurately enough by adapting

the construction of Boyland [7] and using the algorithm of Stark et al [53].

In essence, this algorithm consists of using the graph of a continuous map as the

initial condition for the n-th iterate and then approximating ρ with the calculation of

an integral thorough a trapezoidal rule. Let ψ : S1 → S1 be an arbitrary continuous

map, then the graph of this map consist of all pairs (θ, ψ(θ)). In particular, we may

take ψ(θ) to be a constant K.

The map F induces an action on graphs (θ, ψ(θ)) defined by

(θ, ψ(θ))→ F (θ, ψ(θ))

If we then define ψn to be the result of acting on the graph (θ, ψ(θ)) with F n, we

42

CHAPTER 4. OBSERVED ROTATION INTERVALS 43

have the bounds:

ρ− 1

n≤ 1

n

∫ψn − ψ ≤ ρ+

1

n. (4.1)

To approximate the integral choose a number 1 < m and take θi = im

with

i = 0, 1, . . . ,m then the rotation number is approximated by

1

m

m−1∑i=0

1

n(F n(θi, ψ(θi))− ψ(θi)) (4.2)

This approximation, using trapezoidal rule and equally spaced θi has proved to

be highly accurate and bounded as described in [53].

The monotonic maps, analogous to the construction in [7] and which we used in

the algorithm, were defined as,

F−(x) = infx<y

f(y), F+(x) = supy<x

f(y) (4.3)

Figure 4.1: Monotone maps defined above

Because we knew that in the invertible case a small strength of the quasi-periodic

forcing seemed to retain the shape of rational tongues [18], we started our experiments

with a small c and we chose three different values for b, one for each case described in

section 2.2.2 and one for a even larger value of b, where nontrivial rotation intervals

occur for all a (see figure 2.6).


We look at several different (increasing) values of c and computed the borders

of rotation intervals thorough paths across values of a. We chose the tongue with

ρ = 0 for our experiments. In all three cases, the width of the phase locked region

changed as c increased, particularly in the first two cases we saw the phase locked

region diminishing up to a point before becoming non trivial for every value of a.

We can summarize the observed dynamics in the next sequence of images:

Figure 4.2: Small c value in both pictures (0.01). To the left b = 1.2 and to the rightb = 2.8

• Figure 4.2 As expected, for small values of c we see that in both choices of

b, the dynamics are very similar to that of the unforced map, however, a slight

decrease of length of the phase locked length can be observed very early as c

starts to increase.

• Figure 4.3 As c increases, we see the length of the phase locked region with a

zero rotation number, decreasing up to a point.

• Figure 4.4 Finally, as c gets even larger, we can observe a non-trivial rotation

interval for all values of a, which appears to have a constant length through

values of a.


Figure 4.3: Increasing c value. To the left b = 1.2 and c = 1.85, to the right b = 2.8and c = .33

Figure 4.4: Large c value equal to a hundred in both cases. To the left b = 1.2 andto the right b = 2.8

Moreover, as larger values of c were considered, both ρ+ and ρ− gradually became

parallel lines so that ρ− = a+K1 and ρ+ = a+K2 for some constants K1 and K2.

Given that this behaviour could be seen for all paths, we had evidence to hypoth-

esize that the rotation interval for the non-invertible quasi-periodic map, in the large

coupling case, is always non-trivial and has a fixed length which depend only on the

value of b. Under such suppositions it is possible to work out a scaling for the length

of rotations interval.

The following chapters are devoted to the discussion of such a hypothesis.

The question of how this change occurs is not investigated. Instead, we look at

why they turn into parallel lines.

Chapter 5

A plausible explanation; our

hypothesis

In last chapter we showed that for fixed b and values of c large enough, the graph of

ρ± seems to turn into parallel straight lines.

In the past, Ding et al made a similar observation for the invertible quasi-periodic

case. In that paper the authors consider a simpler system by completely neglecting

the term which includes parameter b and hence studying the maps

xn+1 = xn + a+ sin 2πθn

θn+1 = θn + ω, ω 6∈ Q(5.1)

Straightforward iteration shows that

xn = x0 + na+∑

sin 2πθk (5.2)

Now, since the sequence {θn} mod 1 is given by the iterates of a rigid irrational

rotation it is dense in the unit circle [12].

This sequence is also uniformly distributed in the unit interval, so we built an

histogram each bar should have the same number of occurrences. Observe that the

distance between each pair θi, θi+1 (without mod1) is equal, thus, if we plotted the

number of iterations, i, against the value of the respective θi we would see that all

the iterations fall on a straight line. Taking this line mod1 results in equally spaced

46

CHAPTER 5. A PLAUSIBLE EXPLANATION; OUR HYPOTHESIS 47

parallel lines on which {θn} mod 1 would fall. This plot can be used to build an

histogram by counting the occurrences along horizontal strips (see figure 5). Then,

the average of sin 2πθn (i.e. the sum in equation 5.2) is zero and we obtain the

expression

ρ ≈ a (5.3)

which they confirm numerically in the case c = 6000, b = 1, 0 < a < 0.5 [14].

Figure 5.1: Plot of the sequence {θ0, θ1, . . . , θ150}, staring with θ0 = 0.

We can use this plot to make an histogram by counting the number of dots in each

horizontal stripe. This numbers would get very similar as the number of iterations is

increased.

We can use this plot to make an histogram by counting the number of dots in eachhorizontal stripe. This numbers would get very similar as the number of iterations isincreased.

Our graphics (see figure 4.4) suggest a similar scenario where the borders of ro-

tation intervals are linear functions of a.

Nevertheless, our claim is that rather than neglecting the b-terms, the observed

structure occurs as a result of part of the system having a zero mean. The following

explanation can be found in [26].


5.1 Uniform hypothesis formulation

This chapter explains the observations on the previous chapter and generalizes them

to a broader class of maps.

For our purposes, it is convenient to write the plateau maps corresponding to the

maps with rotation numbers ρ+ and ρ−, as the sum of two maps, one depending on

x and the other dependent of θ. So,

F+(x, θ) = g+(x) + c Φ(θ)

F−(x, θ) = g−(x) + c Φ(θ)

where g± is, respectively, the lift of a map of a circle, so g±(x + 1) = g±(x) + 1,

so henceforth it is denoted by G±. And Φ(θ) is a periodic function, Φ(θ+ 1) = Φ(θ).

We can take the mean of cΦ(θ),∫cΦ(θ)dθ, and add it to G± so we can assume Φ

has zero mean. However, for our choice of Φ(θ), it is straightforward that the mean

tends towards zero

limn→∞

∑Φ(θk)

n→∫ 1

0

sin(2πθ) = 0 (5.4)

Without loss of generality, we show the argument for G+. The other case, G−, is

analogous.

Because G+(x+ 1) = G+(x) + 1 we could think of G+ as

G+(x) = x+ p(x), (5.5)

for some function p of period one.

Additionally, we can split p into its mean(∫

p(x) dx), which we call a, plus

another p(x) (not adding to the mean):

p(x) = a+ p(x) (5.6)


And then G+ (because equation 5.5), becomes:

G+(x) = x+ a+ p(x) (5.7)

So, it makes

F (x, θ) = x+ p(x) + c Φ(θ)

= x+ a+ p(x) + c Φ(θ)

Then, direct iteration yields

xn = na+ x0 +n−1∑j=0

p(xj) +n−1∑j=0

Φ(θj) (5.8)

and so,

ρ = a+ limn→∞

1

n

n−1∑j=0

p(xj) + limn→∞

1

n

n−1∑j=0

Φ(θj) (5.9)

= a+ limn→∞

1

n

n−1∑j=0

p(xj) (5.10)

If p(xj) was uniformly distributed, which we are encouraged to think from our

observations, we would then have:

limn→∞

1

n

n−1∑j=0

p(xj) =

∫ 1

0

p(x) = 0 (5.11)

Because equation 5.5, p(x) = G+(x)− x, and so∫p(x) dx = a =

∫G+(x)− x dx

which would, in turn, make

ρ ≈ a =

∫ 1

0

g(x)− x (5.12)

While we do not formally prove it, in the next chapters we show that is reasonable

to presume that it is the case.


5.2 Ergodicity

In this section we introduce the basics of the theory on which we would like to support

our hypothesis (see equation 5.12). We show strong evidence suggesting that

limn→∞

1

n

n−1∑j=0

p(xj)→ 0 (5.13)

which, in turn, points towards the hypothesis by means of an ergodic result. This

result, which is one of the earliest in ergodic theory, is known as Weyl’s criterion.

Ergodic theory is a branch of mathematics whose goal is to analyze ‘the average

behaviour of systems’ [48]. It aims at describing dynamical systems from a statistical

and qualitative point of view, which often arises in mathematical physics.

Formally, its objects of study are measure preserving transformations for which the

orbit of a point represents the single complete history of the system [48]. Nevertheless,

for this thesis’s purposes, it is not necessary to define concepts such as measures

and probability spaces, and it will be enough just to have an intuitive definition of

distributions.

The main issue in ergodic theory lies in determining when the long term time mean

of a system i.e. the frequency with which an interval is ‘visited’ coincides, almost

everywhere, with the space mean i.e. the length of the interval. When this happens,

the system is called ergodic. For this reason we say that the ergodic approach is

particularly interested in recurrence properties, more specifically, in the frequency

that each region is visited by orbits of points.

One of the three equivalences in Weyl’s criterion relates the distribution of the

orbits around the circle with its mean behavior. It assumes that the function f , in

question, is bounded and integrable [49].

Weyl’s criterion

The following are equivalent,

1. The sequence of iterations of x0, {xn}n=∞n=o is uniformly distributed mod 1, (U(0, 1)),

in the interval.


2. The sum 1n

∑n−1j=0 f(xj) converges to

∫ 1

0f(x) dx as n→∞.

The equivalence 1) ⇒ 2) means that one way to prove the hypothesis that ρ+ =

a + K1 and ρ− = a + K2 is to prove that {xn}n=∞n=0 is uniformly distributed mod1.

Unfortunately, to prove it with all formality is not trivial either. Nevertheless, on a

intuitive level the equivalence 1) seems easier to visualize.

In essence, {xn}∞n=0 is an uniformly distributed sequence if the average number

of elements (i.e. the frequency) that fall into an arbitrary subinterval (a, b) ⊂ [0, 1]

converges to length of the interval, b− a, as n→∞.

We can’t explicitly know the distribution of {xn}n=∞n=0 because we can’t possibly

calculate all elements of the sequence. We can, however, get an intuitive idea of the

distribution from a finite sample of the iterates of {xn}n=mn=0 .

A first, but rather naive approach is to divide the interval [0, 1) in smaller subin-

tervals and count the number of occurrences in each one, in other words, building an

histogram.

Figure 5.2: Histograms showing the change on distribution for a small (0.1) and alarge (50) value of c of the first 10000 iteration

The above histograms make us conjecture that for a large c, the sequence {xn}n=∞n=0

might be approximately uniformly distributed. This observations were confirmed with

normalized histograms (to highlight the differences rather than the occurrences) for

other values of c (see figure 5.3).


Figure 5.3: Normalized histograms for values of c equal to 1.4 and 278.

A question that arises is what value does ‘large c’ refer to? This question is

partly addressed in the following chapter, however using the histograms we could

get an intuitive idea by looking at the average differences between the expected and

the observed occurrences for each bar in the histogram, see figure 5.4. Also, we

can observe that the number of bins considered has little impact on the qualitative

outcome.

Figure 5.4: Difference between the expected and the observed occurrences for manyincreasing values of c. The number of bins considered was 12 to the left, and 18 tothe right.

In the next chapter, we present the statistical theory on which we justify our

hypothesis.

Chapter 6

Statistical test

In this chapter we deal with the distribution of the sequence {xn}n=∞n=0 , in order to

provide evidence to support the observations presented in Chapter 5.

According to [43] ‘one of the two major areas of statistical inference is hypothesis

testing’. It provide us with a method to determine whether there is evidence to

reject a given conjecture, for example a statement about its distribution, regarding a

unknown or inaccessible set of data based solely on a sample of the whole set.

In this case, we want to know the distribution of the sequence {xn}n=∞n=0 just with

the information that we can get from a finite sample {xn}n=mn=0 .

It must be mentioned that the statistical tests we use presupposes each element

of the sequence to be independent. Knowing that xn depends on xn−1 and so forth,

we know that in fact they are not independent.

However, we overcame this obstacle by sampling the sequence {xn}n=∞n=0 using not

consecutive iterations but every 10th iterations, which due to the positive Lyapunov

exponents decreases correlation.

Figure 6 shows the graph of pairs (xj, xj+10), it is evident that no patterns appear

at this stage so no (or perhaps very little) correlation can inferred.

Regarding the application of the test the first step, as with any hypothesis test,

is to set up a null hypothesis H0 which is what we want to investigate, in this case,

that {xn}n=∞n=0 is uniformly distributed in [0, 1).

53

CHAPTER 6. STATISTICAL TEST 54

It is important to note that this kind of hypothesis testing only allows us to

reject a given claim (and not to accept it), based on how likely is it to obtain the

experimental data (i.e. the sample) given a certain hypothesis.

On the basis of a sample of the population we compute a number X, called the

test statistic, which will tell us whether there is evidence to reject the hypothesis. In

reality, what we can say is only whether there is ‘enough’ evidence to say that our

hypothesis is not true or that there’s not evidence to reject the claim, and therefore

it might be true.

There are many different tests depending on the type of statement we wish to

establish. The test we perform is called a simple hypothesis test because we specify

the distribution explicitly [43].

For our purposes, i.e. inferring the distribution of {xn}n=∞n=0 , there are two statis-

tical tests that we can use; one is the Chi-Square Goodness of Fit test and the other

is the Kolmogorov-Smirnov Goodness of Fit test. We used both tests with congruent

results.


6.1 Chi-Square goodness of fit test

This statistical test has been used to study many different physical situations, for

instance, it is known that it was used by E. Rutherford when working on the subject

of radioactive decay [52].

The application of this test necessarily increases the number of variables which

the result depends on by one, so not only we need to set the parameters and decide

the number of iterations to sample, but also we need to chose an arbitrary number

of bins (in which we arrange the data), for this reason we decided to settle for the

Kolmogorov-Smirnov test for publication. However, we include it here because it was

the first evidence we found to support the uniform distribution hypothesis.

The Chi-Square test requires one to divide the interval into an arbitrary number

of equally long subintervals. The main issue with this test lies on how to choose an

optimal number of them given that there are not any rules to do so. However, in

practice it is accepted that a good choice should allow at least 3 elements to be in

each subinterval [44].

In essence, the test considers the average differences between the expected frequen-

cies under the hypothesis of uniformity, and observed frequencies in each subinterval,

this number is called χ2. It is computed in the following way,

χ2 =n∑i=1

(Oi − Ei)2

Ei(6.1)

where Oi is the observed frequency in the ith bin and Ei is the expected frequency

in that same bin.

If the null hypothesis was incorrect, the difference would be large. This result

would tell us that is unlikely that our sample comes from a uniform distributed

sequence.

It is known that this statistic will follow a chi-square distribution, characterized

to accumulate around zero.


Using the statistic χ2 and choosing a statistical significance degree we will set a

threshold depending on the desired degree of confidence. When the statistic larger

than the threshold, the null hypothesis is rejected.

The significance level, α, is the probability of making an error Type I, this is,

rejecting the null hypothesis when the hypothesis is true, therefore we want α to be

small. The values for the threshold are taken from tables (see [44]).

We set the number of intervals to be 30 and fixed parameters a and b. Initially

we considered 100000 iterations and for each experiment we checked that the number

of elements in each subinterval was larger than 5. We used the chi-square table from

[44].

The value of χ2, and the result of the test, depends on four quantities: c, α, the

number of bins and the number of iterations.

We found that for any given choice of parameters a, b there are values for c, such

that the threshold was not reached. In other words, there was not ‘enough’ evidence

to reject the null hypothesis.

Figure (6.2) computes the statistic χ2 for fixed b = 2, a = 0.2 and 30 bins, as the

parameter c increases from 0.1 to 50. The horizontal line corresponds to the value

of the test threshold (for the given choice of iterations and number of bins). The

figure shows that we can reject the test for every point above the line. The computed

statistic does not decreases monotonically, however it seems that as c increases, the

number of points above the threshold decreases.

We computed a second plot to observe whether there was a significant difference

on the value of the statistic using different initial conditions. Figure 6.3 shows that

although there is a quantitative difference, qualitatively it is not relevant.


Figure 6.2: The χ2 statistic for fixed b = 2, a = 0.2, a choice of 30 bins and 100000iterations

Figure 6.3: Computed χ2 statistic for different initial conditions

6.2 Kolmogorov-Smirnov test

The second test we performed was the Kolmogorov-Smirnov test. The aim of this test

is also, to test a hypothesis about the distribution of a partly unknown or unaccessible

set of data.

It has the advantage that it is free of binning, i.e. we no longer have to divide

the interval where the iterations lay. Instead it uses the cumulative distribution. In

short, the cumulative distribution is the probability for a random variable to take a

value which is less or equal to x.


The idea in both tests is similar; to compare what we would expect if the null

hypothesis was correct to what we actually see.

The null hypothesis is once again, that {xn}∞n=0 is uniformly distributed in [0, 1).

This distribution will be the ‘reference distribution’. Under such a hypothesis we

know that the cumulative distribution function should be F (x) = x.

On the other hand, the observed cumulative distribution was obtained by taking

the values of m iterations (mod1) and then rearranging them in increasing order to

get the sequence {yk}k=mk=0 . Again to decrease correlation, every tenth iteration was

taken instead of consecutive ones.

The statistic in this test is the is maximum distance (i.e. the absolute value) of

the difference

yk −k

N(6.2)

for k = 0, ...,m and N being the size of the sample. It is usually denoted with

the letters DN .

It is know that if the sample distribution matches the reference distribution, then

the distribution of√NDN tends to a Kolmogov distribution.

The Kolmogorov distribution is a distribution linked to the Brownian bridge which

is a mathematical model of a Brownian motion [22]. It is given by:

P (K ≤ x) = 1− 2∑∞

i=1(−1)i−1e−2i2x2 (6.3)

=√2πx

∑∞i=1 e

−(2i−1)2 π2

(8x)2 (6.4)

We will reject the null hypothesis if√NDN does not follow a Kolmogorov distri-

bution, that is if√NDN is larger than a threshold Kα, where α is the significance

level (so ideally α is small). Kα is such that the probability P (K ≤ Kα) = 1 − α

[39]. Normally, once we set the value of α, the correspondent statistic Kα is found.

However, in this case we found it easier to set values for Kα and then, using the first


few terms of the sum (and noticing that the terms become smaller) investigate the

value of α. Within a few tries, the value Kα = 5 yielded a convenient α ∼ 0.01

In figure (6.4) we show the computed√NDN for fixed a = .2, b = 1.2 and 100000

iterations, for several increasing values of c.

Figure 6.4: Computed Kolmogorov-Smirnov statistic

The horizontal line represents the value of the threshold Kα so that for points

above it we can reject the null hypothesis.

To explore the convergence to uniformity as a function of c, we used a box-and-

whisker plot that takes a set of eight random initial conditions and calculates for

each of them, the statistic DN , see figure 6.5. From the plot, we could tell that the

convergence is fast at the beginning but then, not much improvement is appreciated.

To investigate this issue further, we decided to approximate DN as a function of

c.

We chose the mean of each box in figure 6.5 as a representative value of DN and

took a logarithmic scale on the vertical axis. By doing so, we implicitly suggested a

function of the form DN(c) = A0 eA1c, where the coefficients A0 and A1 had to be

determined. Using a simple linear regression in Matlab, the coefficients turned out

to be A0 = e0.698 and A1 = −0.0036, so


Figure 6.5: Computed Kolmogorov-Smirnov statistic for a set of eight random initialconditions.

DN ≈ e0.698 −0.0036 c (6.5)

Figure 6.6, shows the mean of DN together with the suggested function for DN .

For large values of c, which is the case that concerns this piece of work, it appears

to be a reasonable approximation, and as expected, DN decreases asymptotically as

the value of c grows larger.

Figure 6.6: Mean of DN for a set of random initial conditions and a together withthe suggested function to approximate it.

Further research is needed regarding this issue.


In conclusion, the two statistical tests performed in this chapter on the distribution

of the sequence {xn}∞n=0, based only on a sample of the sequence, produced similar

results.

The null hypothesis, in both cases, was that the sample was withdrawn from a

uniform distributed sequence.

They both show that there are values of the parameter c (specially as c gets larger)

for which there is not evidence that the null hypothesis can be rejected.

Chapter 7

Borders of rotation intervals

For this chapter we assume the uniform distribution hypothesis to be correct. From

that, it follows that the borders of the rotation interval [α, β] are given by:

α =

∫ 1

0

G−(x)− x dx (7.1)

and

β =

∫ 1

0

G+(x)− x dx (7.2)

In the particular case when b is slightly larger than one, we calculate the integrals,

hence α and β, below. We show only the detailed calculation for equation 7.2 as the

other case is done analogously.

This chapter too, is taken from the recently published paper [26].

From the graph of G+(x) (see figure 7.1), and analogously for G−(x), it is evident

that the integral calculations can be split in three parts; the first part going from 0

to the turning point which we name x−, then from x− to the point where g(x) = x−

which we call y+, and finally from y+ to 1.

To make notation clearer, we will decompose g into two functions:

g(x) = a+ h(x), (7.3)

62

CHAPTER 7. BORDERS OF ROTATION INTERVALS 63

Figure 7.1: Calculation the integral of G+(x).

where

h(x) = x+b

2πsin(2πx) (7.4)

And define the lifts H± in an analogous way to G−(x) (see equation 4.3).

Also, let b = 1 + µ2 with a small positive µ, so h(x) = x+ 1+µ2

2πsin(2πx).

In order to split the integral in the three parts mentioned above, the first step is

to find the turning point x−, which will define the start of the plateau. This point is

one of the solutions of

1 + b cos(2πx) = 0 (7.5)

We pose an asymptotic expansion of x− near 12

in terms of µ. So

x− ≈1

2+ αµ+ βµ2 + γµ3 (7.6)

To determine the coefficients, we use the two following facts. First,

cos(2πx) =−1

b= −(1 + µ2)−1 ≈ −1 + µ2 − µ4 (7.7)

(from Newton’s generalized binomial theorem).

Secondly, we use the sum of cosines and the asymptotic expansion of cos(x):

cos(x) ≈ 1− x2

2!+x4

4!(7.8)


Then, equating the coefficients up to the order µ3, we obtain:

x− ≈1

2− 1

π√

2µ+

5

12π√

2µ3 (7.9)

so α = 1√2π

and β = 5√212π

This point marks the starting point of the plateau, which has a height of h(x−) =

h+.

Substituting x− into h (see equation 7.4)

h(x−) ≈(

1

2+ αµ+ βµ3

)+

(1 + µ2

2πsin(2π

(1

2+ αµ+ βµ3)

))(7.10)

Expanding and then grouping coefficients (up to 3er order) we obtain:

h(x−) ≈(

1

2+ αµ+ βµ3

)+

1 + µ2

2π

(sin(π) cos(2παµ+ 2πβµ3) + cos(π) sin(2παµ+ 2πβµ3)

)=

(1

2+ αµ+ βµ3

)+

1 + µ2

2π

(0 +

(− 1)(

2παµ+ 2πµ3 − (2παµ+ 2πβµ3)3

3!

))=

(1

2+ αµ+ βµ3

)+

(−2παµ− 2πβµ3 + 22+π3α3µ3

3

2π

)+

(−2παµ3

2π

)=

1

2+

(2π2α3

3− α

)µ3 (7.11)

Finally, substituting α in equation 7.11,

h(x−) =1

2+

2

3π√

2µ3 (7.12)

On the other hand, the end of the plateau is marked by the point y+ >12

such that

h(y+) = h(x−). To approximate y+ we now pose y ≈ 12

+ α1µ + α3µ3. Substituting

it into h and equating coefficients,

2

3π2α3

1 − α1 −2

3π√

2= 0 (7.13)


Because the µ coefficient of x− is a double root of equation 7.13, we can factorize

it as

(α1 −

1√2π

)2(2π2α1

3− 4π

3√

2

)(7.14)

and it is simple to check that the other root is√2π

Therefore,

y+ ≈1

2+

√2

πµ+ ... (7.15)

With all this knowledge, we can now calculate ρ+,

ρ+ =

∫ 1

0

(G+(x)− x) dx (7.16)

= a+

∫ 1

0

(H+(x)− x) dx (7.17)

= a+

∫ x−

0

(b

2πsin(2πx)

)dx+

∫ y+

x−

(h+ − xdx+

∫ 1

y+

b

2πsin(2πx)

)dx(7.18)

= a+

∫ y+

x−

(h+ −

(x+

b

2πsin(2πx)

))dx (7.19)

≈ a+

[h+x−

1

2x2 +

b

4π2cos(2πx)

]y+x−

(7.20)

Evaluating on y+ and x−,

ρ+ ≈ a+2

3π√

2µ3

(1

π√

2µ+

√2

π

)

+1

2µ2

((

1

π√

2µ)2 + (

√2

π)2

)

+π2

6

((

1

π√

2µ)4 + (

√2

π)4

)(7.21)

Taking the lowest terms and simplifying, results on,

ρ+ ≈ a+9

8π2µ4 (7.22)


Recalling that µ2 = (b− 1)

ρ+ ≈ a+9

8π2(b− 1)2 (7.23)

An analogous procedure using G+(x) instead of G−(x) yields

ρ− ≈ a− 9

8π2(b− 1)2 (7.24)

Subtracting (7.24) from (7.23), the length of the rotation interval turns out to be,

I ≈ 9

4π2(b− 1)2. (7.25)

We compared this prediction with the numerical computation of the results. There

seems to be a good match, however some oscillatory modulation can be appreciated.

We did not investigate it but in the future it could be a subject of research.

Figure 7.2: Comparison between the numerical and theoretical borders of rotationintervals, with c = 200, b = 1.2 and 100000 iterations.

Chapter 8

Length of rotation interval

Having computed the rotation borders for a particular quasi-periodic map (see chap-

ter 7), we now show how the growth of the length of the rotation interval corresponds

to an unfolding from catastrophe theory.

We use the fact that locally, near the boundary of invertibility, the map G(x)

appears to have a cubic point of inflexion to develop an argument on the growth of

rotation intervals. Later, we confirm it matches with the results from the previous

chapter (i.e. the integral calculations).

From the equations of ρ+ and ρ− (see chapter 7), we know that theoretically the

length of the rotation interval grows like,

|I| = ρ+ − ρ− ≈9

4π2µ2 =

9

4π2(b− 1)2 (8.1)

Using that locally the map appears to have a cubic point of inflexion, local coor-

dinates can be set so g(x) in equation 7.3 is,

g(x) ≈ A+ Cx3 +O(x3) (8.2)

By Taylor, C = 16gxxx evaluated at the point of inflexion, which we adjust to be

at the origin.

A small perturbation of this map produces the map:

67

CHAPTER 8. LENGTH OF ROTATION INTERVAL 68

g(x) = A+Bε1x+ ε2x2 + Cx3 (8.3)

We make another a coordinate change to make the square term zero, and thus

having the standard unfolding of a cubic singularity:

g(x) = A− εBx+ Cx3 +O(x4) (8.4)

Where by Taylor, C remains being 16gxxx and B = −gxε at the inflexion point

with ε = 0.

The two turning points x+ and x− are obtained derivating and equating to zero

equation 8.4, provided that Bε < 0 (otherwise the map is locally invertible),

x± = ±√Bε

3C+O(|ε|

32 ) (8.5)

Let G− and G+ denote the plateau maps, G−(x) = infx<y g(y) and G+(x) =

supy<xg(y), using g(x) as in equation 8.4. And let y+ > 0 be the point with the

same image than the maximum x−, this is g(y+) = g(x−). In other words, y+ is the

solution of,

A− εBx+ Cx3 = g(x−). (8.6)

Since x− = −√

Bε3C

,

g(x−) = A+

√Bε

3C(−2

3Bε)(8.7)

So, to find y+ we equate equations 8.6 and 8.7 and note that x− is a double root,

so we can factorize as (y − x−)2(Cy −R) and solve for R. In the end we obtain,

y+ = 2

√Bε

3C+O(|ε|

32 ) (8.8)


Then, recalling that the origin was set precisely at inflexion point so that the

length of the rotation interval is twice the area between 0 and y+,

1

2|I| =

∫ y+

0

G+(x)−G−(x)dx (8.9)

Figure 8.1: The cubic unfolding, the rotation interval is represented by the areabetween the graphs G+ and G−

We are now in the position to solve the integral in equation 8.9. We proceed by

splitting it into two parts: a rectangle from 0 to x+ of height g(x−)−g(x+) ≈ 43Bε√

Bε3C

and the region from x+ to y+ bounded by g(x) and g(x−) (see figure 8.1).

Therefore,

|I| = 2[

∫ x+

0

g(x−)− g(x+)dx+

∫ y+

x+

g(x−)− g(x)dx] (8.10)

= 2(((4

3Bε

√Bε

3C)(

√Bε

3C) +

∫ y+

x+

g(x−)− g(x)dx) (8.11)

= 2((4

3Bε

√Bε

3C)(

√Bε

3C) + [

2

3Bε

√Bε

3Cx+

1

2Bεx2 − 1

4Cx4]y+x+) (8.12)

= 2(4

9

B2ε2

C+

11

36

B2ε2

C) (8.13)

=3

2

B2ε2

C(8.14)

Substituting B = −gxε = −1, C = 16gxxx = 4π2 and ε = b − 1 we obtain the

expression


|I| = 3

2

B2ε2

C=

9

4π2(b− 1)2 (8.15)

Unsurprisingly, the equation above is the same equation obtained in the previous

chapter (see equation 7.25).

Chapter 9

Conclusions

We described the dynamics of a particular quasi-periodic circle map in terms of

rotation numbers, focusing on the case of ‘large’ coupling force. Especial attention

was given to the presence of rotation intervals given that their occurrence opens the

possibility of chaotic behaviour.

In general, non-invertible quasiperiodic circle maps have not yet been fully un-

derstood. However, results have been obtained on some regions of parameter space

and the overall picture is starting to be uncovered. We contributed to this process

by showing some aspects of the dynamics in the region where the strength of the

coupling is large.

Guided by the graphs of the rotation numbers, we realized that the borders of the

rotation interval had the form ρ− ≈ a + k1 and ρ+ ≈ a + k2 where k1 and k2 could

be approximated.

Based on Weyl’s criterion, we formulated a hypothesis to explain the observed

phenomena (see Chapter 5). The hypothesis is that when the value of the coupling

becomes ‘large’, the average effect of the b-term is zero and thus the iterations of a

point (i.e. the set {xn}∞n=0) are uniformly distributed in S1. This allowed us to work

out the rotation interval by solving an integral, which in turn, produced an expression

for the growth of length of the rotation interval. Interestingly, such an expression

corresponds to a cubic unfolding.

71

CHAPTER 9. CONCLUSIONS 72

The findings are in congruence with previous observations, see for example [14].

Even when we did not formally prove the hypothesis about the distribution of

iterations, we show strong numerical evidence to support it. Specifically, we use a

couple of statistical tests, the Chi-square goodness of fit test and the Kolmogorov-

Smirnov goodness of fit test. Both tests show that in essence the hypothesis about

the distribution of {xn} becomes unlikely to be rejected as the value of c increases.

On the adopted approach, it was of remarkable importance the method developed

by Stark et al [7] which constructs a couple of invertible maps corresponding to the

borders of rotation intervals in circle maps. It should also be noticed that for most

calculations we used the algorithms produced in [53].

In the future we would like to prove that, in fact, the rotation number of the

borders of rotation intervals take the form a+ k1 and a+ k2. One way of proceeding,

would be to prove the quasiperiodic map is ergodic.

We would also like to investigate how the rotation number becomes nontrivial.

Perhaps we could see how much the circle map results can be extended.

Another interesting aspect that we started investigating (but do not include due

to its early stage) is the presence of observable mode locked intervals. This subject

will definitely researched in the near future.

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Appendix A

Using different initial θs

Thorough out this piece of work, we analyzed the dynamics of the quasi-periodically

forced circle map by looking at the rotation numbers in the x− direction for one fixed

initial θ0. This is natural decision considering that the existence of rotation numbers

for this kind of maps is defined in fibers, with the idea of retaining some control over

the relative positions of the iterates.

In this last part we present our last brief investigation which describes how the

rotation number behave as a function of the initial θ. This was motivated by the

observation consistent differences on the computed rotation numbers for different

initial values of θ and large values of c. The argument is valid only for the case where

the b-term is so much smaller that the c-term that it can be neglected, in fact, we

take b = 0.

For numerical simplicity we also assume a = 0.

Consider a fixed value of c and ω 6∈ Q (just as in the previous chapters), then the

n-th iterate is given by:

xn = x0 + c

n∑m=0

sin(2π(θ0 +mω) (A.1)

Following the definition, the rotation number is,

79

APPENDIX A. USING DIFFERENT INITIAL θS 80

ρ = limn→∞

x0 + c∑n

m=0 sin(2π(θ0 +mω)

n(A.2)

Observe that the sum in equation A.1, can be expanded as,

n∑m=0

sin(2π(θ0 +mω) = sin(2πθ0) +n∑

m=0

cos(2πmω) (A.3)

+ cos(2πθ0) +n∑

m=0

sin(2πmω) (A.4)

In its turn, the sums in line (A.3) and line (A.4) are arithmetic progressions,

named k1 and k2 respectively, so:

k1 =n∑

m=0

cos(2πmω) =(sin((n+ 1)ωπ))(cos((πnω)))

sin(πω)(A.5)

k2 =n∑

m=0

sin(2πmω) =(sin((n+ 1)ωπ))(sin((πnω)))

sin(πω)(A.6)

combining equations A.5 and A.6 and taking a large n,

ρθ ≈c(k1 sin(2πθ0) + k2 cos(2πθ0))

n(A.7)

We did not investigate the accuracy of the the approximation, in other words, we

do not know how close the left hand side is to the right hand side on equation A.7.

We can simplify further using trigonometric identities so the n-th approximation

to ρ appears as a function of the initial θ plus a ψ, which is given by tan−1(k2k1

):

ρ ≈ c(√

(k21 + k22))(sin(2πθ0)k1√

(k21+k22)

+ cos(2πθ0)k2√

(k21+k22)

)

n(A.8)

= c(√

(k21 + k22))(sin(2πθ0) cos(ψ) + cos(2πθ0) sin(ψ))

n(A.9)

= c(√

(k21 + k22))(sin(2πθ0 + ψ))

n(A.10)

APPENDIX A. USING DIFFERENT INITIAL θS 81

Where ψ = tan−1( sin(ψ)cos(ψ)

) = tan−1(k2k1

).

Equation A.10 adjusts very well to the approximations by iteration as we can see

in figure A.1.

Figure A.1: Rotation numbers for different initial θ, 10000 iterates and c = 100.

rotation intervals for quasi-periodically forced …

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