Existence and structure of strange non-chaotic attractors [Elektronische Ressource] / vorgelegt von Tobias Henrik Jäger

Existence and structureof strange non-chaotic attractorsDen Naturwissenschaftlichen Fakultaten¨der Friedrich-Alexander-Universitat Erlangen-Nurnberg¨ ¨zur Erlangung des Doktorgradesvorgelegt vonTobias Henrik Ja¨geraus LichiAls Dissertation genehmigt von den NaturwissenschaftlichenFakultaten der Friedrich-Alexander-Universitat Erlangen-Nurnberg¨ ¨ ¨Tag der mu¨ndlichen Pru¨fung: 20.07.2005Vorsitzender der Promotionskommission: Prof. Dr. D.-P. H¨aderErstberichterstatter: Prof. Dr. Gerhard KellerZweitberichterstatter Prof. Dr. Andreas KnaufiiAcknowledgments: First of all, I would like to thank Gerhard Keller, withoutwhose guidance and advice this work would not have been possible. Further, Iwould like to thank Andreas Greven for his support concerning various fundingapplications, Henk Bruin and the Mathematics Department at the University ofSurrey for their hospitality during my visit to Guildford, and all colleagues in theAG Stochastik und Dynamische Systeme at Erlangen. Finally, although not di-rectly related to the thesis the collaboration with Jaroslav Stark, Sylvain Crovisier,Francois Beguin, Frederique LeRoux and Paul Glendinning during the final periodof my PhD studies had also been a great source of inspiration.
Publié le : samedi 1 janvier 2005
Lecture(s) : 15
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Source : WWW.OPUS.UB.UNI-ERLANGEN.DE/OPUS/VOLLTEXTE/2005/242/PDF/TOBIASJAEGERDISSERTATION.PDF
Nombre de pages : 131
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Existence and structure
of strange non-chaotic attractors
Den Naturwissenschaftlichen Fakultaten¨
der Friedrich-Alexander-Universitat Erlangen-Nurnberg¨ ¨
zur Erlangung des Doktorgrades
vorgelegt von
Tobias Henrik Ja¨ger
aus Lich
iAls Dissertation genehmigt von den Naturwissenschaftlichen
Fakultaten der Friedrich-Alexander-Universitat Erlangen-Nurnberg¨ ¨ ¨
Tag der mu¨ndlichen Pru¨fung: 20.07.2005
Vorsitzender der Promotionskommission: Prof. Dr. D.-P. H¨ader
Erstberichterstatter: Prof. Dr. Gerhard Keller
Zweitberichterstatter Prof. Dr. Andreas Knauf
iiAcknowledgments: First of all, I would like to thank Gerhard Keller, without
whose guidance and advice this work would not have been possible. Further, I
would like to thank Andreas Greven for his support concerning various funding
applications, Henk Bruin and the Mathematics Department at the University of
Surrey for their hospitality during my visit to Guildford, and all colleagues in the
AG Stochastik und Dynamische Systeme at Erlangen. Finally, although not di-
rectly related to the thesis the collaboration with Jaroslav Stark, Sylvain Crovisier,
Francois Beguin, Frederique LeRoux and Paul Glendinning during the final period
of my PhD studies had also been a great source of inspiration.
iiiZusammenfassung: Die Existenz seltsamer nicht-chaotischer Attraktoren (SNA)
in quasiperiodisch getriebenen Systemen ist in zweierlei Hinsicht bemerkenswert,
weil dies zeigt daß seltsame Attraktoren sowohl in nicht-hyperbolischen Systemen
als auch in Verbindung mit nicht-chaotischer Dynamik auftreten k¨onnen. Daru¨ber-
hinaus spielen SNA auch eine besondere Rolle fu¨r die Verzweigungen invarianter
Tori, da sie ha¨ufig in sogenannten ‘nicht-glatten’ Sattel- und Heugabelverzweigun-
gen auftreten. Dies ist meist von einem vorhergehenden sehr charakteristischen
Verhalten der invarianten Tori begleitet, welches sich als ‘Exponentielle Entwick-
lung von Peaks’ beschreiben laßt. Ausgehend von dieser Beobachtung sollen in der¨
vorliegenden Arbeit neue Methoden entwickelt werden, um sowohl die Existenz von
SNA nachzuweisen als auch deren Struktur naher zu untersuchen.¨
Dabei werden in der Einleitung zuna¨chst die beno¨tigten Grundbegriffe bereit-
gestellt und dann anhand numerischer Daten belegt, das sich das beschriebene
Ph¨anomen in einer ganzen Reihe verschiedener und auch physikalisch relevanter
Systeme beobachten la¨ßt. In Kapitel 2 werden dann zuna¨chst sogenannte ‘Pinched
skew products’ untersucht. In diesen sehr einfachen Modellsystemen ist die Ex-
istenz von SNA bereits seit la¨ngerem bekannt, so daß es hier darum geht die
Struktur dieser Objekte naher zu beschreiben. Dies wird durch eine quantifizierte¨
Beschreibung der exponentiellen Entwicklung von Peaks ermo¨glicht. Der gesamte
restliche Teil der Arbeit ist dem Nachweis von SNA in bestimmten Parameterfami-
lien quasiperiodisch getriebener Intervallabbildungen gewidmet. Im 3. Kapitel wer-
den dazu einige allgemeine Grundlagen zusammengestellt. Zunachst werden allge-¨
meineSattelverzweigungen indenbetrachteten Systemenbeschrieben,dieabha¨ngig
davon ob es am Verzweigungspunkt zum Auftreten von SNA’s kommt als ‘glatt’
(ohne SNA) oder ‘nicht-glatt’ bezeichnet werden. Der in den numerischen Simula-
tionen beobachtete Ablauf von nicht-glatten Verzweigungen legt zudem nahe, daß
es dabei zum Auftreten eines speziellen Typs sehr ungewohnlicher Orbits kommt,¨
dieals ‘Sink-source-orbits’ bezeichnet werden. Die Existenz dieser Orbitsimpliziert
wiederum die Existenz von SNA’s. Daher genugt es in den folgenden Kapiteln die¨
erstere nachzuweisen, was zu einer wesentlichen Vereinfachung des Problems fu¨hrt.
Die Konstruktion der Sink-source-orbits geschieht dann durch Approximation mit
immer la¨ngeren endlichen Trajektorien. In Kapitel 4 wird zun¨achst die zugrun-
deliegende Strategie erlautert und die erste Stufe der induktiven Konstruktion aus-¨
gefu¨hrt. Zu Ende gefu¨hrt wird diese dann in Kapitel 6, wobei die wesentlichen
Hilfsmittel und Werkzeuge zuvor in Kapitel 5 bereitgestellt werden. Im letzten
Kapitel wird schließlich mit einer leichten Abwandlung der vorhergehenden Kon-
struktion die Existenz von SNA mit speziellen Symmetrieeigenschaften gezeigt, wie
sie in nicht-glatten Heugabelverzweigungen entstehen.
ivAbstract: The existence of strange non-chaotic attractors (SNA) in quasiperiodi-
cally forced systems is remarkable in two ways, as it shows that strange attractors
can occur both in systems which are not hyperbolic and in combination with non-
chaotic dynamics. Further, SNA’s seem to play an important role in the bifurca-
tionsofinvarianttori,astheyoftenoccurinso-called ‘non-smooth’ saddle-nodeand
pitchfork bifurcations. This is accompanied by a very distinctive behaviour, which
can be described as ‘exponential evolution of peaks’. Based on this observation, the
aim of this thesis is to develop a new method by which the existence of SNA can
be proved and their structure can be analyzed.
The introduction first provides some basic terminology. Further, it shows by
meansofnumericalsimulations thatthedescribedphenomenonispresentinanum-
ber of different systems, some of which are directly motivated by physical models.
Afterwards so-called ‘pinched skew products’ are studied in Chapter 2. For these
verysimplemodelsystemstheexistenceofSNAisalreadywell-known,suchthatthe
aimhereistoobtainfurtherinformationaboutthepropertiesandstructureofthese
objects. This is achived by a quantitative description of the exponential evolution
ofpeaks. Theremainderofthisthesisisthendedicated to theproofoftheexistence
of SNA in certain parameter families of interval maps with additive quasiperiodic
forcing. Chapter 3 first contains a description of saddle-node bifurcations in the
considered families. Depending on whether these bifurcations involve the occur-
rence of SNA, they are called either ‘smooth’ (without SNA) or ‘non-smooth’ (with
SNA). The characteristic behaviour during non-smooth bifurcations which can be
observed numerically furthersuggests that there exists a particular type of very un-
usualorbitsatthebifurcationpoint,whichwillbereferredtoas‘sink-source-orbits’.
The existence of such orbits implies the existence of SNA, which leads to a signifi-
cant simplification of the problem. The sink-source-orbits are then constructed by
means of approximation with finite trajectories. Chapter 4 first outlines the overall
strategy and also contains the first step of the inductive construction. Before this is
continued in Chapter 6, the necessary tools and auxiliary statements are provided
inChapter5. Finally, thelastchapter usesaslight modificationofthisconstruction
to show the existence of SNA with a certain kind of symmetry, as they occur in
non-smooth pitchfork bifurcations.
vviContents
1 Introduction 1
1.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Non-smooth saddle-node bifurcations and exponential evolution of
peaks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 A brief sketch of the mechanism behind . . . . . . . . . . . . . . . . 12
1.4 Using the heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.5 Non-smooth pitchfork bifurcations . . . . . . . . . . . . . . . . . . . 19
2 Strange non-chaotic attractors in pinched skew products 20
2.1 The existence of SNA in pinched systems . . . . . . . . . . . . . . . 20
2.2 How to use the exponential evolution of peaks . . . . . . . . . . . . . 21
2.3 Controlling the shape of the iterated upper boundary lines . . . . . . 21
2.4 The topological closure of the upper bounding graph . . . . . . . . . 26
2.5 Upper bounding graphs which contain isolated points . . . . . . . . 32
3 A general setting for the non-smooth saddle-node bifurcation 36
3.1 Equivalence classes of invariant graphs and the essential closure . . . 36
3.2 Saddle-node bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3 Sink-source-orbits and the existence of SNA . . . . . . . . . . . . . . 42
4 The strategy for the construction of the sink-source-orbits 46
4.1 The first stage of the construction . . . . . . . . . . . . . . . . . . . 46
4.2 Dealing with the first close return . . . . . . . . . . . . . . . . . . . . 51
4.3 Admissible and regular times . . . . . . . . . . . . . . . . . . . . . . 55
4.4 Outline of the further strategy . . . . . . . . . . . . . . . . . . . . . 55
5 Tools for the construction 59
5.1 Comparing orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.2 Approximating sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.3 Exceptional intervals and admissible times . . . . . . . . . . . . . . . 68
5.4 Regular times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6 Construction of the sink-source orbits: One-sided forcing 81
6.1 Proof of the induction scheme . . . . . . . . . . . . . . . . . . . . . . 83
6.2 The existence of SNA and some further remarks . . . . . . . . . . . 98
7 Construction of the sink-source-orbits: Symmetric forcing 104
7.1 Proof of the induction scheme . . . . . . . . . . . . . . . . . . . . . . 110
7.2 SNA’s with symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 118
viiviii1 Introduction
In the early 1980’s, Herman [1] and Grebogi et al. [2] independently discovered the
existence ofstrangenon-chaotic attractors (SNA’s)inquasiperiodicallyforced (qpf)
systems. These objects combine a fractal geometry with non-chaotic dynamics,
a combination which is most unusual and has only been observed in a few very
particular cases before. In quasiperiodically forced systems however, they seem to
occur quite frequently and even over whole intervals in parameter space ([2]–[4]).
As a novel phenomenon this provoked considerable interest in theoretical physics,
and in the sequel a large number of numerical studies explored the surprisinglyrich
dynamics of these relatively simple maps. In particular, the widespread existence
of SNA’s was confirmed both numerically (see [5]–[18], just to give a selection) and
even experimentally ([19]–[21]). Further, it turned out that SNA play an important
role in the bifurcations of invariant circles (e.g. [4],[13],[17]). The studied systems
were either discrete time maps, such as the qpf logistic maps ([9],[12],[17]) and the
qpf Arnold circle map ([4],[8],[11],[13],), or skew product flows which are forced
at two or more incommensurate frequencies. Especially the latter underline the
significance of qpf systems for understanding real-world phenomena, as most of
them were derived from models for different physical systems (e.g. quasiperiodically
driven damped pendula and Josephson junctions ([5],[6],[7]) or Duffing oscillators
([20])). Their Poincar´e maps again give rise to discrete-time qpf systems, on which
the present work will focus.
However, despite all efforts there are still only very few mathematically rigor-
ous results about the subject, (with the exception of qpf Schrodinger cocycles, see¨
below). There are some results concerning the regularity of invariant curves ([22],
see also [23]), and there has been some recent progress in carrying over basic results
from one-dimensional dynamics ([24]–[26]). But so far, the two original examples
remain the only ones for which the existence of SNA’s has been proved rigorously.
In both cases, the arguments used were highly specific for the respective class of
mapsanddidnotallow formuchfurthergeneralization, nordidtheygiveverymuch
insightintothegeometrical andstructuralpropertiesoftheattractors. Thesystems
Herman studied were matrix cocycles, with quasiperiodic Schro¨dinger cocycles as
a special case. The linear structure of these systems and their intimate relation
to Schro¨dinger operators with quasiperiodic potentials made it possible to use a
fruitful blend of techniques from operator theory, dynamical systems and complex
analysis, such that by now the mathematical theory is well-developed and deep re-
sults have been obtained. (As we do not want to go into detail here, we just refer
to [27] and [28] for some recent results and further reference.) However, as soon as
the particular class of matrix cocycles is left it seems hard to recover any of these
arguments. On the other hand, for the so-called ‘pinched skew products’ introduced
in [2], establishing the existence of SNA is surprisingly simple and straightforward
(we will briefly sketch this in Section 2; see [3] for a rigorous treatment and [29]
for some slight generalizations). But one has to say that these maps were defined
especially for this purpose and are rather artificial in some aspects. For example, it
1is crucial for the argument that there exists one fibre which is mapped to a single
point. But this means that the maps are not invertible and can therefore not be
the Poincar´e maps of any flow.
In this work we will concentrate on one particular type of SNA, namely ‘strip-
like’ ones, which occur in saddle-node and pitchfork bifurcations of invariant circles
(see Figure 1.1, for a more precise formulation consider the definition of invariant
strips in [24] and [26]). In a saddle-node bifurcation, a stable and an unstable
invariant circle approach each other until they finallycollide and then vanish. How-
ever, there are two different possibilities. In the first case, which is similar to the
one-dimensional one, the two circles merge together to form one single and neutral
invariant circle at thebifurcationpoint. Butit mayalso happenthatthetwo circles
approach each other only on a dense, but (Lebesgue) measure zero set of points.
In this case, instead of a single invariant circle there exists a strange non-chaotic
attractor-repellor pair at the bifurcation point. Attractor and repellor are interwo-
veninsuchaway, thattheyhavethesametopological closure. Thisparticularroute
for the creation of SNA’s has been observed quite frequently ([11],[13],[14],[18], see
also [9]) and was named ‘non-smooth saddle-node bifurcation’ or ‘creation of SNA
via torus collision’. The only rigorous description of this process was also given
by Herman in [1]. In a similar way, the simultaneous collision of two stable and
one unstable invariant circle may lead to the creation of two SNA’s embracing one
strange non-chaotic repellor (e.g. [4], [15]).
1.5(a) (b)
2
1
1
0.5
0 0
−0.5−1
−1
−2
−1.5
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Figure 1.1: Two different types of strange non-chaotic attractors: The left picture shows a
‘strip-like’SNA in the system(θ,x)7!(θ+ω,tanh(5x)+1.2015sin(2πθ)). The topological
closure of this object is bounded above and below by semi-continuous invariant graphs
(compare (1.3)). This is the type of SNA’s that will be studied in the present work. The
right picture shows a different type that occurs for example in the critical Harper map
(Equation (1.10) with λ = 2 and E = 0; more details can be found in [30]), where no such
boundaries exist. In both cases ω is the golden mean.
The crucial observation which starts our investigation here is the fact that the
invariant circles in anon-smooth bifurcation donotapproach each other arbitrarily.
Instead, there is a very distinctive pattern for their behaviour, which we choose to
call ‘exponential evolution of peaks’. This will be exploited in two different ways.
2

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