UvA-DARE (Digital Academic Repository) Bifurcation of random maps Zmarrou, H. Publication date 2008 Document Version Final published version Link to publication Citation for published version (APA): Zmarrou, H. (2008). Bifurcation of random maps. General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. UvA-DARE is a service provided by the library of the University of Amsterdam (https://dare.uva.nl) Download date:06 Oct 2021 Bifurcation of random maps Hicham Zmarrou Bifurcation of random maps Academisch proefschrift ter verkrijging van de graad van doctor aan de Universiteit van Amsterdam op gezag van de Rector Magnificus prof. dr. D.C. van den Boom ten overstaan van een door het college voor promoties ingestelde commissie, in het openbaar te verdedigen in de Agnietenkapel op vrijdag 28 maart 2008, te 14:00 uur door Hicham Zmarrou geboren te Oued Amlil, Marokko Promotiecommissie: Promotor: Prof. dr. A. Doelman Copromotor: dr. A.J. Homburg Overige leden: Prof. dr. S.J. van Strien Prof. dr. H.W. Broer Prof. dr. H.L.J. van der Maas Prof. dr. T. Koornwinder Prof. dr. J.J.O.O. Wiegerinck dr. H. Crauel dr. C. Diks Faculteit der Natuurwetenschappen, Wiskunde en Informatica THOMAS STIELTJES INSTITUTE FOR MATHEMATICS Preface This thesis is the final work of my Ph.D. research at the Korteweg-de Vries Institute for Mathematics, Universiteit van Amsterdam. It serves as documentation of my work during this period, which was performed from spring 2003 until autumn 2007. The research was funded by the Centraal Onderzoeksfonds (COF), of the Universiteit van Amsterdam. This thesis consists of six chapters. In the first chapter, I give a general introduction to random dynamical systems and a survey of the new scientific results presented in this thesis. The next four chapters contain the contents of two papers that have been accepted by international journals. These chapters cover various aspects of stochastic dynamical systems, such as transfer operator theory, stability theory, and random circle diffeomorphism theory. These papers are written jointly with my supervisor Ale Jan Homburg to whom I am extremely grateful for all the time and effort he invested in supervising my work. His support and encouragement built my confidence when I found things difficult, and motivated me to work to the best of my abilities. He has always been happy to answer my questions, even the stupid ones. Most importantly, he is an inspirational mathematician, whose passion for the subject I will always remember. I am also grateful to Arjen Doelman and Robin de Vilder for introducing me to the subject of this thesis and allowed me to get an initial grip on the subject. There are many other persons who deserve, and have, my gratitude. Everyone at the Korteweg-de Vries Institute for Mathematics helped to make my time at the institute enjoyable. They provided stimulating discussions, especially Michel, Ramon, Abdel, Said, Maarten, Geertje, Sjoerd and Ren´e. Other people in the mathematical community who I want to thank for interesting discussions include Guus Balkema and Peter Spreij. Finally, I owe my warmest thanks to my wife Souˆad and our two children Ouissal and Ayman. They have filled my life with their cheerful happiness. My wife Souˆad encouraged me throughout the course of my work. The presence of this invaluable ”team” allowed me to keep my feet always firmly on the ground. Contents Preface V 1 Introduction 1 1.1 Background ................................. 1 1.1.1 Dynamical systems: heuristics .................. 1 1.1.2 Random dynamical systems .................... 2 1.1.3 Equilibria, stationary and random invariant measures ..... 8 1.1.4 Stability and instability: bifurcation theory ........... 9 1.2 Thesis overview ............................... 16 2 Fundamentals of random maps 19 2.1 Some notations and definitions ...................... 19 2.2 Stationary measures of random maps and stability ........... 21 2.2.1 Random maps iterations ...................... 21 2.2.2 Families of random maps ..................... 26 2.3 The skew product approach ........................ 28 2.3.1 Some definitions and a few general facts ............. 29 2.4 Representations of discrete Markov processes .............. 36 3 The dynamics of one-dimensional random maps 41 3.1 Stability of stationary measures ...................... 42 3.2 Bifurcations ................................. 44 4 Random attractors for random circle diffeomorphisms 47 4.1 Random fixed points and random periodicity .............. 47 4.1.1 The random family admits a strictly ergodic stationary measure 47 4.1.2 The random family has cyclic intervals .............. 52 4.2 Random saddle node bifurcation ..................... 53 4.3 Convergence to a random attractor .................... 54 4.4 Corollaries for the skew products ..................... 56 5 Regularity of the stationary densities and applications 59 5.1 Smoothness of the stationary densities .................. 59 5.2 Stable random diffeomorphisms ...................... 62 5.3 Auxiliary parameters ............................ 64 5.4 Conditionally (almost) stationary measures ............... 65 VIII Contents 5.5 Escape times ................................ 69 5.6 Decay of correlations ............................ 73 5.7 Regularity of solutions of integral equations ............... 77 6 Case studies 79 6.1 Random circle diffeomorphisms ...................... 79 6.1.1 The standard circle map ...................... 79 6.1.2 The skew product system ..................... 83 6.2 Random unimodal maps .......................... 85 Bibliography 89 Nederlandse samenvatting 95 English summary 97 1 Introduction ”Probability theory has always generated its problems by its contact with other areas. There are very few problems that are generated by its own internal structure. This is partly because, one stripped of everything else, a probability space is essentially the unit interval with the Lebesgue mea- sure.” S.R.S. Varadhan, AMS Bulletin January (2003) 1.1 Background In this section we provide general background to the theory of random dynamical systems and their bifurcations, the object of this thesis. Our discussion is broader then strictly needed for this thesis, in the sense that we mention e.g. iterated function systems and stochastic differential equations which are not the central topic of this thesis. The issues discussed at length in the following chapters in this thesis are summarized in Section 1.2. 1.1.1 Dynamical systems: heuristics Dynamical systems have been studied for many years, in a variety of different ways, and with many different motivations and applications. The essence of a dynamical system is to tell how some state changes with time under the action of mathemat- ical laws. These are important things to study as they give information about the working of the real world, physical instruments. For example, swinging pendulums, hydrology, changing populations and genetics, climate forecast, financial markets and neural networks in the brain can all be treated using dynamical systems models. Some dynamical systems can be described as unforced systems - that is, they propa- gate in isolation from external influences. However, very few systems in the real world are genuinely isolated, and so it is often appropriate to describe dynamical systems as perturbed systems. In this case we have the dynamics of a system (perturbation) driving, or influencing, the dynamics of another. Regularly forced systems have been extensively studied for a long time. A simple example is a pendulum which experi- ences at regular intervals an external ’push’ of the same magnitude each time. By contrast, forcing with irregularly (random) frequencies has received comparatively less attention and is in general much less understood. It is thus widely accepted that understanding the effect of random perturbations on 2 1 Introduction nonlinear dynamical systems is an important challenge. Random perturbations can be small and irrelevant, or they can be so large as to overwhelm the underlying dy- namics. The middle ground is the most interesting, the perturbations can be small, yet contribute non-trivially to the overall dynamics, so that one must consider the interaction of the deterministic dynamics with the stochastic perturbation. This is the approach we take in this thesis. To respond to this need stochastic differential equations (SDE) were introduced by Itˆo at the beginning of the 1940s. He gives an explicit construction of diffusion processes which were introduced in the 1930s by Kolmogorov via partial differential equations and measures in their path spaces. For more than half a century it was customary x ω to consider solutions Xt ( ) at time t > 0 of an SDE for some fixed initial condi- tion, x, and the distribution of corresponding random