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Engel-M-2018-Phd-Thesis.Pdf Imperial College London, Department of Mathematics Local phenomena in random dynamical systems: bifurcations, synchronisation, and quasi-stationary dynamics A thesis presented for the degree of Doctor of Philosophy of Imperial College London by Maximilian Engel Supervisors: Prof. Jeroen S. W. Lamb, Dr. Martin Rasmussen January 2018 Declaration I declare that this thesis is entirely my own work and that where any material could be construed as the work of others, it is fully cited and referenced, and/or with appropriate acknowledgement given. Maximilian Engel 2 Copyright The copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or redistribution, researchers must make clear to others the licence terms of this work. 3 Acknowledgements I would like to express my gratitude to the following people who have supported me in accomplishing the present work. First of all, I would like to thank my supervisors Jeroen Lamb and Martin Rasmussen for the numerous passionate discussions we have had throughout the past three years, for encouraging me in difficult times and being enthusiastic but still critical in times of apparent breakthroughs. I am extremely grateful for their guidance, support and trust in me as a PhD student. In particular, I have to emphasize their willingness to enable many inspiring research connections and visits to different parts of the world. I would like to thank Julian Newman and Doan Thai Son for many fruitful discussions about the theory of random dynamical systems. Furthermore, I would like to thank my great friend and office colleague Nikolas N¨usken for the many mathematical debates, valu- able comments and suggestions. My thanks go also to my friends Elisa Leroy, Waldemar Marz and Guillermo Olic´onM´endezfor their helpful comments. Additionally, I would like to thank Alexis Arnaudon, Maik Gr¨oger,Darryl Holm, Aleksandar Mijatovic, Grigorios Pavliotis and Sebastian Wieczorek for useful discussions. I also thank everyone in the dynamical systems group for providing an enriching re- search environment. My thanks go to my fellow PhD students for the unforgettable time in London, especially my office mates in room 6M09. I thank Lai-Sang Young for inspiring discussions and for being so welcoming during my time visiting Courant Institute at NYU. A big thanks also to Hans Crauel for the interesting discussions and his hospitality in Frankfurt and in Iran. Finally, I give my warmest thanks to my wonderful wife Ginka who is always supportive and gives me confidence and strength. My warmest thanks go also to my parents Susanne and Rolf, and their spouses Guido and Sabine, my sisters Charlotte, Josephine and Judith and my brother Julian, for all their support and encouragement. 4 Publications The material in Sections 3.1, 3.2, 3.3 and 3.4 of this thesis is joint work with Doan Thai Son, Jeroen Lamb and Martin Rasmussen, and has been submitted for publication in the paper [38]. The material in Chapter 4 is joint work with Jeroen Lamb and Martin Rasmussen, and has been submitted for publication in the paper [45]. A paper comprising the main results in Chapter 6 is currently in preparation. 5 Abstract We consider several related topics in the bifurcation theory of random dynamical systems: synchronisation by noise, noise-induced chaos, qualitative changes of finite-time behaviour and stability of systems surviving in a bounded domain. Firstly, we study the dynamics of a two-dimensional ordinary differential equation ex- hibiting a Hopf bifurcation subject to additive white noise. Depending on the determin- istic Hopf bifurcation parameter and a phase-amplitude coupling parameter called shear, three dynamical phases can be identified: a random attractor with uniform synchronisa- tion of trajectories, a random attractor with non-uniform synchronisation of trajectories and a random attractor without synchronisation of trajectories. We prove the existence of the first two phases which both exhibit a random equilibrium with negative top Lyapunov exponent but differ in terms of finite-time and uniform stability properties. We provide numerical results in support of the existence of the third phase which is characterised by a so-called random strange attractor with positive top Lyapunov exponent implying chaotic behaviour. Secondly, we reduce the model of the Hopf bifurcation to its linear components and study the dynamics of a stochastically driven limit cycle on the cylinder. In this case, we can prove the existence of a bifurcation from an attractive random equilibrium to a random strange attractor, indicated by a change of sign of the top Lyapunov exponent. By establishing the existence of a random strange attractor for a model with white noise, we extend results by Qiudong Wang and Lai-Sang Young on periodically kicked limit cycles to the stochastic context. Furthermore, we discuss a characterisation of the invariant measures associated with the random strange attractor and deduce positive measure- theoretic entropy for the random system. Finally, we study the bifurcation behaviour of unbounded noise systems in bounded domains, exhibiting the local character of random bifurcations which are usually hidden in the global analysis. The systems are analysed by being conditioned to trajectories which do not hit the boundary of the domain for asymptotically long times. The notion of a stationary distribution is replaced by the concept of a quasi-stationary distribution and the average limiting behaviour can be described by a so-called quasi-ergodic distribution. Based on the well-explored stochastic analysis of such distributions, we develop a dynami- cal stability theory for stochastic differential equations within this context. Most notably, 6 7 we define conditioned average Lyapunov exponents and demonstrate that they measure the typical stability behaviour of surviving trajectories. We analyse typical examples of random bifurcation theory within this environment, in particular the Hopf bifurcation with additive noise, with reference to whom we also study (numerically) a spectrum of conditioned Lyapunov exponents. Furthermore, we discuss relations to dynamical systems with holes. Contents Declaration 2 Copyright 3 Acknowledgements 4 Publications 5 Abstract 6 List of Figures 11 1 Introduction 12 2 Random dynamical systems 30 2.1 Invariant measures . 32 2.2 Random attractors . 33 2.3 Spectral theories of random dynamical systems . 35 3 Hopf bifurcation with additive noise 42 3.1 Statement of the main results . 43 3.1.1 Generation of a random dynamical system with a random attractor 44 3.1.2 Negativity of top Lyapunov exponent and synchronisation . 45 3.1.3 Qualitative changes in the finite-time behaviour indicated by the dichotomy spectrum . 47 3.2 Generation of RDS and existence of a random attractor . 48 3.3 Synchronisation . 53 3.3.1 Negativity of the sum of the Lyapunov exponents . 53 3.3.2 Negative top Lyapunov exponent implies synchronisation . 56 3.3.3 Small shear implies synchronisation . 59 3.4 Random Hopf bifurcation . 61 3.4.1 Bifurcation for small shear . 62 3.4.2 Shear intensity as bifurcation parameter . 68 8 CONTENTS 9 3.5 Chaos via shear: approaches to proving the conjecture . 71 3.5.1 Formula for the top Lyapunov exponent . 71 3.5.2 Studying the stationary Fokker-Planck equation . 73 3.5.3 Using the finite element method . 76 3.6 Duffing-van der Pol equation and stochastic Brusselator . 78 3.7 Discussion . 81 4 Bifurcation of a stochastically driven limit cycle 82 4.1 Analysis of the top Lyapunov exponent . 85 4.1.1 The model as a random dynamical system . 85 4.1.2 The Furstenberg{Khasminskii formula . 88 4.1.3 Explicit formula for the top Lyapunov exponent . 90 4.2 The bifurcation result . 95 4.2.1 Bifurcation from negative to positive Lyapunov exponent . 95 4.2.2 Numerical exploitation of the results . 99 4.2.3 Robustness of the result . 101 4.3 Discussion . 103 5 Ergodic theory of chaotic random attractors 104 5.1 Positive Lyapunov exponents imply atomless invariant measures . 105 5.2 Pesin's formula . 109 5.3 SRB measures . 116 5.4 Discussion . 118 6 Quasi-stationary dynamics and bifurcations 120 6.1 Quasi-stationary and quasi-ergodic distributions . 122 6.1.1 General setting . 122 6.1.2 Stochastic differential equations . 130 6.2 Lyapunov exponents and local stability . 133 6.3 Local synchronisation for nonlinear systems . 148 6.4 Relation to pathwise random dynamics . 153 6.4.1 Conditionally invariant measures on the skew product space . 153 6.4.2 Two-sided time and relation to the survival process . 157 6.5 Exponential dichotomies and the dichotomy spectrum . 159 6.6 Bifurcations . 165 6.6.1 Pitchfork bifurcation of a killed process . 165 6.6.2 Hopf bifurcation of a killed process . 171 6.7 Discussion . 175 References 177 CONTENTS 10 Appendix 184 A.1 Multidimensional conversion formula from Stratonovich to It^ointegral . 184 A.2 The Fokker{Planck equation . 184 List of Figures 1.1 .......................................... 14 1.2 .......................................... 15 1.3 .......................................... 20 1.4 .......................................... 21 1.5 .......................................... 25 1.6 .......................................... 26 1.7 .........................................
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