Imperial College London Department of Mathematics Non-autonomous Random Dynamical Systems: Stochastic Approximation and Rate-Induced Tipping Michael Hartl Supervised by Prof Sebastian van Strien and Dr Martin Rasmussen A thesis presented for the degree of Doctor of Philosophy at Imperial College London. Declaration I certify that the research documented in this thesis is entirely my own. All ideas, theories and results that originate from the work of others are marked as such and fully referenced, and ideas originating from discussions with others are also acknowledged as such. 1 Copyright The copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non-Commercial No Derivatives license. 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 license terms of this work. 2 Abstract In this thesis we extend the foundational theory behind and areas of application of non- autonomous random dynamical systems beyond the current state of the art. We generalize results from autonomous random dynamical systems theory to a non-autonomous realm. We use this framework to study stochastic approximations from a different point of view. In particular we apply it to study noise induced transitions between equilibrium points and prove a bifurcation result. Then we turn our attention to parameter shift systems with bounded additive noise. We extend the framework of rate induced tipping in deterministic parameter shifts for this case and introduce tipping probabilities. Finally we perform a case study by developing and applying a numerical method for calculating tipping probabilities and examining the results thereof. 3 Acknowledgments I consider myself very lucky that I was included in the Innovative Training Network CRITICS1, funded entirely by the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No 643073. Special thanks to: Hassan Alkhayuon, Peter Ashwin, Michel Benaïm, Jens Bendel, Daniele Castellana, Andrew Clarke, Michael Collins, Peter Deiml, Gabriela Depetri, Maximilian En- gel, Gabriel Fuhrmann, Tobias Jäger, Gerhard Keller, Jeroen Lamb, Johannes Lohmann, Iacopo Longo, Usman Mirza, Karl Nyman, Christian Oertel, Guillermo Olicon, Greg Pavli- otis, Courtney Quinn, Martin Rasmussen, Flavia Remo, Chris Richley, Paul Ritchie, Pablo Rodríguez-Sánchez, Edmilson Roque, Anderson Santos, Cristina Sargent, Tobias Schwedes, Jakob Seifert, Jan Sieber, Leif Stolberg, Sebastian van Strien, Damian Smug, Kalle Timperi, Dmitry Turaev, and my family. Dedicated to Rudi Wutz. 1Critical Transitions in Complex Systems 4 Contents 1 Introduction6 1.1 Non-autonomous random dynamical systems...................6 1.2 Stochastic approximations..............................8 1.3 Rate-induced tipping................................. 10 1.4 Structure of the thesis................................ 13 1.5 Notation........................................ 15 2 Random dynamical systems 17 2.1 Background on autonomous RDS.......................... 17 2.1.1 Skew product systems............................ 17 2.1.2 Random invariant sets and measures.................... 18 2.1.3 Attractors and repellers........................... 22 2.2 Non-autonomous random dynamical systems................... 24 2.2.1 Non-autonomous sets and measures.................... 24 2.2.2 Attractors and repellers........................... 32 3 Stochastic Approximations 35 3.1 Setup and notation.................................. 35 3.2 Examples....................................... 36 3.2.1 Urn models and market competition.................... 36 3.2.2 Learning in games.............................. 37 3.2.3 Stochastic gradient descent......................... 40 3.3 The Limit Set Theorem............................... 41 3.4 Noise induced tipping in one dimension...................... 43 3.4.1 Preliminaries................................. 43 3.4.2 The invertible case.............................. 46 3.4.3 The non-invertible case........................... 57 3.4.4 Further remarks............................... 60 3.5 A bifurcation arising from touchpoints....................... 62 3.6 Conclusions and outlook............................... 66 4 Rate-induced tipping in random systems 68 4.1 Deterministic R-tipping............................... 68 4.2 Asymptotically autonomous NRDS......................... 69 4.3 Random parameter shift systems and rate induced tipping............ 75 4.4 An example...................................... 83 4.5 Further remarks and outlook............................ 94 A Hausdorff distance 104 B Conditional expectation, martingales and stopping times 105 C Chain recurrence and asymptotic pseudotrajectories 109 5 1 Introduction 1.1 Non-autonomous random dynamical systems The theory of dynamical systems dates back to at least the 17th century, when Isaac Newton and contemporaries began exploring the motion of celestial bodies, which would later develop into the field of classical mechanics. Very loosely speaking, a dynamical system is a space with a prescribed set of motion laws. The classical theory of dynamical systems assumes that those laws are known and fixed for all times. In many areas of applications however this setting is far too restrictive. This gives rise to the notion of non-autonomous dynamical systems. There have been efforts towards developing a unified theory of non-autonomous dynamical systems. A foundational account of this is the book [49] by Kloeden and Rasmussen, which also contains a comprehensive historical overview of the developments in this field. The basic idea is to describe non-autonomy by a base flow on some abstract parameter space which influences the observed dynamics on the phase space. Generally there are two main categories of such base flows. Firstly, one may think of situations where a system is subject to some driving force, which does not follow any known or predescribed motion laws. This can be simply due to the fact that a system is way too complex in for every detail of it to be included in the model, as it is the case for example in climate studies. Another possibility is that a system is subject to a truly random influence, like many economic or financial markets who are subject to human decisions. Models in those cases are often based on stochastic differential equations driven by a random process, often a Brownian motion; examples include [23, 34, 58]. In systems with multiple time scales, a fast variable can seem random from the viewpoint of a slow timescale, such that a stochastic differential equation describes the motion of slow variables fairly well, see e.g. [57] for a theoretical approach and [43] for practical examples. Discrete time random systems are also of interest, where points in a space are iterated according to a map depending on some random parameter, such as random circle diffeomorphisms [78] or iterated function systems [10, 42]. The abstract parameter space in this case is usually a measure space, equipped with a measure preserving transformation; a setup popularized by Arnold in his seminal book [3]. On the other hand, motion laws can explicitly depend on time, for example through a varying parameter. Examples include the FitzHugh–Naguro model for the firing of neurons in the brain (cf. [39]), where the parameter is the level of stimulation. An early theoretical account is [72, 71]. In this setting, the abstract parameter space is the real “time” line, and the dynamics are given by a right shift. In the case of a periodic time dependency one can also use the unit circle as the abstract parameter space. Despite the two types of systems being included within the umbrella term non-autonomous dynamical system, a big part of the existing literature treats them separately. Random dynamical systems with their measure preserving base transformations give rise to a lot of measure theoretic tools that are not available for the time shift case. We will present a few highlights in Chapter 2.1. On the other hand this limits the way a non-autonomous system can depend on the driving force; the right shift on the real line does not possess an invariant probability measure. Only very recently, mathematicians have turned their attention to simultaneously time and noise dependent systems. To the best of our knowledge, the non-autonomous random 6 dynamical system formalism as presented in this thesis appeared for the first time in [30] under the name partial-random dynamical system, as the solution operator of a stochastic partial differential equation with a time varying domain. The term “partial-random” refers to the fact that the non-autonomous aspect of the dynamical system is only partially due to a random influence. The authors introduce a concept of non-autonomous random attractors and prove the existence of a global attractor in their sense for an example class of systems. Their ideas were followed on by Wang [73, 74]. Other work, such as [1, 24, 74], is particularly concerned with existence of pullback attractors under periodic deterministic forcing. Cui and Langa [32] discuss various concepts of non-autonomous random attractors with compact deterministic component of the base flow. Remarkably this paper does not necessarily assume that their NRDS is induced by a non-autonomous SDE. A NRDS on