Extremes and Recurrence in Dynamical Systems

TL;DR

This book introduces the study of extreme events in dynamical systems, covering theoretical foundations, derivation of extreme value laws, and applications for understanding complex systems across sciences.

Contribution

It provides a comprehensive framework linking extreme events, stochastic processes, and dynamical systems, with explicit derivations and practical software tools.

Findings

Derivation of extreme value laws for selected systems

Relationship between hitting times and extreme events

Use of extreme events to infer system properties

Abstract

This book provides a comprehensive introduction for the study of extreme events in the context of dynamical systems. The introduction provides a broad overview of the interdisciplinary research area of extreme events, underlining its relevance for mathematics, natural sciences, engineering, and social sciences. After exploring the basics of the classical theory of extreme events, the book presents a careful examination of how a dynamical system can serve as a generator of stochastic processes, and explores in detail the relationship between the hitting and return time statistics of a dynamical system and the possibility of constructing extreme value laws for given observables. Explicit derivation of extreme value laws are then provided for selected dynamical systems. The book then discusses how extreme events can be used as probes for inferring fundamental dynamical and geometrical properties of a dynamical system and for providing a novel point of view in problems of physical and geophysical relevance. A final summary of the main results is then presented along with a discussion of open research questions. Finally, an appendix with software in Matlab programming language allows the readers to develop further understanding of the presented concepts.