Metastability and transient dynamics from a dynamical systems perspective

joint work with

Kalel L. Rossi¹, Roberto C. Budzinski², Everton S. Medeiros^1,3, Bruno R.R. Boaretto⁴, Lyle Mueller², Gerrit Ansmann⁵, Klaus Lehnertz⁵

¹ICBM, Carl von Ossietzky University Oldenburg, Germany

²Western University Ontario, Canada

³State University of Sao Paulo, Rio Claro, Brazil

⁴Universidade Federal de Sao Paulo, Sao Jose dos Campos, Brazil

⁵University Bonn, Germany

Metastability, characterized typically as a variability of dynamical regimes over time, is a ubiquitous behavior in many systems, particularly in the earth system and in the brain. We examine this behavior using dynamical systems theory to provide a consistent and accessible framework for explaining different forms of metastability. We provide a general and clear definition of the behavior, which is often missing in the literature dealing with specific applications. Furthermore, we discuss many possible dynamical mechanisms that can generate metastability and show, how known experimental and theoretical studies can be explained by those various mechanisms. Specifically, we explain the differences between multistability and metastability, which is particularly important to distinguish the two main manifestations of metastability: (1) in multistable systems subject to noise and (2) in deterministic systems in which the dynamics involves different dynamical regimes with saddle character. Special emphasis is given to transient dynamics which is one of the key features of metastability since it focusses on the transitions from one metastable state to another. Thereby chaotic saddles play a particular role often being the backbone of such transitions. We explain the mechanisms how chaotic saddles appear, their dynamical and statistical properties and their possible role in the dynamics of metastable systems. To demonstrate the associated dynamics, we employ a network of relaxation oscillators and show how metastability can occur as a hopping dynamics between different space-time patterns like low-amplitude oscillations, nonlinear waves and extreme events.