Global continuation as a complement to traditional continuation and bifurcation analysis

Abstract

Multistable dynamical systems are ever-prevalent, used to model for example ecosystems, power grids, climate elements, neurons, and more. When perturbed, such systems may ``tip'' from one state of operation to another, often with abrupt, irreversible, and high-impact consequences in each context. Traditionally, these systems are analysed via bifurcation diagrams, the result of a process we refer to as \emph{local continuation}, as it only captures the linear (local) system response to infinitesimal perturbations. Local continuation requires substantial expertise, constant interventions, and may yield inaccurate assessment of the system's response to large perturbations that is crucial for tipping analysis. To address some inherent challenges of local continuation and to provide fundamentally new information during a continuation, this paper introduces \emph{global continuation} as a complement suitable for the study of multistability, critical transitions and real-world-oriented applications. Global continuation finds and continues in parallel (practically) all system attractors and their response to finite perturbations by synthesising information from the whole state space, while placing a focus on the qualities or observables of a dynamical system that the practitioner cares about in context. Global continuation does not require deep expertise and is effortless to use and troubleshoot, making it attractive to applied scientists from different disciplines. We highlight several unique advantages that allow global continuation to complement the status quo and exemplify them through a plethora of representative examples. Global continuation is also implemented as open source software in DynamicalSystems.jl, enhancing its accessibility.