Rate-Induced Tipping In NonAutonomous Systems

Lead Research Organisation: Imperial College London
Department Name: Mathematics

Abstract

This project aims at making crucial contributions to the study of critical transitions for complex systems. In particular, it addresses the phenomenon of rate-induced tipping for systems with nonautonomous dynamics and applications to climate. The research efforts will focus on identifying the mathematical mechanisms generating the critical transition classifying them in the context of nonautonomous bifurcation theory, and testing the efficacy of the resulting theory on non-autonomous climate models which are not treatable with the state-of-the-art mathematical tools. The applicant's pioneering results on the subject, his interdisciplinary background and contributions to the theory of non-autonomous dynamical systems together with the planned training in random and set-valued dynamical systems at the host institution and the support of the supervisor's world-leading expertise in the field put him in a unique position to effectively take on the challenge. The researcher will also consolidate his expertise in deterministic nonautonomous dynamics and bifurcation theory and complement it with new skills in random and set-valued dynamical systems. Thus, obtaining a highly competitive profile in the field of nonautonomous dynamical systems. On the other hand, he will bring his expertise on Carathéodory differential equations and rate-induced tipping to the host institution and, by mean of his collaborations with the University of Valladolid and Technical University Munich, contribute to the big international research network at Imperial College London. Finally the long-term impact of the action includes interdisciplinary key sectors like the investigation of the effect of the growth of emissions of greenhouse gasses on the naturally time-forced (thus nonautonomous) climate of the planet.
 
Description The research results of this award focus on the dynamical theory for nonautonomous dsystems, that is, systems whose evolution depends explicitly on some external time-dependent forcing. Four significant achievements are highlighted below:

1. Characterising and foreseeing Rate-Induced Transitions: A current challenge in dynamical systems is understanding how rapidly changing conditions can push a system towards a fundamentally different state-a phenomenon known as rate-induced tipping. Applications include the study of climate change as well as several critical transitions in nature and society. In our work, we have developed new tools to analyse when and how such tipping occurs in some general classes of scalar differential equations. By applying finite-time Lyapunov exponents, we have also shown that a sudden transition for a moving equilibrium from a desirable to un undesirable future behaviour can be fully characterised and even anticipated. Long-term implications include predicting shifts in ecosystems, financial markets, and engineered systems subject to external forcing.

2. Advancing Understanding of Nonautonomous Attractors and Nonautonomous Bifurcations: Traditional bifurcation theory describes how autonomous systems-which are not subjected to time-dependent forcing-change their behaviour when parameters are varied in a slow fashion. Unlike the autonomous theory, which is markedly developed and advanced in many aspects, the nonautonomous theory still presents fundamental open challenges. This attains both, the objects that bifurcate (the attractors), and the larger variety of possible bifurcation patterns which are possible already in low dimension. Our work has focused on bifurcations of hyperbolic solutions, i.e. globally defined bounded trajectories that have special properties of local exponential attractivity. We have completely characterised the dynamical scenarios and bifurcations of scalar concave differential equations as well as identified new bifurcations of saddle hyperbolic solutions with a special behaviour in the past and in the future, so-called homoclinic solutions.

3. Extending Singular Perturbation Theory to Nonautonomous Systems: The analysis of singularly perturbed problems and especially fast-slow systems can rely on a rich literature, provided the systems is autonomous or it features at most a dependence on the slow timescale. However, the available results for fast-slow systems with an explicit dependence on the fast timescale are few and hard to apply, substantially hindering the understanding of fundamental problems climate science, neuroscience and epidemiology, to name a few. Our results shade a new light on some fundamental results by Artstein and Vigodner which, in turn, attempted a first generalisation of the perturbation results for autonomous systems originally due to Tichonov. Importantly, we showed that the weak convergence of measures provided by Artstein and Vigodner can be upgraded to a convergence of solutions to a slowly changing family of attractors where the parameter is now varied on a slow timescale. As a consequence, a more fitting dynamical understanding of the transient and asymptotic qualitative behaviour of a fast-slow system becomes possible.
Exploitation Route Academic Impact: Our achievements provide a clear pathway towards a comprehensive theory for tipping points in time-dependent systems via nonautonomous dynamical systems theory. In view of the very mild underlying assumptions on regularity and recurrency, our results are applicable to a vast number of applications in theoretical climate science, population dynamics, mathematical ecology and engineering. While our results are mostly developed for low-dimensional systems, they are a proof of concept for the further extension of the theory to more complicated and realistic problems.
Our achievements also opened new possible lines of research. For example, we are currently investigating the role that bounded and unbounded noise play in the transitions of time-dependent systems, based on the bifurcation results we have achieved in the deterministic case.

Non-Academic Impact: Our results allow a rigorous understanding of some types tipping points which are not possible for autonomous systems. Although our work is mostly constrained to low-dimensional systems, where an analytical approach is still feasible, qualitatively comparable features and behaviours have already been identified in exceedingly more complicated models in fields as diverse as climate science, economics and ecology, to name a few. Hence, our work offers a concrete benchmark for the rigorous development of new tools for risk assessment and management strategies of more realistic problems which, however, are not amenable to analytical treatment.
Sectors Environment