In this thesis, we investigate a number of topics related to the concept of rate-induced tipping. We use the more general term rate-induced phenomena to refer to any dynamical behaviour which is caused by the rate at which an external parameter is varied over time.
First, we consider rate-induced phenomena in systems where the underlying autonomous dynamics have attracting limit cycles, extending the work of Alkhayuon and Ashwin [1]. We introduce a new concept of rate-induced phase sensitivity, where the rate at which a parameter changes can trigger finite time unpredictability in the dynamics. We also find that this new phenomenon interacts in an interesting way with the already established concept of rate-induced tipping.
Secondly, we investigate rate-induced phenomena in continuous time dynamical systems perturbed by bounded Markovian noise. We introduce a well-defined rate-induced tipping probability function and investigate its properties. In order to do this, we develop a general theory for studying continuous time systems perturbed by bounded noise using three complimentary perspectives: set-valued dynamical systems, Markov processes and random dynamical systems. We believe this work will also be useful in a much wider context than our application to rate-induced tipping.
Finally, we discuss the possibility of obtaining statistical early warning signs for rate-induced tipping from time series data. We identify conditions under which this is theoretically possible and discuss when it is reliably possible in practice.