# Rate-induced Tipping in Discrete-time Dynamical Systems

## TL;DR

This paper defines and analyzes rate-induced tipping (R-tipping) in discrete-time systems, establishing conditions for its occurrence, and contrasting behavior with continuous systems, with applications to the Ikeda map.

## Contribution

It introduces a formal definition of R-tipping in discrete maps, proves conditions for its occurrence, and compares discrete and continuous dynamics, including an application to the Ikeda map.

## Key findings

- Local pullback attractors stay near stable paths under slow parameter change.
- Fast parameter change can cause local pullback attractors to diverge, leading to R-tipping.
- Discretization can cause differences in R-tipping behavior between maps and flows.

## Abstract

We develop a definition of rate-induced tipping (R-tipping) in discrete-time dynamical systems (maps) and prove results giving conditions under which R-tipping will or will not happen. Specifically, we study (possibly non-invertible) maps with a time-varying parameter subject to a parameter shift. We show that each stable path has a unique associated solution (a local pullback attractor) which stays near the path for all negative time. When the parameter changes slowly, this local pullback attractor stays near the path for all time, but if the parameter changes quickly, the local pullback attractor may move away from the path in positive time; this is the phenomenon of R-tipping. We demonstrate that forward basin stability is an insufficient condition to prevent R-tipping in maps of any dimension but that forward inflowing stability is sufficient. Furthermore, we show that R-tipping will happen when there is a certain kind of forward basin instability, and we prove precisely what happens to the local pullback attractor as the rate of the parameter change approaches infinity. We then highlight the differences between discrete- and continuous-time systems by showing that when a map is obtained by discretizing a flow, the pullback attractors for the map and flow can have dramatically different behavior; there may be R-tipping in one system but not in the other. We finish by applying our results to demonstrate R-tipping in the 2-dimensional Ikeda map.