Fractals in rate-induced tipping

TL;DR

This paper explores how non-attracting fractal sets in dynamical systems influence rate-induced tipping, revealing fractal structures in parameter space that determine critical transition rates.

Contribution

It introduces a novel theory linking fractals in phase space to fractals in parameter space, affecting rate-induced tipping in nonlinear systems.

Findings

Fractals in phase space induce fractals in parameter space.

Fractal dimensions relate to tipping behavior.

Theory validated on three dynamical systems.

Abstract

When parameters of a dynamical system change sufficiently fast, critical transitions can take place even in the absence of bifurcations. This phenomenon is known as rate-induced tipping and has been reported in a variety of systems, from simple ordinary differential equations and maps to mathematical models in climate sciences and ecology. In most examples, the transition happens at a critical rate of parameter change, a rate-induced tipping point, and is associated with a simple unstable orbit (edge state). In this work, we show how this simple picture changes when non-attracting fractal sets exist in the autonomous system, a ubiquitous situation in non-linear dynamics. We show that these fractals in phase space induce fractals in parameter space, which control the rates and parameter changes that result in tipping. We explain how such rate-induced fractals appear and how the fractal dimensions of the different sets are related to each other. We illustrate our general theory in three paradigmatic systems: a piecewise linear one-dimensional map, the two-dimensional H\'enon map, and a forced pendulum.