Fractals in rate-induced tipping.

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 explore 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 Hénon map, and a forced pendulum.