[Paper Review] Fractals in rate-induced tipping

Jason Qianchuan Wang, Yi Zheng+1 more|arXiv (Cornell University)|Jan 23, 2026

Ecosystem dynamics and resilience0 citations

TL;DR

The paper investigates how non-attracting fractal sets in autonomous dynamics create fractal structures in rate-induced tipping, showing that tipping behavior becomes highly sensitive and governed by fractal dimensions linking saddles, edges, and tipping boundaries. It demonstrates this via three paradigmatic systems and provides a general mechanism connecting fractal dimensions.

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 Hénon map, and a forced pendulum.

Motivation & Objective

• Motivate and formalize rate-induced tipping (R-tipping) within non-autonomous dynamics with time-varying parameters.
• Show how non-attracting fractal sets (saddles) influence tipping by creating fractal structures in parameter space.
• Establish relationships between fractal dimensions of saddles, edge states, and tipping boundaries.
• Illustrate the general theory through three canonical dynamical systems and discuss implications for R-tipping theory.

Proposed method

• Define R-tipping within a discrete-time framework with autonomous map F and time-dependent parameter path Lambda(s).
• Introduce tipping function phi that classifies trajectories as tracking or tipping based on long-term behavior.
• Analyze systems with fractal saddles to reveal fractal dependence of tipping on rate and parameter path.
• Provide numerical demonstrations on a piecewise-linear map, the two-dimensional Hénon map, and a forced pendulum.
• Discuss the co-dimension of fractal boundaries and derive connections between fractal dimensions of edge states, saddles, and tipping boundaries.

Experimental results

Research questions

• RQ1 How do fractal (non-attracting) saddles influence rate-induced tipping in non-autonomous dynamics?
• RQ2 What is the relationship between fractal dimensions of saddles, edge states, and the boundary between tracking and tipping?
• RQ3 Can fractal tipping be observed in simple paradigmatic systems and how does it depend on rate and parameter-path choices?
• RQ4 How do these fractal mechanisms extend the understanding of R-tipping beyond traditional single-edge-state scenarios?

Key findings

• Fractals in phase space induce fractals in parameter space that govern tipping rates and parameter changes.
• The fractal co-dimension of the track/tip boundary can match the fractal co-dimension of the saddles, suggesting a linked dimensional structure.
• Fractal Saddles induce transient chaotic dynamics that create extreme sensitivity of tracking versus tipping to rate and protocol path.
• Numerical evidence is provided in three systems: a piecewise-linear one-dimensional map, the two-dimensional Hénon map, and a forced pendulum.
• The paper develops a general mechanism explaining how fractal dimensions relate across the dynamical objects involved (edge states, saddles, and tipping boundaries).