Fractals in rate-induced tipping

Area of Science:

• Non-linear dynamics
• Complex systems theory
• Mathematical modeling

Background:

• Critical transitions, like tipping points, can occur in dynamical systems when parameters change rapidly.
• Rate-induced tipping describes these transitions, often linked to unstable orbits (edge states).
• Non-attracting fractal sets are common in non-linear dynamics but their role in rate-induced tipping is less understood.

Purpose of the Study:

• To investigate how non-attracting fractal sets influence rate-induced tipping.
• To explore the emergence of fractal structures in parameter space due to phase-space fractals.
• To analyze the relationship between fractal dimensions and tipping dynamics.

Main Methods:

• Theoretical analysis of dynamical systems with non-attracting fractal sets.
• Mathematical modeling to derive fractal structures in parameter space.
• Numerical simulations using paradigmatic systems: 1D map, Hénon map, and forced pendulum.

Main Results:

• Non-attracting fractal sets in phase space induce fractal structures in parameter space.
• These parameter-space fractals dictate the critical rates and parameter changes leading to tipping.
• The fractal dimensions of phase-space and parameter-space sets are interconnected.

Conclusions:

• The presence of fractal sets fundamentally alters the understanding of rate-induced tipping.
• Fractal geometry plays a crucial role in determining tipping points and dynamics in complex systems.
• The findings offer a more comprehensive framework for studying critical transitions in non-linear systems.