Inverse-square law between time and amplitude for crossing tipping thresholds.

Area of Science:

• Dynamical systems theory
• Nonlinear dynamics
• Complex systems analysis

Background:

• Classical tipping points occur with slow parameter drift and positive feedback loops.
• Understanding system behavior after crossing a threshold is critical for stability analysis.

Purpose of the Study:

• To investigate criteria for avoiding tipping points when a system parameter reverses direction after crossing a threshold.
• To develop approximations for tipping probability in systems with stochastic forcing and reversing parameters.
• To validate theoretical findings using a high-dimensional model of the Indian summer monsoon.

Main Methods:

• Derivation of an inverse-square law relating parameter overshoot and time above threshold to tipping avoidance.
• Approximation of tipping probability for systems with time-varying parameters and stochastic forcing.
• Numerical simulations of a higher-dimensional system (Indian summer monsoon model) to verify theoretical predictions.

Main Results:

• A simple criterion, an inverse-square law, predicts tipping avoidance based on maximum parameter overshoot and duration above threshold.
• Approximations for tipping probability are derived for slowly changing parameters near a fold tipping point.
• Numerical results from the Indian summer monsoon model converge to the derived asymptotic expressions.
• The inverse-square law is observable in probability level curves under random disturbances.

Conclusions:

• The study provides a quantitative understanding of how to avoid tipping points in dynamical systems by reversing parameter drift.
• The findings offer insights into system resilience and predictability, particularly in the context of climate modeling.
• The inverse-square law serves as a valuable, observable indicator of system stability near tipping points.