Constraining safe and unsafe overshoots in saddle-node bifurcations.

Area of Science:

• Nonlinear Dynamics
• Complex Systems Analysis
• Mathematical Modeling

Background:

• Dynamical systems can exhibit bifurcations, critical points where system behavior changes qualitatively.
• Saddle-node bifurcations are characterized by the collision and annihilation of equilibrium points.
• Time-dependent parameters introduce complexities, especially when crossing bifurcation thresholds.

Purpose of the Study:

• To investigate the behavior of a dynamical system undergoing a saddle-node bifurcation with a time-dependent parameter featuring an overshoot.
• To determine the conditions under which parameter overshoots lead to stable recovery versus runaway trajectories (tipping).
• To establish new criteria for distinguishing safe from unsafe overshoots, particularly for larger amplitude events.

Main Methods:

• Analytical investigation of a dynamical system with an explicitly time-dependent parameter undergoing a saddle-node bifurcation.
• Focus on parameter overshoots that cross the bifurcation threshold for a finite duration and amplitude.
• Numerical simulations to substantiate analytical findings and explore various overshoot profiles.

Main Results:

• Confirmed the inverse square-root relationship (te∝R-1/2) for discriminating safe/unsafe overshoots in shallow overshoot regimes.
• Established a crossover to a new power law (te∝R-1) for larger overshoots, dependent on the parameter's asymptotic behavior.
• Demonstrated that overshoots with finite support follow te∝R-1, with exponents ranging from -1 to -1/2.

Conclusions:

• The study provides a refined understanding of tipping phenomena in systems with transiently varying parameters.
• The identified power-law relationships offer improved analytical tools for risk assessment in nonlinear dynamics.
• Results have implications for predicting and mitigating catastrophic shifts in climate, ecological, and engineering systems.