Crisis in Time-Dependent Dynamical Systems

Area of Science:

• Physics
• Nonlinear Dynamics
• Complex Systems

Background:

• Dynamical systems often operate in fluctuating environments, leading to complex behaviors.
• Transitions and bifurcations in these systems remain incompletely understood, particularly crises.

Purpose of the Study:

• To investigate the occurrence and mechanisms of crises in nonautonomous dynamical systems.
• To derive and verify a scaling law for escape probability near crisis points.

Main Methods:

• Analysis of crises in low-dimensional dynamical systems.
• Derivation of the escape probability scaling law near the critical point.
• Numerical verification in various systems, including the Kuramoto model with inertia.

Main Results:

• Crises, characterized by phase space flooding, occur in nonautonomous systems.
• Escape probability near the critical point scales as exp[-α(lnδ)^{2}].
• The parameter α is dependent on the specific dynamical system model.

Conclusions:

• The derived scaling law provides a quantitative description of crisis dynamics in nonautonomous systems.
• The findings are relevant to understanding phenomena like the loss of stability in chimera states.