Critical transitions and perturbation growth directions

Area of Science:

• Dynamical Systems Theory
• Nonlinear Dynamics
• Chaos Theory

Background:

• Critical transitions signify abrupt shifts in dynamical systems.
• Identifying precursors to these transitions is crucial for prediction and control.
• Traditional indicators like variance may not always capture complex system dynamics.

Purpose of the Study:

• To explore quantifiers of chaos for identifying critical transitions.
• To investigate covariant Lyapunov vectors as predictive indicators.
• To develop novel methods for predicting critical transitions in complex systems.

Main Methods:

• Analysis of dynamical structure using quantifiers of chaos.
• Examination of growth rates and directions of covariant Lyapunov vectors.
• Application to models of fast-slow systems: FitzHugh-Nagumo oscillators, Josephson junctions, and Hindmarsh-Rose model.
• Development of a new method for estimating covariant Lyapunov vectors from past data.

Main Results:

• Tangencies between covariant Lyapunov vectors were identified as a common feature preceding critical transitions.
• Deviation from hyperbolic dynamics, indicated by these tangencies, successfully predicted critical transitions.
• Alignment of covariant Lyapunov vectors and changes in finite-time Lyapunov exponents outperformed variance-based indicators in noisy systems.
• Approximated covariant Lyapunov vectors proved effective for predicting critical transitions.

Conclusions:

• Covariant Lyapunov vectors offer robust indicators for critical transitions in complex dynamical systems.
• The proposed method for approximating covariant Lyapunov vectors enhances predictive capabilities, even without future trajectory knowledge.
• These findings advance the understanding and prediction of abrupt changes in diverse scientific fields.