Polynomial Lyapunov exponents: A practical scaling tool for weak instability in Rn

Area of Science:

• Dynamical Systems and Chaos Theory
• Nonlinear Dynamics
• Mathematical Physics

Background:

• Classical Lyapunov exponents quantify exponential divergence of trajectories, indicating system stability.
• Weak instability, characterized by subexponential perturbation growth and vanishing maximal Lyapunov exponents, poses challenges for traditional analysis.
• Existing methods struggle to distinguish diverse long-time behaviors in systems with weak instability.

Purpose of the Study:

• Introduce the polynomial Lyapunov exponent (PLE) as a novel metric for systems with vanishing classical Lyapunov exponents.
• Develop a scaling-based quantity to capture power-law separation of trajectories.
• Provide a practical tool for analyzing weak instability where conventional methods fail.

Main Methods:

• Define the polynomial Lyapunov exponent (PLE) based on power-law scaling of trajectory separation.
• Establish theoretical properties, including norm invariance and a comparison principle.
• Develop a numerical estimator using linear regression on log-transformed data.
• Apply the PLE to benchmark systems like nonnormal linear flows and marginally stable dynamics.

Main Results:

• The polynomial Lyapunov exponent (PLE) effectively quantifies polynomial growth rates in weakly unstable systems.
• The PLE provides an interpretable measure distinguishing different long-time behaviors where classical Lyapunov exponents are inconclusive.
• Numerical experiments demonstrate the PLE's robustness to perturbation magnitude and direction.

Conclusions:

• The polynomial Lyapunov exponent (PLE) is a valuable tool for analyzing systems exhibiting weak instability.
• This method offers a practical and interpretable approach to characterizing polynomial scaling in trajectory separation.
• The PLE complements classical Lyapunov exponents, extending stability analysis to previously intractable regimes.