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

Classical Lyapunov exponents characterize stability through exponential separation of nearby trajectories. However, many dynamical systems exhibit weak instability, in which perturbations grow subexponentially and the maximal Lyapunov exponent vanishes. In such cases, exponential diagnostics are unable to distinguish qualitatively different long-time behaviors. We introduce the polynomial Lyapunov exponent (PLE) as a scaling-based quantity that captures power-law separation of trajectories in systems where λ = 0. We establish basic properties of the PLE, including invariance under equivalent norms and a two-sided comparison principle that relates a limiting PLE to polynomial upper and lower bounds on perturbation growth. We also present a simple numerical estimator based on linear regression in logarithmic time and apply it to benchmark systems exhibiting weak instability, including nonnormal linear flows and marginally stable dynamics. These examples show that the PLE provides an interpretable and practical measure of polynomial growth rates in settings where classical Lyapunov exponents are inconclusive. The approach is tailored to polynomial scaling and is not intended to capture more general subexponential behavior. Numerical experiments demonstrate robustness with respect to perturbation magnitude and direction.