Chaos in a Noisy World: New Methods and Evidence from Time-Series Analysis
Stephen Ellner, Peter Turchin
Chaos is usually regarded as a distinct alternative to random effects such as environmental fluctuations or external disturbances. We argue that strict separation between chaotic and stochastic dynamics in ecological systems is unnecessary and misleading, and we present a more comprehensive approach for systems subject to stochastic perturbations. The defining property of chaos is sensitive dependence on initial conditions. Chaotic systems are "noise amplifiers" that magnify perturbations; nonchaotic systems are "noise mufflers" that dampen perturbations. We also present statistical methods for detecting chaos in time-series data, based on using nonlinear time-series modeling to estimate the Lyapunov exponent λ, which gives the average rate at which perturbation effects grow (λ > 0) or decay (λ < 0). These methods allow for dynamic noise and can detect low-dimensional chaos with realistic amounts of data. Results for natural and laboratory populations span the entire range from noise-dominated and strongly stable dynamics through weak chaos. The distribution of estimated Lyapunov exponents is concentrated near the transition between stable and chaotic dynamics. In such borderline cases the fluctuations in short-term Lyapunov exponents may be more informative than the average exponent λ for characterizing nonlinear dynamics.