Local Lyapunov Exponents : Predictability depends on where you are

The dominant Lyapunov exponent of a dynamical system measures the average rate at which nearby trajectories of a system diverge. Even though a positive exponent provides evidence for chaotic dynamics and upredictability, there may predictability of the time series over some finite time periods. In this paper one version of a local Lyapunov exponent is defined for a dynamic system perturbed by noise. These local Lyapunov exponents are used to detect the parts of the time series that may be more predictable than others. An examination of the fluctuations of the local Lyapunov exponents about the average exponent may provide important information in understanding the heterogeneity of a system. We will discuss the theoretical properties of these local exponents and propose a method of estimating these quantities using nonparametric regression. Also we will present an application of local exponents for interpreting surface pressure data.