New Nonlinear Adaptive Filters with Applications to Chaos

Abstract

We present a new approach to nonlinear adaptive filtering based on Successive Linearization. Our approach provides a simple, modular and unified implementation for a broad class of polynomial filters. We refer to this implementation as the layered structure and note that it offers substantial computational efficiency over previous methods. A new class of Polynomial Autoregressive filters is introduced which can model limit cycle and chaotic dynamics. Existing geometric methods for modeling and characterizing chaotic processes suffer from several drawbacks. They require a huge number of data points to reconstruct the attractor geometry and performance is severely limited by noisy experimental measurements. We present a new method for processing chaotic signals using nonlinear adaptive filters. We demonstrate the modeling, prediction and filtering of these signals. We also show how the prediction error growth rate can be used to estimate the effective Lyapunov exponent of the chaotic map. Our approach requires orders of magnitude fewer data points and is robust to noise in the experimental data. Although reconstruction of the attractor geometry is unnecessary, the adaptive filter contains most of the geometric information.