Gradient-descent methods for parameter estimation in chaotic systems

The rich nonlinear dynamics of chaos allows to model a broad variety of systems, including complex biological ones. The system of interest is usually observed through some time series and the modelization problem consists of adjusting the parameters of a model chaotic system until its dynamics is matched to the reference time series. In this paper, we describe a general methodology to adaptively select the values of the model parameters. Specifically, we assume that the observed time series are originated by a primary chaotic system with unknown parameters and we use it to drive a secondary chaotic system, so that both systems be coupled. The parameters of the secondary system are adaptively optimized (by a gradient-descent optimization of a suitable cost function) to make it follow the dynamics of the primary system. In this way, the secondary parameters are interpreted as estimates of the primary ones. We illustrate the application of the method by jointly estimating the complete parameter vector of a Lorenz system.