Estimation of parameters in nonlinear systems using balanced synchronization

Using synchronization between observations and a model with undetermined parameters is a natural way to complete the specification of the model. The quality of the synchronization, a cost function to be minimized, typically is evaluated by a least squares difference between the data time series and the model time series. If the coupling between the data and the model is too strong, this cost function is small for any data and any model and the variation of the cost function with respect to the parameters of interest is too small to permit selection of an optimal value of the parameters. We introduce two methods for balancing the competing desires of a small cost function for the quality of the synchronization and the numerical ability to determine parameters accurately. One method of "balanced" synchronization adds to the synchronization cost function a requirement that the conditional Lyapunov exponent of the model system, conditioned on being driven by the data remain negative but small in magnitude. The other method allows the coupling between the data and the model to vary in time according to the error in synchronization. This method succeeds because the data and the model exhibit generalized synchronization in the region where the parameters of the model are well determined. Examples are explored which have deterministic chaos with and without noise in the data signal.