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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.

We examine the use of synchronization as a mechanism for extracting parameter and state information from experimental systems. We focus on important aspects of this problem that have received little attention previously and we explore them using experiments and simulations with the chaotic Colpitts oscillator as an example system. We explore the impact of model imperfection on the ability to extract valid information from an experimental system. We compare two optimization methods: an initial value method and a constrained method. Each of these involves coupling the model equations to the experimental data in order to regularize the chaotic motions on the synchronization manifold. We explore both time-dependent and time-independent coupling and discuss the use of periodic impulse coupling. We also examine both optimized and fixed (or manually adjusted) coupling. For the case of an optimized time-dependent coupling function u(t) we find a robust structure which includes sharp peaks and intervals where it is zero. This structure shows a strong correlation with the location in phase space and appears to depend on noise, imperfections of the model, and the Lyapunov direction vectors. For time-independent coupling we find the counterintuitive result that often the optimal rms error in fitting the model to the data initially increases with coupling strength. Comparison of this result with that obtained using simulated data may provide one measure of model imperfection. The constrained method with time-dependent coupling appears to have benefits in synchronizing long data sets with minimal impact, while the initial value method with time-independent coupling tends to be substantially faster, more flexible, and easier to use. We also describe a method of coupling which is useful for sparse experimental data sets. Our use of the Colpitts oscillator allows us to explore in detail the case of a system with one positive Lyapunov exponent. The methods we explored are easily extended to driven systems such as neurons with time-dependent injected current. They are expected to be of value in nonchaotic systems as well. Software is available on request.

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