Building dynamical models from data and prior knowledge: the case of the first period-doubling bifurcation.

This paper reviews some aspects of nonlinear model building from data with (gray box) and without (black box) prior knowledge. The model class is very important because it determines two aspects of the final model, namely (i) the type of nonlinearity that can be accurately approximated and (ii) the type of prior knowledge that can be taken into account. Such features are usually in conflict when it comes to choosing the model class.

Evaluation of dynamical models: dissipative synchronization and other techniques.

Some recent developments for the validation of nonlinear models built from data are reviewed. Besides giving an overall view of the field, a procedure is proposed and investigated based on the concept of dissipative synchronization between the data and the model, which is very useful in validating models that should reproduce dominant dynamical features, like bifurcations, of the original system. In order to assess the discriminating power of the procedure, four well-known benchmarks have been used: namely, Duffing-Ueda, Duffing-Holmes, and van der Pol oscillators, plus the Hénon map.

Piecewise affine models of chaotic attractors: the Rossler and Lorenz systems.

This paper proposes a procedure by which it is possible to synthesize Rossler [Phys. Lett. A 57, 397-398 (1976)] and Lorenz [J.

Using data-driven discrete-time models and the unscented Kalman filter to estimate unobserved variables of nonlinear systems.

This paper addresses the problem of state estimation for nonlinear systems by means of the unscented Kalman filter (UKF). Compared to the traditional extended Kalman filter, the UKF does not require the local linearization of the system equations used in the propagation stage. Important results using the UKF have been reported recently but in every case the system equations used by the filter were considered known.

Stability analysis of sleep apnea time series using identified models: a case study.

This paper investigates the use of identified nonlinear multivariable autonomous models in the classification of breathing patterns of a patient with sleep apnea. Details about the identification procedure are provided and the results reported for the case study at hand suggest that identified models could be useful in computer-based monitoring.

Constraining the topology of neural networks to ensure dynamics with symmetry properties.

This paper addresses the training of network models from data produced by systems with symmetry properties. It is argued that although general networks are global approximators, in practice some properties such as symmetry are very hard to learn from data. In order to guarantee that the final network will be symmetrical, constraints are developed for two types of models, namely, the multilayer perceptron (MLP) network and the radial basis function (RBF) network.