Modeling parameter dependence from time series

Two approaches for modeling of parameter dependence of dynamical systems from time series are investigated and applied to different examples. For both methods it is assumed that a few time series are available that have been measured for different (known) parameter values of the underlying (experimental) dynamical system. The objective is to model the changing dynamics of the system as a function of its parameters and to use this for experimental bifurcation analysis. Using parametrized families the tasks of modeling the dynamics and of modeling its parameter dependence are separated. Technical difficulties that may occur with this approach are discussed and illustrated. An alternative are extended state space models where both modeling tasks are treated simultaneously. To obtain reliable models from a few time series only, ensembles of models are employed that show very good extrapolation and generalization properties.