Reconstructing bifurcation diagrams of dynamical systems using measured time series

We present an algorithm for reconstructing the bifurcation structure of a dynamical system from time series. The method consists in finding a parameterized predictor function whose bifurcation structure is similar to that of the given system. Nonlinear autoregressive (NAR) models with polynomial terms are employed as predictor functions. The appropriate terms in the NAR models are obtained using a fast orthogonal search scheme. This scheme eliminates the problem of multiparameter optimization and makes the approach robust to noise. The algorithm is applied to the reconstruction of the bifurcation diagram (BD) of a neuron model from the simulated membrane potential waveforms. The reconstructed BD captures the different behaviors of the given system. Moreover, the algorithm also works well even for a limited number of time series.