Generalized one-parameter bifurcation diagram reconstruction using time series

In this paper, we look into the use of time series in reconstructing bifurcation diagrams of systems that cannot be modeled directly from first principles. In this reconstruction problem, time series at different unknown parameter values are available. These time series are then used to obtain a suitable family of nonlinear predictor functions which exhibits qualitatively similar bifurcations as the original system. To solve the reconstruction problem, we propose a generalized one-parameter reconstruction algorithm based on principal curves. We illustrate its feasibility numerically by reconstructing the bifurcation diagrams of the FitzHugh-Nagumo equations and the Lorenz equations with parameter values mostly in the fixed point regime. Using the proposed algorithm, we obtain a reconstruction which preserved the important features of the bifurcation diagrams of the original systems such as the Hopf bifurcations of the two systems, the limit cycle of the FitzHugh-Nagumo equations, and the symmetry of the fixed points of the Lorenz equations, among others. Moreover, the reconstructed systems also exhibit the same sequence of behavior as the original systems with respect to the changes in the bifurcation parameters. (C) 1998 Elsevier Science B.V.