Tracing initial conditions, historical evolutionary path and parameters of chaotic processes from a short segment of scalar time series

An iterative optimization method is used to uncover unobserved initial state (t = 0), historical evolutionary path (t < t0) and parameters of a chaotic process from a segment of scalar time series (t0 ⩽ t ⩽ t1, t0 > 0). Given the system structure, we can precisely estimate the model parameters, recover the trajectory components unobserved, identify the state of all variables at the beginning (t = t0) of the observed time series, and trace the historical evolution of the system back to a long time interval (0 ⩽ t < t0). Chaotic time series of Lorenz system and Rössler system are utilized for illustration. The results show that the method is effective and tolerant to large mismatches between the guessed and actual values of the initial state and parameters.