Dynamical parameter identification from a scalar time series

If a drive system with unknown parameters represents "reality" and the response system a "computational model," the unidirectional coupling can be used to change model parameters, as well as the model state, such that both systems synchronize with each other and model parameters coincide with their true values of "reality." Such a parameter identification method is called adaptive synchronization (also autosynchronization) method and is widely used in the literature. Because one usually cannot find proper parameter update rules by exploiting information obtained from only a scalar time series, parameter identification with adaptive synchronization from a scalar time series is not well understood and still remains challenging until now. In this paper we introduce a novel adaptive synchronization approach with an effective guidance parameter to update rule design. This method includes three steps: (i) finding some proper control signals such that the "computational model" synchronizes with the "real" system if no parameter mismatch exists (that is, both systems have identical parameters); (ii) designing parameter update rules in terms of a necessary condition for ensuring local synchronization; and (iii) determining the value for each parameter update rate for ensuring the local stability of autosynchronization manifold according to the conditional Lyapunov exponents method. The reliability of the suggested technique is illustrated with the Lorenz system and a unified chaotic model.