Use of synchronization and adaptive control in parameter estimation from a time series

A technique is introduced for estimating unknown parameters when a time series of only one variable from a multivariate nonlinear dynamical system is given. The technique employs a combination of two different control methods, linear feedback for synchronizing system variables and adaptive control, and is based on dynamic minimization of synchronization error. The technique is shown to work even when the unknown parameters appear in the evolution equations of the variables other than the one for which the time series is given. The technique not only establishes that explicit detailed information about all system variables and parameters is contained in a scalar time series, but also gives a way to extract it out under suitable conditions. Illustrations are presented for Lorenz and R\"ossler systems and a nonlinear dynamical system in plasma physics. Also it is found that the technique is reasonably stable against noise in the given time series and the estimated value of a parameter fluctuates around the correct value, with the error of estimation growing linearly with the noise strength, for small noise.