On the application of adaptive control and phase synchronization in non-linear fluid dynamics

Abstract

In this paper we study certain aspects of non-linear dynamical systems arising in atmospheric physics from the viewpoint of their synchronization. Apart from phase synchronization, a new method using an adaptive controller is formulated. The model under investigation was originally obtained by Stenflo (Phys. Scr. 53 (1996) 83) in case of acoustic gravity waves and its chaotic properties were studied by Liu (Phys. Scr. 61 (2000) 526), Zhou (J. Math. Phys. 38 (1997) 5225) and Banerjee et al. (Phys. Scripta 63 (2001) 177). Here, we have shown how an adaptive controller can be constructed for the synchronization of two such systems by analysing the error equation and utilizing the properties of the corresponding Hurwitz matrix. As a result, suitable criteria are obtained for synchronization. On the other hand, the phase of such a chaotic time series is defined from the view of Hilbert transform and is shown that the technique of phase synchronization can be equally applied to such systems.