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**On the application of adaptive control and phase synchronization in non-linear fluid dynamics.**
*(English)*
Zbl 1225.76137

Summary: In this paper we study certain aspects of nonlinear 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 L. Stenflo [Phys. Scr. 53, 83–84 (1996)] in case of acoustic gravity waves and its chaotic properties were studied by Z. Liu [Phys. Scr. 61, 526 (2000)], C. Zhou et al. [J. Math. Phys. 38, No. 10, 5225–5239 (1997; Zbl 0897.76038)] and S. Banerjee et al. [Phys. Scr. 63, 177 (2001)]. 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.

### MSC:

76E30 | Nonlinear effects in hydrodynamic stability |

37N10 | Dynamical systems in fluid mechanics, oceanography and meteorology |

93C40 | Adaptive control/observation systems |

65P20 | Numerical chaos |

37M10 | Time series analysis of dynamical systems |

86A10 | Meteorology and atmospheric physics |