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By Albert R. Baswell

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Mat. , Moscow, 1959 (Russian) (translated by F. , New York, 1987). [32] F. Verhulst, Nonlinear Differential Equations and Dynamical Systems. Springer, Berlin, 1996. [33] Yu. N. Skiba, Linear instability of ideal flows on a sphere. Mathematical Methods in the Applied Sciences 32 (2009), 284–306. Liapunov and Normal-Mode Instability of the Rossby-Haurwitz Wave 27 [34] L. A. Dikiy, Hydrodynamic Stability and Atmosphere Dynamics. Gidrometeoizdat, Leningrad, 1976 (in Russian). [35] B. Machenhauer, Spectral methods.

Using this principle, the Cerebellar Model Arithmetic Computer, or CMAC–neural network has been built [2, 48] and implemented in robotics [63], using trial-and-error learning to produce bursts of muscular activity for controlling robot arms. So, this ‘cerebellar control’ works for simple robotic problems, like non-redundant manipulation. However, comparing the number of its neurons (107 − 108 ), to the size of conventional neural networks (including CMAC), suggests that artificial neural nets cannot satisfactorily model the function of this sophisticated ‘super–bio–computer’, as its dimensionality is virtually infinite.

The RH waves form the main features of daily meteorological fields, and their stability study is of considerable interest for deeper understanding of the low-frequency variability of the atmosphere. Conservation laws for perturbations to the RH-wave are obtained and used to divide all the perturbations into four invariant non-intersecting sets: M−n , M+n , Hn and M0n − Hn (Theorem 2). The main conservation law asserts that any perturbation evolves in such a way that its energy and enstrophy decrease, remain constant or increase simultaneously (Theorem 1).

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