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45 (1988), 2789–2810. [22] Yu. N. Skiba, Liapunov Instability of the Rossby-Haurwitz Waves and Dipole Modons. Sov. J. Numer. Analysis & Math. Modelling 6 (1991), 515–534. [23] Yu. N. Skiba, Rossby-Haurwitz Wave Stability. Izvestiya, Atmos. Ocean. Physics 28 (1992), 388–394. [24] Yu. N. Skiba, Nonlinear and linear instability of the Rossby-Haurwitz wave. Journal of Mathematical Sciences 149 (2008), 1708–1725. [25] Yu. N. Skiba, On the normal mode instability of harmonic waves on a sphere. Geophys.

The growth rate of the unstable modes is estimated (Theorem 8), and the orthogonality of the normal mode amplitude to the RH wave is proved in the two inner products defined by (21) for s = 0 (the L2 -inner product) and s = 1 (the energy inner product). In the normal-mode instability study of a zonal RH wave (55), one can use the condition by Rayleigh-Kuo [8] related only with the structure of the basic flow. Unfortunately, the utility of Rayleigh-Kuo condition is rather limited, since it does not give any information about the growing perturbations.

2. Hamiltonian Action and Neural Path Integral Here, we propose a quantum–like adaptive control approach to modeling the ‘cerebellar mystery’. Corresponding to the affine Hamiltonian control function (5) we define the affine Hamiltonian control action, tout Saf f [q, p] = tin dτ pi q˙i − Haf f (q, p) . (8) From the affine Hamiltonian action (8) we further derive the associated expression for the neural phase–space path integral (in normal units), representing the cerebellar sensory– motor amplitude out|in , i i in qout , pout i |qin , pi = D[w, q, p] ei Saf f [q,p] = D[w, q, p] exp i (9) tout tin dτ pi q˙i − Haf f (q, p) , n with D[w, q, p] = wi (τ )dpi (τ )dq i (τ ) , 2π τ =1 where wi = wi (t) denote the cerebellar synaptic weights positioned along its neural pathways, being continuously updated using the Hebbian–like self–organizing learning rule (3).