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Extra resources for Advances in mathematics research, Vol.10
Theorem 7 asserts that unstable modes may exist only in subset M0n − Hn . Thus, exponential growth (as well as exponential decay) of perturbations is impossible in the sets M−n , M+n and Hn , and hence, all the modes of these three sets are neutral. In practice, the probability of the fulfilment of instability condition (68) is quite small, because the set M0n is only the boundary hypersurface separating the basic sets M−n and M+n . Therefore it is difficult to expect that real atmospheric disturbances will frequently meet the precise instability condition (68).
In this way, RHB describes the excitation/contraction dynamics for the ith equivalent muscle–joint actuator, using the simple impulse–hyperbolic product relation Fimuscle (t, x, x) ˙ = Fiimp × FiHill . Now, for the purpose of biomedical engineering and rehabilitation, RHB has developed the so–called hybrid rotational actuator. C. motor drives, as used in robotics [65, 43, 28] Fkrobo = ik (t) − Jk x ¨k (t) − Bk x˙ k (t), ∞−Dimensional Cerebellar Controller for Realistic Human Biodynamics 33 with lk ik (t) + Rk ik (t) + Ck x˙ k (t) = uk (t), where k = 1, .
I. Zubov, Methods of A. M. Liapunov and Their Application. Leningradsky University Press, Leningrad (1957) (English translation - 1964, Noordhoff, Groningen).  Yu. N. Skiba, J. Adem, On the linear stability study of zonal incompressible flows on a sphere. Numer. Methods Partial Differential Equations 14 (1998), 649–665.  A. A. Andronov, A. A. Vitt, S. E. Khaikin, Theory of Oscillators. Fiz. Mat. , Moscow, 1959 (Russian) (translated by F. , New York, 1987).  F. Verhulst, Nonlinear Differential Equations and Dynamical Systems.