Root-mean-square deviation of atomic positions

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Zubov's method is a technique for computing the basin of attraction for a set of ordinary differential equations (a dynamical system). The domain of attraction is the set {x:v(x)<1}, where v(x) is the solution to a partial differential equation known as the Zubov equation. 'Zubov's method' can be used in a number of ways.

Zubov's theorem states that:

If x=f(x),t is an ordinary differential equation in n with f(0)=0, a set A containing 0 in its interior is the domain of attraction of zero if and only if there exist continuous functions v,h such that:

If f is continuously differentiable, then the differential equation has at most one continuously differentiable solution satisfying v(0)=0.

References

Vladimir Ivanovich Zubov, Methods of A.M. Lyapunov and their application, Izdatel'stvo Leningradskogo Universiteta, 1961. (Translated by the United States Atomic Energy Commission, 1964.) ASIN B0007F2CDQ.

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