Zubov's method explained

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.^[1] Zubov's method can be used in a number of ways.

Statement

Zubov's theorem states that:

If

x' = f(x), t \in \R is an ordinary differential equation in \R ^{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:v( 0 ) = h( 0 ) = 0 , 0 < v(x) < 1 for x \in A \setminus \{ 0 \} , h > 0 on \R ^{n} \setminus \{ 0 \}

• for every

\gamma _{2} > 0 there exist \gamma _{1} > 0 , \alpha _{1} > 0 such that v(x) > \gamma _{1,} h(x) > \alpha _{1} , if ||x||>\gamma _{2} v(x _{n)} → 1 for x _{n} → \partial A or ||x _{n||} → inf ty \nabla v(x) ⋅ f(x) = - h(x)( 1- v(x)) \sqrt{ 1+ ||f(x)|| ^{2}}

If f is continuously differentiable, then the differential equation has at most one continuously differentiable solution satisfying

v( 0 ) = 0 .

Notes and References

1. 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.