Zubov's method

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.

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 $$||x||>\gamma_2$$
 * $$v(x_n) \rightarrow 1$$ for $$x_n \rightarrow \partial A$$ or $$||x_n|| \rightarrow \infty$$
 * $$ \nabla v(x) \cdot f(x) = -h(x)(1-v(x)) \sqrt{1+||f(x)||^2}$$

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