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 $x^{\prime }=f(x),t\in \mathbb{ℝ}$ is an ordinary differential equation in ${\mathbb{ℝ}}^n$ 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 ${\mathbb{ℝ}}^n\setminus \{0\}$
• for every ${\gamma }_2>0$ there exist ${\gamma }_1>0,{\alpha }_1>0$ such that $v(x)>{\gamma }_1,h(x)>{\alpha }_1$ , if $||x||>{\gamma }_2$
• $v(x_n)\to 1$ for $x_n\to \partial A$ or $||x_n||\to \mathrm{\infty }$
• $\mathrm{\nabla }v(x)\cdot f(x)=-h(x)(1-v(x))\sqrt{1+||f(x)|{|}^2}$

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