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 ${\displaystyle \{x:\,v(x)<1\}}$, where ${\displaystyle v(x)}$ 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 ${\displaystyle x'=f(x),t\in \mathbb {R} }$ is an ordinary differential equation in ${\displaystyle \mathbb {R} ^{n}}$ with ${\displaystyle f(0)=0}$, a set ${\displaystyle A}$ containing 0 in its interior is the domain of attraction of zero if and only if there exist continuous functions ${\displaystyle v,h}$ such that:

• ${\displaystyle v(0)=h(0)=0}$, ${\displaystyle 0<v(x)<1}$ for ${\displaystyle x\in A\setminus \{0\}}$, ${\displaystyle h>0}$ on ${\displaystyle \mathbb {R} ^{n}\setminus \{0\}}$
• for every ${\displaystyle \gamma _{2}>0}$ there exist ${\displaystyle \gamma _{1}>0,\alpha _{1}>0}$ such that ${\displaystyle v(x)>\gamma _{1},h(x)>\alpha _{1}}$ , if ${\displaystyle ||x||>\gamma _{2}}$
• ${\displaystyle v(x_{n})\rightarrow 1}$ for ${\displaystyle x_{n}\rightarrow \partial A}$ or ${\displaystyle ||x_{n}||\rightarrow \infty }$
• ${\displaystyle \nabla v(x)\cdot f(x)=-h(x)(1-v(x)){\sqrt {1+||f(x)||^{2}}}}$

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

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.