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.

Zubov's theorem states that:

If $x'=f(x),t\in \mathbb {R}$ 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 $||x||>\gamma _{2}$
• $v(x_{n})\rightarrow 1$ 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 $v(0)=0$.

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.