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

• [math]\displaystyle{ v(0) = h(0) = 0 }[/math], [math]\displaystyle{ 0 \lt v(x) \lt 1 }[/math] for [math]\displaystyle{ x \in A \setminus \{0\} }[/math], [math]\displaystyle{ h \gt 0 }[/math] on [math]\displaystyle{ \R^n \setminus \{0\} }[/math]
• for every [math]\displaystyle{ \gamma_2 \gt 0 }[/math] there exist [math]\displaystyle{ \gamma_1 \gt 0, \alpha_1 \gt 0 }[/math] such that [math]\displaystyle{ v(x) \gt \gamma_1, h(x) \gt \alpha_1 }[/math] , if [math]\displaystyle{ ||x||\gt \gamma_2 }[/math]
• [math]\displaystyle{ v(x_n) \rightarrow 1 }[/math] for [math]\displaystyle{ x_n \rightarrow \partial A }[/math] or [math]\displaystyle{ ||x_n|| \rightarrow \infty }[/math]
• [math]\displaystyle{ \nabla v(x) \cdot f(x) = -h(x)(1-v(x)) \sqrt{1+||f(x)||^2} }[/math]

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