en>Michael Hardy at 04:25, 3 September 2013

2013-09-03T04:25:37Z

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'''Zubov's method''' is a technique for computing the [[basin of attraction]] for a set of [[ordinary differential equation]]s (a [[dynamical system]]). The domain of attraction is the set <math>\{x:\, v(x)<1\}</math>, where <math>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.

Zubov's theorem states that:

:If <math>x' = f(x), t \in \R</math> is an ordinary differential equation in <math>\R^n</math> with <math>f(0)=0</math>, a set <math>A</math> containing 0 in its interior is the domain of attraction of zero if and only if there exist continuous functions <math>v, h</math> such that:
:* <math>v(0) = h(0) = 0</math>, <math>0 < v(x) < 1</math> for <math>x \in A \setminus \{0\}</math>, <math>h > 0</math> on <math>\R^n \setminus \{0\}</math>
:* for every <math>\gamma_2 > 0</math> there exist <math>\gamma_1 > 0, \alpha_1 > 0</math> such that <math>v(x) > \gamma_1, h(x) > \alpha_1</math> , if <math>||x||>\gamma_2</math>
:* <math>v(x_n) \rightarrow 1</math> for <math>x_n \rightarrow \partial A</math> or <math>||x_n|| \rightarrow \infty</math>
:* <math> \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>v(0) = 0</math>.

==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.

[[Category:Ordinary differential equations]]

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en>Michael Hardy at 04:25, 3 September 2013