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Construction of a local and global Lyapunov function for discrete dynamical systems using radial basis functions. (English) Zbl 1162.39006
The author investigates a method of determining the basin of attraction of an asymptotically stable fixed point $\overline x$ of a discrete time autonomous dynamical system $x_{n+1}=g(x_n)$ (where $g\in C^\sigma(R^d,R^d)$, using Lyapunov functions constructed by approximating the solution $V(x)$ of the equation $V(g(x))-V(x)=-\|x-\overline x\|^2$. The author makes reference to {\it P. Giesl} [J. Difference Equ. Appl. 13, No. 6, 523--546 (2007; Zbl 1120.39018)] for a constructive existence theorem for a smooth solution $U$ of the above difference equation. However, he seems unaware of the result of {\it St. Balint, E. Kaslik, A. M. Balint} and {\it A. Grigis} [Adv. Difference Equ., Article ID23939 (2006; Zbl 1134.39013)] (and the references within) which addresses a similar problem. Considering the solution $V(x)$ of the above difference equation and its Taylor polynomial like functions $n(x)$, the function $W(x)=\frac{V(x)}{n(x)}$ is constructed and its properties are given. Approximations of $w(x)$ and $W(x)$ are constructed, using radial basis functions, and local and global error estimates are provided. It is shown that the function $v$ defined by $v(x)=n(x)\cdot w(x)$ is a local and global Lyapunov function. Two examples confirming the effectiveness of proposed method are presented.

39A11Stability of difference equations (MSC2000)
37B25Lyapunov functions and stability; attractors, repellers
37C25Fixed points, periodic points, fixed-point index theory
39A12Discrete version of topics in analysis
Full Text: DOI
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