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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 ${x}_{n+1}=g\left({x}_{n}\right)$ (where $g\in {C}^{\sigma }\left({R}^{d},{R}^{d}\right)$, using Lyapunov functions constructed by approximating the solution $V\left(x\right)$ of the equation $V\left(g\left(x\right)\right)-V\left(x\right)=-\parallel x-\overline{x}{\parallel }^{2}$. The author makes reference to 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 St. Balint, E. Kaslik, A. M. Balint and A. Grigis [Adv. Difference Equ., Article ID23939 (2006; Zbl 1134.39013)] (and the references within) which addresses a similar problem. Considering the solution $V\left(x\right)$ of the above difference equation and its Taylor polynomial like functions $n\left(x\right)$, the function $W\left(x\right)=\frac{V\left(x\right)}{n\left(x\right)}$ is constructed and its properties are given. Approximations of $w\left(x\right)$ and $W\left(x\right)$ are constructed, using radial basis functions, and local and global error estimates are provided. It is shown that the function $v$ defined by $v\left(x\right)=n\left(x\right)·w\left(x\right)$ is a local and global Lyapunov function. Two examples confirming the effectiveness of proposed method are presented.

##### MSC:
 39A11 Stability of difference equations (MSC2000) 37B25 Lyapunov functions and stability; attractors, repellers 37C25 Fixed points, periodic points, fixed-point index theory 39A12 Discrete version of topics in analysis