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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 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 \(V(x)\) 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.

MSC:
39A11 Stability of difference equations (MSC2000)
37B25 Stability of topological dynamical systems
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics
39A12 Discrete version of topics in analysis
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