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Methods for determination and approximation of domains of attraction in the case of autonomous discrete dynamical systems | | St. Balint
; E. Kaslik
; A.M. Balint
; A. Grigis
; | | Date: |
6 Oct 2004 | | Subject: | Dynamical Systems; Optimization and Control MSC-class: 34D20, 39A11 | math.DS math.OC | | Abstract: | A method for determination and two methods for approximation of the domain of attraction $D_{a}(0)$ of an asymptotically stable steady state of an autonomous, $mathbb{R}$-analytical, discrete system is presented. The method of determination is based on the construction of a Lyapunov function $V$, whose domain of analyticity is $D_{a}(0)$. The first method of approximation uses a sequence of Lyapunov functions $V_{p}$, which converges to the Lyapunov function $V$ on $D_{a}(0)$. Each $V_{p}$ defines an estimate $N_{p}$ of $D_{a}(0)$. For any $xin D_{a}(0)$ there exists an estimate $N_{p^{x}}$ which contains $x$. The second method of approximation uses a ball $B(R)subset D_{a}(0)$ which generates the sequence of estimates $M_{p}=f^{-p}(B(R))$. For any $xin D_{a}(0)$ there exists an estimate $M_{p^{x}}$ which contains $x$. The cases $|partial_{0}f|<1$ and $
ho(partial_{0}f)<1$ are treated separately (even though the second case includes the first one) because significant differences occur. | | Source: | arXiv, math.DS/0410146 | | Services: | Forum | Review | PDF | Favorites | |
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