Mathematics – Dynamical Systems
Scientific paper
2004-10-06
Mathematics
Dynamical Systems
17 pages, 6 figures, submitted to "Discrete and Continuous Dynamical Systems" (AIMS)
Scientific paper
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 $x\in D_{a}(0)$ 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 $x\in D_{a}(0)$ there exists an estimate $M_{p^{x}}$ which contains $x$. The cases $\|\partial_{0}f\|<1$ and $\rho(\partial_{0}f)<1$ are treated separately (even though the second case includes the first one) because significant differences occur.
Balint Agneta-Maria
Balint St.
Grigis A.
Kaslik Eva
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