Methods for determination and approximation of domains of attraction in the case of autonomous discrete dynamical systems

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 $x\in D_a(0)$ 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 $\Vert {\partial }_{0}f\Vert <1$ and $\rho ({\partial }_{0}f)<1$ are treated separately (even though the second case includes the first one) because significant differences occur.

17 pages, 6 figures, submitted to "Discrete and Continuous Dynamical Systems" (AIMS)

References in corpus (1)

• An extension of the characterization of the domain of attraction of an asymptotically stable fixed point in the case of a nonlinear discrete dynamical system