Methods for Determination and Approximation of the Domain of Attraction in the Case of Autonomous Discrete Dynamical Systems

Amethod for determination and twomethods for approximation of the domain of attraction Da(0) of the asymptotically stable zero steady state of an autonomous, R-analytical, discrete dynamical system are presented. The method of determination is based on the construction of a Lyapunov function V , whose domain of analyticity is Da(0). The first method of approximation uses a sequence of Lyapunov functions Vp, which converge to the Lyapunov function V onDa(0). Each Vp defines an estimateNp ofDa(0). For any x ∈ Da(0), there exists an estimate Npx which contains x. The second method of approximation uses a ball B(R)⊂Da(0) which generates the sequence of estimatesMp = f −p(B(R)). For any x ∈Da(0), there exists an estimateMpx which contains x. The cases ‖∂0 f ‖ < 1 and ρ(∂0 f ) < 1≤ ‖∂0 f ‖ are treated separately because significant differences occur.