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

A method for determination and two methods for approximation of the domain of attraction Da(0) of an asymptotically stable steady state of an autonomous, 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 Da(0). The first method of approximation uses a sequence of Lyapunov functions Vp, which converges to the Lyapunov function V onDa(0). Each Vp defines an estimate Np 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 estimates Mp = f(B(R)). For any x ∈ Da(0) there exists an estimate Mpx which contains x. The cases ‖∂0f‖ < 1 and ρ(∂0f) < 1 are treated separately (even though the second case includes the first one) because significant differences occur. 2000 Mathematics Subject Classification. 34D20, 39A11.