Methods for determination and approximation of the domain of attraction in the case of autonomous discrete dynamical systems
© Hindawi Publishing Corporation. 2006
Received: 15 October 2004
Accepted: 18 October 2004
Published: 12 February 2006
A method for determination and two methods for approximation of the domain of attraction D a (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 D a (0). The first method of approximation uses a sequence of Lyapunov functions V p , which converge to the Lyapunov function V on D a (0). Each V p defines an estimate N p of D a (0). For any x ∈ D a (0), there exists an estimate which contains x. The second method of approximation uses a ball B(R) ⊂ D a (0) which generates the sequence of estimates M p = f-p(B(R)). For any x ∈ D a (0), there exists an estimate which contains x. The cases ||∂0f||<1 and ρ(∂0f) < 1 ≤||∂0f|| are treated separately because significant differences occur.
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