Methods for determination and approximation of the domain of attraction

Abstract

In this paper, an R-analytical function and the sequence of its Taylor polynomials (which are Lyapunov functions different from those of Vanelli & Vidyasagar (1985, Automatica, 21(1):6 9-80)) is presented, in order to determine and approximate the domain of attraction of the exponentially asymptotically stable zero steady state of an autonomous, R-analytical system of differential equations. The analytical function and the sequence of its Taylor polynomials are constructed by recurrence formulae using the coefficients of the power series expansion of f at 0.

The domain of attraction for the endemic equilibrium of an SIRS epidemic model

“…For V p , we have the following results (see [13]): Theorem 4. For any p ≥ 2, there exists r p > 0 such that for any x ∈B(r p ) \ {0} one has:…”

Modelling of growth effects of pressure changes in an edge-defined film-fed growth system

“…After each change there is a transition process with duration of t i ðsÞ; at the end of which the crystal radius and the meniscus height become constant again [36]. These values are presented in Table 4.…”

Approximation on attraction domain of Cohen–Grossberg neural networks

“…Since 1985, some important approaches on attraction domain estimation were presented [6][7][8]. Under the assumption of diagonalizability of Jacobian matrix at the equilibrium point, the optimal Lyapunov function method was proposed by Kaslik and Balint [8], which provides a way in approximating the attraction domain.…”

“…Under the assumption of diagonalizability of Jacobian matrix at the equilibrium point, the optimal Lyapunov function method was proposed by Kaslik and Balint [8], which provides a way in approximating the attraction domain. In 2009, an iterative expansion approach for improving the approximation of attraction domain was presented [9].…”

“…V p (u) is proved to be a Lyapunov function of system (2) (see [8,Proposition 13]). Suppose that V p ðuÞ > 0 and _ V p ðuÞ ¼: hrV P ðuÞ; f ðuÞi < 0 ð8Þ are satisfied in L v ¼ G V P ðu 0 Þ n U 1 .…”