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Basin of attraction through invariant curves and dominant functions. (English) Zbl 1418.39013
Summary: We study a second-order difference equation of the form \(z_{n + 1} = z_n F(z_{n - 1}) + h\), where both \(F(z)\) and \(z F(z)\) are decreasing. We consider a set of invariant curves at \(h = 1\) and use it to characterize the behaviour of solutions when \(h > 1\) and when \(0 < h < 1\). The case \(h > 1\) is related to the Y2K problem. For \(0 < h < 1\), we study the stability of the equilibrium solutions and find an invariant region where solutions are attracted to the stable equilibrium. In particular, for certain range of the parameters, a subset of the basin of attraction of the stable equilibrium is achieved by bounding positive solutions using the iteration of dominant functions with attracting equilibria.
39A30 Stability theory for difference equations
39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type
Full Text: DOI
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