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Analytic invariant curves for an iterative equation related to mosquito model
Author(s) -
Zhao Houyu
Publication year - 2013
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.3009
Subject(s) - mathematics , invertible matrix , invariant (physics) , unit circle , iterative method , mathematical analysis , complex plane , pure mathematics , mathematical physics , mathematical optimization
In this paper, we study the existence of analytic invariant curves of a iterative equation f ( f ( x ) ) = ( af ( x ) + bx e − x)e − f ( x )which from mosquito model. By constructing an invertible analytic solution g ( x ) of an auxiliary equation of the form g (α 2 x )e g ( x ) + g ( αx )= ag ( αx )e g ( x )+ bg ( x )invertible analytic solutions of the form g ( αg − 1 ( x )) for the original iterative functional equation are obtained. Besides the hyperbolic case 0 < | α | < 1, we focus on those α on the unit circle S 1 , that is, | α | = 1. We discuss not only those α at resonance, that is at a root of the unity, but also those α near resonance under the Brjuno condition. Copyright © 2013 John Wiley & Sons, Ltd.