Small islands of stability in the chaotic sea

摘要

Quadratic map is well investigated. The main island of stability for this map exists for values of parameter C in interval (-3,1). We can show that an island of stability for antiquadratic map exists when 0 < c < 1 too. Therefore, at sufficiently big p, we can find special areas in the phase space for Carleson map and intervals for parameter g, when the islands of stability exist, just by specifying areas in the phase space of models maps, intervals for parameter C and using the formulae above. For illustration of our method we represent two phase portraits. The phase portrait of quadratic map when c = -0.6 in the window [-2,2]×[-2,2] is represented on the left picture. The phase portrait of Carleson map when g = 200.4197414901 in the window [-0.2081827596·10{sup}(-6), 0.6245482788·10{sup}(-6)]×[0.20987074422873, 0.20987074559753] is represented in special coordinates described in Statement 1 on the right picture.