Åbo Akademi, Finland

\documentclass[12pt]{article} \usepackage[cp1251]{inputenc} \usepackage[english]{babel} \usepackage{amsmath} \usepackage{amscd} \usepackage{amssymb,amsfonts} \usepackage{graphicx,graphics} \renewcommand{\baselinestretch}{1.5} \textwidth=158mm \textheight=232mm \voffset=-24mm \pagestyle{empty} \begin{document} \centerline{\large\bf Coexistence of attractors of low period } \centerline{\large\bf for some families of one-dimensional maps.} \centerline{{\bf Gunnar S\"oderbacka} (Finland, Turku)} \centerline{Docent} \centerline{\it gs091953@gmail.com} We look at the dynamcis to the bimodal map $g$ defined by $g( v ) = b + v -\frac{k }{ 1 + e^v } $, where $k<b<0$, and at the dynamics of its approximations. In Reference 1, they introduced the name EOS-map for this map. The EOS-map can be considered as something between the quadratic (logistic) and the cubic polynomial map. The quadratic map can have only one attractor for fixed parameters. The cubic map often has two attractors. The EOS-map can have coexistence of attractors, which may be both chaotic or periodic or one of them is chaotic and the other periodic, but in very small parameters regions. It has two critical points and is increasing at the ends with small slope and decreasing between the critical points. Because there is a closed globally attracting interval, it can be approximated by continuous piece-wise linear maps, where more properties can be analytically proved. The interesting shapes of parameter regions for low periodic attractors, and especially, where two attractors coexist, are described. We present many open questions, some of them connected with bifurcations. For more about the EOS-map and for to understand how it can arise from a population dynamics model we refer to Reference 2 and references therein. \begin{center} {\bf References} \end{center} {[1]} J. Bielawski, T. Chotibut, F. Falniowski, M. Misiurewicz; G. Piliouras. Interval maps mimicking circle rotations, Communications in Nonlinear Science and Numerical Simulation, Volume {\bf 150}, November 2025, 108963. .\\ {[2]} S Kryzhevich, G S\"oderbacka. Multiattractors in a Several-Predators-One-Prey System, Nonlinear Phenomena in Complex System, vol. 28, no. 3 (2025), pp. 289 - 297. \\ \end{document}