In this paper we describe some sequences of global bifurcations involving attracting and repelling closed invariant curves of two-dimensional maps that have a fixed point which may lose stability both via a supercritical Neimark bifurcation and a supercritical pitchfork or flip bifurcation. These bifurcations, characterized by the creation of heteroclinic and homoclinic connections or homoclinic tangles, are first described through qualitative phase diagrams and then by several numerical examples. Similar bifurcation phenomena can also be observed when the parameters in a two-dimensional parameter plane cross through many overlapping Arnold's tongues.

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An adaptive oligopoly model, where the demand function is isoelastic and the competitors operate under constant marginal costs, is considered. The Cournot equilibrium point then loses stability through a subcritical Neimark bifurcation. The present paper focuses some global bifurcations, which precede the Neimark bifurcation, and produce other attractors which coexist with the still attractive Cournot fixed point.

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