Milnor and Topological Attractors in a Family of Two-Dimensional Lotka–Volterra Maps

Abstract

In this work, we consider a family of Lotka–Volterra maps [Formula: see text] for [Formula: see text] and [Formula: see text] which unfold a map originally proposed by Sharkosky for [Formula: see text] and [Formula: see text]. Multistability is observed, and attractors may exist not only in the positive quadrant of the plane, but also in the region [Formula: see text]. Some properties and bifurcations are described. The [Formula: see text]-axis is invariant, on which the map reduces to the logistic. For any [Formula: see text] an interval of values for [Formula: see text] exists for which all the cycles on the [Formula: see text]-axis are transversely attracting. This invariant set is the source of several kinds of bifurcations. Riddling bifurcations lead to attractors in Milnor sense, not topological but with a stable set of positive measure, which may be the unique attracting set, or coexisting with other topological attractors. The riddling and blowout bifurcations are described related to chaotic intervals on the invariant set, and these global bifurcations have different dynamic results. Chaotic intervals which are not topological attractors may have all the cycles transversely attracting and as Milnor attractors. We show that Milnor attractors may also be related to attracting cycles on the [Formula: see text]-axis at the bifurcation associated with the transverse and parallel eigenvalues. We show particular examples related to topological attractors with very narrow basins of attraction, when the majority of the trajectories are divergent.