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

Laura Gardini

http://orcid.org/0000-0002-3386-6543

Department DESP, University of Urbino, Italy

E-mail Address: laura.gardini@uniurb.it

Corresponding author.

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and

Wirot Tikjha

Faculty of Science and Technology, Pibulsongkram Rajabhat University, Phitsanulok 65000, Thailand

Centre of Excellence in Mathematics, PERDO, CHE, Thailand

E-mail Address: wirottik@psru.ac.th

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https://doi.org/10.1142/S0218127420300402Cited by:1 (Source: Crossref)

Abstract

In this work, we consider a family of Lotka–Volterra maps $(x^{'}, y^{'}) = (x (a - x - y), b x y)$ for $a > 1$ and $b > 0$ which unfold a map originally proposed by Sharkosky for $a = 4$ and $b = 1$ . Multistability is observed, and attractors may exist not only in the positive quadrant of the plane, but also in the region $y < 0$ . Some properties and bifurcations are described. The $x$ -axis is invariant, on which the map reduces to the logistic. For any $a > 1$ an interval of values for $b$ exists for which all the cycles on the $x$ -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 $x$ -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.

Keywords:

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