Bifurcations in a one-parameter family of Lotka-Volterra 2D transformations

Abstract

A particular system of two-dimensional Lotka-Volterra maps, Ta:(x′,y′)=(x(a−x−y),xy), unfolding a map originally proposed by Sharkovsky for a=4, is considered. We show the routes to chaos leading to the dynamics of map T4. For map T4 we show that even if the stable set of the origin O includes a set dense in an invariant area, the only homoclinic points of O belong to the x−axis, as well as the cycles leading to heteroclinic connections, while many internal cycles are snap-back repellers. We also show that a particular 6-cycle known analytically for map T4 exists, and is known explicitly in closed form, for any a∈(3,4] appearing at a supercritical Neimark-Sacker bifurcation of the positive fixed point. Moreover, we show the existence of infinitely many k−cycles on the x−axis (for any k>3), which are topological attractors of map Ta for a∈(3.96,4) and saddle cycles transversely attracting at a=4.