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

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.

Finitely many physical measures for sectional-hyperbolic attracting sets and statistical stability

We show that a sectional-hyperbolic attracting set for a Hölder- $C^{1}$ vector field admits finitely many physical/SRB measures whose ergodic basins cover Lebesgue almost all points of the basin of topological attraction. In addition, these physical measures depend continuously on the flow in the $C^{1}$ topology, that is, sectional-hyperbolic attracting sets are statistically stable. To prove these results we show that each central-unstable disk in a neighborhood of this class of attracting sets is eventually expanded to contain a ball whose inner radius is uniformly bounded away from zero.

Polynomial dissipativity of proportional delayed BAM neural networks

This paper pays close attention to the global polynomial dissipativity (GPD) for proportional delayed BAM neural networks (PDBAMNNs). The global exponential dissipativity (GED) and the global dissipativity (GD) are also talked about. Under the help of novel Lyapunov functionals and a generalized Halanay inequality, a set of dissipative criteria for such systems are led out, together with the global polynomial attracting set (GPAS) and the global attracting set (GAS). Further, the relationship among GPD, GED and GD is unveiled. Finally, a proposed theoretical condition is validated through a simulation experiment.

Attracting Sets Of Nonlinear Difference Equations With Time-Varying Delays

In this paper, a class of nonlinear difference equations with time-varying delays is considered. Based on a generalized discrete Halanay inequality, some sufficient conditions for the attracting set and the global asymptotic stability of the nonlinear difference equations with time-varying delays are obtained.