Chaotic and stable perturbed maps: 2-cycles and spatial models

Abstract

As the growth rate parameter increases in the Ricker, logistic and some other maps, the models exhibit an irreversible period doubling route to chaos. If a constant positive perturbation is introduced, then the Ricker model (but not the classical logistic map) experiences period doubling reversals; the break of chaos finally gives birth to a stable two-cycle. We outline the maps which demonstrate a similar behavior and also study relevant discrete spatial models where the value in each cell at the next step is defined only by the values at the cell and its nearest neighbors. The stable 2-cycle in a scalar map does not necessarily imply 2-cyclic-type behavior in each cell for the spatial generalization of the map.

Global stabilization of periodic orbits using a proportional feedback control with pulses

“…For the general model (1), with f satisfying (A1), try to loosen the necessary conditions for global stabilization using other types of control, such as the predictive control considered in [11] or the constant feedback method addressed in [9]. For very high reproduction rates, a positive constant perturbation can lead to a stable cycle, see [4] and references therein. 4.…”

Stabilization of two cycles of difference equations with stochastic perturbations

“…The present paper continues and generalizes the previous results on stabilization by a constant perturbation, two-cyclic behaviour and bubbling, for example, [7,21,22] in the following directions:…”

“…In addition, all our three theorems not only extend local stability results of [7] to the stochastic case but also deal with global stability, outlining some new effects.…”

“…In this section, we prove three theorems which can be considered as generalizations of Theorems II.1 -3 from [7] to the case when the right-hand side of the corresponding equation contains a stochastic perturbation r, which is supposed to vanish at infinity. In addition, all our three theorems not only extend local stability results of [7] to the stochastic case but also deal with global stability, outlining some new effects.…”

“…This study was begun in [7], however, in addition to more general (stochastic) models, we analyse global asymptotic stability (compared to local in [7]) and discover some new effects for some types of maps, such as existence of both an attracting equilibrium and an attracting two-cycle for r large enough.…”