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

Abstract

We investigate the stabilization of periodic orbits of one-dimensional discrete maps by using a proportional feedback method applied in the form of pulses. We determine a range of the parameter μ values representing the strength of the feedback for which all positive solutions of the controlled equation converge to a periodic orbit.An important feature of our approach is that the required assumptions for which our results hold are met by a general class of maps, improving in this way some previous results. We discuss the applicability of our scheme to some models of population dynamics, and give numerical simulations to illustrate our analytical results.

Stabilization of difference equations with noisy proportional feedback control

“…In this section, we consider two examples: the Beverton-Holt chaotic map (5.1) f (x) = 2.5x 1 + x 5 , x ≥ 0, considered in [6] and a particular case of (1.7)…”

“…Sometimes reduction boosts population sizes [17,21], this is called the hydra effect, see, for example, [15] and its literature list. Stabilization with proportional feedback control was recently studied in [6,11,14], see also references therein. Since in many cases there is stochasticity involved in the control, it is reasonable to model population dynamics processes with stochastic equations.…”

“…These conditions imply asymptotic stability [6] and are more restrictive than Assumption 1.1, which, with the appropriate control, also ensures asymptotic stability [4] of an arbitrarily chosen point x * ∈ (0, b), for an appropriate ν. Thus, in the present paper we stick up to less restrictive Assumption 1.1.…”

Dynamic analysis of unilateral diffusion Gompertz model with impulsive control strategy

“…With this in mind, the mathematical model of diffusion with pulse has been established, which makes the research more significant [6][7][8][9][10][11][12][13][14]. In particular, impulsive differential equations are very important in the research of population migration phenomenon [15][16][17][18][19][20][21][22][23][24]. Jiao et al [25] established a SIR model with pulse vaccination and proved the existence of a disease-free periodic solution, and a large pulse vaccination rate was a sufficient condition to eradicate the disease.…”

Stabilization of two cycles of difference equations with stochastic perturbations

“…Thus (see, e.g [8,. Lemma 1] and also[9, Lemma 2.4]) the points x r and x r þ 1/r are globally asymptotically stable for x [ (x r 2 1, x r þ 1/(2r)) and x [ (x r þ 1/(2r), x r þ 1/r þ 1), and all the solutions with the initial (or some intermediate) values in (x r 2 1, x r þ 1/r þ 1), but not x r þ 1/(2r), tend to either x r or x r þ 1/r, which is confirmed by the bifurcation diagram (see the two 'middle lines' inFigure 2).Next, let us fix a number M .…”