Advances in Mathematical Physics

Volume 2016 (2016), Article ID 2028037, 8 pages

http://dx.doi.org/10.1155/2016/2028037

## Controlling Neimark-Sacker Bifurcation in Delayed Species Model Using Feedback Controller

School of Mathematics and Computer Science, Zunyi Normal College, Zunyi 563002, China

Received 16 June 2016; Accepted 10 July 2016

Academic Editor: Pavel Kurasov

Copyright © 2016 Jie Ran et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Based on the stability and orthogonal polynomial approximation theory, the ordinary, dislocated, enhancing, and random feedback control methods are used to suppress the Neimark-Sacker bifurcation to fixed point in this paper. It is shown that the convergence rate of enhancing feedback control and random feedback control can be faster than those of dislocated and ordinary feedback control. The random feedback control method, which does not require any adjustable control parameters of the model, just only slightly changes the random intensity. Finally, numerical simulations are presented to verify the effectiveness of the proposed controllers.

#### 1. Introduction

The studies of biological models gradually become one of hot spots in nonlinear dynamics. The biological models have great research background and actual significance; therefore, a growing number of researchers have shown great interests in the research of biological models. In many biological models and practical problems, bifurcation and chaos are undesirable behaviors. Thus, we need to control them. In 1976, a population model, is given by Ecologist May for the first time. A one-dimensional deterministic delayed population model, is investigated by Sun et al. [1].

Recently, the Hopf bifurcation has been given much attention, and those works about bifurcation mainly include the validated existence of bifurcation and its control [2–5]. The aim of bifurcation control is to design a controller to modify the bifurcation properties of a given nonlinear system and then achieve the other desirable dynamical behaviors. OGY feedback control method is studied by Ott et al. [6]. Chen et al. have investigated the feedback control in continuous-time systems [7, 8]. The control of Hopf bifurcation in time-delayed neural network system is investigated by Zhou et al. [9]. Bifurcation analysis and tracking control of an epidemic model with nonlinear incidence rate are investigated by Yi et al. [10]. Wen and Xu studied feedback control of Hopf–Hopf interaction bifurcation with development of torus solutions in high-dimensional maps [11]. Feedback control of bifurcation and chaos in dynamical systems is investigated by Abed and Wang [12]. The Hopf bifurcation control via dynamic state-feedback control is studied by Nguyen and Hong in [13]. Amplitude control of limit cycle from Hopf bifurcation is studied in [14, 15]. Hopf bifurcation control of the system based on washout filter controller is investigated by Wu and Sun [16]. Liu and Xiao have studied complex dynamic behaviors of a discrete-time predator-prey system [17].

However, owing to the uncertain factors of external environment, manufacture, material, and installation, some parameters in practical model are not constant and will be characterized as bound random parameters [18]. The stochastic system can accurately represent the original system better. Therefore the study of stochastic system is more meaningful than deterministic systems. The Hopf bifurcation control is investigated in stochastic system with random parameter [18–20]. It is of interest to examine the stochastic method in biological system and explore its implications.

The rest of this letter is organized as follows. In Section 2, the conditions for the emergence of Neimark-Sacker bifurcation are reviewed. In Section 3, the ordinary, dislocated, enhancing, and random feedback controls for controlling Neimark-Sacker bifurcation are proposed. And numerical simulations are presented to verify the effectiveness of the proposed bifurcation control methods. Finally, conclusions are given in Section 4.

#### 2. Neimark-Sacker Bifurcation

Let us consider the logistic population model [1] for a single species:where stands for the population size at time and is the growth rate. In the real environment, the population size is determined not only by the current population size but also by its size in the past. So, we consider where stands for the population size at time and is the growth rate. If we introduce in model (4), a two-dimensional discrete-time dynamical model [2] can be rewritten as

By a simple computation with mathematical software, it is straightforward to obtain the following proposition.

Proposition 1. *(a) For all parameter values, model (5) has one fixed point, .**(b) If , then model (5) has, additionally, a nontrivial positive fixed point, , where .**The Jacobian matrix of model (5) evaluated at the fixed point is given byand the characteristic equation of Jacobian matrix of model (5) can be written aswhere , .*

Next, according to the point of view of biology, we study the stability of the nonzero fixed points. Note that the local stability of a fixed point is determined by the modules of eigenvalues of the characteristic equation at the fixed point. From the mathematical software and Lemma 2.2 [17], the following proposition shows the local stability of the fixed point .

Proposition 2. *(a) is a sink if .**(b) is a source if .**(c) is not hyperbolic if .**When the term (c) of Proposition 2 holds, we can obtain that the eigenvalues of the matrix at the fixed point are a pair of conjugate complex numbers, the modules of which are one. The condition in term (c) of Proposition 2 can be written as the setthe fixed point can undergo Neimark-Sacker bifurcation when parameters vary in the small neighborhood of . By a simple computation, all eigenvalues of (7) arewhen , we can obtain eigenvalues**Obviously, the transversality condition, the nondegeneracy condition, and the additional nondegeneracy condition of Neimark-Sacker bifurcation hold (see [3]). Thus, the nontrivial fixed point loses stability in the small neighborhood of . The bifurcation diagram and phase portrait for model (5) are depicted in Figure 1.*