# Research Article: Hybrid control of the Neimark-Sacker bifurcation in a delayed Nicholson’s blowflies equation

Date Published: October 6, 2015

Publisher: Springer International Publishing

Author(s): Yuanyuan Wang, Lisha Wang.

http://doi.org/10.1186/s13662-015-0640-2

Abstract

In this article, for delayed Nicholson’s blowflies equation, we propose a hybrid control nonstandard finite-difference (NSFD) scheme in which state feedback and parameter perturbation are used to control the Neimark-Sacker bifurcation. Firstly, the local stability of the positive equilibria for hybrid control delay differential equation is discussed according to Hopf bifurcation theory. Then, for any step-size, a hybrid control numerical algorithm is introduced to generate the Neimark-Sacker bifurcation at a desired point. Finally, numerical simulation results confirm that the control strategy is efficient in controlling the Neimark-Sacker bifurcation. At the same time, the results show that the NSFD control scheme is better than the Euler control method.

Partial Text

The delay differential equation (DDE) 1.1documentclass[12pt]{minimal}
usepackage{amsmath}
usepackage{wasysym}
usepackage{amsfonts}
usepackage{amssymb}
usepackage{amsbsy}
usepackage{mathrsfs}
usepackage{upgreek}
setlength{oddsidemargin}{-69pt}
begin{document}\$\$ dot{x}(t)=ax(t-tau)e^{-bx(t-tau)}-cx(t), \$\$end{document}x˙(t)=ax(t−τ)e−bx(t−τ)−cx(t), which is one of the important ecological systems, describes the dynamics of Nicholson’s blowflies equation. Here documentclass[12pt]{minimal}
usepackage{amsmath}
usepackage{wasysym}
usepackage{amsfonts}
usepackage{amssymb}
usepackage{amsbsy}
usepackage{mathrsfs}
usepackage{upgreek}
setlength{oddsidemargin}{-69pt}
begin{document}\$x(t)\$end{document}x(t) is the size of the population at time t, a is the maximum per capita daily egg production rate, documentclass[12pt]{minimal}
usepackage{amsmath}
usepackage{wasysym}
usepackage{amsfonts}
usepackage{amssymb}
usepackage{amsbsy}
usepackage{mathrsfs}
usepackage{upgreek}
setlength{oddsidemargin}{-69pt}
begin{document}\$1/b\$end{document}1/b is the size at which the population reproduces at the maximum rate, c is the per capita daily adult death rate, and τ is the generation time, the positive equilibrium documentclass[12pt]{minimal}
usepackage{amsmath}
usepackage{wasysym}
usepackage{amsfonts}
usepackage{amssymb}
usepackage{amsbsy}
usepackage{mathrsfs}
usepackage{upgreek}
setlength{oddsidemargin}{-69pt}
begin{document}\$x_{ast}=(1/b)ln(a/c)\$end{document}x∗=(1/b)ln(a/c). Equation (1.1) has been extensively studied in the literature. The majority of the results on (1.1) deal with the global attractiveness of the positive equilibrium and oscillatory behaviors of solutions [1, 2].

In the original delay differential equation model (1.1), the time delay τ acts as a bifurcation parameter. As the delay τ passes through some critical value documentclass[12pt]{minimal}
usepackage{amsmath}
usepackage{wasysym}
usepackage{amsfonts}
usepackage{amssymb}
usepackage{amsbsy}
usepackage{mathrsfs}
usepackage{upgreek}
setlength{oddsidemargin}{-69pt}
begin{document}\$tau_{k}\$end{document}τk, a couple of complex conjugating eigenvalues of the system pass the imaginary axis at some pure imaginary points, and stable periodic Hopf bifurcating solutions occur. Then, when τ passes documentclass[12pt]{minimal}
usepackage{amsmath}
usepackage{wasysym}
usepackage{amsfonts}
usepackage{amssymb}
usepackage{amsbsy}
usepackage{mathrsfs}
usepackage{upgreek}
setlength{oddsidemargin}{-69pt}
begin{document}\$tau_{k}\$end{document}τk, the real parts of these eigenvalues pass to the positive real axis causing the Hopf bifurcating solution to be unstable. We summarize these features of the solution via the existence and stability of a positive equilibrium following the works in [19], Theorem 2.3.

Let documentclass[12pt]{minimal}
usepackage{amsmath}
usepackage{wasysym}
usepackage{amsfonts}
usepackage{amssymb}
usepackage{amsbsy}
usepackage{mathrsfs}
usepackage{upgreek}
setlength{oddsidemargin}{-69pt}
begin{document}\$u(t)=x(tau t)\$end{document}u(t)=x(τt). Then Eq. (1.1) can be rewritten as 3.1documentclass[12pt]{minimal}
usepackage{amsmath}
usepackage{wasysym}
usepackage{amsfonts}
usepackage{amssymb}
usepackage{amsbsy}
usepackage{mathrsfs}
usepackage{upgreek}
setlength{oddsidemargin}{-69pt}
begin{document}\$\$ dot{u}(t)=atau u(t-1)e^{-bu(t-1)}-c tau u(t). \$\$end{document}u˙(t)=aτu(t−1)e−bu(t−1)−cτu(t). One can see that if documentclass[12pt]{minimal}
usepackage{amsmath}
usepackage{wasysym}
usepackage{amsfonts}
usepackage{amssymb}
usepackage{amsbsy}
usepackage{mathrsfs}
usepackage{upgreek}
setlength{oddsidemargin}{-69pt}
begin{document}\$u_{ast}\$end{document}u∗ is a positive fixed point to Eq. (3.1), then documentclass[12pt]{minimal}
usepackage{amsmath}
usepackage{wasysym}
usepackage{amsfonts}
usepackage{amssymb}
usepackage{amsbsy}
usepackage{mathrsfs}
usepackage{upgreek}
setlength{oddsidemargin}{-69pt}
begin{document}\$u_{ast}\$end{document}u∗ satisfies 3.2documentclass[12pt]{minimal}
usepackage{amsmath}
usepackage{wasysym}
usepackage{amsfonts}
usepackage{amssymb}
usepackage{amsbsy}
usepackage{mathrsfs}
usepackage{upgreek}
setlength{oddsidemargin}{-69pt}
begin{document}\$\$ c=ae^{-bu_{ast}}, \$\$end{document}c=ae−bu∗, here documentclass[12pt]{minimal}
usepackage{amsmath}
usepackage{wasysym}
usepackage{amsfonts}
usepackage{amssymb}
usepackage{amsbsy}
usepackage{mathrsfs}
usepackage{upgreek}
setlength{oddsidemargin}{-69pt}
begin{document}\$u_{ast}=x_{ast}\$end{document}u∗=x∗. Apply both parameter perturbation and state feedback to system (3.1) as follows: 3.3documentclass[12pt]{minimal}
usepackage{amsmath}
usepackage{wasysym}
usepackage{amsfonts}
usepackage{amssymb}
usepackage{amsbsy}
usepackage{mathrsfs}
usepackage{upgreek}
setlength{oddsidemargin}{-69pt}
begin{document}\$\$ dot{u}(t)=alpha bigl[atau u(t-1)e^{-bu(t-1)}-ctau u(t) bigr]+(1-alpha)tau bigl(u(t-1)-u_{ast} bigr), quad 0< alphaleq1. \$\$end{document}u˙(t)=α[aτu(t−1)e−bu(t−1)−cτu(t)]+(1−α)τ(u(t−1)−u∗),0<α≤1. Set documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$z(t)=u(t)-u_{ast}\$end{document}z(t)=u(t)−u∗. Equation (3.3) becomes 3.4documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$\$ dot{z}(t)=alpha bigl[atau bigl(z(t-1)+u_{ast} bigr)e^{-b(z(t-1)+u_{ast})}-ctau bigl(z(t)+u_{ast} bigr) bigr]+(1-alpha) tau z(t-1). \$\$end{document}z˙(t)=α[aτ(z(t−1)+u∗)e−b(z(t−1)+u∗)−cτ(z(t)+u∗)]+(1−α)τz(t−1). The linearization of Eq. (3.4) at documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$z=0\$end{document}z=0 is 3.5documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$\$ dot{z}(t)=alpha ctau bigl[(1-bu_{ast})z(t-1)-z(t) bigr]+(1-alpha)tau z(t-1), \$\$end{document}z˙(t)=αcτ[(1−bu∗)z(t−1)−z(t)]+(1−α)τz(t−1), whose characteristic equation is 3.6documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$\$ lambda=alpha ctau bigl[(1-bu_{ast})e^{-lambda}-1 bigr]+(1-alpha)tau e^{-lambda}. \$\$end{document}λ=αcτ[(1−bu∗)e−λ−1]+(1−α)τe−λ. When documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$bu_{ast}>frac{1-alpha}{alpha c}\$end{document}bu∗>1−ααc, documentclass[12pt]{minimal}
usepackage{amsmath}
usepackage{wasysym}
usepackage{amsfonts}
usepackage{amssymb}
usepackage{amsbsy}
usepackage{mathrsfs}
usepackage{upgreek}
setlength{oddsidemargin}{-69pt}
begin{document}\$lambda<0\$end{document}λ<0. For documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$omega neq0\$end{document}ω≠0, iω is a root of Eq. (3.6) if and only if documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$\$ mathrm{i}omega=alpha ctau bigl[(1-bu_{ast})e^{-mathrm{i}omega }-1 bigr]+(1-alpha)tau e^{-mathrm{i}omega}. \$\$end{document}iω=αcτ[(1−bu∗)e−iω−1]+(1−α)τe−iω. Separating the real and imaginary parts, we obtain 3.7documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$\$ textstylebegin{cases} [alpha ctau(1-bu_{ast})+(1-alpha)tau]cosomega=alpha ctau,\ {[}alpha ctau(1-bu_{ast})+(1-alpha)tau]sinomega=-omega, end{cases} \$\$end{document}{[αcτ(1−bu∗)+(1−α)τ]cosω=αcτ,[αcτ(1−bu∗)+(1−α)τ]sinω=−ω, which leads to documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$\$ omega^{2}+(alpha ctau)^{2}= bigl[alpha c tau(1-bu_{ast})+(1-alpha)tau bigr]^{2}, \$\$end{document}ω2+(αcτ)2=[αcτ(1−bu∗)+(1−α)τ]2, that is, 3.8documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$\$ omega=pmtausqrt{ bigl[(1-alpha)-alpha cbu_{ast} bigr] bigl[(1- alpha)-alpha c(2-bu_{ast}) bigr]}. \$\$end{document}ω=±τ[(1−α)−αcbu∗][(1−α)−αc(2−bu∗)]. This is possible if and only if documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$bu_{ast}>2+frac{1-alpha}{alpha c}\$end{document}bu∗>2+1−ααc.

In this section, we mainly discuss the stability and bifurcation of the numerical discrete hybrid control system. We implement the hybrid control strategy [9–11]. Set documentclass[12pt]{minimal}
usepackage{amsmath}
usepackage{wasysym}
usepackage{amsfonts}
usepackage{amssymb}
usepackage{amsbsy}
usepackage{mathrsfs}
usepackage{upgreek}
setlength{oddsidemargin}{-69pt}
begin{document}\$v(t)=u(t)-u_{ast}\$end{document}v(t)=u(t)−u∗. Equation (3.3) becomes 4.1documentclass[12pt]{minimal}
usepackage{amsmath}
usepackage{wasysym}
usepackage{amsfonts}
usepackage{amssymb}
usepackage{amsbsy}
usepackage{mathrsfs}
usepackage{upgreek}
setlength{oddsidemargin}{-69pt}
begin{document}\$\$ dot{v}(t)=alpha bigl[atau bigl(v(t-1)+u_{ast} bigr)e^{-b(v(t-1)+u_{ast})}-ctau bigl(v(t)+u_{ast} bigr) bigr]+(1-alpha) tau v(t-1),quad 0< alphaleq1. \$\$end{document}v˙(t)=α[aτ(v(t−1)+u∗)e−b(v(t−1)+u∗)−cτ(v(t)+u∗)]+(1−α)τv(t−1),0<α≤1. When documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$alpha=1\$end{document}α=1, Eq. (4.1) is the uncontrolled system. The differential equation documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$\$ frac{dv}{dt}=-alpha ctau v(t) \$\$end{document}dvdt=−αcτv(t) has the general solution documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$v(t)=bar{C}e^{-alpha ctau t}\$end{document}v(t)=C¯e−αcτt. We consider step-size of the form documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$h=1/m\$end{document}h=1/m, where documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$min Z_{+}\$end{document}m∈Z+. The solution can be written as documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$\$ frac{v(t+h)-v(t)}{frac{1-e^{-alpha c tau h}}{alpha ctau }}=-alpha ctau v(t). \$\$end{document}v(t+h)−v(t)1−e−αcτhαcτ=−αcτv(t). This is an exact finite difference numerical method: documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document} \$\$begin{aligned} v(t+h)-v(t)&=bar{C}e^{-alpha ctau(t+h)}- bar{C}e^{-alpha ctau t} \ & =bar{C}e^{-alpha ctau t} bigl(e^{-alpha ctau h}-1 bigr) \ & =-alpha ctau v(t)frac {1-e^{-alpha ctau h}}{alpha ctau}. end{aligned}\$\$ end{document}v(t+h)−v(t)=C¯e−αcτ(t+h)−C¯e−αcτt=C¯e−αcτt(e−αcτh−1)=−αcτv(t)1−e−αcτhαcτ. Employ the NSFD scheme [13, 14, 17] to Eq. (4.1) and choose the ‘denominator function’ ψ as 4.2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$\$ psi(h)=frac{1-e^{-alpha ctau h}}{alpha ctau}. \$\$end{document}ψ(h)=1−e−αcτhαcτ. It yields the difference equation 4.3documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document} \$\$begin{aligned} v_{n+1}={}&e^{-alpha ctau h}v_{n}+ bigl(e^{-alpha ctau h}-1 bigr)u_{ast }+ bigl(1-e^{-alpha ctau h} bigr) (v_{n-m}+u_{ast})e^{-bv_{n-m}} \ &{}+frac {(1-e^{-alpha ctau h})(1-alpha)}{alpha c}v_{n-m}. end{aligned}\$\$ end{document}vn+1=e−αcτhvn+(e−αcτh−1)u∗+(1−e−αcτh)(vn−m+u∗)e−bvn−m+(1−e−αcτh)(1−α)αcvn−m. Introducing a new variable documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$V_{n}=(v_{n},v_{n-1},ldots,v_{n-m})^{T}\$end{document}Vn=(vn,vn−1,…,vn−m)T, we can rewrite (4.3) as 4.4documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$\$ V_{n+1}=bar{F}(V_{n},tau), \$\$end{document}Vn+1=F¯(Vn,τ), where documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$bar{F}=(bar{F}_{0},bar{F}_{1},ldots,bar{F}_{m})^{T}\$end{document}F¯=(F¯0,F¯1,…,F¯m)T, and 4.5documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$\$ bar{F}_{k}= textstylebegin{cases} e^{-alpha ctau h}v_{n-k}+(e^{-alpha ctau h}-1)u_{ast} +(1-e^{-alpha ctau h})(v_{n-m-k}+u_{ast})e^{-bv_{n-m-k}}\ quad{}+frac{(1-e^{-alpha ctau h})(1-alpha)}{alpha c}v_{n-m-k},quad k=0,\ v_{n-k+1}, quad 1leq k leq m. end{cases} \$\$end{document}F¯k={e−αcτhvn−k+(e−αcτh−1)u∗+(1−e−αcτh)(vn−m−k+u∗)e−bvn−m−k+(1−e−αcτh)(1−α)αcvn−m−k,k=0,vn−k+1,1≤k≤m. Clearly the linear part of map (4.4) is 4.6documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$\$ V_{n+1}=tilde{A}V_{n}. \$\$end{document}Vn+1=A˜Vn. Here 4.7documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$\$ tilde{A}= begin{bmatrix} e^{-alpha ctau h}&0&cdots&0&0&(1-e^{-alpha ctau h})(1-bu_{ast }+frac{1-alpha}{alpha c}) \ 1&0&cdots&0&0&0\ 0&1&cdots&0&0&0\ dots & dots & dots &dots &dots & dots\ 0&0&cdots&1&0&0\ 0&0&cdots&0&1&0 end{bmatrix}. \$\$end{document}A˜=[e−αcτh0⋯00(1−e−αcτh)(1−bu∗+1−ααc)10⋯00001⋯000………………00⋯10000⋯010]. The characteristic equation of Ã is 4.8documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$\$ lambda^{m+1}-e^{-alpha ctau h}lambda^{m}- bigl(1-e^{-alpha ctau h} bigr) biggl(1-bu_{ast}+frac{1-alpha}{alpha c} biggr)=0. \$\$end{document}λm+1−e−αcτhλm−(1−e−αcτh)(1−bu∗+1−ααc)=0. In this section, we discuss direction and stability of the Neimark-Sacker bifurcation in a discrete control system. In Section 4, we obtained conditions for the Neimark-Sacker bifurcation to occur when documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$tau=tau_{k}\$end{document}τ=τk for documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$k=0,1,2,ldots,[frac{m-1}{2}]\$end{document}k=0,1,2,…,[m−12]. In this section we study the direction of the Neimark-Sacker bifurcation and the stability of the bifurcating periodic solutions when documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$tau=tau_{0}\$end{document}τ=τ0, using techniques from normal form and center manifold theory [21, 22]. documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document} \$\$begin{aligned} v_{n+1}={}&e^{-alpha ctau h}v_{n}+ bigl(1-e^{-alpha ctau h} bigr) biggl(1+frac {1-alpha}{alpha c}-bu_{ast} biggr) v_{n-m} \ &{}+frac{1}{2} bigl(1-e^{-alpha ctau h} bigr) bigl(b^{2}u_{ast}-2b bigr)v_{n-m}^{2}+frac{1}{6} bigl(1-e^{-alpha ctau h} bigr) bigl(3b^{2}-b^{3}u_{ast} bigr)v_{n-m}^{3}+O bigl(|x_{n-m}|^{4} bigr). end{aligned}\$\$ end{document}vn+1=e−αcτhvn+(1−e−αcτh)(1+1−ααc−bu∗)vn−m+12(1−e−αcτh)(b2u∗−2b)vn−m2+16(1−e−αcτh)(3b2−b3u∗)vn−m3+O(|xn−m|4). So, we can write system (4.3) as documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$\$ V_{n+1}=tilde{A}V_{n}+frac{1}{2}B(V_{n},V_{n})+ frac {1}{6}C(V_{n},V_{n},V_{n})+O bigl( |V_{n} |^{4} bigr), \$\$end{document}Vn+1=A˜Vn+12B(Vn,Vn)+16C(Vn,Vn,Vn)+O(∥Vn∥4), where documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document} \$\$begin{aligned}& B(V_{n},V_{n})= bigl(b_{0}(V_{n},V_{n}),0, ldots,0 bigr)^{T}, qquad C(V_{n},V_{n},V_{n})= bigl(c_{0}(V_{n},V_{n},V_{n}),0, ldots,0 bigr)^{T}, end{aligned}\$\$ end{document}B(Vn,Vn)=(b0(Vn,Vn),0,…,0)T,C(Vn,Vn,Vn)=(c0(Vn,Vn,Vn),0,…,0)T, and 5.1documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$\$ begin{aligned} &tilde{a}_{0}=e^{-alpha ctau h}, \ &tilde{a}_{1}= bigl(1-e^{-alpha ctau h} bigr) biggl(1+ frac{1-alpha}{alpha c}-bu_{ast} biggr), \ &b_{0}(phi,phi)= bigl(1-e^{-alpha ctau h} bigr) bigl(b^{2}u_{ast}-2b bigr)phi _{m}^{2}= widetilde{b}cdotphi_{m}^{2}, \ & c_{0}(phi,phi,phi)= bigl(1-e^{-alpha ctau h} bigr) bigl(3b^{2}-b^{3}u_{ast } bigr) phi_{m}^{3} =widetilde{c}cdotphi_{m}^{3}. end{aligned} \$\$end{document}a˜0=e−αcτh,a˜1=(1−e−αcτh)(1+1−ααc−bu∗),b0(ϕ,ϕ)=(1−e−αcτh)(b2u∗−2b)ϕm2=b˜⋅ϕm2,c0(ϕ,ϕ,ϕ)=(1−e−αcτh)(3b2−b3u∗)ϕm3=c˜⋅ϕm3. Let documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$q=q(tau_{0})in mathbb{C}^{m+1}\$end{document}q=q(τ0)∈Cm+1 be an eigenvector of Ã corresponding to documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$e^{mathrm{i}omega_{0}}\$end{document}eiω0, then documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$\$tilde{A}q=e^{mathrm{i}omega_{0}}q, qquad tilde{A}overline {q}=e^{-mathrm{i}omega_{0}} overline{q}. \$\$end{document}A˜q=eiω0q,A˜q‾=e−iω0q‾. We also introduce an adjoint eigenvector documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$q^{ast}=q^{ast}(tau)in {mathbb{C}^{m+1}}\$end{document}q∗=q∗(τ)∈Cm+1 having the properties documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$\$tilde{A}^{T}q^{ast}=e^{-mathrm{i}omega_{0}}q^{ast},qquad tilde{A}^{T}overline{q}^{ast}=e^{mathrm{i}omega_{0}} overline {q}^{ast}, \$\$end{document}A˜Tq∗=e−iω0q∗,A˜Tq‾∗=eiω0q‾∗, and satisfying the normalization documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$langle q^{ast},qrangle=1\$end{document}〈q∗,q〉=1, where documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}\$langle q^{ast},qrangle=sum_{i=0}^{m}overline{q}_{i}^{ast}q_{i}\$end{document}〈q∗,q〉=∑i=0mq‾i∗qi. One of the purposes of this section is to test the results in Sections 3-5 by numerical examples; the second one is to show that the hybrid control NSFD numerical algorithm is better than the Euler control method. In this paper, we have developed a hybrid control nonstandard finite-difference (NSFD) scheme by combining state feedback and parameter perturbation for controlling the Neimark-Sacker bifurcation in a discrete nonlinear dynamical system. In Section 3, by applying hybrid control Nicholson’s blowflies equation with delay, we obtain the Hopf bifurcation. In Section 4, compared with the results in Section 3, for any step-size, the hybrid control numerical strategy can delay the onset of an inherent bifurcation when such a bifurcation is undesired (desired) by choosing an appropriate control parameter α. For any step-size, we obtain the consistent dynamical results of the corresponding continuous-time model. In Section 6, numerical examples are provided to illustrate the theoretical results. Applying the Euler control method for sufficiently small step-size, we can also prove the result. We obtain that the NSFD control scheme is better than the Euler control method. There are lots of good prospects in bifurcation and control area. In the future, we can further design a better controller to control the bifurcation of Nicholson’s blowflies equation with delay.   Source: http://doi.org/10.1186/s13662-015-0640-2