Bifurcation Analysis of a Discrete Food Chain Model with Harvesting

We explore existence of fixed points, topological classifications around fixed points, existence of periodic points and prime period, and bifurcation analysis of a three-species discrete food chain model with harvesting. Finally, theoretical results are numerically verified.

Global and Local Behavior of the System of Piecewise Linear Difference Equations xn+1 = |xn| − yn − b and yn+1 = xn − |yn| + 1 Where b ≥ 4

The aim of this article is to study the system of piecewise linear difference equations xn+1=|xn|−yn−b and yn+1=xn−|yn|+1 where n≥0. A global behavior for b=4 shows that all solutions become the equilibrium point. For a large value of |x0| and |y0|, we can prove that (i) if b=5, then the solution becomes the equilibrium point and (ii) if b≥6, then the solution becomes the periodic solution of prime period 5.

Supercritical Neimark–Sacker Bifurcation and Hybrid Control in a Discrete-Time Glycolytic Oscillator Model

We study the local dynamical properties, Neimark–Sacker bifurcation, and hybrid control in a glycolytic oscillator model in the interior of ℝ+2. It is proved that, for all parametric values, Pxy+α/β+α2,α is the unique positive equilibrium point of the glycolytic oscillator model. Further local dynamical properties along with different topological classifications about the equilibrium Pxy+α/β+α2,α have been investigated by employing the method of linearization. Existence of prime period and periodic points of the model under consideration are also investigated. It is proved that, about the fixed point Pxy+α/β+α2,α, the discrete-time glycolytic oscillator model undergoes no bifurcation, except Neimark–Sacker bifurcation. A further hybrid control strategy is applied to control Neimark–Sacker bifurcation in the discrete-time model. Finally, theoretical results are verified numerically.

Global Asymptotic Stability of a Nonautonomous Difference Equation

We study the following nonautonomous difference equation:xn+1=(xnxn-1+pn)/(xn+xn-1),n=0,1,…, wherepn>0is a period-2 sequence and the initial valuesx-1,x0∈(0,∞). We show that the unique prime period-2 solution of the equation above is globally asymptotically stable.

COMPLETE PERIODICITY ANALYSIS FOR A DISCONTINUOUS RECURRENCE EQUATION

A nonlinear three-term recurrence relation arising from seeking the steady states of a cellular neural network with bang bang control is studied. A complete analysis of its periodic behavior is given. In particular, we show that each solution is periodic and its prime period can be determined by two of its consecutive terms. By means of our periodicity analysis, we may then solve the steady state problem which to our knowledge is not solved by other means.

On the Period-Two Cycles ofxn+1=(α+βxn+γxn-k)/(A+Bxn+Cxn-k)

We consider the higher order nonlinear rational difference equationxn+1=(α+βxn+γxn-k)/(A+Bxn+Cxn-k),n=0,1,2,…, where the parametersα,β,γ,A,B,Care positive real numbers and the initial conditionsx-k,…,x-1,x0are nonnegative real numbers,k∈{1,2,…}. We give a necessary and sufficient condition for the equation to have a prime period-two solution. We show that the period-two solution of the equation is locally asymptotically stable.