This paper investigates an autonomous discrete-time glycolytic oscillator model with a unique positive equilibrium point which exhibits chaos in the sense of Li–Yorke in a certain region of the parameters. We use Marotto’s theorem to prove the existence of chaos by finding a snap-back repeller. The illustration of the results is presented by using numerical simulations.

This paper investigates the local and global character of the unique positive equilibrium of a mixed monotone fractional second-order difference equation with quadratic terms. The corresponding associated map of the equation decreases in the first variable, and it can be either decreasing or increasing in the second variable depending on the corresponding parametric values. We use the theory of monotone maps to study global dynamics. For local stability, we use the center manifold theory in the case of the non-hyperbolic equilibrium point. We show that the observed equation exhibits three types of global behavior characterized by the existence of the unique positive equilibrium, which can be locally stable, non-hyperbolic when there also exist infinitely many non-hyperbolic and stable minimal period-two solutions, and a saddle. Numerical simulations are carried out to better illustrate the results.

We investigate the local and global character of the unique equilibrium point and boundedness of the solutions of certain homogeneous fractional difference equation with quadratic terms. Also, we consider Neimark–Sacker bifurcations and give the asymptotic approximation of the invariant curve.

We investigate global dynamics of the equation xn+1 = xn−1+F axn+ f , n= 0,1,2, ..., where the parameters a,F and f are positive numbers and the initial conditions x−1,x0 are arbitrary nonnegative numbers such that x−1+x0 > 0. The existence and local stability of the unique positive equilibrium are a nalyzed algebraically. We characterize the global dynamics of this equation with th e basins of attraction of its equilibrium point and periodic solutions.

We investigate the local and global character of the unique equilibrium point of certain homogeneous fractional difference equation with quadratic terms. The existence of the period-two solution in one special case is given. Also, in this case the local and global stability of the minimal period-two solution for some special values of the parameters are given. AMS Subject Classifications: 39A10, 39A20, 39A23, 39A30.

By using the Kolmogorov–Arnold–Moser theory, we investigate the stability of the equilibrium solution of the difference equation un+1=A+Bun+un2(1+Dun)un−1,n=0,1,2,… where A,B,D > 0,u−1,u0>0. We also use the symmetries to find effectively the periodic solutions with feasible periods. Copyright © 2016 John Wiley & Sons, Ltd.

We investigate the basins of attraction of equilibrium points and minimal period-two solutions of the difference equation of the form x n+1 = x n−1 2/(ax n 2 + bx n x n−1 + cx n−1 2), n = 0,1, 2,…, where the parameters a,  b, and  c are positive numbers and the initial conditions x −1 and x 0 are arbitrary nonnegative numbers. The unique feature of this equation is the coexistence of an equilibrium solution and the minimal period-two solution both of which are locally asymptotically stable.