In this paper, a rational difference equation with positive parameters and non-negative conditions is used to determine the presence and direction of the Neimark–Sacker bifurcation. The neimark–Sacker bifurcation of the system is first studied using the characteristic equation. In addition, we study bifurcation invariant curves from the perspective of normal form theory. A computer simulation is used to illustrate the analytical results.

This research investigates the synchronization of distributed delayed discrete-time fractional-order complex-valued neural networks. The necessary conditions have been established for the stability of the proposed networks using the theory of discrete fractional calculus, the discrete Laplace transform, and the theory of fractional-order discrete Mittag–Leffler functions. In order to guarantee the global asymptotic stability, adequate criteria are determined using Lyapunov’s direct technique, the Lyapunov approach, and some novel analysis techniques of fractional calculation. Thus, some sufficient conditions are obtained to guarantee the global stability. The validity of the theoretical results are finally shown using numerical examples.

We examine the unboundedness, persistence, boundedness, uniqueness, and existence of non-negative equilibrium of an exponential symmetric difference equations system: ${\Omega }_{n+1}={\alpha }_1+{\beta }_1{\Omega }_n+{\gamma }_1{\Omega }_{n-1}{e}^{-({\Omega }_n+{\varpi }_n)}$, ${\varpi }_{n+1}={\alpha }_2+{\beta }_2{\varpi }_n+{\gamma }_2{\varpi }_{n-1}{e}^{-({\Omega }_n+{\varpi }_n)},$$n=0,1,\cdots$, whereby initial values ${\Omega }_{-1},{\varpi }_{-1},{\Omega }_0,{\varpi }_0$ and parameters ${\alpha }_1,{\alpha }_2$ are non-negative real numbers and ${\beta }_1,{\beta }_2\in (0,1)$ and ${\gamma }_1,{\gamma }_2\le 0$. We will discuss asymptotic global and local stability and the convergence rate of this system. Ultimately, to check our results, we set out some numerical explanations.