The importance of difference equations cannot be overemphasized. These equations model discrete physical phenomena on one hand and are integral part of numerical schemes used to solve differential equations, on the other hand. This widens the applicability of such equations to many branches of scientific knowledge. Discrete dynamical systems are described by difference equations and potentially have applications in probability theory, economics, biology, computer science, control engineering, genetics, signal processing, population dynamics, health sciences, ecology, physiology, physics, etc.

This special issue provides a platform to disseminate original research in the fields of difference equations, discrete dynamical systems, and bifurcation theory. This was an excellent opportunity for researchers to share their findings with the scientific community.

All manuscripts submitted to this special issue have been reviewed through peer-reviewing process. Based on the reviewers’ reports, 11 out of 26 original research articles have been accepted for publication in this well-reputed Journal. A brief summary of each published article in this special issue by providing a short editorial note has also been presented as follows.

In the paper “Dynamics and Stability Analysis of a Brucellosis Model with Two Discrete Delays,” P. O. Lolika and S. Mushayabasa have investigated the dynamics and stability analysis of a Brucellosis Model with two discrete delays in which first delay represents the incubation period while the second accounts for the time needed to detect and cull infectious animals. Feasibility and stability of the model steady states have been investigated both analytically and numerically. Furthermore, the occurrence of Neimark-Sacker bifurcation has also been investigated.

In the research article “Dynamics Analysis and Control of a Five-Term Fractional-Order System,” L. Yang and X. Liu have proposed a new fractional-order chaotic system with five terms. They have studied the stability about equilibria of the model. Moreover, rich dynamics with interesting characteristics have been demonstrated by phase portraits and bifurcation diagrams numerically.

In the research article “An Improved Computationally Efficient Method for Finding the Drazin Inverse,” H. B. Jebreen and Y. Chalco-Cano have proposed a computationally effective iterative scheme for finding the Drazin inverse. The convergence has been investigated analytically by applying a suitable initial matrix.

In the research article “Stability Analysis and Control Optimization of a Prey-Predator Model with Linear Feedback Control,” Y. Li et al. have provided an appropriate balance between the chemical and biological control and therefore a Smith predator-prey system has also been established for integrated pest management. They have studied the existence and uniqueness of the order-one periodic solution by means of the subsequent function method to confirm the feasibility of the biological and chemical control strategy of pest management. Furthermore, the stability of the system has also been proved by the limit method of the successor points’ sequences and the analogue of the Poincare criterion.

In the original research article “Global Asymptotic Stability and Naimark-Sacker Bifurcation of Certain Mix Monotone Difference Equation,” M. R. S. Kulenovic et al. have investigated the local and global dynamics about equilibrium point of second-order rational difference equations with positive parameters and initial conditions. Finally it has also been investigated that the difference equation has undergone a Neimark-Sacker bifurcation resulting in the existence of the locally stable periodic solution of unknown period.

In the article “Stochastic P-Bifurcation of a Bistable Viscoelastic Beam with Fractional Constitutive Relation under Gaussian White Noise,” Y. Li et al. have studied the stochastic P-bifurcation problem for axially moving of a bistable viscoelastic beam with fractional derivatives of high order nonlinear terms under Gaussian white noise excitation. They have shown that the fractional derivative term is equivalent to a linear combination of the damping force and restoring force so that the original system can be simplified to an equivalent system by principle for minimum mean square error. They have also obtained the stationary Probability Density Function (PDF) of the system’s amplitude by stochastic averaging and critical parametric condition for stochastic P-bifurcation of amplitude by singularity theory. Finally, Y. Li et al. have analyzed the types of the stationary PDF curves of the system qualitatively by choosing parameters corresponding to each region within the transition set curve.

In the paper titled “Exponential Stability and Robust Control for Discrete-Time Time-Delay Infinite Markov Jump Systems,” Y. Liu and T. Hou have investigated the exponential stability and robust control problem for a class of discrete-time time-delay stochastic systems with infinite Markov jump and multiplicative noises where the jumping parameters are modeled as an infinite-state Markov chain. They have also derived the new sufficient condition in terms of matrix inequalities to guarantee the mean square exponential stability of the equilibrium point by using a novel Lyapunov-Krasovskii functional, and then some sufficient conditions for the existence of feedback control have also been presented to guarantee that the resulting closed-loop system has mean square exponential stability for the zero exogenous disturbance and satisfies a prescribed performance level. Numerical simulations have also been presented to verify theoretical results.

In the article “New Qualitative Results for Solutions of Functional Differential Equations of Second Order,” C. Tunç and S. Erdur have studied the existence of periodic solutions, stability of zero solution, asymptotic stability of zero solution, square integrability of the first derivative of solutions, and boundedness of solutions of nonlinear functional differential equations of second-order by the second method of Lyapunov. They have also obtained the sufficient conditions guaranteeing the existence of periodic solutions, stability of zero solution, asymptotic stability of zero solution, square integrability of the first derivative of solutions, and boundedness of solutions of the equations thus considered.

In the article “Bifurcations of a New Fractional-Order System with a One-Scroll Chaotic Attractor,” X. Liu et al. have presented a new fractional-order system which has shown a chaotic attractor of the one-scroll structure. They have also investigated the stability analysis about equilibrium points and determined the generation conditions of the one-scroll structure for the attractor based on the stability analysis. Furthermore, in a commensurate-order case, bifurcations with the variation of a system parameter have also been investigated as derivative orders decrease from 0.99. In an incommensurate-order case, bifurcations with the variation of a derivative order have also been analyzed as other orders are decreased from 1.

In the article “The Impact of User Behavior on Information Diffusion in D2D Communications: A Discrete Dynamical Model,” C. Gan et al. have explored the impact of user behavior on information diffusion in D2D (Device-to-Device) communications. A discrete dynamical model, which combined network metrics and user behaviors, including social relationship, user influence, and interest, has been proposed and analyzed. Specifically, combined with social tie and user interest, the success rate of data dissemination between D2D users has been described, and the interaction factor, user influence, and stability factor have also been defined. Furthermore, the state transition process of user has been depicted by a discrete-time Markov chain, and global stability analysis of the proposed model has also been performed.

Finally, in “Double Delayed Feedback Control of a Non-Linear Finance System,” Z. Jiang et al. have investigated a class of chaotic finance system with double delayed feedback control. Specifically, they have studied the stability analysis about equilibrium, existence of periodic solutions, and properties of the branching periodic solutions by using center manifold theory.

Conflicts of Interest

The editors declare that they have no conflicts of interest regarding the publication of this Special Issue.


We are greatly acknowledging the contribution of Professor Dr. Muhammad Naeem Qureshi (Late) who improved the quality of the initial proposal for this Special Issue. We would like to pay a great homage to all of the authors for their valuable contributions rendered in this respect and also to the reviewers for their valuable suggestions made in the evaluation of the papers during the reviewing process.

A. Q. Khan
Tarek F. Ibrahim