Advances in Dynamical Systems and Control

1. Introduction

In this Editorial, we present “Advances in Dynamical Systems and Control”, a Special Issue of Axioms. This Special Issue comprises 10 articles contributing frontier research in the areas of dynamical systems and control, both in theoretical and application advances. The study of dynamical systems and control is crucial for advancing engineering. It encompasses a wide range of topics, including chaos and bifurcations, complex systems, fractional difference and differential equations, fuzzy control and systems, linear and non-linear control systems, mathematical education in science and engineering, matrix and spectral analysis, modeling, stability and robust stability, as well as the stability of pseudo-polynomials and quasi-polynomials. Therefore, this Special Issue aims to address issues pertaining to the above fields through articles concerned with a variety of related topics.

2. Overview of the Published Papers

In contribution 1, entitled “Chaotic Steady States of the Reinartz Oscillator: Mathematical Evidence and Experimental Confirmation”, the Reinartz sinusoidal oscillator is analyzed to study its chaotic steady states and solve the chaos and hyperchaos localization. The oscillator is considered in its conventional topology. The results show that a pair of positive Lyapunov exponents are sufficient to verify that physically reasonable circuit values yield robust dynamical behavior. All the necessary fingerprints of structural stable chaos are proven via the numerical results, and the dynamics are compared with the strange attractor captured as oscilloscope screenshots.

In contribution 2, entitled “Fractal Fractional Derivative Models for Simulating Chemical Degradation in a Bioreactor”, a three-equation differential mathematical model is presented to describe the degradation of a phenol and p-cresol combination in a continually agitated bioreactor. The authors conducted a stability analysis of the model’s equilibrium points and used three alternative kernels to analyze the model with fractal–fractional derivatives, exploring the effects of the fractal size and fractional order. They developed highly efficient numerical techniques for the concentration of biomass, phenol, and p-cresol. To complete the study, numerical simulations were used to illustrate the accuracy of the suggested method.

In contribution 3, entitled “On Population Models with Delays and Dependence on Past Values”, the authors present a study on methods for adding dependence onto past values in population dynamics models. The studied methods include the following: (i) populations at earlier time units, (ii) the use of non-local operators in the model descriptions, and (iii) the introduction of exposed population groups. The authors conclude that modeling assumptions should be clearly stated when using fractional derivatives.

In contribution 4, entitled “Simpson’s Variational Integrator for Systems with Quadratic Lagrangians”, the authors proposed a variational symplectic integrator, which is then compared with the Newmark’s variational integrator. The proposed scheme is implicit, symplectic, and conditionally stable. The precision and convergence of the proposed integrator are illustrated via simulations.

In contribution 5, entitled “The Existence of Li–Yorke Chaos in a Discrete-Time Glycolytic Oscillator Model”, the existence of chaos is proven by finding a snap-back repeller, using Marotto’s theorem. The study was performed for an autonomous discrete-time glycolytic oscillator model, which exhibits chaos in the Li-Yorke sense.

In contribution 6, entitled “Dynamical Behaviors of Stochastic SIS Epidemic Model with Ornstein–Uhlenbeck Process”, for an incomplete inoculation stochastic SIS epidemic model perturbed by the Ornstein–Uhlenbeck and Brownian motion, the existence of a unique global solution is established and control conditions for extinction are derived. The authors established sufficient conditions for the existence of stationary distribution via two Lyapunov functions and the ergodicity of the Ornstein–Uhlenbeck process.

Contribution 7, entitled “Robust State Feedback Control with D-Admissible Assurance for Uncertain Discrete Singular Systems”, addresses the state feedback control associated with D-admissible assurance for discrete singular systems subjected to parameter uncertainties in both the difference term and system matrices. A refined analysis criterion of D-admissible assurance is reported, where the distinct form embraces multiple slack matrices and reduces linear matrix inequality (LMI) constraints, which may be beneficial for reducing conservatism in admissibility analysis.

In contribution 8, entitled “Parameters Determination via Fuzzy Inference Systems for the Logistic Populations Growth Model”, the problem of determining parameters for the logistic population growth model is addressed. Unlike traditional schemes, the proposed approach incorporates ecosystem variables as inputs into a fuzzy inference system designed to capture the inherent uncertainties of population dynamics. As the resulting model uses fuzzy numbers as coefficients, it is represented by a fuzzy differential equation.

In contribution 9, entitled “Dynamics of a Fractional-Order Eco-Epidemiological Model with Two Disease Strains in a Predator Population Incorporating Harvesting”, the authors formulated and analyzed a fractional-order eco-epidemical model, which considers two disease strains in a predator population. They examined the positivity, boundedness, existence, and uniqueness of the solutions. In the model’s formulation, the population is considered to comprise three groups: susceptible predators infected by the first disease, predators infected by the second disease, and a prey population.

Contribution 10, entitled “Exponential Stability for a Degenerate/Singular Beam-Type Equation in Non-Divergence Form”, presents a stability analysis for a degenerate/singular beam equation in non-divergence form. The authors employed energy methods to derive stability conditions for the problem under consideration.