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Nonlinear Dynamics, Chaos, and Mathematical Physics
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Nonlinear Dynamics, Chaos, and Mathematical Physics

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C2: Dynamical Systems".

Deadline for manuscript submissions: closed (31 January 2026) | Viewed by 5803

Special Issue Editor

Prof. Dr. Masoud Asadi-Zeydabadi

E-Mail Website
Guest Editor
Department of Physics, University of Colorado Denver, 1201 Larimer St., Denver, CO 80204, USA
Interests: nonlinear dynamics and chaos; biological modeling and medical physics

Special Issue Information

Dear Colleagues, 

The applications of nonlinear dynamics and chaos theory span a wide range of disciplines, offering valuable insights into complex systems and phenomena. In physics, chaotic behavior is observed in systems such as turbulent fluid flow and celestial mechanics. In engineering, the study of chaos aids in designing robust and efficient systems, particularly in fields like control theory and signal processing. Biological systems, such as neural networks, exhibit nonlinear dynamics, and chaos theory helps unravel the intricate patterns underlying these phenomena. Additionally, economics and finance use chaos theory, as it provides a framework for understanding the unpredictable nature of markets and economic systems. Disease dynamics, weather forecasting, ecological dynamics, and even social systems can be analyzed through the lens of nonlinear dynamics, highlighting the versatility and applicability of this theoretical framework across diverse scientific and practical domains.

Topics include, but are not limited to, the following:

  • Control Systems and Robotics;
  • Hamiltonian Chaos;
  • Fluid Dynamics and Turbulence;
  • Weather Forecasting and Climate Modeling;
  • Neural Networks and Brain Dynamics;
  • Communication Systems and Signal Processing;
  • Coupled Nonlinear Oscillators and Synchronization;
  • Epidemiology and Disease Dynamics;
  • Ecological Systems and Population Dynamics;
  • Financial Markets and Economic Systems;
  • Nonlinear Data  Analysis;
  • Topology and Chaos.

Prof. Dr. Masoud Asadi-Zeydabadi
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 250 words) can be sent to the Editorial Office for assessment.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • control systems
  • robotics
  • chaos
  • dynamics
  • signal processing
  • data analysis
  • topology

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Published Papers (7 papers)

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Research

20 pages, 375 KB  
Article
Higher-Order Fuzzy Difference Equations: Existence, Stability, and Illustrative Numerical Examples
by Hashem Althagafi and Ahmed Ghezal
Mathematics 2026, 14(6), 1051; https://doi.org/10.3390/math14061051 - 20 Mar 2026
Viewed by 226
Abstract
This paper examines the dynamics of positive solutions to a system of fuzzy difference equations, which provide effective tools for modeling dynamical systems with uncertain or imprecise parameters. The main objective is to establish the existence, uniqueness, and qualitative properties of positive solutions [...] Read more.
This paper examines the dynamics of positive solutions to a system of fuzzy difference equations, which provide effective tools for modeling dynamical systems with uncertain or imprecise parameters. The main objective is to establish the existence, uniqueness, and qualitative properties of positive solutions within a fuzzy framework. After recalling some fundamental notions from fuzzy set theory, we analyze the dynamics of the proposed system. The main results prove the existence of a unique positive fuzzy solution under suitable conditions and establish the boundedness, continuity, and convergence of the solutions. In particular, all solutions converge to a unique positive equilibrium point. Numerical experiments for (l1,l2)=(2,3) and (l1,l2)=(4,1) with uncertainty levels γ=0.2 and γ=0.8 illustrate the theoretical results and confirm the convergence toward the unique positive equilibrium. Full article
(This article belongs to the Special Issue Nonlinear Dynamics, Chaos, and Mathematical Physics)
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18 pages, 334 KB  
Article
Three-Dimensional Second-Order Rational Difference Equations: Explicit Formulas and Simulations
by Ahmed Ghezal, Hassan J. Al Salman and Ahmed A. Al Ghafli
Mathematics 2026, 14(5), 876; https://doi.org/10.3390/math14050876 - 5 Mar 2026
Viewed by 309
Abstract
This paper studies a three-dimensional second-order rational difference system, which generalizes earlier scalar and bidimensional models. We derive explicit closed-form solutions for general initial conditions and identify special cases that simplify the system’s structure. These explicit solutions are particularly significant, as they not [...] Read more.
This paper studies a three-dimensional second-order rational difference system, which generalizes earlier scalar and bidimensional models. We derive explicit closed-form solutions for general initial conditions and identify special cases that simplify the system’s structure. These explicit solutions are particularly significant, as they not only enable rigorous stability analysis but also provide a precise analytical characterization of the system’s long-term behavior, offer deeper insight into its underlying periodic structure, and establish a solid theoretical foundation for potential future applications in the control, prediction, and optimization of multivariable dynamical systems. The analysis carefully addresses conditions ensuring well-defined solutions, avoiding singularities. Numerical simulations illustrate various dynamic behaviors, including oscillations, convergence, and sensitivity to parameters, confirming the richness and robustness of the system’s temporal evolution. Full article
(This article belongs to the Special Issue Nonlinear Dynamics, Chaos, and Mathematical Physics)
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33 pages, 6069 KB  
Article
Stability and Bifurcation Analysis of a Discrete Tumor-Immune System with Allee Effects
by Messaoud Berkal, Mohammed Bakheet Almatrafi, Samir Azioune and Mohammed-Salah Abdelouahab
Mathematics 2026, 14(4), 713; https://doi.org/10.3390/math14040713 - 18 Feb 2026
Viewed by 343
Abstract
Differential equations are usually employed to accurately represent the ongoing relationships between tumor cells and immune effector populations, enabling scientists to discover how variation in growth and response rates affects tumor development or elimination. The essential objective of this work is to analyze [...] Read more.
Differential equations are usually employed to accurately represent the ongoing relationships between tumor cells and immune effector populations, enabling scientists to discover how variation in growth and response rates affects tumor development or elimination. The essential objective of this work is to analyze the dynamical development of a discrete tumor-immune interaction model, with a particular focus on finding out how the combined effects of tumor growth and immune response influence tumor progression. The forward Euler approach is effectively used to discretize the governed system. The bifurcation theory is used to establish the fixed points of the considered system, the stability about the fixed points, and Neimark–Sacker and period-doubling bifurcations. We identify parameter domains that result in tumor existence, restricted oscillations, or full-tumor elimination utilizing stability evaluation, bifurcation examination, and computational simulations. In addition, the 0–1 test is presented. Chaos control is also developed. This article successfully discusses some numerical simulations to verify the results obtained. In general, the research gives an overall insight into this interaction and highlights the circumstances under which the immune system is capable of suppressing or removing tumor cells. Full article
(This article belongs to the Special Issue Nonlinear Dynamics, Chaos, and Mathematical Physics)
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13 pages, 267 KB  
Article
Solvability of Three-Dimensional Nonlinear Difference Systems via Transformations and Generalized Fibonacci Recursions
by Yasser Almoteri and Ahmed Ghezal
Mathematics 2025, 13(24), 3904; https://doi.org/10.3390/math13243904 - 5 Dec 2025
Cited by 1 | Viewed by 448
Abstract
This paper presents closed-form solutions for a three-dimensional system of nonlinear difference equations with variable coefficients. The approach employs functional transformations and leverages generalized Fibonacci sequences to construct the solutions explicitly. These solutions reveal a profound connection to generalized Fibonacci recursions. The proposed [...] Read more.
This paper presents closed-form solutions for a three-dimensional system of nonlinear difference equations with variable coefficients. The approach employs functional transformations and leverages generalized Fibonacci sequences to construct the solutions explicitly. These solutions reveal a profound connection to generalized Fibonacci recursions. The proposed method is based on sophisticated mathematical transformations that reduce the complex nonlinear system to a solvable linear form, followed by the derivation of general solutions through iterative techniques and harmonic analysis. Furthermore, we extend our results to a generalized class of systems by introducing flexible functional transformations, while rigorously maintaining the required regularity conditions. The findings demonstrate the effectiveness of this methodology in addressing a broad class of complex nonlinear systems and open new perspectives for modeling multivariate dynamical phenomena. The analysis further reveals two distinct dynamical regimes—an unbounded oscillatory growth phase and a bounded cyclic equilibrium—arising from the relative magnitude of the variable coefficients, thereby highlighting the method’s capacity to characterize both amplifying and self-regulating behaviors within a unified analytical framework. Full article
(This article belongs to the Special Issue Nonlinear Dynamics, Chaos, and Mathematical Physics)
23 pages, 498 KB  
Article
On the Existence and Uniqueness of Two-Dimensional Nonlinear Fuzzy Difference Equations with Logarithmic Interactions
by Yasser Almoteri and Ahmed Ghezal
Mathematics 2025, 13(21), 3532; https://doi.org/10.3390/math13213532 - 4 Nov 2025
Cited by 2 | Viewed by 590
Abstract
This paper investigates a new class of two-dimensional fuzzy difference equations that integrate logarithmic nonlinearities with interaction effects between system variables. Motivated by the need to model complex dynamical systems influenced by uncertainty and interdependencies, we propose a system that extends existing one-dimensional [...] Read more.
This paper investigates a new class of two-dimensional fuzzy difference equations that integrate logarithmic nonlinearities with interaction effects between system variables. Motivated by the need to model complex dynamical systems influenced by uncertainty and interdependencies, we propose a system that extends existing one-dimensional models to capture more realistic interactions within a discrete-time framework. Our approach employs the characterization theory to transform the fuzzy system into an equivalent family of classical difference equations, thereby facilitating a rigorous analysis of the existence, uniqueness, and boundedness of positive solutions. To support the theoretical findings, two numerical examples are provided, illustrating the model’s capacity to capture complex dynamical patterns under fuzzy conditions. An application to a fuzzy population growth model illustrates how the model captures both interaction effects and uncertainty while ensuring well-defined and stable solutions. Numerical simulations show that, for instance, with α=0.10, β=δ=1.0, γ=0.08, and ρx=ρy=0.10, the trajectories of (xt,yt) rapidly converge toward a stable fuzzy equilibrium, with uncertainty bands confirming the positivity and boundedness of the solutions. Full article
(This article belongs to the Special Issue Nonlinear Dynamics, Chaos, and Mathematical Physics)
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17 pages, 283 KB  
Article
Closed-Form Solutions of a Nonlinear Rational Second-Order Three-Dimensional System of Difference Equations
by Messaoud Berkal, Taha Radwan, Mehmet Gümüş, Raafat Abo-Zeid and Karim K. Ahmed
Mathematics 2025, 13(20), 3327; https://doi.org/10.3390/math13203327 - 18 Oct 2025
Viewed by 644
Abstract
In this paper, we investigate the behavior of solutions to a nonlinear system of rational difference equations of order two, defined by [...] Read more.
In this paper, we investigate the behavior of solutions to a nonlinear system of rational difference equations of order two, defined by xn+1=xnyn−1yn(a+bxnyn−1),yn+1=ynzn−1zn(c+dynzn−1),zn+1=znxn−1xn(e+fznxn−1), where n denotes a nonzero integer; the parameters a,b,c,d,e,f are real constants; and the initial conditions x−1,x0,y−1,y0,z−1,z0 are nonzero real numbers. By applying a suitable variable transformation, we reduce the original coupled system to three independent rational difference equations. This allows for separate analysis using established methods for second-order nonlinear recurrence relations. We derive explicit solutions and examine the qualitative behavior, including boundedness and periodicity, under different conditions. Our findings contribute to the theory of rational difference equations and offer insights for higher-order systems in applied sciences. Full article
(This article belongs to the Special Issue Nonlinear Dynamics, Chaos, and Mathematical Physics)
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20 pages, 312 KB  
Article
Analytical Study of Nonlinear Systems of Higher-Order Difference Equations: Solutions, Stability, and Numerical Simulations
by Hashem Althagafi and Ahmed Ghezal
Mathematics 2024, 12(8), 1159; https://doi.org/10.3390/math12081159 - 12 Apr 2024
Cited by 7 | Viewed by 1935
Abstract
This paper aims to derive analytical expressions for solutions of fractional bidimensional systems of difference equations with higher-order terms under specific parametric conditions. Additionally, formulations of solutions for one-dimensional equations derived from these systems are explored. Furthermore, rigorous proof is provided for the [...] Read more.
This paper aims to derive analytical expressions for solutions of fractional bidimensional systems of difference equations with higher-order terms under specific parametric conditions. Additionally, formulations of solutions for one-dimensional equations derived from these systems are explored. Furthermore, rigorous proof is provided for the local stability of the unique positive equilibrium point of the proposed systems. The theoretical findings are validated through numerical examples using MATLAB, facilitating graphical illustrations of the results. Full article
(This article belongs to the Special Issue Nonlinear Dynamics, Chaos, and Mathematical Physics)
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