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Mathematics
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Advances in Dynamical Systems: Stability, Bifurcation, and Chaos with Applications
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Advances in Dynamical Systems: Stability, Bifurcation, and Chaos with Applications

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

Deadline for manuscript submissions: 30 September 2026 | Viewed by 1981

Special Issue Editors

Prof. Dr. Mohammed Salah Abdelouahab

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Guest Editor
Laboratory of Mathematics and Their Interactions, Abdelhafid Boussouf University Center, Mila 43000, Algeria
Interests: dynamical system; chaos; chaos control; chaos synchronization; fractional calculus; biomathematics
Prof. Dr. José Salvador Cánovas

E-Mail Website
Guest Editor
Department of Applied Mathematics and Statisitcs, Technical University of Cartagena, Cartagena, Spain
Interests: dynamical systems; time series and signal analysis; economic dynamics
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. René Lozi

E-Mail Website
Guest Editor
Department of Mathematics, Laboratoire J. A. Dieudonné, Côte d’Azur University, 06103 Nice, CEDEX 2, France
Interests: chaos; dynamical systems; strange attractors; cryptography-based chaos; pseudo random number generators; optimization-based chaos; memristors
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

We are delighted to present this Special Issue that explores the future of nonlinear dynamical systems. It aims to link new theoretical developments with challenging applications spanning all of science and engineering. The notions of stability, bifurcation, and chaos remain as vital today as they have ever been: they help us to predict, understand, and control complex phenomena that evolve over time.

The ideas are now extending, formalizing, and uniting - linking abstract theory with some of the most substantive and practical problems in science, engineering, and beyond.

The topic solicits submissions proposing new method of analysis, computational algorithms, and related rigorous frameworks for describing and manipulating dynamical phenomena.

The interested topics include, but are not limited to:

New Theoretical Developments: New tools to study stability, critical transitions, and chaotic attractors in, stochastic, and fractional-order systems and so on.

Computational and Data-Driven Frontiers: Numerical methods for bifurcation analysis, machine learning techniques for chaos detection, and big-data-based modeling of complex dynamics.

Interdisciplinary Applications:

  • Engineering: Autonomous systems control, aeroelastic phenomena, stability of power grids, and robotics.
  • Biology & Medicine: Modeling Neural Dynamics, Cardiac Rhythms, Ecosystem Resilience, And Infectious Disease Dynamics.
  • Economics & Finance: Analysis of market volatility, economic various tipping points, and agent-based modeling.
  • Climate & Environmental Science: Prediction of climate patterns, forecasting of extreme events, and modeling of ecological shift.

The proposed special issue, combining state-of-the-art theory with significant applications, will be useful to the mathematicians, scientists and engineers. It promotes the exchange of ideas across all boundaries and promotes the advances in predicting, designing and controlling the complex dynamic systems in our world.

Prof. Dr. Mohammed Salah Abdelouahab
Prof. Dr. René Lozi
Prof. Dr. José Salvador Cánovas
Guest Editors

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

  • dynamical systems
  • stability
  • bifurcation
  • chaos
  • computational methods
  • interdisciplinary modeling

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Further information on MDPI's Special Issue policies can be found here.

Published Papers (3 papers)

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Research

25 pages, 1915 KB  
Article
Memristive Hopfield Neural Network with Hidden Multiple Attractors and Its Application in Color Image Encryption
by Zhenhua Hu and Zhuanzheng Zhao
Mathematics 2025, 13(24), 3972; https://doi.org/10.3390/math13243972 - 12 Dec 2025
Abstract
Memristor is widely used to construct various memristive neural networks with complex dynamical behaviors. However, hidden multiple attractors have never been realized in memristive neural networks. This paper proposes a novel chaotic system based on a memristive Hopfield neural network (HNN) capable of [...] Read more.
Memristor is widely used to construct various memristive neural networks with complex dynamical behaviors. However, hidden multiple attractors have never been realized in memristive neural networks. This paper proposes a novel chaotic system based on a memristive Hopfield neural network (HNN) capable of generating hidden multiple attractors. A multi-segment memristor model with multistability is designed and serves as the core component in constructing the memristive Hopfield neural network. Dynamical analysis reveals that the proposed network exhibits various complex behaviors, including hidden multiple attractors and a super multi-stable phenomenon characterized by the coexistence of infinitely many double-chaotic attractors—these dynamical features are reported for the first time in the literature. This encryption process consists of three key steps. Firstly, the original chaotic sequence undergoes transformation to generate a pseudo-random keystream immediately. Subsequently, based on this keystream, a global permutation operation is performed on the image pixels. Then, their positions are disrupted through a permutation process. Finally, bit-level diffusion is applied using an Exclusive OR(XOR) operation. Relevant research shows that these phenomena indicate a high sensitivity to key changes and a high entropy level in the information system. The strong resistance to various attacks further proves the effectiveness of this design. Full article
(This article belongs to the Special Issue Advances in Dynamical Systems: Stability, Bifurcation, and Chaos with Applications)
19 pages, 4702 KB  
Article
How Far Can We Trust Chaos? Extending the Horizon of Predictability
by Alexandros K. Angelidis, Georgios C. Makris, Evangelos Ioannidis, Ioannis E. Antoniou and Charalampos Bratsas
Mathematics 2025, 13(23), 3851; https://doi.org/10.3390/math13233851 - 1 Dec 2025
Viewed by 332
Abstract
Chaos reveals a fundamental paradox in the scientific understanding of Complex Systems. Although chaotic models may be mathematically deterministic, they are practically non-determinable due to the finite precision that is inherent in all computational machines. Beyond the horizon of predictability, numerical computations accumulate [...] Read more.
Chaos reveals a fundamental paradox in the scientific understanding of Complex Systems. Although chaotic models may be mathematically deterministic, they are practically non-determinable due to the finite precision that is inherent in all computational machines. Beyond the horizon of predictability, numerical computations accumulate errors, often undetectable. We investigate the possibility of reliable (error-free) time series of chaos. We prove that this is feasible for two well-studied isomorphic chaotic maps, namely the Tent map and the Logistic map. The generated chaotic time series have an unlimited horizon of predictability. A new linear formula for the horizon of predictability of the Analytic Computation of the Logistic map, for any given precision and acceptable error, is obtained. Reliable (error-free) time series of chaos serve as the “gold standard” for chaos applications. The practical significance of our findings include: (i) the ability to compare the performance of neural networks that predict chaotic time series; (ii) the reliability and numerical accuracy of chaotic orbit computations in encryption, maintaining high cryptographic strength; and (iii) the reliable forecasting of future prices in chaotic economic and financial models. Full article
(This article belongs to the Special Issue Advances in Dynamical Systems: Stability, Bifurcation, and Chaos with Applications)
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18 pages, 7332 KB  
Article
On Fractional Discrete-Time Computer Virus Model: Stability, Bifurcation, Chaos and Complexity Analysis
by Omar Kahouli, Imane Zouak, Adel Ouannas, Lilia El Amraoui and Mohamed Ayari
Mathematics 2025, 13(20), 3272; https://doi.org/10.3390/math13203272 - 13 Oct 2025
Viewed by 393
Abstract
Computer viruses continue to threaten the security of digital networks, and their complex propagation dynamics require refined modelling tools. Most existing models rely on integer-order dynamics or assume uniform memory effects, which limit their ability to capture heterogeneous behaviours observed in practice. To [...] Read more.
Computer viruses continue to threaten the security of digital networks, and their complex propagation dynamics require refined modelling tools. Most existing models rely on integer-order dynamics or assume uniform memory effects, which limit their ability to capture heterogeneous behaviours observed in practice. To address this gap, we propose a discrete incommensurate fractional-order virus model based on Caputo-like delta differences, where each compartment is assigned a distinct fractional order to represent mismatched time scales. The model’s dynamics are analysed in terms of stability, bifurcation, and chaos. Numerical results reveal the emergence of rich chaotic attractors, emphasizing the impact of fractional memory on system evolution. To quantify complexity, we employ Approximate Entropy and Spectral Entropy and relate these indicators to the maximum Lyapunov exponent, confirming the system’s sensitivity and unpredictability. All numerical simulations and visualizations were performed using MATLAB (R2015a). The findings highlight the importance of heterogeneous memory in computer-virus modeling and offer new insights for developing theoretical foundations of robust cybersecurity strategies. Full article
(This article belongs to the Special Issue Advances in Dynamical Systems: Stability, Bifurcation, and Chaos with Applications)
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