Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
4.6
2.2
Journals
Mathematics
Special Issues
Advances in Dynamical Systems: Stability, Bifurcation, and Chaos with Applications
share announcement

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 5745

Special Issue Editors

Prof. Dr. Mohammed Salah Abdelouahab

E-Mail Website
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. 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
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
Dr. Erivelton Nepomuceno

E-Mail Website
Guest Editor
Department of Electronic Engineering, Maynooth University, Maynooth, Ireland
Interests: chaotic cryptography; circuits and systems; green computing; ocean energy; system identification

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
Dr. Erivelton Nepomuceno
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

Benefits of Publishing in a Special Issue

  • Ease of navigation: Grouping papers by topic helps scholars navigate broad scope journals more efficiently.
  • Greater discoverability: Special Issues support the reach and impact of scientific research. Articles in Special Issues are more discoverable and cited more frequently.
  • Expansion of research network: Special Issues facilitate connections among authors, fostering scientific collaborations.
  • External promotion: Articles in Special Issues are often promoted through the journal's social media, increasing their visibility.
  • Reprint: MDPI Books provides the opportunity to republish successful Special Issues in book format, both online and in print.

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

Published Papers (5 papers)

Download All Papers
Order results
Result details
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:

Research

28 pages, 14954 KB  
Article
Time-Reversible Synchronization of Chua Circuits for Edge Intelligent Sensors
by Artur Karimov, Kirill Shirnin, Ivan Babkin, Pavel Burundukov, Vyacheslav Rybin and Denis Butusov
Mathematics 2026, 14(8), 1359; https://doi.org/10.3390/math14081359 - 18 Apr 2026
Viewed by 197
Abstract
Time-reversible synchronization (TRS) of nonlinear oscillators is a recently proposed technique that ensures super-exponential convergence of dynamics between master and slave systems, which is beneficial in many real-time applications. Nevertheless, this approach has not been demonstrated in any real-time embedded system to practically [...] Read more.
Time-reversible synchronization (TRS) of nonlinear oscillators is a recently proposed technique that ensures super-exponential convergence of dynamics between master and slave systems, which is beneficial in many real-time applications. Nevertheless, this approach has not been demonstrated in any real-time embedded system to practically verify it and quantitatively estimate its advantages. Furthermore, previous studies did not consider the application of time-reversible synchronization to a wide, practically relevant class of chaotic systems with piecewise-linear nonlinearity. To fill these gaps, in this work, we developed an FPGA-based time-reversible synchronization controller for the analog Chua circuit and its digital counterpart. To achieve complete synchronization, we first reconstructed dynamical equations of the circuit. Then, we performed a rigorous theoretical analysis of synchronization possibility between analog and digital systems by each single variable. Next, we implemented the digital model of the Chua circuit in the MyRIO-1900 FPGA using the reconstructed dynamical model and showed its capability of digital-to-analog and analog-to-digital conventional Pecora–Carroll (PC) synchronization. Then, an algorithm of time-reversible synchronization on MyRIO-1900 was tested, achieving complete synchronization at the predefined normalized RMSE level of 0.01, requiring an average of 8.0 fewer points and a median of 10.1 fewer points than the PC synchronization. Finally, we implemented a proof-of-concept version of a capacitive sensor based on the analog Chua circuit with an FPGA-based observer using PC synchronization or the TRS algorithm with a heuristic selection of a starting point. Our experiments reveal that when using the TRS algorithm, the time needed to detect a pre-selected 3% level of capacitance change is reduced by a mean factor of 4 and a median factor of 4.9 in comparison with the conventional PC synchronization. This allows for using the developed solution in applications where the synchronization rate is crucial, including chaos-based sensing, communication, and monitoring. Full article
(This article belongs to the Special Issue Advances in Dynamical Systems: Stability, Bifurcation, and Chaos with Applications)
►▼ Show Figures

Figure 1

38 pages, 812 KB  
Article
Basin of Attraction Analysis in Piecewise-Linear Systems with Big-Bang Bifurcation for the Period-Increment Phenomenon
by Juan Carlos Vargas Bernal, Simeón Casanova Trujillo and Diego A. Londoño Patiño
Mathematics 2026, 14(2), 379; https://doi.org/10.3390/math14020379 - 22 Jan 2026
Viewed by 459
Abstract
This paper investigates the basins of attraction of periodic orbits arising in one-dimensional piecewise-linear discrete dynamical systems as the system parameters vary in a neighborhood of a Big-Bang bifurcation point associated with the period-increment phenomenon. In this setting, the Big-Bang point corresponds to [...] Read more.
This paper investigates the basins of attraction of periodic orbits arising in one-dimensional piecewise-linear discrete dynamical systems as the system parameters vary in a neighborhood of a Big-Bang bifurcation point associated with the period-increment phenomenon. In this setting, the Big-Bang point corresponds to a parameter value through which infinitely many bifurcation curves pass, leading to the successive emergence of periodic orbits whose periods increase incrementally. The analysis is carried out using a fully analytical approach, exploiting the one-dimensional nature of the system and the occurrence of border-collision bifurcations. Within this framework, we construct analytical sequences that characterize the convergence of any initial condition on the real line toward a periodic point belonging to a periodic orbit, either isolated or coexisting with another periodic orbit. As the main results, we explicitly characterize the basins of attraction of periodic orbits generated in the period-increment Big-Bang scenario and provide explicit analytical conditions on the system parameters for the existence of these periodic orbits. Moreover, we show that, in certain regions of the parameter plane, at most two periodic orbits can coexist, and we describe explicitly the structure of their corresponding basins of attraction. This work provides a new analytical perspective on basin organization in piecewise-linear systems exhibiting the period-increment phenomenon. Full article
(This article belongs to the Special Issue Advances in Dynamical Systems: Stability, Bifurcation, and Chaos with Applications)
►▼ Show Figures

Figure 1

24 pages, 7742 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
Viewed by 616
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)
►▼ Show Figures

Figure 1

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 1958
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)
►▼ Show Figures

Figure 1

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
Cited by 2 | Viewed by 716
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)
►▼ Show Figures

Figure 1

Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:
 
Displaying articles 1-5
Back to TopTop