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Mathematics
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Nonlinear Dynamics and Chaos Theory, 2nd Edition
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Nonlinear Dynamics and Chaos Theory, 2nd Edition

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E: Applied Mathematics".

Deadline for manuscript submissions: 31 July 2026 | Viewed by 2034

Special Issue Editors

Prof. Dr. Youming Lei

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Northwestern Polytechnical University, 710072 Xi'an, China
Interests: chaos control and synchronization; random dynamical systems; complex system; molecular dynamics simulation
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Lijun Pei

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China
Interests: delay differential equations; nonlinear time series analysis and predication; bifurcation; stability; chaos; synchronization; nonlinear dynamics
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Lei Hou

E-Mail Website
Guest Editor
School of Astronautics, Harbin Institute of Technology, Harbin 150001, China
Interests: nonlinear dynamics and vibration control; intelligent modeling; intelligent fault diagnosis; digital twins
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Nonlinear dynamics and chaos theory plays an ever-important role in the exploration of science and technology. It aims to study the mechanisms and applications of rich dynamical phenomena in the real world. Applications and examples of nonlinear dynamical systems are ubiquitous over a diverse set of areas. The growing demand for determining the laws and mechanisms of rich nonlinear dynamical phenomena as well as their applications to diverse fields is inducing growth in the demand for nonlinear dynamics and chaos theory.  

In this Special Issue, we aim to present the recent developments in the theory and applications of nonlinear dynamics and chaos, with a special emphasis on stability, bifurcation, chaos, hidden and multi-scroll attractors, image encryption, chaotic circuits, high-speed trains, hydraulic cylinders, neural networks, and environmental protection.

This Special Issue will accept high-quality papers containing original research results and review articles of exceptional merit in the following fields:

  • Stability, bifurcation and chaos;
  • Extreme multi-stability, and hidden and multi-scroll attractors;
  • Image encryption and chaotic circuits;
  • Neural networks;
  • High-speed trains;
  • Hydraulic cylinders;
  • Mathematical biology.

Prof. Dr. Youming Lei
Prof. Dr. Lijun Pei
Prof. Dr. Lei Hou
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

  • stability
  • bifurcation
  • chaos
  • hidden and multi-scroll attractors
  • singularities

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Related Special Issue

Published Papers (3 papers)

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Research

16 pages, 2290 KB  
Article
Chaos and Bifurcations in the Dynamics of the Variable-Order Fractional Rössler System
by Athar I. Ahmed, Mohamed Elbadri, Naseam Al-kuleab, Dalal M. AlMutairi, Nidal E. Taha and Mohammed E. Dafaalla
Mathematics 2025, 13(22), 3695; https://doi.org/10.3390/math13223695 - 18 Nov 2025
Viewed by 264
Abstract
This article investigates the chaotic features of a novel variable-order fractional Rössler system built with Liouville–Caputo derivatives of variable order. Variable-order fractional (VOF) operators incorporated in the system render its dynamics more flexible and richer with memory and hereditary effects. We run numerical [...] Read more.
This article investigates the chaotic features of a novel variable-order fractional Rössler system built with Liouville–Caputo derivatives of variable order. Variable-order fractional (VOF) operators incorporated in the system render its dynamics more flexible and richer with memory and hereditary effects. We run numerical simulations to see how different fractional-order functions alter the qualitative behavior of the system. We demonstrate this via phase portraits and time-series responses. The research analyzes bifurcation development, chaotic oscillations, and stability transition and demonstrates dynamic patterns impossible to describe with integer-order models. Lyapunov exponent analysis also demonstrates system sensitivity to initial conditions and small disturbances. The outcomes confirm that the variable-order procedure provides a faithful representation of nonlinear and intricate processes of engineering and physical sciences, pointing out the dominant role of memory effects on the transitions among periodic, quasi-periodic, and chaotic regimes. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Chaos Theory, 2nd Edition)
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19 pages, 8707 KB  
Article
Dynamic, Bifurcation, and Lyapunov Analysis of Fractional Rössler Chaos Using Two Numerical Methods
by Reem Allogmany and S. S. Alzahrani
Mathematics 2025, 13(22), 3642; https://doi.org/10.3390/math13223642 - 13 Nov 2025
Viewed by 218
Abstract
In this paper, we first used a Modified Numerical Approximation Method (NAM) and then a fractional Laplace Decomposition Method (LDM) to find the solution to the symmetric Rössler attractor. The newly proposed NAM is obtained through a nuanced discretization of the Caputo derivative, [...] Read more.
In this paper, we first used a Modified Numerical Approximation Method (NAM) and then a fractional Laplace Decomposition Method (LDM) to find the solution to the symmetric Rössler attractor. The newly proposed NAM is obtained through a nuanced discretization of the Caputo derivative, rendering it exceptionally effective in emulating the inherent sensitivity and memory-dependent characteristics of fractional-order systems. Second, a comprehensive analysis is conducted to examine how variations in the fractional parameters ρ1, ρ2, and ρ3 influence the dynamic response of the system. Third, the simulation results, which include time series, bifurcation diagrams, and Lyapunov exponent spectra, show that the proposed method works well to find changes in system behavior that integer-order or lower-accuracy schemes cannot find. The fractional Laplace Decomposition Method (LDM) is straightforward to implement, computationally efficient, and exhibits outstanding accuracy. Other widely used approximation approaches achieve comparable results. The comparisons between NAM and LDM reveal that these two methodologies are not only highly consistent but also mutually reinforcing. Their straightforward application and robust consistency of numerical solutions indicate that these methods can be effectively utilized in the majority of fractional-order systems, resulting in more accurate and practical answers. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Chaos Theory, 2nd Edition)
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22 pages, 7609 KB  
Article
Bidirectional Conservative–Dissipative Transitions in a Five-Dimensional Fractional Chaotic System
by Yiming Wang, Fengjiao Gao and Mingqing Zhu
Mathematics 2025, 13(15), 2477; https://doi.org/10.3390/math13152477 - 1 Aug 2025
Viewed by 563
Abstract
This study investigates a modified five-dimensional chaotic system by incorporating structural term adjustments and Caputo fractional-order differential operators. The modified system exhibits significantly enriched dynamic behaviors, including offset boosting, phase trajectory rotation, phase trajectory reversal, and contraction phenomena. Additionally, the system exhibits bidirectional [...] Read more.
This study investigates a modified five-dimensional chaotic system by incorporating structural term adjustments and Caputo fractional-order differential operators. The modified system exhibits significantly enriched dynamic behaviors, including offset boosting, phase trajectory rotation, phase trajectory reversal, and contraction phenomena. Additionally, the system exhibits bidirectional transitions—conservative-to-dissipative transitions governed by initial conditions and dissipative-to-conservative transitions controlled by fractional order variations—along with a unique chaotic-to-quasiperiodic transition observed exclusively at low fractional orders. To validate the system’s physical realizability, a signal processing platform based on Digital Signal Processing (DSP) is implemented. Experimental measurements closely align with numerical simulations, confirming the system’s feasibility for practical applications. Full article
(This article belongs to the Special Issue Nonlinear Dynamics and Chaos Theory, 2nd Edition)
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