A Real-World Application of Chaos Theory

Topic Information

Dear Colleagues,

Chaos theory is one of the most elegant theories in mathematics which deals with how small changes in initial conditions can lead to vastly different outcomes in complex systems, making long-term predictions difficult. Several different techniques are used to predict the chaos in a dynamical system, some of which are sensitivity, phase portraits, bifurcation theory, Lyapunov exponents, etc. Each of these techniques helps mathematicians analyze problems more deeply.

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

• (i) Bifurcation analysis;
• (ii) multi-stability analysis;
• (iii) sensitivity analysis;
• (iv) Lyapunov exponents;
• (v) attractors;
• (vi) basin theory;
• (vii) Poincare map.

Dr. Adil Jhangeer
Dr. Mudassar Imran
Topic Editors

Keywords

Participating Journals

Journal Name	Impact Factor	CiteScore	Launched Year	First Decision (median)	APC
AppliedMath appliedmath	0.7	1.1	2021	20.6 Days	CHF 1200	Submit
Axioms axioms	1.6	-	2012	21.7 Days	CHF 2400	Submit
Computation computation	1.9	4.1	2013	14.8 Days	CHF 1800	Submit
Mathematics mathematics	2.2	4.6	2013	17.3 Days	CHF 2600	Submit
Symmetry symmetry	2.2	5.3	2009	15.8 Days	CHF 2400	Submit

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Published Papers (7 papers)

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20 pages, 1129 KB  

Open AccessArticle

Solving the Synthesis Problem Self-Organizing Control System in the Class of Elliptical Accidents Optics for Objects with One Input and One Output

by Maxot Rakhmetov, Ainagul Adiyeva, Balaussa Orazbayeva, Shynar Yelezhanova, Raigul Tuleuova and Raushan Moldasheva

Computation 2026, 14(1), 21; https://doi.org/10.3390/computation14010021 - 14 Jan 2026

Viewed by 118

Abstract

Nonlinear single-input single-output (SISO) systems operating under parametric uncertainty often exhibit bifurcations, multistability, and deterministic chaos, which significantly limit the effectiveness of classical linear, adaptive, and switching control methods. This paper proposes a novel synthesis framework for self-organizing control systems based on catastrophe [...] Read more.

Nonlinear single-input single-output (SISO) systems operating under parametric uncertainty often exhibit bifurcations, multistability, and deterministic chaos, which significantly limit the effectiveness of classical linear, adaptive, and switching control methods. This paper proposes a novel synthesis framework for self-organizing control systems based on catastrophe theory, specifically within the class of elliptic catastrophes. Unlike conventional approaches that stabilize a predefined system structure, the proposed method embeds the control law directly into a structurally stable catastrophe model, enabling autonomous bifurcation-driven transitions between stable equilibria. The synthesis procedure is formulated using a Lyapunov vector-function gradient–velocity method, which guarantees aperiodic robust stability under parametric uncertainty. The definiteness of the Lyapunov functions is established using Morse’s lemma, providing a rigorous stability foundation. To support practical implementation, a data-driven parameter tuning mechanism based on self-organizing maps (SOM) is integrated, allowing adaptive adjustment of controller coefficients while preserving Lyapunov stability conditions. Simulation results demonstrate suppression of chaotic regimes, smooth bifurcation-induced transitions between stable operating modes, and improved transient performance compared to benchmark adaptive control schemes. The proposed framework provides a structurally robust alternative for controlling nonlinear systems in uncertain and dynamically changing environments. Full article

(This article belongs to the Topic A Real-World Application of Chaos Theory)

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44 pages, 20298 KB  

Open AccessArticle

Stochastic Dynamics and Control in Nonlinear Waves with Darboux Transformations, Quasi-Periodic Behavior, and Noise-Induced Transitions

by Adil Jhangeer and Mudassar Imran

Mathematics 2026, 14(2), 251; https://doi.org/10.3390/math14020251 - 9 Jan 2026

Viewed by 251

Abstract

Stochastically forced nonlinear wave systems are commonly associated with complex dynamical behavior, although little is known about the general interaction of nonlinear dispersion, irrational forcing frequencies, and multiplicative noise. To fill this gap, we consider a generalized stochastic SIdV equation and examine the [...] Read more.

Stochastically forced nonlinear wave systems are commonly associated with complex dynamical behavior, although little is known about the general interaction of nonlinear dispersion, irrational forcing frequencies, and multiplicative noise. To fill this gap, we consider a generalized stochastic SIdV equation and examine the effects of deterministic and stochastic influences on the long-term behavior of the equation. The PDE was modeled using a stochastic traveling-wave transformation that simplifies it into a planar system, which was studied using Darboux-seeded constructions, Poincaré maps, bifurcation patterns, Lyapunov exponents, recurrence plots, and sensitivity diagnostics. We discovered that natural, implicit, and unique seeds produce highly diverse transformed wave fields exhibiting both irrational and golden-ratio forcing, controlling the transition from quasi-periodicity to chaos. Stochastic perturbation is demonstrated to suppress as well as to amplify chaotic states, based on noise levels, altering attractor geometry, predictability, and multistability. Meanwhile, OGY control is demonstrated to be able to stabilize chosen unstable periodic orbits of the double-well regime. A stochastic bifurcation analysis was performed with respect to noise strength $\sigma$, revealing that the attractor structure of the system remains robust under stochastic excitation, with noise inducing only bounded fluctuations rather than qualitative dynamical transitions within the investigated parameter regime. These findings demonstrate that the emergence, deformation, and controllability of complex oscillatory patterns of stochastic nonlinear wave models are jointly controlled by nonlinear structure, external forcing, and noise. Full article

(This article belongs to the Topic A Real-World Application of Chaos Theory)

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25 pages, 6358 KB  

Open AccessArticle

A Novel Chaotic Encryption Algorithm Based on Fuzzy Rule-Based Sugeno Inference: Theory and Application

by Aydin Muhurcu and Gulcin Muhurcu

Mathematics 2026, 14(2), 243; https://doi.org/10.3390/math14020243 - 8 Jan 2026

Viewed by 244

Abstract

This study proposes a robust chaotic encryption framework based on a Fuzzy Rule-Based Sugeno Inference (FRBSI) system, integrated with high-level security analyses. The algorithm employs a dynamic mixture of Lorenz chaotic state variables, which are numerically modeled using the Euler-Forward method to ensure [...] Read more.

This study proposes a robust chaotic encryption framework based on a Fuzzy Rule-Based Sugeno Inference (FRBSI) system, integrated with high-level security analyses. The algorithm employs a dynamic mixture of Lorenz chaotic state variables, which are numerically modeled using the Euler-Forward method to ensure computational accuracy. Unlike conventional methods, the carrier signal’s characteristics are not static; instead, its amplitude and dynamic behavior are continuously adapted through the FRBSI mechanism, driven by the instantaneous thresholds of the information signal. The security of the proposed system was rigorously evaluated through Histogram analysis, Number of Pixels Change Rate (NPCR), and Unified Average Changing Intensity (UACI) metrics, which confirmed the algorithm’s high sensitivity to plaintext variations and resistance against differential attacks. Furthermore, Key Sensitivity tests demonstrated that even a single-bit discrepancy in the receiver-side Sugeno rule base leads to a total failure in signal reconstruction, providing a formidable defense against brute-force attempts. The system’s performance was validated in the MATLAB/Simulink of R2021a version environment, where frequency and time-domain analyses were performed via oscilloscope and Fourier transforms. The results indicate that the proposed multi-layered fuzzy-chaotic structure significantly outperforms traditional encryption techniques in terms of unpredictability, structural security, and robustness. Full article

(This article belongs to the Topic A Real-World Application of Chaos Theory)

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16 pages, 9419 KB  

Open AccessArticle

Initial-Offset-Control and Amplitude Regulation in Memristive Neural Network

by Hua Liu, Haijun Wang, Wenhui Zhang and Suling Zhang

Symmetry 2025, 17(10), 1682; https://doi.org/10.3390/sym17101682 - 8 Oct 2025

Viewed by 608

Abstract

Traditional Hopfield neural networks (HNNs) suffer from limitations in generating controllable chaotic dynamics, which are essential for applications in neuromorphic computing and secure communications. Memristors, with their memory-dependent nonlinear characteristics, provide a promising approach to regulate neuronal activities, yet systematic studies on attractor [...] Read more.

Traditional Hopfield neural networks (HNNs) suffer from limitations in generating controllable chaotic dynamics, which are essential for applications in neuromorphic computing and secure communications. Memristors, with their memory-dependent nonlinear characteristics, provide a promising approach to regulate neuronal activities, yet systematic studies on attractor offset behaviors remain scarce. In this study, we propose a fully memristive electromagnetic radiation neural network by incorporating three distinct memristors as external electromagnetic stimuli into an HNN. The parameters of the memristors were tuned to modulate chaotic oscillations, while variations in initial conditions were employed to explore multistability through bifurcation and basin stability analyses. The results demonstrate that the system enables large-scale amplitude control of chaotic signals via memristor parameter adjustments, allowing arbitrary scaling of attractor amplitudes. Various offset behaviors emerge, including parameter-driven symmetric double-scroll relocations in phase space and initial-condition-induced offset boosting that leads to extreme multistability. These dynamics were experimentally validated using an STM32-based electronic circuit, confirming precise amplitude and offset control. Furthermore, a multi-channel pseudo-random number generator (PRNG) was implemented, leveraging the initial-boosted offset to enhance security entropy. This offers a hardware-efficient chaotic solution for encrypted communication systems, demonstrating strong application potential. Full article

(This article belongs to the Topic A Real-World Application of Chaos Theory)

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17 pages, 7815 KB  

Open AccessArticle

Design and Analysis of Memristive Electromagnetic Radiation in a Hopfield Neural Network

by Zhimin Gu, Bin Hu, Hongxin Zhang, Xiaodan Wang, Yaning Qi and Min Yang

Symmetry 2025, 17(8), 1352; https://doi.org/10.3390/sym17081352 - 19 Aug 2025

Cited by 1 | Viewed by 820

Abstract

This study introduces a memristive Hopfield neural network (M-HNN) model to investigate electromagnetic radiation impacts on neural dynamics in complex electromagnetic environments. The proposed framework integrates a magnetic flux-controlled memristor into a three-neuron Hopfield architecture, revealing significant alterations in network dynamics through comprehensive [...] Read more.

This study introduces a memristive Hopfield neural network (M-HNN) model to investigate electromagnetic radiation impacts on neural dynamics in complex electromagnetic environments. The proposed framework integrates a magnetic flux-controlled memristor into a three-neuron Hopfield architecture, revealing significant alterations in network dynamics through comprehensive nonlinear analysis. Numerical investigations demonstrate that memristor-induced electromagnetic effects induce distinctive phenomena, including coexisting attractors, transient chaotic states, symmetric bifurcation diagrams and attractor structures, and constant chaos. The proposed system can generate more than 12 different attractors and extends the chaotic region. Compared with the chaotic range of the baseline Hopfield neural network (HNN), the expansion amplitude reaches 933%. Dynamic characteristics are systematically examined using phase trajectory analysis, bifurcation mapping, and Lyapunov exponent quantification. Experimental validation via a DSP-based hardware implementation confirms the model’s operational feasibility and consistency with numerical predictions, establishing a reliable platform for electromagnetic–neural interaction studies. Full article

(This article belongs to the Topic A Real-World Application of Chaos Theory)

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25 pages, 6742 KB  

Open AccessArticle

Reservoir Computing with a Single Oscillating Gas Bubble: Emphasizing the Chaotic Regime

by Hend Abdel-Ghani, A. H. Abbas and Ivan S. Maksymov

AppliedMath 2025, 5(3), 101; https://doi.org/10.3390/appliedmath5030101 - 7 Aug 2025

Cited by 1 | Viewed by 1421

Abstract

The rising computational and energy demands of artificial intelligence systems urge the exploration of alternative software and hardware solutions that exploit physical effects for computation. According to machine learning theory, a neural network-based computational system must exhibit nonlinearity to effectively model complex patterns [...] Read more.

The rising computational and energy demands of artificial intelligence systems urge the exploration of alternative software and hardware solutions that exploit physical effects for computation. According to machine learning theory, a neural network-based computational system must exhibit nonlinearity to effectively model complex patterns and relationships. This requirement has driven extensive research into various nonlinear physical systems to enhance the performance of neural networks. In this paper, we propose and theoretically validate a reservoir-computing system based on a single bubble trapped within a bulk of liquid. By applying an external acoustic pressure wave to both encode input information and excite the complex nonlinear dynamics, we showcase the ability of this single-bubble reservoir-computing system to forecast a Hénon benchmarking time series and undertake classification tasks with high accuracy. Specifically, we demonstrate that a chaotic physical regime of bubble oscillation—where tiny differences in initial conditions lead to wildly different outcomes, making the system unpredictable despite following clear rules, yet still suitable for accurate computations—proves to be the most effective for such tasks. Full article

(This article belongs to the Topic A Real-World Application of Chaos Theory)

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24 pages, 1751 KB  

Open AccessArticle

Chaotic-Based Shellcode Encryption: A New Strategy for Bypassing Antivirus Mechanisms

by Gang-Cheng Huang, Ko-Chin Chang and Tai-Hung Lai

Symmetry 2024, 16(11), 1526; https://doi.org/10.3390/sym16111526 - 14 Nov 2024

Cited by 2 | Viewed by 5430

Abstract

This study employed chaotic systems as an innovative approach for shellcode obfuscation to evade current antivirus detection methods. Standard AV solutions primarily rely on static signatures and heuristic analysis to identify malicious code. However, chaotic systems employ dynamic and unpredictable encryption methods, significantly [...] Read more.

This study employed chaotic systems as an innovative approach for shellcode obfuscation to evade current antivirus detection methods. Standard AV solutions primarily rely on static signatures and heuristic analysis to identify malicious code. However, chaotic systems employ dynamic and unpredictable encryption methods, significantly obstructing detection efforts. The utilization of various chaotic maps for shellcode encryption facilitates the generation of multiple unique variations from the same functional code, each exhibiting distinct unpredictability due to the inherent nonlinearity and sensitivity of chaotic systems to initial conditions. The unpredictability of these situations poses a considerable challenge for antivirus software in recognizing consistent patterns, resulting in decreased detection rates. The findings from our experiments demonstrate that chaos-driven encryption methods significantly outperform traditional encryption techniques in terms of evading detection. This paper emphasizes the potential of chaos theory to enhance malware evasion strategies, offering a sophisticated approach to bypassing modern antivirus protections while ensuring the effectiveness of malicious payloads. Full article

(This article belongs to the Topic A Real-World Application of Chaos Theory)

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