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Applied Mathematics in Nonlinear Dynamics and Chaos

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

Deadline for manuscript submissions: 30 September 2025 | Viewed by 17151

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Special Issue Editors

Dr. Inna Samuilik

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Guest Editor

Department of Engineering Mathematics, Riga Technical University, LV-1048 Riga, Latvia
Interests: complex networks; nonlinear systems; nonlinear dynamics; mathematical modelling; chaos theory

Prof. Dr. Felix Sadyrbaev

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Guest Editor

Institute of Mathematics and Computer Science, University of Latvia, Riga, Latvia
Interests: nonlinear boundary problems; ordinary differential equations; theory of oscillations; asymmetric problems; solutions of nonlinear differential equations; approximation of solutions; calculus of variations; spectral problems; analysis of three dimensional systems

Special Issue Information

Dear Colleagues,

We are pleased to announce this Special Issue of the journal Mathematics, entitled "Applied Mathematics in Nonlinear Dynamics and Chaos". This collection is focused on works devoted to new ideas in applied mathematics. It is assumed that these ideas can be formalized using the apparatus of the theory of dynamical systems. We welcome papers that consider processes that lead the regular to chaotic behavior of solutions. Chaotic behavior can be controlled by changing parameters and refining the model used. Of particular interest are articles that describe the implementation of the control of chaotic behavior. Is there order in chaos? Is it possible to imagine that the control of any chaotic systems is, in principle, impossible? To what extent, in this case, is this still possible? Since chaotic behavior can be present in systems arising in various fields of knowledge, this collection welcomes articles that study chaos in specific models used, for example, in engineering, mechanics, chemistry, biology, and the social sciences. Of particular note is mathematical modeling with the help of dynamic systems of processes in biological populations, not excluding the human community. Perhaps successful models will suggest ways to solve pressing problems in society, and shed light on some seemingly incomprehensible and unsolvable conflict situations of our time. All of the above do not exclude, but on the contrary, make desirable to some extent, standard forms of studying phenomena, their evolution, and development.

We would like to receive contributions from experts in their field (and simply interested beginner, but already skilled, mathematical workers) including the results of work in the following areas:

The development of a dynamic mathematical model from a set of experimental data in some areas of production and/or natural science;
Theoretical work in the field of formalization of the phenomenon of chaos in terms inherent in the theory of dynamic systems;
“crazy” ideas regarding dynamic chaos and its connection with traditional theory, from real-minded specialists;
New ideas regarding the invasion of spaces of higher dimensions from the point of view of chaos and ways of translating the realities of these spaces into the realm of feelings, and not just logical formal thinking;
The stabilization of chaos in mathematical models, and recommendations for the stabilization of simulated real processes;
Vivid examples of the importance of understanding chaotic processes and descriptions of these processes that are accessible to the average person;
Chaos in number theory from the point of view of dynamical systems;
Proof of the possibility or refutations of the possibility of managing large social systems with the guaranteed limitation of uncontrolled processes;
Uncontrolled processes whether or not they are necessary, and how they emerge from controllable ones (in connection with artificial intelligence);
Everything related to objects and phenomena that seem interesting from the point of view of theory and practice, to which the methods of the theory of dynamic systems and those not specified above are applicable.

Dr. Inna Samuilik
Prof. Dr. Felix Sadyrbaev
Guest Editors

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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 100 words) can be sent to the Editorial Office for announcement on this website.

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

nonlinear dynamics
bifurcation theory
chaos theory
irregular attractors
control theory
complex systems
numerical methods for dynamic systems
modeling and technology for dynamic systems in science and engineering

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Published Papers (14 papers)

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Research

25 pages, 2066 KiB

Open AccessFeature PaperArticle

Is π a Chaos Generator?

by Natalia Petrovskaya

Mathematics 2025, 13(7), 1126; https://doi.org/10.3390/math13071126 - 29 Mar 2025

Viewed by 290

Abstract

We consider a circular motion problem related to blind search in confined space. A particle moves in a unit circle in discrete time to find the escape channel and leave the circle through it. We first explain how the exit time depends on the initial position of the particle when the channel width is fixed. We then investigate how narrowing the channel moves the system from discrete changes in the exit time to the ultimate ‘countable chaos’ state that arises in the problem when the channel width becomes infinitely small. It will be shown in the paper that inherent randomness exists in the problem due to the nature of circular motion as the number

π

acts as a random number generator in the system. Randomness of the decimal digits of

π

results in sensitive dependence on initial conditions in the system with an infinitely narrow channel, and we argue that even a simple linear dynamical system can exhibit features of chaotic behaviour, provided that the system has inherent noise. Full article

(This article belongs to the Special Issue Applied Mathematics in Nonlinear Dynamics and Chaos)

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19 pages, 5273 KiB

Open AccessArticle

Norm-Based Adaptive Control with a Novel Practical Predefined-Time Sliding Mode for Chaotic System Synchronization

by Huan Ding, Jing Qian, Danning Tian and Yun Zeng

Mathematics 2025, 13(5), 748; https://doi.org/10.3390/math13050748 - 25 Feb 2025

Viewed by 364

Abstract

This paper proposes a novel, practical, predefined-time control theory for chaotic system synchronization under external disturbances and modeling uncertainties. Based on this theory, a robust sliding mode surface is designed to minimize chattering on a sliding surface, enhancing system stability. Additionally, a norm-based adaptive control strategy is developed to dynamically adjust control gains, ensuring system convergence to the equilibrium point within the predefined time. Theoretical analysis guarantees predefined-time stability using a Lyapunov framework. Numerical simulations on the Chen and multi-wing chaotic Lu systems demonstrate the proposed method’s superior convergence speed, precision, and robustness, highlighting its applicability to complex systems. Full article

(This article belongs to the Special Issue Applied Mathematics in Nonlinear Dynamics and Chaos)

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24 pages, 693 KiB

Open AccessArticle

Neuro Adaptive Command Filter Control for Predefined-Time Tracking in Strict-Feedback Nonlinear Systems Under Deception Attacks

by Jianhua Zhang, Zhanyang Yu, Quanmin Zhu and Xuan Yu

Mathematics 2025, 13(5), 742; https://doi.org/10.3390/math13050742 - 25 Feb 2025

Viewed by 355

Abstract

This paper presents a neural network enhanced adaptive control scheme tailored for strict-feedback nonlinear systems under the influence of deception attacks, with the aim of achieving precise tracking within a predefined time frame. Such studies are crucial as they address the increasing complexity of modern systems, particularly in environments where data integrity is at risk. Traditional methods, for instance, often struggle with the inherent unpredictability of nonlinear systems and the need for real-time adaptability in the presence of deception attacks, leading to compromised robustness and control instability. Unlike conventional approaches, this study adopts a Practical Predefined-Time Stability (PPTS) criterion as the theoretical foundation for predefined-time control design. By utilizing a novel nonlinear command filter, the research develops a command filter-based predefined-time adaptive back stepping control scheme. Furthermore, the incorporation of a switching threshold event-triggered mechanism effectively circumvents issues such as “complexity explosion” and control singularity, resulting in significant savings in computational and communication resources, as well as optimized data transmission efficiency. The proposed method demonstrates a 30% improvement in tracking accuracy and a 40% reduction in computational load compared to traditional methods. Through simulations and practical application cases, the study verifies the effectiveness and practicality of the proposed control method in terms of predefined-time stability and resilience against deception attacks. Full article

(This article belongs to the Special Issue Applied Mathematics in Nonlinear Dynamics and Chaos)

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15 pages, 3668 KiB

Open AccessArticle

Dragon Intermittency at the Transition to Synchronization in Coupled Rulkov Neurons

by Irina A. Bashkirtseva, Lev B. Ryashko and Alexander N. Pisarchik

Mathematics 2025, 13(3), 415; https://doi.org/10.3390/math13030415 - 26 Jan 2025

Viewed by 1168

Abstract

We investigate the synchronization dynamics of two non-identical, mutually coupled Rulkov neurons, emphasizing the effects of coupling strength and parameter mismatch on the system’s behavior. At low coupling strengths, the system exhibits multistability, characterized by the coexistence of three distinct 3-cycles. As the coupling strength is increased, the system becomes monostable with a single 3-cycle remaining as the sole attractor. A further increase in the coupling strength leads to chaos, which we identify as arising through a novel type of intermittency. This intermittency is characterized by alternating dynamics between two low-dimensional invariant subspaces: one corresponding to synchronization and the other to asynchronous behavior. We show that the system’s phase-space trajectory spends variable durations near one subspace before being repelled into the other, revealing non-trivial statistical properties near the onset of intermittency. Specifically, we find two key power-law scalings: (i) the mean duration of the synchronization interval scales with the coupling parameter, exhibiting a critical exponent of

- 0.5

near the onset of intermittency, and (ii) the probability distribution of synchronization interval durations follows a power law with an exponent of

- 1.7

for short synchronization intervals. Intriguingly, for each fixed coupling strength and parameter mismatch, there exists a most probable super-long synchronization interval, which decreases as either parameter is increased. We term this phenomenon “dragon intermittency” due to the distinctive dragon-like shape of the probability distribution of synchronization interval durations. Full article

(This article belongs to the Special Issue Applied Mathematics in Nonlinear Dynamics and Chaos)

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24 pages, 5567 KiB

Open AccessArticle

The Discrete Ueda System and Its Fractional Order Version: Chaos, Stabilization and Synchronization

by Louiza Diabi, Adel Ouannas, Amel Hioual, Giuseppe Grassi and Shaher Momani

Mathematics 2025, 13(2), 239; https://doi.org/10.3390/math13020239 - 12 Jan 2025

Cited by 2 | Viewed by 682

Abstract

The Ueda oscillator is one of the most popular and studied nonlinear oscillators. Whenever subjected to external periodic excitation, it exhibits a fascinating array of nonlinear behaviors, including chaos. This research introduces a novel fractional discrete Ueda system based on

Y

-th Caputo [...] Read more.

Y

-th Caputo fractional difference and thoroughly investigates its chaotic dynamics via commensurate and incommensurate orders. Applying numerical methods like maximum Lyapunov exponent spectra, bifurcation plots, and phase plane. We demonstrate the emergence of chaotic attractors influenced by fractional orders and system parameters. Advanced complexity measures, including approximation entropy (

A p E n

) and

C_{0}

complexity, are applied to validate and measure the nonlinear and chaotic nature of the system; the results indicate a high level of complexity. Furthermore, we propose a control scheme to stabilize and synchronize the introduced Ueda map, ensuring the convergence of trajectories to desired states. MATLAB R2024a simulations are employed to confirm the theoretical findings, highlighting the robustness of our results and paving the way for future works. Full article

(This article belongs to the Special Issue Applied Mathematics in Nonlinear Dynamics and Chaos)

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11 pages, 1720 KiB

Open AccessArticle

One More Thing on the Subject: Generating Chaos via x|x|^a−1, Melnikov’s Approach Using Simulations

by Nikolay Kyurkchiev, Anton Iliev, Vesselin Kyurkchiev and Asen Rahnev

Mathematics 2025, 13(2), 232; https://doi.org/10.3390/math13020232 - 11 Jan 2025

Cited by 1 | Viewed by 545

Abstract

In this article, we propose a new hypothetical differential model with many free parameters, which makes it attractive to users. The motivation is as follows: an extended model is proposed that allows us to investigate classical and newer models appearing in the literature at a “higher energy level”, as well as the generation of high–order Melnikov polynomials (corresponding to the proposed extended model) with possible applications in the field of antenna feeder technology. We present a few specific modules for examining these oscillators’ behavior. A much broader Web-based application for scientific computing will incorporate this as a key component. Full article

(This article belongs to the Special Issue Applied Mathematics in Nonlinear Dynamics and Chaos)

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23 pages, 41060 KiB

Open AccessArticle

Chaotic Dynamics Analysis and FPGA Implementation Based on Gauss Legendre Integral

by Li Wen, Li Cui, Hairong Lin and Fei Yu

Mathematics 2025, 13(2), 201; https://doi.org/10.3390/math13020201 - 9 Jan 2025

Viewed by 608

Abstract

In this paper, we first design the corresponding integration algorithm and matlab program according to the Gauss–Legendre integration principle. Then, we select the Lorenz system, the Duffing system, the hidden attractor chaotic system and the Multi-wing hidden chaotic attractor system for chaotic dynamics analysis. We apply the Gauss–Legendre integral and the Runge–Kutta algorithm to the solution of dissipative chaotic systems for the first time and analyze and compare the differences between the two algorithms. Then, we propose for the first time a chaotic basin of the attraction estimation method based on the Gauss–Legendre integral and Lyapunov exponent and the decision criterion of this method. This method can better obtain the region of chaotic basin of attraction and can better distinguish the attractor and pseudo-attractor, which provides a new way for chaotic system analysis. Finally, we use FPGA technology to realize four corresponding chaotic systems based on the Gauss–Legendre integration algorithm. Full article

(This article belongs to the Special Issue Applied Mathematics in Nonlinear Dynamics and Chaos)

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20 pages, 7002 KiB

Open AccessArticle

Dynamics and Stabilization of Chaotic Monetary System Using Radial Basis Function Neural Network Control

by Muhamad Deni Johansyah, Aceng Sambas, Fareh Hannachi, Seyed Mohamad Hamidzadeh, Volodymyr Rusyn, Monika Hidayanti, Bob Foster and Endang Rusyaman

Mathematics 2024, 12(24), 3977; https://doi.org/10.3390/math12243977 - 18 Dec 2024

Cited by 1 | Viewed by 797

Abstract

In this paper, we investigated a three-dimensional chaotic system that models key aspects of a monetary system, including interest rates, investment demand, and price levels. The proposed system is described by a set of autonomous quadratic ordinary differential equations. We analyze the dynamic behavior of this system through equilibrium points and their stability, Lyapunov exponents (LEs), and bifurcation diagrams. The system demonstrates a variety of behaviors, including chaotic, periodic, and equilibrium states depending on parameter values. Additionally, we explore the multistability of the system and present a radial basis function neural network (RBFNN) controller design to stabilize the chaotic behavior. The effectiveness of the controller is validated through numerical simulations, highlighting its potential applications in economic and financial modeling. Full article

(This article belongs to the Special Issue Applied Mathematics in Nonlinear Dynamics and Chaos)

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16 pages, 14452 KiB

Open AccessArticle

Disconnected Stationary Solutions in 3D Kolmogorov Flow and Their Relation to Chaotic Dynamics

by Nikolay M. Evstigneev, Taisia V. Karamysheva, Nikolai A. Magnitskii and Oleg I. Ryabkov

Mathematics 2024, 12(21), 3389; https://doi.org/10.3390/math12213389 - 30 Oct 2024

Viewed by 784

Abstract

This paper aims to investigate the nonlinear transition to turbulence in generalized 3D Kolmogorov flow. The difference between this and classical Kolmogorov flow is that the forcing term in the x direction

sin (y)

is replaced with [...] Read more.

sin (y)

is replaced with

sin (y) cos (z)

. This drastically complicates the problem. First, a stability analysis is performed by deriving the analog of the Orr–Sommerfeld equation. It is shown that for infinite stretching, the flow is stable, contrary to classical forcing. Next, a neutral curve is constructed, and the stability of the main solution is analyzed. It is shown that for the cubic domain, the main solution is linearly stable, at least for

0 < R \leq 100

. Next, we turn our attention to the numerical investigation of the solutions in the cubic domain. The main feature of this problem is that it is spatially periodic, allowing one to apply a relatively simple pseudo-spectral numerical method for its investigation. We apply the method of deflation to find distinct solutions in the discrete system and the method of arc length continuation to trace the bifurcation solution branches. Such solutions are called disconnected solutions if these are solutions not connected to the branch of the main solution. We investigate the influence of disconnected solutions on the dynamics of the system. It is demonstrated that when disconnected solutions are formed, the nonlinear transition to turbulence is possible, and dangerous initial conditions are these disconnected solutions. Full article

(This article belongs to the Special Issue Applied Mathematics in Nonlinear Dynamics and Chaos)

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19 pages, 3563 KiB

Open AccessArticle

Free Vibration of Graphene Nanoplatelet-Reinforced Porous Double-Curved Shells of Revolution with a General Radius of Curvature Based on a Semi-Analytical Method

by Aiwen Wang and Kairui Zhang

Mathematics 2024, 12(19), 3060; https://doi.org/10.3390/math12193060 - 30 Sep 2024

Cited by 1 | Viewed by 964

Abstract

Based on domain decomposition, a semi-analytical method (SAM) is applied to analyze the free vibration of double-curved shells of revolution with a general curvature radius made from graphene nanoplatelet (GPL)-reinforced porous composites. The mechanical properties of the GPL-reinforced composition are assessed with the Halpin–Tsai model. The double-curvature shell of revolution is broken down into segments along its axis in accordance with first-order shear deformation theory (FSDT) and the multi-segment partitioning technique, to derive the shell’s functional energy. At the same time, interfacial potential is used to ensure the continuity of the contact surface between neighboring segments. By applying the least-squares weighted residual method (LWRM) and modified variational principle (MVP) to relax and achieve interface compatibility conditions, a theoretical framework for analyzing vibrations is developed. The displacements and rotations are described through Fourier series and Chebyshev polynomials, accordingly, converting a two-dimensional issue into a suite of decoupled one-dimensional problems. The obtained solutions are contrasted with those achieved using the finite element method (FEM) and other existing results, and the current formulation’s validity and precision are confirmed. Example cases are presented to demonstrate the free vibration of GPL-reinforced porous composite double-curved paraboloidal, elliptical, and hyperbolical shells of revolution. The findings demonstrate that the natural frequency of the shell is related to pore coefficients, porosity, the mass fraction of the GPLs, and the distribution patterns of the GPLs. Full article

(This article belongs to the Special Issue Applied Mathematics in Nonlinear Dynamics and Chaos)

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25 pages, 2056 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Zhang Neuro-PID Control for Generalized Bi-Variable Function Projective Synchronization of Nonautonomous Nonlinear Systems with Various Perturbations

by Meichun Huang and Yunong Zhang

Mathematics 2024, 12(17), 2715; https://doi.org/10.3390/math12172715 - 30 Aug 2024

Cited by 1 | Viewed by 783

Abstract

Nonautonomous nonlinear (NN) systems have broad application prospects and significant research value in nonlinear science. In this paper, a new synchronization type—namely, generalized bi-variable function projective synchronization (GBVFPS)—is proposed. The scaling function matrix of GBVFPS is not one-variable but bi-variable. This indicates that the GBVFPS can be transformed into various synchronization types such as projective synchronization (PS), modified PS, function PS, modified function PS, and generalized function PS. In order to achieve the GBVFPS in two different NN systems with various perturbations, by designing a novel Zhang neuro-PID controller, an effective and anti-perturbation GBVFPS control method is proposed. Rigorous theoretical analyses are presented to prove the convergence performance and anti-perturbation ability of the GBVFPS control method, especially its ability to suppress six different perturbations. Besides, the effectiveness, superiority, and anti-perturbation ability of the proposed GBVFPS control method are further substantiated through two representative numerical simulations, including the synchronization of two NN chaotic systems and the synchronization of two four-dimensional vehicular inverted pendulum systems. Full article

(This article belongs to the Special Issue Applied Mathematics in Nonlinear Dynamics and Chaos)

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18 pages, 1208 KiB

Open AccessArticle

Oscillator with Line of Equilibiria and Nonlinear Function Terms: Stability Analysis, Chaos, and Application for Secure Communications

by Othman Abdullah Almatroud, Ali A. Shukur, Viet-Thanh Pham and Giuseppe Grassi

Mathematics 2024, 12(12), 1874; https://doi.org/10.3390/math12121874 - 16 Jun 2024

Cited by 6 | Viewed by 814

Abstract

We explore an oscillator with nonlinear functions and equilibrium lines that displays chaos. The equilibrium stability and complexity of the oscillator have been analysed and investigated. The presence of multiple equilibrium lines sets it apart from previously reported oscillators. The synchronization of the oscillator is considered as an application for secure communications. An observer is designed by considering a transmitted signal as a state, in other words, by injecting a linear function satisfying Lipschitz’s condition to the proposed oscillator. Moreover, the adaptive control of the new oscillator is obtained. Full article

(This article belongs to the Special Issue Applied Mathematics in Nonlinear Dynamics and Chaos)

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20 pages, 2332 KiB

Open AccessArticle

Dynamic Behavior and Bifurcation Analysis of a Modified Reduced Lorenz Model

by Mohammed O. Al-Kaff, Ghada AlNemer, Hamdy A. El-Metwally, Abd-Elalim A. Elsadany and Elmetwally M. Elabbasy

Mathematics 2024, 12(9), 1354; https://doi.org/10.3390/math12091354 - 29 Apr 2024

Cited by 1 | Viewed by 1422

Abstract

This study introduces a newly modified Lorenz model capable of demonstrating bifurcation within a specified range of parameters. The model demonstrates various bifurcation behaviors, which are depicted as distinct structures in the diagram. The study aims to discover and analyze the existence and stability of fixed points in the model. To achieve this, the center manifold theorem and bifurcation theory are employed to identify the requirements for pitchfork bifurcation, period-doubling bifurcation, and Neimark–Sacker bifurcation. In addition to theoretical findings, numerical simulations, including bifurcation diagrams, phase pictures, and maximum Lyapunov exponents, showcase the nuanced, complex, and diverse dynamics. Finally, the study applies the Ott–Grebogi–Yorke (OGY) method to control the chaos observed in the reduced modified Lorenz model. Full article

(This article belongs to the Special Issue Applied Mathematics in Nonlinear Dynamics and Chaos)

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19 pages, 452 KiB

Open AccessArticle

A Mathematical Model for Dynamic Electric Vehicles: Analysis and Optimization

by Khalid Khan, Inna Samuilik and Amir Ali

Mathematics 2024, 12(2), 224; https://doi.org/10.3390/math12020224 - 10 Jan 2024

Cited by 4 | Viewed by 5557

Abstract

In this article, we introduce a flexible and reliable technique to simulate and optimize the characteristics of a Dynamic Electrical Vehicle (DEV). The DEV model is a discrete event-based modeling technique used in electrical vehicle research to improve the effectiveness and performance of various electrical vehicles (EVs) components. Here, the DEVS model is applied to EV research in several ways, including battery management optimization, evaluation of power train design and control strategy, and driver behavior analysis. The essential power train elements, including the battery, motor, generator, internal combustion engine, and power electronics are included in the mathematical model for the dynamic electric vehicle. The model is derived using the conservation of energy principle. The model includes mathematical equations for the electrical power output, battery charge level, motor torque, motor power output, generator power output, internal combustion engine torque, mechanical power delivered to the generator, and the efficiencies of the power electronics, motor, generator, and engine. The model is examined by using a numerical method called the Runge–Kutta Method of order 4 for dynamic electric vehicle’s performance under various driving states for maximum effectiveness and performance. It is revealed that the DEV model provides a systematic method to simulate and optimize the behavior of complex EV systems. Full article

(This article belongs to the Special Issue Applied Mathematics in Nonlinear Dynamics and Chaos)

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