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Mathematical Modelling of Nonlinear Dynamical Systems
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Mathematical Modelling of Nonlinear Dynamical Systems

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

Deadline for manuscript submissions: closed (28 February 2026) | Viewed by 5760

Special Issue Editors

Prof. Dr. Guillermo Huerta-Cuellar

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Guest Editor
Centro Universitario de Los Lagos (CULAGOS), Universidad de Guadalajara, Guadalajara 44100, Mexico
Interests: nonlinear dynamical systems; numerical modeling; laser dynamics; chaos theory; deterministic Brownian motion; fractional calculus
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Jesus M. Munoz-Pacheco

E-Mail Website
Guest Editor
Faculty of Electronics Sciences, Benemérita Universidad Autónoma de Puebla, Av. San Claudio y 18 Sur, Puebla 72570, Mexico
Interests: nonlinear signal processing; fractional-order systems; control theory; complexity
Special Issues, Collections and Topics in MDPI journals
Dr. José Luis Echenausía-Monroy

E-Mail Website
Guest Editor
Department of Electronics and Telecommunications, Centro de Investigación Científica y de Educación Superior de Ensenada (CICESE), Carr. Ensenada-Tijuana 3918, Zona Playitas, Ensenada 22860, Mexico
Interests: nonlinear dynamical systems; numerical analysis; communication and signal processing; control theory; Matlab simulation; control systems engineering; communication engineering; optical engineering; estimation; instrumentation; virtual instrumentation; LabVIEW programming

Special Issue Information

Dear Colleagues,

We are pleased to announce a call for submissions to a Special Issue titled "Mathematical Modelling of Nonlinear Dynamic Systems". We invite our colleagues, friends, and other researchers in the field of nonlinear dynamical systems to contribute their latest research findings, reviews, and insightful perspectives. The aim is to bring together cutting-edge research that addresses the complexities and challenges associated with the mathematical modeling of nonlinear dynamical systems across various fields. Additionally, this Special Issue welcomes works on fractional calculus, which has emerged as a powerful tool for modeling complex phenomena characterized by nonlocality and long-memory effects, as well as its wide-ranging applications in engineering, physics, biology, economics, and other disciplines. This issue aims to provide a platform for disseminating innovative research that advances our understanding of the behavior, control, and optimization of nonlinear dynamical systems.

Prof. Dr. Guillermo Huerta-Cuellar
Prof. Dr. Jesus M. Munoz-Pacheco
Dr. José Luis Echenausía-Monroy
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

  • theoretical developments
  • computational methods
  • control and optimization
  • interdisciplinary research
  • artificial intelligence
  • machine learning
  • big data
  • fractional-order dynamical systems
  • dynamical systems
  • bifurcation analysis
  • chaotic behavior
  • fractional order systems
  • iterative dynamics
  • ordinary differential equations
  • partial differential equations
  • numerical methods
 

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Related Special Issue

Published Papers (7 papers)

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Research

18 pages, 7856 KB  
Article
An Investigation of Variable Segmental Inertial Parameters in Manual Load Lifting: A Genetic Algorithm-Based Inverse Dynamics Approach
by Muhammed Çil, Bilal Usanmaz and Ömer Gündoğdu
Mathematics 2026, 14(6), 1065; https://doi.org/10.3390/math14061065 - 21 Mar 2026
Viewed by 121
Abstract
This study investigates the common assumption that segmental inertial parameters remain constant during manual lifting using a model-based experimental approach. The primary objective was to evaluate the variability in these parameters and the subsequent effects on biomechanical calculations. The research was conducted with [...] Read more.
This study investigates the common assumption that segmental inertial parameters remain constant during manual lifting using a model-based experimental approach. The primary objective was to evaluate the variability in these parameters and the subsequent effects on biomechanical calculations. The research was conducted with 20 participants (10 females and 10 males) who performed lifting tasks in the two-dimensional sagittal plane under three distinct load conditions: 2.5 kg, 5.0 kg, and 7.5 kg. Angular variations of the hand, arm, and leg joints were recorded using video-based image processing techniques. These kinematic data, integrated with anthropometric measurements, were incorporated into Newton–Euler-based equations of motion to determine joint reaction forces and net joint moments. During the initial forward dynamics stage, the solvability of the problem was tested using constant mass ratios from the established literature. In the following inverse dynamics stage, genetic algorithms were utilized to overcome solution diversity and identify the variable inertial parameters responsible for the observed motion. The results indicate that changes in segment moments of inertia reached 18–37%, leading to variations of 0–19% in net joint moments. These findings highlight the critical necessity of incorporating dynamic inertial parameters into accurate biomechanical moment calculations for manual materials handling. Full article
(This article belongs to the Special Issue Mathematical Modelling of Nonlinear Dynamical Systems)
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26 pages, 2035 KB  
Article
Stability Dependence on Inertia in the Driven Damped Pendulum: A Master Control Parameter Analysis
by Alexander N. Pisarchik
Mathematics 2026, 14(6), 1060; https://doi.org/10.3390/math14061060 - 20 Mar 2026
Viewed by 226
Abstract
The driven damped pendulum is a foundational model in nonlinear dynamics, with applications ranging from Josephson junctions to MEMS oscillators. Conventional dimensionless treatments obscure the common physical origin of damping and driving in the inertia coefficient. Here we restore this dependence and establish [...] Read more.
The driven damped pendulum is a foundational model in nonlinear dynamics, with applications ranging from Josephson junctions to MEMS oscillators. Conventional dimensionless treatments obscure the common physical origin of damping and driving in the inertia coefficient. Here we restore this dependence and establish inertia as a master control parameter governing stability, resonance, and bifurcations. Through linear analysis and perturbation theory, we derive universal scaling laws revealing a fundamental dichotomy: quantities at resonance—peak amplitude and nonlinear frequency shift—are independent of inertia due to exact algebraic cancellation between the inertia dependence of the effective driving amplitude and effective damping coefficient. Off resonance, however, amplitude scales inversely with inertia, bandwidth narrows proportionally, and the bistability threshold exhibits an even steeper dependence. A critical inertia separates underdamped from overdamped regimes, yielding non-monotonic relaxation times that maximize attractor memory at extreme inertia values. These scaling laws provide design guidelines: low inertia promotes broadband response for energy harvesting; high inertia suppresses off-resonant vibrations for precision timing and quantum applications. By establishing inertia as a physically realizable path through parameter space, this work unifies disparate phenomena and provides a framework for understanding stability in inertial-driven systems. Full article
(This article belongs to the Special Issue Mathematical Modelling of Nonlinear Dynamical Systems)
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31 pages, 5845 KB  
Article
Gait Dynamics Classification with Criticality Analysis and Support Vector Machines
by Shadi Eltanani, Tjeerd V. olde Scheper, Johnny Collett, Helen Dawes and Patrick Esser
Mathematics 2026, 14(1), 177; https://doi.org/10.3390/math14010177 - 2 Jan 2026
Viewed by 483
Abstract
Classifying demographic groups of humans from gait patterns is desirable from several long-standing diagnostic and monitoring perspectives. IMU recorded gait patterns are mapped into a nonlinear dynamic representation space using criticality analysis and subsequently classified using standard Support Vector Machines. Inertial-only gait recordings [...] Read more.
Classifying demographic groups of humans from gait patterns is desirable from several long-standing diagnostic and monitoring perspectives. IMU recorded gait patterns are mapped into a nonlinear dynamic representation space using criticality analysis and subsequently classified using standard Support Vector Machines. Inertial-only gait recordings were found to readily classify in the CA representations. Accuracies across age categories for female versus male were 72.77%, 78.95%, and 80.11% for σ=0.1, 1, and 10, respectively; within the female group, accuracies were 73.36%, 76.70%, and 78.90%; and within the male group, 77.65%, 81.48%, and 81.05%. These results show that dynamic biological data are easily classifiable when projected into the nonlinear space, while classifying the data without this is not nearly as effective. Full article
(This article belongs to the Special Issue Mathematical Modelling of Nonlinear Dynamical Systems)
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18 pages, 19597 KB  
Article
The Shape of Chaos: A Geometric Perspective on Characterizing Chaos
by José Luis Echenausía-Monroy, Luis Javier Ontañón-García, Daniel Alejandro Magallón-García, Guillermo Huerta-Cuellar, Hector Eduardo Gilardi-Velázquez, José Ricardo Cuesta-García, Raúl Rivera-Rodríguez and Joaquín Álvarez
Mathematics 2026, 14(1), 15; https://doi.org/10.3390/math14010015 - 20 Dec 2025
Viewed by 728
Abstract
Chaotic dynamical systems are ubiquitous in nature and modern technology, with applications ranging from secure communications and cryptography to the design of chaos-based sensors and modeling biological phenomena such as arrhythmias and neuronal behavior. Given their complexity, precise analysis of these systems is [...] Read more.
Chaotic dynamical systems are ubiquitous in nature and modern technology, with applications ranging from secure communications and cryptography to the design of chaos-based sensors and modeling biological phenomena such as arrhythmias and neuronal behavior. Given their complexity, precise analysis of these systems is crucial for both theoretical understanding and practical implementation. The characterization of chaotic dynamical systems typically relies on conventional measures such as Lyapunov exponents and fractal dimensions. While these metrics are fundamental for describing dynamical behavior, they are often computationally expensive and may fail to capture subtle changes in the overall geometry of the attractor, limiting comparisons between systems with topologically similar structures and similar values in common chaos metrics such as the Lyapunov exponent. To address this limitation, this work proposes a geometric framework that treats chaotic attractors as spatial objects, using topological tools—specifically the α-sphere—to quantify their shape and spatial extent. The proposed method was validated using Chua’s system (including two reported variations), the Rössler system (standard and piecewise-linear), and a fractional-order multi-scroll system. A parametric characterization of the Rössler system was also performed by varying parameter b. Experimental results show that this geometric approach successfully distinguishes between attractors where classical metrics reveal no perceptible differences, in addition to being computationally simpler. Notably, we observed geometric variations of up to 80% among attractors with similar dynamics and introduced a specific index to quantify these global discrepancies. Although this geometric analysis serves as a complement rather than a substitute for chaos detection, it provides a reliable and interpretable metric for differentiating systems and selecting attractors based on their spatial properties. Full article
(This article belongs to the Special Issue Mathematical Modelling of Nonlinear Dynamical Systems)
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29 pages, 2082 KB  
Article
Vibration Analysis of Laminated Composite Beam with Magnetostrictive Layers Flexibly Restrained at the Ends
by Bogdan Marinca, Nicolae Herisanu and Vasile Marinca
Mathematics 2025, 13(23), 3856; https://doi.org/10.3390/math13233856 - 1 Dec 2025
Viewed by 372
Abstract
The dynamic model and nonlinear forced vibration of a laminated beam with magnetostrictive layers, embedded on a nonlinear elastic Winkler–Pasternak foundation, in the presence of an electromagnetic actuator, mechanical impact, dry friction, a longitudinal magnetic field, and van der Waals force is investigated [...] Read more.
The dynamic model and nonlinear forced vibration of a laminated beam with magnetostrictive layers, embedded on a nonlinear elastic Winkler–Pasternak foundation, in the presence of an electromagnetic actuator, mechanical impact, dry friction, a longitudinal magnetic field, and van der Waals force is investigated in the present work. The dynamic equations of this complex system are established based on von Karman theory and Hamilton’s principle. Then, by means of the Galerkin–Bubnov procedure, the partial differential equations are transformed into ordinary differential equations. The Optimal Auxiliary Functions Method (OAFM) is applied to solve the nonlinear differential equation. The results obtained are validated by comparisons with numerical results given by the Runge–Kutta procedure. Local stability in the neighborhood of the primary resonance is examined by means of the homotopy perturbation method, the Jacobian matrix, and the Routh–Hurwitz criteria. Global stability is studied by introducing the control law input function and using the approximate solution obtained by the OAFM in the construction of the Lyapunov function. La Salle’s invariance principle and Potryagin’s principle complete our study. The effects of some parameters are graphically presented. Our paper reveals the immense potential of the OAFM in the study of complex nonlinear dynamical systems. Full article
(This article belongs to the Special Issue Mathematical Modelling of Nonlinear Dynamical Systems)
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23 pages, 2651 KB  
Article
Asymptotic Analysis of Poverty Dynamics via Feller Semigroups
by Lahcen Boulaasair, Mehmet Yavuz and Hassane Bouzahir
Mathematics 2025, 13(13), 2120; https://doi.org/10.3390/math13132120 - 28 Jun 2025
Cited by 1 | Viewed by 661
Abstract
Poverty is a multifaceted phenomenon impacting millions globally, defined by a deficiency in both material and immaterial resources, which consequently restricts access to satisfactory living conditions. Comprehensive poverty analysis can be accomplished through the application of mathematical and modeling techniques, which are useful [...] Read more.
Poverty is a multifaceted phenomenon impacting millions globally, defined by a deficiency in both material and immaterial resources, which consequently restricts access to satisfactory living conditions. Comprehensive poverty analysis can be accomplished through the application of mathematical and modeling techniques, which are useful in understanding and predicting poverty trends. These models, which often incorporate principles from economics, stochastic processes, and dynamic systems, enable the assessment of the factors influencing poverty and the effectiveness of public policies in alleviating it. This paper introduces a mathematical compartmental model to investigate poverty within a population (ψ(t)), considering the effects of immigration, crime, and incarceration. The model aims to elucidate the interconnections between these factors and their combined impact on poverty levels. We begin the study by ensuring the mathematical validity of the model by demonstrating the uniqueness of a positive solution. Next, it is shown that under specific conditions, the probability of poverty persistence approaches certainty. Conversely, conditions leading to an exponential reduction in poverty are identified. Additionally, the semigroup associated with our model is proven to possess the Feller property, and its distribution has a unique invariant measure. To confirm and validate these theoretical results, interesting numerical simulations are performed. Full article
(This article belongs to the Special Issue Mathematical Modelling of Nonlinear Dynamical Systems)
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17 pages, 2842 KB  
Article
A Comparative Study of Brownian Dynamics Based on the Jerk Equation Against a Stochastic Process Under an External Force Field
by Adriana Ruiz-Silva, Bahia Betzavet Cassal-Quiroga, Rodolfo de Jesus Escalante-Gonzalez, José A. Del-Puerto-Flores, Hector Eduardo Gilardi-Velazquez and Eric Campos
Mathematics 2025, 13(5), 804; https://doi.org/10.3390/math13050804 - 28 Feb 2025
Cited by 1 | Viewed by 1531
Abstract
Brownian motion has been studied since 1827, leading to numerous important advances in many branches of science and to it being studied primarily as a stochastic dynamical system. In this paper, we present a deterministic model for the Brownian motion for a particle [...] Read more.
Brownian motion has been studied since 1827, leading to numerous important advances in many branches of science and to it being studied primarily as a stochastic dynamical system. In this paper, we present a deterministic model for the Brownian motion for a particle in a constant force field based on the Ornstein–Uhlenbeck model. By adding one degree of freedom, the system evolves into three differential equations. This change in the model is based on the Jerk equation with commutation surfaces and is analyzed in three cases: overdamped, critically damped, and underdamped. The dynamics of the proposed model are compared with classical results using a random process with normal distribution, where despite the absence of a stochastic component, the model preserves key Brownian motion characteristics, which are lost in stochastic models, giving a new perspective to the study of particle dynamics under different force fields. This is validated by a linear average square displacement and a Gaussian distribution of particle displacement in all cases. Furthermore, the correlation properties are examined using detrended fluctuation analysis (DFA) for compared cases, which confirms that the model effectively replicates the essential behaviors of Brownian motion that the classic models lose. Full article
(This article belongs to the Special Issue Mathematical Modelling of Nonlinear Dynamical Systems)
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