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Analytical Approaches to Nonlinear Dynamical Systems and Applications, 3rd Edition
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Analytical Approaches to Nonlinear Dynamical Systems and Applications, 3rd Edition

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

Deadline for manuscript submissions: 31 December 2025 | Viewed by 2498

Special Issue Editor

Prof. Dr. Nicolae Herisanu

E-Mail Website
Guest Editor
1. Department of Mechanics and Strength of Materials, University Politehnica Timisoara, 300222 Timisoara, Romania
2. Department of Electromechanics and Vibration, Center for Advanced and Fundamental Technical Research, Romanian Academy, 300223 Timisoara, Romania
Interests: nonlinear dynamical systems; rotating electric machines
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue of Mathematics aims to explore the analytical methodologies that provide profound insights into the complexities of nonlinear dynamical systems. We invite contributions that highlight recent advancements and emerging trends in the analytical study of these systems, particularly those described by nonlinear differential equations.

We encourage submissions that present innovative analytical treatments of nonlinear dynamical systems, showcasing applications across diverse fields such as physics, applied mathematics, mechanics, engineering, and life sciences. Interdisciplinary approaches that illustrate potential future research directions are also highly welcome. This Special Issue seeks to foster a deeper understanding of nonlinear phenomena and to inspire further exploration in this vital area of study.

Dr. Nicolae Herisanu
Guest Editor

Manuscript Submission Information

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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
  • nonlinear phenomena
  • analytical methods
  • nonlinear differential equations
  • approximate analytical solutions

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Published Papers (4 papers)

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Research

15 pages, 2758 KiB  
Article
Vibration of the Liénard Oscillator with Quadratic Damping and Constant Excitation
by Livija Cveticanin, Nicolae Herisanu, Gamal Mohamed Ismail and Miodrag Zukovic
Mathematics 2025, 13(6), 937; https://doi.org/10.3390/math13060937 - 12 Mar 2025
Viewed by 368
Abstract
In this paper, the Liénard oscillator with nonlinear deflection, quadratic damping, and constant excitation is considered. In general, there is no analytic solution for the Liénard equation. However, for certain parameter values, the exact analytic solution exists and has the form of the [...] Read more.
In this paper, the Liénard oscillator with nonlinear deflection, quadratic damping, and constant excitation is considered. In general, there is no analytic solution for the Liénard equation. However, for certain parameter values, the exact analytic solution exists and has the form of the Ateb function. In addition, for the oscillator with weakly perturbed parameters, the approximate analytic solution is obtained. For the considered Liénard equation, independently of parameter values, the first integral is found. The main advantage of the first integral is that after simple analysis and without solving the equation of motion, it gives important data about oscillation: the dependence of vibration on initial conditions and on the variation of the constant of excitation. In addition, by integration of the first integral, the period of vibration follows. The results of the research on the Liénard equation are applied for optimization of the properties of a sieve in the process industry. For the sieve with mass variation, dependent on the displacement function, the influence of excitation force on the system vibration is analyzed, and the optimal value is suggested. Full article
(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications, 3rd Edition)
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18 pages, 1149 KiB  
Article
Applications of the Kosambi–Cartan–Chern Theory to Hamiltonian Systems on a Cotangent Bundle: Linking Geometric Quantities to the Self-Similar Motions of Three Point Vortices
by Yuma Hirakui and Takahiro Yajima
Mathematics 2025, 13(1), 126; https://doi.org/10.3390/math13010126 - 31 Dec 2024
Viewed by 552
Abstract
This study presents a differential geometric framework for Hamiltonian systems expressed in terms of first-order differential equations. For systems governed by second-order ordinary differential equations on tangent bundles, such as Euler–Lagrange systems, the stability of trajectories under perturbations is analyzed based on the [...] Read more.
This study presents a differential geometric framework for Hamiltonian systems expressed in terms of first-order differential equations. For systems governed by second-order ordinary differential equations on tangent bundles, such as Euler–Lagrange systems, the stability of trajectories under perturbations is analyzed based on the eigenvalue of the deviation curvature tensor. Building upon this Jacobi stability analysis approach, four geometric quantities for Hamiltonian systems are derived considering perturbations to trajectories on a cotangent bundle. As a specific Hamiltonian system, a hydrodynamic three-point vortex system is examined, and its four geometric quantities are computed using the Hamiltonian equation. The eigenvalues of these geometric quantities are then used to classify the divergent and collapsing trajectories of point vortices. Specifically, for the divergent trajectories of vortices, the eigenvalues of the geometric quantities converge to zero over time. Conversely, for their collapsing trajectories, the eigenvalues increase with time. This result implies that at the point of vortex collapse, the system becomes geometrically unstable, with diverging trajectory perturbations. Full article
(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications, 3rd Edition)
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24 pages, 6146 KiB  
Article
On the Nonlinear Forced Vibration of the Magnetostrictive Laminated Beam in a Complex Environment
by Nicolae Herisanu, Bogdan Marinca and Vasile Marinca
Mathematics 2024, 12(23), 3836; https://doi.org/10.3390/math12233836 - 4 Dec 2024
Viewed by 638
Abstract
The present study dealt with a comprehensive mathematical model to explore the nonlinear forced vibration of a magnetostrictive laminated beam. This system was subjected to mechanical impact, a nonlinear Winkler–Pasternak foundation, and an electromagnetic actuator considering the thickness effect. The expressions of the [...] Read more.
The present study dealt with a comprehensive mathematical model to explore the nonlinear forced vibration of a magnetostrictive laminated beam. This system was subjected to mechanical impact, a nonlinear Winkler–Pasternak foundation, and an electromagnetic actuator considering the thickness effect. The expressions of the nonlinear differential equations were obtained for the pinned–pinned boundary conditions with the help of the Galerkin–Bubnov procedure and Hamiltonian approach. The nonlinear differential equations were studied using an original, explicit, and very efficient technique, namely the optimal auxiliary functions method (OAFM). It should be emphasized that our procedure assures a rapid convergence of the approximate analytical solutions after only one iteration, without the presence of a small parameter in the governing equations or boundary conditions. Detailed results are presented on the effects of some parameters, among them being analyzed were the damping, frequency, electromagnetic, and nonlinear elastic foundation coefficients. The local stability of the equilibrium points was performed by introducing two variable expansion method, the homotopy perturbation method, and then applying the Routh–Hurwitz criteria and eigenvalues of the Jacobian matrix. Full article
(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications, 3rd Edition)
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13 pages, 2088 KiB  
Article
Computational Insights into the Unstable Fixed Point of the Fractional Difference Logistic Map
by Ernestas Uzdila, Inga Telksniene, Tadas Telksnys and Minvydas Ragulskis
Mathematics 2024, 12(23), 3635; https://doi.org/10.3390/math12233635 - 21 Nov 2024
Viewed by 581
Abstract
Thedivergence from the unstable fixed point of the fractional difference logistic map is investigated in this paper. In contrary to the classical logistic map, the memory horizon of the fractional difference logistic map reaches the initial condition. And though higher order orbits do [...] Read more.
Thedivergence from the unstable fixed point of the fractional difference logistic map is investigated in this paper. In contrary to the classical logistic map, the memory horizon of the fractional difference logistic map reaches the initial condition. And though higher order orbits do not exist in the fractional difference logistic map, a trajectory started at the unstable fixed point may continuously remain at the fixed point as the number of iterations tends to infinity. Such an effect is well known for the classical logistic map, but less so in the fractional difference logistic map. It appears that this effect depends on the accuracy of the floating point arithmetic. It is demonstrated that the divergence from the unstable fixed point of the fractional difference logistic map is a completely computational artifact. Using double precision, approximately 32% values of a from the interval 2.7<a≤3.7 diverge from the unstable fixed point. Full article
(This article belongs to the Special Issue Analytical Approaches to Nonlinear Dynamical Systems and Applications, 3rd Edition)
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