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Mathematical Modeling, Asymptotic Analysis and Stability of Solutions of Nonlinear...
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Mathematical Modeling, Asymptotic Analysis and Stability of Solutions of Nonlinear Dynamical Systems

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (15 April 2024) | Viewed by 4577

Special Issue Editor

Prof. Dr. Vladislav Vasilievich Lyubimov

E-Mail Website
Guest Editor
Department of Further Mathematics, Samara National Research University, 443086 Samara, Russia
Interests: system dynamics; mathematical modeling; control system synthesis; approximation theory; asymptotic methods; optimal control; aerospace control; space vehicles

Special Issue Information

Dear Colleagues,

As a rule, modern dynamical systems can be described by means of essential nonlinear differential equations and their systems. In most cases, analytical solutions to such systems cannot be found. In this regard, it is of interest to develop new and modernize known methods for the study of the stability of dynamic systems. In particular, it should be noted that the use of asymptotic methods makes it possible to study the evolution of slow variables and greatly simplifies the analysis of the stability of dynamical systems.

The intention of this Special Issue is to collect original and review articles devoted to the latest achievements in the field of mathematical modeling and the development of methods for the analysis of the stability of nonlinear dynamical systems.

Prof. Dr. Vladislav Vasilievich Lyubimov
Guest Editor

Manuscript Submission Information

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Keywords

  • mathematical modeling
  • nonlinear dynamic systems
  • discrete dynamic systems
  • asymptotic methods
  • stability of dynamic systems

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Published Papers (4 papers)

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Research

20 pages, 5549 KiB  
Article
Mathematical Modeling of the Displacement of a Light-Fuel Self-Moving Automobile with an On-Board Liquid Crystal Elastomer Propulsion Device
by Yunlong Qiu, Jiajing Chen, Yuntong Dai, Lin Zhou, Yong Yu and Kai Li
Mathematics 2024, 12(9), 1322; https://doi.org/10.3390/math12091322 - 26 Apr 2024
Cited by 9 | Viewed by 954
Abstract
The achievement and control of desired motions in active machines often involves precise manipulation of artificial muscles in a distributed and sequential manner, which poses significant challenges. A novel motion control strategy based on self-oscillation in active machines offers distinctive benefits, such as [...] Read more.
The achievement and control of desired motions in active machines often involves precise manipulation of artificial muscles in a distributed and sequential manner, which poses significant challenges. A novel motion control strategy based on self-oscillation in active machines offers distinctive benefits, such as direct energy harvesting from the ambient environment and the elimination of complex controllers. Drawing inspiration from automobiles, a self-moving automobile designed for operation under steady illumination is developed, comprising two wheels and a liquid crystal elastomer fiber. To explore the dynamic behavior of this self-moving automobile under steady illumination, a nonlinear theoretical model is proposed, integrating with the established dynamic liquid crystal elastomer model. Numerical simulations are conducted using the Runge-Kutta method based on MATLAB software, and it is observed that the automobile undergoes a supercritical Hopf bifurcation, transitioning from a static state to a self-moving state. The sustained periodic self-moving is facilitated by the interplay between light energy and damping dissipation. Furthermore, the conditions under which the Hopf bifurcation occurs are analyzed in detail. It is worth noting that increasing the light intensity or decreasing rolling resistance coefficient can improve the self-moving average velocity. The innovative design of the self-moving automobile offers advantages such as not requiring an independent power source, possessing a simple structure, and being sustainable. These characteristics make it highly promising for a range of applications including actuators, soft robotics, energy harvesting, and more. Full article
(This article belongs to the Special Issue Mathematical Modeling, Asymptotic Analysis and Stability of Solutions of Nonlinear Dynamical Systems)
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17 pages, 674 KiB  
Article
Asymptotic Antipodal Solutions as the Limit of Elliptic Relative Equilibria for the Two- and n-Body Problems in the Two-Dimensional Conformal Sphere
by Rubén Darío Ortiz Ortiz, Ana Magnolia Marín Ramírez and Ismael Oviedo de Julián
Mathematics 2024, 12(7), 1025; https://doi.org/10.3390/math12071025 - 29 Mar 2024
Viewed by 724
Abstract
We consider the two- and n-body problems on the two-dimensional conformal sphere MR2, with a radius R>0. We employ an alternative potential free of singularities at antipodal points. We study the limit of relative equilibria under [...] Read more.
We consider the two- and n-body problems on the two-dimensional conformal sphere MR2, with a radius R>0. We employ an alternative potential free of singularities at antipodal points. We study the limit of relative equilibria under the SO(2) symmetry; we examine the specific conditions under which a pair of positive-mass particles, situated at antipodal points, can maintain a state of relative equilibrium as they traverse along a geodesic. It is identified that, under an appropriate radius–mass relationship, these particles experience an unrestricted and free movement in alignment with the geodesic of the canonical Killing vector field in MR2. An even number of bodies with pairwise conjugated positions, arranged in a regular n-gon, all with the same mass m, move freely on a geodesic with suitable velocities, where this geodesic motion behaves like a relative equilibrium. Also, a center of mass formula is included. A relation is found for the relative equilibrium in the two-body problem in the sphere similar to the Snell law. Full article
(This article belongs to the Special Issue Mathematical Modeling, Asymptotic Analysis and Stability of Solutions of Nonlinear Dynamical Systems)
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10 pages, 2838 KiB  
Article
A Novel Chaotic System with Only Quadratic Nonlinearities: Analysis of Dynamical Properties and Stability
by Othman Abdullah Almatroud, Karthikeyan Rajagopal, Viet-Thanh Pham and Giuseppe Grassi
Mathematics 2024, 12(4), 612; https://doi.org/10.3390/math12040612 - 19 Feb 2024
Cited by 1 | Viewed by 1126
Abstract
In nonlinear dynamics, there is a continuous exploration of introducing systems with evidence of chaotic behavior. The presence of nonlinearity within system equations is crucial, as it allows for the emergence of chaotic dynamics. Given that quadratic terms represent the simplest form of [...] Read more.
In nonlinear dynamics, there is a continuous exploration of introducing systems with evidence of chaotic behavior. The presence of nonlinearity within system equations is crucial, as it allows for the emergence of chaotic dynamics. Given that quadratic terms represent the simplest form of nonlinearity, our study focuses on introducing a novel chaotic system characterized by only quadratic nonlinearities. We conducted an extensive analysis of this system’s dynamical properties, encompassing the examination of equilibrium stability, bifurcation phenomena, Lyapunov analysis, and the system’s basin of attraction. Our investigations revealed the presence of eight unstable equilibria, the coexistence of symmetrical strange repeller(s), and the potential for multistability in the system. Full article
(This article belongs to the Special Issue Mathematical Modeling, Asymptotic Analysis and Stability of Solutions of Nonlinear Dynamical Systems)
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12 pages, 597 KiB  
Article
A Method of Qualitative Analysis for Determining Monotonic Stability Regions of Particular Solutions of Differential Equations of Dynamic Systems
by Vladislav V. Lyubimov
Mathematics 2023, 11(14), 3142; https://doi.org/10.3390/math11143142 - 16 Jul 2023
Cited by 1 | Viewed by 1128
Abstract
Developing stability analysis methods for modern dynamical system solutions has been a significant challenge in the field. This study aims to formulate a qualitative analysis approach for the monotone stability region of a specific solution to a single differential equation within a dynamical [...] Read more.
Developing stability analysis methods for modern dynamical system solutions has been a significant challenge in the field. This study aims to formulate a qualitative analysis approach for the monotone stability region of a specific solution to a single differential equation within a dynamical system. The system in question comprises two first-order nonlinear ordinary differential equations of a particular kind. The method proposed hinges on applying elements of combinatorics to the traditional mathematical investigation of a function with a single independent variable. This approach enables the exact determination of the different qualitative scenarios in which the desired solution changes, under the assumption that the function values monotonically diminish from a specified value down to a finite zero. This paper outlines the creation and decomposition of the monotone stability region associated with the solution under consideration. Full article
(This article belongs to the Special Issue Mathematical Modeling, Asymptotic Analysis and Stability of Solutions of Nonlinear Dynamical Systems)
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