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16 pages, 667 KB

Open AccessArticle

Jacobi Stability Analysis of Liu System: Detecting Chaos

by Qinghui Liu and Xin Zhang

Mathematics 2024, 12(13), 1981; https://doi.org/10.3390/math12131981 - 26 Jun 2024

Cited by 1 | Viewed by 1649

By utilizing the Kosambi–Cartan–Chern (KCC) geometric theory, this paper is dedicated to providing novel insights into the Liu dynamical system, which stands out as one of the most distinctive and noteworthy nonlinear dynamical systems. Firstly, five important geometrical invariants of the system are obtained by associating the nonlinear connection with the Berwald connection. Secondly, in terms of the eigenvalues of the deviation curvature tensor, the Jacobi stability of the Liu dynamical system at fixed points is investigated, which indicates that three fixed points are Jacobi unstable. The Jacobi stability of the system is analyzed and compared with that of Lyapunov stability. Lastly, the dynamical behavior of components of the deviation vector is studied, which serves to geometrically delineate the chaotic behavior of the system near the origin. The onset of chaos for the Liu dynamical system is obtained. This work provides an analysis of the Jacobi stability of the Liu dynamical system, serving as a useful reference for future chaotic system research. Full article

(This article belongs to the Section C2: Dynamical Systems)

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16 pages, 571 KB

Open AccessArticle

KCC Theory of the Oregonator Model for Belousov-Zhabotinsky Reaction

by M. K. Gupta, Abha Sahu, C. K. Yadav, Anjali Goswami and Chetan Swarup

Axioms 2023, 12(12), 1133; https://doi.org/10.3390/axioms12121133 - 18 Dec 2023

Cited by 1 | Viewed by 3081

Abstract

The behavior of the simplest realistic Oregonator model of the BZ-reaction from the perspective of KCC theory has been investigated. In order to reduce the complexity of the model, we initially transformed the first-order differential equation of the Oregonator model into a system of second-order differential equations. In this approach, we describe the evolution of the Oregonator model in geometric terms, by considering it as a geodesic in a Finsler space. We have found five KCC invariants using the general expression of the nonlinear and Berwald connections. To understand the chaotic behavior of the Oregonator model, the deviation vector and its curvature around equilibrium points are studied. We have obtained the necessary and sufficient conditions for the parameters of the system in order to have the Jacobi stability near the equilibrium points. Further, a comprehensive examination was conducted to compare the linear stability and Jacobi stability of the Oregonator model at its equilibrium points, and We highlight these instances with a few illustrative examples. Full article

(This article belongs to the Special Issue Differential Geometry and Its Application, 2nd Edition)

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30 pages, 558 KB

Open AccessArticle

On the Jacobi Stability of Two SIR Epidemic Patterns with Demography

by Florian Munteanu

Symmetry 2023, 15(5), 1110; https://doi.org/10.3390/sym15051110 - 18 May 2023

Cited by 4 | Viewed by 2649

Abstract

In the present work, two SIR patterns with demography will be considered: the classical pattern and a modified pattern with a linear coefficient of the infection transmission. By reformulating of each first-order differential systems as a system with two second-order differential equations, we will examine the nonlinear dynamics of the system from the Jacobi stability perspective through the Kosambi–Cartan–Chern (KCC) geometric theory. The intrinsic geometric properties of the systems will be studied by determining the associated geometric objects, i.e., the zero-connection curvature tensor, the nonlinear connection, the Berwald connection, and the five KCC invariants: the external force

ε^{i}

—the first invariant; the deviation curvature tensor

P_{j}^{i}

—the second invariant; the torsion tensor

P_{j k}^{i}

—the third invariant; the Riemann–Christoffel curvature tensor

P_{j k l}^{i}

—the fourth invariant; the Douglas tensor

D_{j k l}^{i}

—the fifth invariant. In order to obtain necessary and sufficient conditions for the Jacobi stability near each equilibrium point, the deviation curvature tensor will be determined at each equilibrium point. Furthermore, we will compare the Jacobi stability with the classical linear stability, inclusive by diagrams related to the values of parameters of the system. Full article

(This article belongs to the Special Issue Geometric Algebra and Its Applications)

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23 pages, 1063 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Jacobi and Lyapunov Stability Analysis of Circular Geodesics around a Spherically Symmetric Dilaton Black Hole

by Cristina Blaga, Paul Blaga and Tiberiu Harko

Symmetry 2023, 15(2), 329; https://doi.org/10.3390/sym15020329 - 24 Jan 2023

Cited by 9 | Viewed by 2621

Abstract

We analyze the stability of the geodesic curves in the geometry of the Gibbons–Maeda–Garfinkle–Horowitz–Strominger black hole, describing the space time of a charged black hole in the low energy limit of the string theory. The stability analysis is performed by using both the linear (Lyapunov) stability method, as well as the notion of Jacobi stability, based on the Kosambi–Cartan–Chern theory. Brief reviews of the two stability methods are also presented. After obtaining the geodesic equations in spherical symmetry, we reformulate them as a two-dimensional dynamic system. The Jacobi stability analysis of the geodesic equations is performed by considering the important geometric invariants that can be used for the description of this system (the nonlinear and the Berwald connections), as well as the deviation curvature tensor, respectively. The characteristic values of the deviation curvature tensor are specifically calculated, as given by the second derivative of effective potential of the geodesic motion. The Lyapunov stability analysis leads to the same results. Hence, we can conclude that, in the particular case of the geodesic motion on circular orbits in the Gibbons–Maeda–Garfinkle–Horowitz–Strominger, the Lyapunov and the Jacobi stability analysis gives equivalent results. Full article

(This article belongs to the Special Issue BGL 2022: Dedicated to the Creators of the Non-euclidean Geometry: János Bolyai (220th Anniversary) and N.I. Lobachevski (230th Anniversary))

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16 pages, 352 KB

Open AccessArticle

A Study of the Jacobi Stability of the Rosenzweig–MacArthur Predator–Prey System through the KCC Geometric Theory

by Florian Munteanu

Symmetry 2022, 14(9), 1815; https://doi.org/10.3390/sym14091815 - 1 Sep 2022

Cited by 9 | Viewed by 2312

Abstract

In this paper, we consider an autonomous two-dimensional ODE Kolmogorov-type system with three parameters, which is a particular system of the general predator–prey systems with a Holling type II. By reformulating this system as a set of two second-order differential equations, we investigate the nonlinear dynamics of the system from the Jacobi stability point of view using the Kosambi–Cartan–Chern (KCC) geometric theory. We then determine the nonlinear connection, the Berwald connection, and the five KCC invariants which express the intrinsic geometric properties of the system, including the deviation curvature tensor. Furthermore, we obtain the necessary and sufficient conditions for the parameters of the system in order to have the Jacobi stability near the equilibrium points, and we point these out on a few illustrative examples. Full article

(This article belongs to the Special Issue Geometric Algebra and Its Applications)

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12 pages, 291 KB

Open AccessArticle

On the Non Metrizability of Berwald Finsler Spacetimes

by Andrea Fuster, Sjors Heefer, Christian Pfeifer and Nicoleta Voicu

Universe 2020, 6(5), 64; https://doi.org/10.3390/universe6050064 - 1 May 2020

Cited by 18 | Viewed by 2952

Abstract

We investigate whether Szabo’s metrizability theorem can be extended to Finsler spaces of indefinite signature. For smooth, positive definite Finsler metrics, this important theorem states that, if the metric is of Berwald type (i.e., its Chern–Rund connection defines an affine connection on the underlying manifold), then it is affinely equivalent to a Riemann space, meaning that its affine connection is the Levi–Civita connection of some Riemannian metric. We show for the first time that this result does not extend to general Finsler spacetimes. More precisely, we find a large class of Berwald spacetimes for which the Ricci tensor of the affine connection is not symmetric. The fundamental difference from positive definite Finsler spaces that makes such an asymmetry possible is the fact that generally, Finsler spacetimes satisfy certain smoothness properties only on a proper conic subset of the slit tangent bundle. Indeed, we prove that when the Finsler Lagrangian is smooth on the entire slit tangent bundle, the Ricci tensor must necessarily be symmetric. For large classes of Finsler spacetimes, however, the Berwald property does not imply that the affine structure is equivalent to the affine structure of a pseudo-Riemannian metric. Instead, the affine structure is that of a metric-affine geometry with vanishing torsion. Full article

(This article belongs to the Special Issue Finsler Modification of Classical General Relativity)

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