KCC theory and its application in a tumor growth model

Gupta, M. K.; Yadav, C. K.

doi:10.1002/mma.4542

Cited by 15 publications

(8 citation statements)

References 26 publications

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“…where Γ 1 , Γ 2 , and Γ 3 are circulations of first, second, and third point vortices, respectively. Thus, from (11), (12), (13), and ( 14), equations of motions of point vortices are expressed as the following Hamilton's canonical form:…”

Section: Review Of Point Vortex and Its Self-similaritymentioning

confidence: 99%

Geometrical Classification of Self-Similar Motion of Two-Dimensional Three Point Vortex System by Deviation Curvature on Jacobi Field

Hirakui

Yajima

2021

Advances in Mathematical Physics

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In this study, we geometrically analyze the relation between a point vortex system and deviation curvatures on the Jacobi field. First, eigenvalues of deviation curvatures are calculated from relative distances of point vortices in a three point vortex system. Afterward, based on the assumption of self-similarity, time evolutions of eigenvalues of deviation curvatures are shown. The self-similar motions of three point vortices are classified into two types, expansion and collapse, when the relative distances vary monotonously. Then, we find that the eigenvalues of self-similarity are proportional to the inverse fourth power of relative distances. The eigenvalues of the deviation curvatures monotonically convergent to zero for expansion, whereas they monotonically diverge for collapse, which indicates that the strengths of interactions between point vortices related to the time evolution of spatial geometric structure in terms of the deviation curvatures. In particular, for collapse, the collision point becomes a geometric singularity because the eigenvalues of the deviation curvature diverge. These results show that the self-similar motions of point vortices are classified by eigenvalues of the deviation curvature. Further, nonself-similar expansion is numerically analyzed. In this case, the eigenvalues of the deviation curvature are nonmonotonous but converge to zero, suggesting that the motion of the nonself-similar three point vortex system is also classified by eigenvalues of the deviation curvature.

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Section: Review Of Point Vortex and Its Self-similaritymentioning

confidence: 99%

Geometrical Classification of Self-Similar Motion of Two-Dimensional Three Point Vortex System by Deviation Curvature on Jacobi Field

Hirakui

Yajima

2021

Advances in Mathematical Physics

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“…For example, in biology, the Volterra-Hamilton system for modeling the dynamics of modular populations of a forest has been studied using Jacobi stability [18]. In other studies of biology, the KCC theory was applied to a tumor growth model and the Jacobi stability of the model was investigated [23]. For an oscillating system, a study regarding the Brusselator compared the Jacobi and linear stabilities [5].…”

Section: Introductionmentioning

confidence: 99%

Application of Jacobi stability analysis to a first-order dynamical system: relation between nonlinearizability of one-dimensional differential equation and Jacobi stable region

Hirakui,

Yajima

2024

J. Phys. Commun.

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In this study, we discuss Jacobi stability in equilibrium and nonequilibrium regions for a first-order one-dimensional system using deviation curvatures. The deviation curvature is calculated using the Kosambi-Cartan-Chern theory, which is applied to second-order differential equations. The deviation curvatures of the first-order one-dimensional differential equations are calculated using two methods as follows. Method 1 is only differentiating both sides of the equation. Additionally, Method 2 is differentiating both sides of the equation and then substituting the original equation into the second-order system. From the general form of the deviation curvatures calculated using each method, the analytical results are obtained as (A), (B), and (C). (A) Equilibrium points are Jacobi unstable for both methods; however, the type of equilibrium points is different. In Method 1, the equilibrium point is a nonisolated fixed point. Conversely, the equilibrium point is a saddle point in Method 2. (B) When there is a Jacobi stable region, the size of the Jacobi stable region in the Method 1 is different from that in Method 2. Especially, the Jacobi stable region in Method 1 is always larger than that in Method 2. (C) When there are multiple equilibrium points, the Jacobi stable region always exists in the nonequilibrium region located between the equilibrium points. These results are confirmed numerically using specific dynamical systems, which are given by the logistic equation and its evolution equation with the Hill function. From the results of (A) and (B), differences in types of equilibrium points affect the size of the Jacobi stable region. From (C), the Jacobi stable regions appear as nonequilibrium regions where the equations cannot be linearized.

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“…The second KCC invariants which is also known as a deviation curvature tensor gives us the Jacobi stability of the trajectories which measures the robustness of the second-order differential equation [18]. The Jacobi stability studies the robustness of second order differential equation which is analyzed by calculating deviation curvature tensor (second KCC invariant) by KCC theory [19][20][21][22]. The Lyapunov and Jacobi stability of circular orbits in the SBH spacetime has already been investigated in detail by Hossein [4].…”

Section: Introductionmentioning

confidence: 99%

Stability of circular geodesics in equatorial plane of Kerr spacetime

Singh,

Nandan,

Joshi

et al. 2022

Preprint

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We analyze the stability of circular geodesics for timelike as well as null geodesics of the Kerr BH spacetime with rotation parameter on the equatorial plane by Lyapunov stability analysis. Also, we verify the results of stability by presenting the phase portrait for both timelike and null geodesics. Further, by reviewing the Kosambi-Cartan-Chern (KCC) theory, we analyze the Jacobi stability for Kerr spacetime and present a comparative study of the methods used for stability analysis of geodesics.

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KCC theory and its application in a tumor growth model

Cited by 15 publications

References 26 publications

Geometrical Classification of Self-Similar Motion of Two-Dimensional Three Point Vortex System by Deviation Curvature on Jacobi Field

Geometrical Classification of Self-Similar Motion of Two-Dimensional Three Point Vortex System by Deviation Curvature on Jacobi Field

Application of Jacobi stability analysis to a first-order dynamical system: relation between nonlinearizability of one-dimensional differential equation and Jacobi stable region

Stability of circular geodesics in equatorial plane of Kerr spacetime

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