Differential geometric structure of non-equilibrium dynamics in competition and predation: Finsler geometry and KCC theory

Yamasaki, Kazuhito; Yajima, Takahiro

doi:10.1080/1726037x.2016.1250500

Cited by 14 publications

(8 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Using Equations (11), (12), and (13), the motions of the three point vortices are shown on a two-dimensional plane. Further, to describe a three point vortex system as Hamilton's canonical form, the Hamiltonian of three point vortices is defined as follows:…”

Section: Review Of Point Vortex and Its Self-similaritymentioning

confidence: 99%

See 1 more Smart Citation

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

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Review Of Point Vortex and Its Self-similaritymentioning

confidence: 99%

“…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

Self Cite

View full text Add to dashboard Cite

show abstract

“…Indeed, Jacobi stability analysis has a good connection with the robustness of some biological systems. For example, it can reveal the fragility and robustness of a cell model with arrest states [29] and reflect the geometric structure of interactions between the competition and predation [36,37]. Jacobi stability analysis was seen as a powerful method used for detecting a kind of stability artifact [8].…”

mentioning

confidence: 99%

Jacobi stability analysis and impulsive control of a 5D self-exciting homopolar disc dynamo

Wei

Wang

et al. 2022

DCDS-B

View full text Add to dashboard Cite

<p style='text-indent:20px;'>In this paper, we make a thorough inquiry about the Jacobi stability of 5D self-exciting homopolar disc dynamo system on the basis of differential geometric methods namely Kosambi-Cartan-Chern theory. The Jacobi stability of the equilibria under specific parameter values are discussed through the characteristic value of the matrix of second KCC invariants. Periodic orbit is proved to be Jacobi unstable. Then we make use of the deviation vector to analyze the trajectories behaviors in the neighborhood of the equilibria. Instability exponent is applicable for predicting the onset of chaos quantitatively. In addition, we also consider impulsive control problem and suppress hidden attractor effectively in the 5D self-exciting homopolar disc dynamo.</p>

show abstract

“…Robustness is a measure of insensitivity and adaptation to change of the system internal parameters and the environment. KCC theory has been applied to the dynamical system in cosmology [9,22,15,8], gravitation [42,1,47], biology or ecology [4,48,49,17,44,5,6]. Nowadays, Jacobi stability analysis has become a useful tool in the study of the complexity of typical chaotic systems.…”

mentioning

confidence: 99%

Dynamics at infinity and Jacobi stability of trajectories for the Yang-Chen system

Liu¹,

Huang²,

Wei³

2021

DCDS-B

View full text Add to dashboard Cite

The present work is devoted to giving new insights into a chaotic system with two stable node-foci, which is named Yang-Chen system. Firstly, based on the global view of the influence of equilibrium point on the complexity of the system, the dynamic behavior of the system at infinity is analyzed. Secondly, the Jacobi stability of the trajectories for the system is discussed from the viewpoint of Kosambi-Cartan-Chern theory (KCC-theory). The dynamical behavior of the deviation vector near the whole trajectories (including all equilibrium points) is analyzed in detail. The obtained results show that in the sense of Jacobi stability, all equilibrium points of the system, including those of the two linear stable node-foci, are Jacobi unstable. These studies show that one might witness chaotic behavior of the system trajectories before they enter in a neighborhood of equilibrium point or periodic orbit. There exists a sort of stability artifact that cannot be found without using the powerful method of Jacobi stability analysis.

show abstract

Differential geometric structure of non-equilibrium dynamics in competition and predation: Finsler geometry and KCC theory

Cited by 14 publications

References 28 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

Jacobi stability analysis and impulsive control of a 5D self-exciting homopolar disc dynamo

Dynamics at infinity and Jacobi stability of trajectories for the Yang-Chen system

Contact Info

Product

Resources

About