Kosambi–Cartan–Chern Stability in the Intermediate Nonequilibrium Region of the Brusselator Model

Yamasaki, Kazuhito; Yajima, Takahiro

doi:10.1142/s021812742250016x

Cited by 3 publications

(1 citation statement)

References 49 publications

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“…Jacobi stability has been analyzed by a large number of authors in the past years as an effective method for predicting chaotic behaviour of the systems [24][25][26]. Yamasaki and Yajima [27,28] has discussed the KCC stability in the intermediate nonequilibrium region of the Catastrophe and Brusselator model.…”

Section: Scaling Factor Of the Oregonator Modelmentioning

confidence: 99%

KCC Theory of the Oregonator Model for Belousov-Zhabotinsky Reaction

Gupta,

Sahu,

Yadav

et al. 2023

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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.

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Section: Scaling Factor Of the Oregonator Modelmentioning

confidence: 99%

KCC Theory of the Oregonator Model for Belousov-Zhabotinsky Reaction

Gupta,

Sahu,

Yadav

et al. 2023

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Elementary Catastrophe’s Chaos in One-Dimensional Discrete Systems Based on Nonlinear Connections and Deviation Curvature Statistics

Yamasaki

2024

Int. J. Bifurcation Chaos

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This study shows, by means of numerical analysis, that the characteristics of discrete dynamical systems, in which chaos and catastrophe coexist, are closely related to the geometric statistics in Finsler geometry. The two geometric statistics introduced are nonlinear connections information, denoted as [Formula: see text], and the mean deviation curvature, denoted as [Formula: see text]. The quantity [Formula: see text] can be used to determine the occurrence of chaos in terms of nonequilibrium stability. The resulting chaos is characterized by [Formula: see text] in terms of the trajectory’s robustness, which is related to the localization or globalization of chaos. The characteristics of catastrophe-induced chaos are clearly visualized through the contour topography of [Formula: see text], in which an abrupt change is represented by cliff topography (i.e. a line of critical points); initial dependence is reflected in the reversibility of topographic patterns. On overlaying the contour topography with the singularity pattern, it is evident that chaos does not arise around the singular point. Furthermore, the extensive development of cusp and butterfly chaos demands information on the nonlinear connections within the singularity pattern. The asymmetry in swallowtail chaos is less distinguishable in an equilibrated state, but becomes more evident when the system is in a state of nonequilibrium. In many analyses, chaos and catastrophe are examined separately. However, these results demonstrate that when both are present, the two have a complex relationship constrained by the singularity.

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KCC theory for a type of nonlinear damped SD oscillators

Li,

Liu,

Wang

2024

Int. J. Geom. Methods Mod. Phys.

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Smooth and discontinuous (SD) oscillators are the nonlinear models that will exhibit SD dynamics due to continuous variation of the smooth parameter. Kosambi–Cartan–Chern (KCC) theory is a differential geometric theory that describes the deviations from the entire trajectory of the variational equation to the nearby trajectory. This paper presents a completely new dynamical analysis of a type of SD oscillators with nonlinear damping based on the KCC theory. First, the paper gives five KCC geometrical invariants of SD oscillators in the smooth and non-smooth cases, respectively. The results show that the geometric quantities in the non-smooth case cannot be derived directly from the geometric quantities in the smooth case. Second, from these geometric quantities, KCC stability of SD oscillators trajectory at any point except [Formula: see text] is analyzed. Lastly, numerical results show that in some regions that deviate from the equilibrium point of system, the system will exhibit complex and variable dynamical behavior due to small changes in parameters. This paper shows that the KCC theory is also a useful tool in the dynamical analysis of non-smooth systems.

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Kosambi–Cartan–Chern Stability in the Intermediate Nonequilibrium Region of the Brusselator Model

Cited by 3 publications

References 49 publications

KCC Theory of the Oregonator Model for Belousov-Zhabotinsky Reaction

KCC Theory of the Oregonator Model for Belousov-Zhabotinsky Reaction

Elementary Catastrophe’s Chaos in One-Dimensional Discrete Systems Based on Nonlinear Connections and Deviation Curvature Statistics

KCC theory for a type of nonlinear damped SD oscillators

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