KCC-theory and geometry of the Rikitake system

Yajima, Takahiro; Nagahama, Hiroyuki

doi:10.1088/1751-8113/40/11/011

Cited by 35 publications

(19 citation statements)

References 34 publications

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“…The second invariant gives the Jacobi stability of the system [23,24]. The KCC theory has been applied for the study of different physical, biochemical or technical systems (see [23,24,1,2,27] and references therein).…”

mentioning

confidence: 99%

Jacobi stability analysis of dynamical systems—applications in gravitation and cosmology

Böhmer

Harkó

Sabau

2012

Advances in Theoretical and Mathematical Physics

128

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The Kosambi-Cartan-Chern (KCC) theory represents a powerful mathematical method for the analysis of dynamical systems. In this approach one describes the evolution of a dynamical system in geometric terms, by considering it as a geodesic in a Finsler space. By associating a non-linear connection and a Berwald type connection to the dynamical system, five geometrical invariants are obtained, with the second invariant giving the Jacobi stability of the system. The Jacobi (in)stability is a natural generalization of the (in)stability of the geodesic flow on a differentiable manifold endowed with a metric (Riemannian or Finslerian) to the non-metric setting. In the present paper we review the basic mathematical formalism of the KCC theory, and present some specific applications of this method in general relativity, cosmology and astrophysics. In particular we investigate the Jacobi stability of the general relativistic static fluid sphere with a linear barotropic equation of state, of the vacuum in the brane world models, of a dynamical dark energy model, and of the Lane-Emden equation, respectively. It is shown that the Jacobi stability analysis offers a powerful and simple method for constraining the physical properties of different systems, described by second order differential equations.

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mentioning

confidence: 99%

Jacobi stability analysis of dynamical systems—applications in gravitation and cosmology

Böhmer

Harkó

Sabau

2012

Advances in Theoretical and Mathematical Physics

128

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“…Then, we obtain

x^{0} = t

y^{0} = d x^{0} / d t = 1

. In this case, a function

G^{α}

is defined as follows:

\begin{matrix} true \\ right 2 G^{α} & = & left Γ_{β γ}^{α} \frac{d x^{β}}{d t} \frac{d x^{γ}}{d t} \\ = & left Γ_{j k}^{α} \frac{d x^{j}}{d t} \frac{d x^{k}}{d t} + 2 Γ_{false(j 0 false)}^{α} \frac{d x^{j}}{d t} + Γ_{00}^{α}, \end{matrix}

where the notation ( j 0) in

Γ_{false(j 0 false)}^{α}

expresses an exchange of j for 0:

{normalΓ}_{(j 0)}^{α} \equiv \frac{1}{2} ({normalΓ}_{j 0}^{α} + {normalΓ}_{0 j}^{α}) .

Equation comprising function describes the behavior of an integer‐order dynamical system with an external force . In this case, we consider the general case that the connection coefficients

Γ_{β γ}^{α}

of the function

G^{α}

in are asymmetric.…”

Section: Differential Geometry Of Fractional‐order Differential Equatmentioning

confidence: 99%

“…The systems of second‐order differential equations are closely related to differential geometries . From the applied perspective, various dynamical systems such as ecological systems, the Rikitake system in geophysics and the Lorenz system in meteorology have been studied via the geometrical treatments of the systems of second‐order differential equations . In astrophysics, the geometric objects of second‐order differential equations called the Emden–Fowler equation is used to describe the equilibrium of a star .…”

Section: Introductionmentioning

confidence: 99%

Geometric Structures of Fractional Dynamical Systems in Non‐Riemannian Space: Applications to Mechanical and Electromechanical Systems

Yajima

Nagahama

2018

Annalen der Physik

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Based on a non-Riemannian treatment of geometric objects, the geometric structures of fractional-order dynamical systems are investigated. A fractional derivative describes non-local effects across a space or a history encoded in memory features of the system. A system of fractional-order differential equations is formulated in film space that includes fictitious forces. Film space is a geometric space whose coordinates comprise time, and the geometric quantities vary in time. Fractional-order torsion tensors that appear are related to the dissipated energy and the energy conversions between subsystems and power of the system. The geometric treatment is then applied to damped-harmonic and fractional oscillators and the hybrid electromechanical Rikitake system. The damped-harmonic oscillator is characterized by two torsion tensors, whereas the fractional oscillator is characterized by one torsion tensor. Herein, the fractional order of the derivative of the metric tensor is used to characterize the damping of the fractional oscillator. The energy conversions between electromechanical subsystems in the Rikitake system are characterized by the torsion tensor. These results suggest that the non-Riemannian geometric objects can represent the non-local properties of fractional-order dynamical systems.

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“…This yields to a manifold (Finsler space) similar to phase-space, whose curvature properties determine the behavior of all solutions in a non-linear setting. The KCC theory has been applied for the study of different physical, biochemical or technical systems (see [2,1,4,3,5,32,33,37,16] and references therein). Moreover, to complete our study, we also use the Lyapunov function method to analyze the stability properties.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear Stability Analysis of the Emden–Fowler Equation

Böhmer¹,

Harkó²

2021

JNMP

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In this paper we qualitatively study radial solutions of the semilinear elliptic equation ∆u + u n = 0 with u(0) = 1 and u ′ (0) = 0 on the positive real line, called the Emden-Fowler or Lane-Emden equation. This equation is of great importance in Newtonian astrophysics and the constant n is called the polytropic index.By introducing a set of new variables, the Emden-Fowler equation can be written as an autonomous system of two ordinary differential equations which can be analyzed using linear and nonlinear stability analysis. We perform the study of stability by using linear stability analysis, the Jacobi stability analysis (Kosambi-Cartan-Chern theory) and the Lyapunov function method. Depending on the values of n these different methods yield different results. We identify a parameter range for n where all three methods imply stability.

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KCC-theory and geometry of the Rikitake system

Cited by 35 publications

References 34 publications

Jacobi stability analysis of dynamical systems—applications in gravitation and cosmology

Jacobi stability analysis of dynamical systems—applications in gravitation and cosmology

Geometric Structures of Fractional Dynamical Systems in Non‐Riemannian Space: Applications to Mechanical and Electromechanical Systems

Nonlinear Stability Analysis of the Emden–Fowler Equation

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