Some remarks on Jacobi stability

Sabau, Sorin V.

doi:10.1016/j.na.2005.02.061

Cited by 60 publications

(125 citation statements)

References 2 publications

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“…Lyapunov stability analysis involves the linearization of the dynamical system via the Jacobian matrix of a non-linear system, while the KCC theory addresses the Lyapunov stability of a whole trajectory in a tubular region [24].…”

Section: Discussion and Final Remarksmentioning

confidence: 99%

“…if ||ξ (t)|| < t 2 , t ≈ 0 + , the trajectories are bunching together, while if ||ξ (t)|| > t 2 , t ≈ 0 + , the trajectories are dispersing [24,25]. In terms of the deviation curvature tensor the focusing tendency of the trajectories can be described as follows: The trajectories of the system of equations (31) [24,25].…”

Section: Kosambi-cartan-chern (Kcc) Theory and Jacobi Stabilitymentioning

confidence: 99%

“…In terms of the deviation curvature tensor the focusing tendency of the trajectories can be described as follows: The trajectories of the system of equations (31) [24,25]. Based on these considerations we can define the Jacobi stability for a dynamical system as follows [24,25,29]:…”

Section: Kosambi-cartan-chern (Kcc) Theory and Jacobi Stabilitymentioning

confidence: 99%

“…By associating a non-linear connection and a Berwald type connection to the differential system, five geometrical invariants are obtained. The second invariant gives the Jacobi stability of the system [24,25]. The KCC theory has been applied for the study of different physical, biochemical or technical systems (see [24,25,28,29] and references therein).…”

Section: Introductionmentioning

confidence: 99%

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Jacobi stability of the vacuum in the static spherically symmetric brane world models

Harkó

Sabau

2008

Phys. Rev. D

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We analyze the stability of the structure equations of the vacuum in the brane world models, by using both the linear (Lyapunov) stability analysis, and the Jacobi stability analysis, the KosambiCartan-Chern (KCC) theory. In the brane world models the four dimensional effective Einstein equations acquire extra terms, called dark radiation and dark pressure, respectively, which arise from the embedding of the 3-brane in the bulk. Generally, the spherically symmetric vacuum solutions of the brane gravitational field equations, have properties quite distinct as compared to the standard black hole solutions of general relativity. We close the structure equations by assuming a simple linear equation of state for the dark pressure. In this case the vacuum is Jacobi stable only for a small range of values of the proportionality constant relating the dark pressure and the dark radiation. The unstable trajectories on the brane behave chaotically, in the sense that after a finite radial distance it would be impossible to distinguish the trajectories that were very near each other at an initial point. Hence the Jacobi stability analysis offers a powerful method for constraining the physical properties of the vacuum on the brane.

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Section: Discussion and Final Remarksmentioning

confidence: 99%

Section: Kosambi-cartan-chern (Kcc) Theory and Jacobi Stabilitymentioning

confidence: 99%

Section: Kosambi-cartan-chern (Kcc) Theory and Jacobi Stabilitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Jacobi stability of the vacuum in the static spherically symmetric brane world models

Harkó

Sabau

2008

Phys. Rev. D

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“…The concept of Jacobi stability initially appeared in the context of differential geometry for geodesics of Riemann or Finsler manifold and then was extended to trajectories of second order ODE ([1]- [11]). This is the differential geometric theory of the variational equations for deviation of whole trajectories to nearby ones.…”

Section: Introductionmentioning

confidence: 99%

Jacobi Stability of Linearized Geometric Dynamics

Udrişte¹,

Nicola²

2009

Journal of Dynamical Systems and Geometric Theories

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This paper is dedicated to the study of geometric dynamics (an Euler-Lagrange prolongation of a flow on a Riemannian manifold) from the point of view of KCC theory, Jacobi stability and Lyapounov stability. Section 1 recalls the geometncal roots of Jacobi stability and announces the subject of the paper. Section 2 introduces the variational ODEs (Jacobi fields ODEs), the KCC differential mvanants for a second order ODE system, and defines the Jacobi stability. SectIOn 3 studies the KCC differential mvanants asSOCiated to geometnc dynamiCs. SectIOn 4 descnbes vanous ImeanzatlOns of geometnc dynamiCs. SectIOn 5 studies the Jacobi stability for lineanzed geometnc dynamiCs around a statIOnary pomt of the field. SectIOn 6 shows that the lmeanzed geometnc dynamiCs around a cntlcal pomt of the energy can be Jacobi stable or unstable. SectIOn 7 proves the Lyapounov mstabllity of Jacobi fields ODEs along geometnc dynamiCs traJectones.

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Two geometrical invariants for three‐dimensional systems

Liu,

2024

Math Methods in App Sciences

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The subject of KCC theory is a second‐order ordinary differential equation, it is sometimes difficult to convert the high dimensional system into an equivalent second‐order system because of the analytical requirements of KCC theory. By means of the Euler‐Lagrange extension of a flow on a Riemannian manifold, this paper gives five geometric invariants of some three‐dimensional systems with great convenience, and focus on the analysis of two of them. The results show that the hyperbolic equilibria corresponding to the seven standard forms of three‐dimensional linear systems are Jacobi unstable. This is completely different from what we got before in two‐dimensional systems, where Jacobi stable and Jacobi unstable correspond to focus and node, respectively. All equilibria of classical Lü chaotic system and Yang‐Chen chaotic system are Jacobi unstable. Meanwhile, in three‐dimensional linear case, the torsion tensors at any point of the trajectory are identically equal to zero, but the two nonlinear systems have nonzero torsion tensors components.

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Some remarks on Jacobi stability

Cited by 60 publications

References 2 publications

Jacobi stability of the vacuum in the static spherically symmetric brane world models

Jacobi stability of the vacuum in the static spherically symmetric brane world models

Jacobi Stability of Linearized Geometric Dynamics

Two geometrical invariants for three‐dimensional systems

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