Jacobi Stability for Geometric Dynamics

Udrişte, Constantin; Nicola, Ileana Rodica

doi:10.1080/1726037x.2007.10698528

Cited by 11 publications

(13 citation statements)

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“…The trajectories of the system (5) are Jacobi stable if and only if the real parts of the eigenvalues of the deviation tensor P i j are strictly negative everywhere, and Jacobi unstable, otherwise. Now, we can write a rigorous definition of the Jacobi stability for a geodesic on a manifold endowed with an Euclidean, Riemannian or Finslerian metric or, even for a trajectory x i = x i (s) of the dynamical system corresponding to (5) [19][20][21][22]: Definition 3.3. A trajectory x i = x i (s) of ( 5) is said to be Jacobi stable if for any ε > 0, there exists δ(ε) > 0 such that ∥ xi (s) − x i (s)∥ < ε holds for all s ≥ s 0 and for all trajectories xi = xi (s) for which ∥ xi (s 0 )…”

Section: Kosambi-cartan-chern (Kcc) Geometric Theory and Jacobi Stabi...mentioning

confidence: 99%

“…According with [19][20][21][22], we take the trajectories of ( 5) as curves in a Euclidean space R n , where the norm ∥ • ∥ is the induced norm by the canonical inner product < •, • > on R n . More that, we will assume that the deviation vector ξ from (13) verify the initial conditions ξ(s 0 ) = O and ξ(s 0 ) = W ̸ = O, where O is the null vector from R n .…”

Section: Kosambi-cartan-chern (Kcc) Geometric Theory and Jacobi Stabi...mentioning

confidence: 99%

“…In the present paper we will study for the first time another kind of stability for this system, namely Jacobi stability. The Jacobi stability is a natural generalization of the stability of the geodesic flow on a differentiable manifold equipped with a Riemannian or Finslerian metrics to a manifold without a metric [14][15][16][17][18][19]. The Jacobi stability examines the robustness of a dynamical system defined by a system of second-order differential equations (SODEs), where the robustness is a measure of insensitivity and adaptation to change of the system internal parameters and the environment.…”

Section: Introductionmentioning

confidence: 99%

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A Study of the Jacobi Stability of the Rosenzweig–MacArthur Predator–prey System by the KCC Geometric Theory

Munteanu¹

2022

Preprint

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In this paper, we will consider an autonomous two-dimensional ODE Kolmogorov type 1 system with three parameters, which is a particular system of the general predator–prey systems with 2 a Holling type II. By reformulating this system as a set of two second order differential equations, we 3 will investigate the nonlinear dynamics of the system from the Jacobi stability point of view, using 4 the Kosambi-Cartan-Chern (KCC) geometric theory. We will determine the nonlinear connection, the 5 Berwald connection and the five KCC-invariants which express the intrinsic geometric properties 6 of the system, including the deviation curvature tensor. Furthermore, we will obtain necessary and 7 sufficient conditions on the parameters of the system in order to have the Jacobi stability near the 8 equilibrium points and we will point out these on a few examples.

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Section: Kosambi-cartan-chern (Kcc) Geometric Theory and Jacobi Stabi...mentioning

confidence: 99%

Section: Kosambi-cartan-chern (Kcc) Geometric Theory and Jacobi Stabi...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Study of the Jacobi Stability of the Rosenzweig–MacArthur Predator–prey System by the KCC Geometric Theory

Munteanu¹

2022

Preprint

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“…This approach of Jacobi stability appears as a prolongation of the geometric stability approach of the geodesic flow, from a Riemann or Finsler manifold to a differentiable manifold without any metric [24][25][26][27][28][29]. More precisely, the concept of Jacobi stability plays the role of proof of the resilience of a dynamical system defined by a system of second-order differential equations (semi-spray or SODE), where this resilience reflects the adaptability and the preservation of the system's basic behavior to changes in internal parameters and to influences from outside circumstances.…”

Section: Introductionmentioning

confidence: 99%

Jacobi Stability for T-System

Munteanu

2024

Symmetry

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In this paper will be considered a three-dimensional autonomous quadratic polynomial system of first-order differential equations with three real parameters, the so-called T-system. This system is symmetric relative to the Oz-axis and represents a special type of the generalized Lorenz system. The approach of this work will consist of the study of the nonlinear dynamics of this system through the Kosambi–Cartan–Chern (KCC) geometric theory. More exactly, we will focus on the associated system of second-order differential equations (SODE) from the point of view of Jacobi stability by determining the five invariants of the KCC theory. These invariants determine the internal geometrical characteristics of the system, and particularly, the deviation curvature tensor is decisive for Jacobi stability. Furthermore, we will look for necessary and sufficient conditions that the system parameters must satisfy in order to have Jacobi stability for every equilibrium point.

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“…Using the Euclidean metric gij = 6ij, we have seen in[17] that the matrix of the tensor P associated to(3.4) is or Theorem 5.1. 1} The deviation curvature tensor P is symmetric with respect to the Riemannian metric g. 2} The eigenvalues of deviation curvature tensor Pare I real numbers.…”

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confidence: 98%

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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Jacobi Stability for Geometric Dynamics

Cited by 11 publications

References 10 publications

A Study of the Jacobi Stability of the Rosenzweig–MacArthur Predator–prey System by the KCC Geometric Theory

A Study of the Jacobi Stability of the Rosenzweig–MacArthur Predator–prey System by the KCC Geometric Theory

Jacobi Stability for T-System

Jacobi Stability of Linearized Geometric Dynamics

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