Jacobi stability analysis of modified Chua circuit system

Gupta, M. K.; Yadav, C. K.

doi:10.1142/s021988781750089x

Cited by 25 publications

(13 citation statements)

References 28 publications

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“…Research found in other studies 46,48,49 that the curvature of the deviation vector can reveal the chaotic behavior of the system to some extent, and their numerical results also show this relation. In this subsection, we analyze qualitatively the curvature k 0 of the curve (t) = 1 (t), 2 (t) in the system (1.1).…”

Section: Curvature Of the Deviation Vectormentioning

confidence: 65%

“…44,45 In other words, the Jacobi unstable trajectories of a dynamical system behave chaotically, because it is impossible to distinguish the trajectories that are very close at the initial moment after a finite time interval. 43 The Jacobi stability of some typical dynamical systems (Lorenz system, 46 Rössler system, 47 Chen system, 48 Chua circuit system, 49 Navier-Stokes system, 50 Rikitake system, 51 and others (see, e.g.,other studies [52][53][54][55] as well as their references) has been studied. They qualitatively described the chaotic evolution of the dynamical systems by analyzing the dynamics of the deviation vector.…”

Section: Figurementioning

confidence: 99%

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New insights into a chaotic system with only a Lyapunov stable equilibrium

Chen

Liu

Wei

et al. 2020

Math Methods in App Sciences

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This paper gives some new insights into a chaotic system. The considered system has only one Lyapunov stable equilibrium and positive Lyapunov exponent in some certain parameter range, which means there coexists chaotic attractors and Lyapunov stable equibirium in system. First, from the perspective of analyzing the global structure of the system, based on the Poincaré compactification technique, the complete description of its dynamical behavior on the sphere at infinity is presented. The obtaining results show that its global dynamical behavior is very complex. Second, from a geometric perspective, the Jacobi stability of the system is investigated, including the unique equilibrium and a periodic orbit. Interestingly, we find that the unique Lyapunov stable equilibrium is always Jacobi unstable, a Lyapunov stable periodic orbit falls into both Jacobi stable and Jacobi unstable regions. It is shown that we might witness chaotic behavior of the system before the trajectories enter a neighborhood of the equilibrium point. It is hoped that the investigation of this paper can contribute to better understand and study the complex chaotic system with stable equilibria.

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Section: Curvature Of the Deviation Vectormentioning

confidence: 65%

Section: Figurementioning

confidence: 99%

New insights into a chaotic system with only a Lyapunov stable equilibrium

Chen

Liu

Wei

et al. 2020

Math Methods in App Sciences

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“…The geometric background of this study is explained based on some related literatures. () Let

false(x^{1}, x^{2}, ⋯, x^{n} false) = false(x false), false(\frac{d x^{1}}{d t}, \frac{d x^{2}}{d t}, ⋯, \frac{d x^{n}}{d t} false) = false(\frac{d x}{d t} false) = y

and t be 2 n +1 coordinates in an open connected subset Ω of the Euclidean (2 n +1)‐dimensional space R n × R n × R 1 . Let us consider a second‐order differential equation of the form

\frac{d^{2} x^{i}}{d t^{2}} + 2 G^{i} false(x, y, t false) = 0, i = 1, 2, . . . n .

for which each G i is

C^{\infty}

in a neighborhood of initial conditions (( x ) 0 ,( y ) 0 , t 0 )∈Ω.…”

Section: Kcc Theory and Jacobi Stabilitymentioning

confidence: 99%

KCC theory and its application in a tumor growth model

Gupta

Yadav

2017

Math Methods in App Sciences

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In this study, we consider the stability of tumor model by using the standard differential geometric method that is known as Kosambi-Cartan-Chern (KCC) theory or Jacobi stability analysis. In the KCC theory, we describe the time evolution of tumor model in geometric terms. We obtain nonlinear connection, Berwald connection and KCC invariants. The second KCC invariant gives the Jacobi stability properties of tumor model. We found that the equilibrium points are Jacobi unstable for positive coordinates. We also discussed the time evolution of components of deviation tensor and the behavior of deviation vector near the equilibrium points.

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“…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. Jacobi stability analysis of dynamical systems has been recently studied by several authors in [15], [16], [20][21][22][23][24][25][26][27], using the Kosambi-Cartan-Chern (KCC) theory [28][29][30]. More exactly, the dynamics of the system is studied with the help of the geometric objects associated to the system of second order differential equations obtained from the initial first order differential system.…”

Section: Introductionmentioning

confidence: 99%

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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Jacobi stability analysis of modified Chua circuit system

Cited by 25 publications

References 28 publications

New insights into a chaotic system with only a Lyapunov stable equilibrium

New insights into a chaotic system with only a Lyapunov stable equilibrium

KCC theory and its application in a tumor growth model

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

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