Analyzing the Jacobi Stability of Lü’s Circuit System

Munteanu, Florian

doi:10.3390/sym14061248

Cited by 8 publications

(11 citation statements)

References 24 publications

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“…This indicates that such over-determined nonlinear affine systems could be good alternatives for objective plants in sparsity-based control schemes. C. Use Case 3: Lü's Attractor System Consider the following famous Lü's attractor system [52] with three additional control input variables:…”

Section: B Use Case 2: a Nonlinear Control System With Third-order Ov...mentioning

confidence: 99%

A Sparse Neural Network-Based Control Method for Saturated Nonlinear Affine Systems

Zhang,

Yin,

Huo

et al. 2024

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Saturated nonlinear affine systems are widely encountered in many engineering fields. Currently, most control methods on saturated nonlinear affine systems are not specifically designed based on sparsity-based control methodologies, and they might require sparse identification at the beginning stage and applying tracking control afterwards. In this paper, a sparse neural network (SNN)-based control method from an lp-norm (1 ≤ p < 2) optimization perspective is proposed for saturated nonlinear affine systems by taking advantage of the nice properties of primal dual neural networks for optimization. In particular, when p = 1, a new alternative controller based on SNN is derived, encountering computational difficulties distinct from those of another solution set in the basic dual neural network. The convergence properties of such SNN-based controllers are investigated and analyzed to find a control solution. Five illustrative examples further are shown to demonstrate the efficiency of the proposed SNN-based control method for tracking the desired references of saturated nonlinear affine systems. In the practical application scenario involving the UR5 robot control, the trajectory’s average errors are consistently confined to a minimal magnitude of 10−4 m. These findings substantiate the efficacy of the SNN-based control approach proposed for precise tracking control in saturated nonlinear affine systems.

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Section: B Use Case 2: a Nonlinear Control System With Third-order Ov...mentioning

confidence: 99%

A Sparse Neural Network-Based Control Method for Saturated Nonlinear Affine Systems

Zhang,

Yin,

Huo

et al. 2024

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“…According to [26] and [27], in this section we will made a brief presentation of the basic notions and main results from Kosambi-Cartan-Chern (KCC) theory [15], [16], [20], [21], [28][29][30]. The Kosambi-Cartan-Chern (KCC) theory is a modern and nice geometric approach of the dynamical systems which associates a semispray, a nonlinear connection and a Berwald connection to the SODE corresponding to the dynamical system.…”

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

confidence: 99%

“…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%

“…The KCC theory originated from the works of D. D. Kosambi [28], E. Cartan [29] and S. S. Chern [30], and hence the abbreviation KCC (Kosambi-Cartan-Chern). This geometric theory has a lot of applications in engineering, physics, chemistry and biology [20], [23,[25][26][27], [34]. Also, new developments and applications of KCC theory in gravitation and cosmology was done in [35], [36].…”

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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“…All these systems have an analytical form very close to the T-system, but with a different complex dynamics. A geometric approach to the Jacobi stability of these systems was recently studied in [18][19][20][21].…”

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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Analyzing the Jacobi Stability of Lü’s Circuit System

Cited by 8 publications

References 24 publications

A Sparse Neural Network-Based Control Method for Saturated Nonlinear Affine Systems

A Sparse Neural Network-Based Control Method for Saturated Nonlinear Affine Systems

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

Jacobi Stability for T-System

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