Jacobi Stability Research Articles

Jacobi Stability Analysis of Liu System: Detecting Chaos

By utilizing the Kosambi–Cartan–Chern (KCC) geometric theory, this paper is dedicated to providing novel insights into the Liu dynamical system, which stands out as one of the most distinctive and noteworthy nonlinear dynamical systems. Firstly, five important geometrical invariants of the system are obtained by associating the nonlinear connection with the Berwald connection. Secondly, in terms of the eigenvalues of the deviation curvature tensor, the Jacobi stability of the Liu dynamical system at fixed points is investigated, which indicates that three fixed points are Jacobi unstable. The Jacobi stability of the system is analyzed and compared with that of Lyapunov stability. Lastly, the dynamical behavior of components of the deviation vector is studied, which serves to geometrically delineate the chaotic behavior of the system near the origin. The onset of chaos for the Liu dynamical system is obtained. This work provides an analysis of the Jacobi stability of the Liu dynamical system, serving as a useful reference for future chaotic system research.

Generalization of KCC-theory to fractional dynamical systems and application to viscoelastic oscillations

As a geometrical method for analyzing the stability of dynamical systems, Kosambi–Cartan–Chern (KCC)-theory determines the Jacobi stability of a perturbation given to a trajectory based on the deviation curvature tensor. In this study, KCC-theory is generalized to dynamical systems with fractional derivatives, and the deviation curvature tensor is obtained from the fractional differential equation under the condition that the order of the fractional derivatives is close to an integer. Fractional KCC-theory is then applied to a one-dimensional spring–mass–damper model in which the damping force is expressed as a fractional derivative. In this case, three solution types are obtained depending on the damping coefficient and the fractional derivative order. One is damped oscillation with overshoot, which also appears in models with integer order derivatives, and another is damped oscillation without overshoot. The remaining one is overdamping, which neither oscillates nor overshoots. The deviation curvature tensor obtained from the fractional oscillatory model shows that both the damped oscillation without overshoot and the overdamping are Jacobi stable. When a discontinuous Jacobi instability appears on the solution curve, the solution type is classified as damped oscillation with overshoot. This result indicates that damped oscillation with overshoot and other solutions can be classified according to Jacobi stability.

Geometrical and Numerical Analysis of Predator–Prey System Based on the Allee Effect in Predator

This study explores the complex dynamics of the predator–prey interactions, with a specific emphasis on the influence of the Allee effect on the predator population. We examined the fundamental mathematical characteristics of the model under consideration, such as the positivity of the system and the boundedness of the solutions. We investigated the equilibrium points and analyzed their stability using the Jacobi and Lyapunov methods. A comprehensive examination was carried out on the geometric properties of the dynamical system to compute the five invariants of the KCC theory. In particular, the deviation curvature tensor and its eigenvalues are investigated to demonstrate the behavior of the system stability. We have also obtained the necessary and sufficient conditions for the given set of parameters of the system in order to have the Jacobi stability (instability) near the equilibrium point. To visualize the dynamical behavior of the predator–prey model with the Allee effect in the predator density, numerical simulations were conducted. The investigation encompasses an examination of the system’s behavior from both geometric and numerical standpoints, with the objective of attaining a thorough comprehension using few examples.

Application of Jacobi stability analysis to a first-order dynamical system: relation between nonlinearizability of one-dimensional differential equation and Jacobi stable region

In this study, we discuss Jacobi stability in equilibrium and nonequilibrium regions for a first-order one-dimensional system using deviation curvatures. The deviation curvature is calculated using the Kosambi-Cartan-Chern theory, which is applied to second-order differential equations. The deviation curvatures of the first-order one-dimensional differential equations are calculated using two methods as follows. Method 1 is only differentiating both sides of the equation. Additionally, Method 2 is differentiating both sides of the equation and then substituting the original equation into the second-order system. From the general form of the deviation curvatures calculated using each method, the analytical results are obtained as (A), (B), and (C). (A) Equilibrium points are Jacobi unstable for both methods; however, the type of equilibrium points is different. In Method 1, the equilibrium point is a nonisolated fixed point. Conversely, the equilibrium point is a saddle point in Method 2. (B) When there is a Jacobi stable region, the size of the Jacobi stable region in the Method 1 is different from that in Method 2. Especially, the Jacobi stable region in Method 1 is always larger than that in Method 2. (C) When there are multiple equilibrium points, the Jacobi stable region always exists in the nonequilibrium region located between the equilibrium points. These results are confirmed numerically using specific dynamical systems, which are given by the logistic equation and its evolution equation with the Hill function. From the results of (A) and (B), differences in types of equilibrium points affect the size of the Jacobi stable region. From (C), the Jacobi stable regions appear as nonequilibrium regions where the equations cannot be linearized.

Classification of Time-Optimal Paths Under an External Force Based on Jacobi Stability in Finsler Space

Classification of Time-Optimal Paths Under an External Force Based on Jacobi Stability in Finsler Space

Jacobi Stability for T-System

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.

KCC Theory of the Oregonator Model for Belousov-Zhabotinsky Reaction

The behavior of the simplest realistic Oregonator model of the BZ-reaction from the perspective of KCC theory has been investigated. In order to reduce the complexity of the model, we initially transformed the first-order differential equation of the Oregonator model into a system of second-order differential equations. In this approach, we describe the evolution of the Oregonator model in geometric terms, by considering it as a geodesic in a Finsler space. We have found five KCC invariants using the general expression of the nonlinear and Berwald connections. To understand the chaotic behavior of the Oregonator model, the deviation vector and its curvature around equilibrium points are studied. We have obtained the necessary and sufficient conditions for the parameters of the system in order to have the Jacobi stability near the equilibrium points. Further, a comprehensive examination was conducted to compare the linear stability and Jacobi stability of the Oregonator model at its equilibrium points, and We highlight these instances with a few illustrative examples.

Анализ устойчивости по Якоби и восстановление параметров двойного маятника с демпфированием

Jacobi stability for a double damped pendulum was analyzed on the basis of the Kosambi–Cartan–Chern theory making it possible to determine geometric structures and five geometric invariants of the dynamical system. Eigenvalues of the second invariant (deviation curvature tensor) provided the estimate of the Jacobi stability related to the insensitivity measure against disturbances of the system itself and the environment. Such studies are relevant in applications, where it is required to determine the system stability regions according to Lyapunov and Jacobi simultaneously. Inverse problem of restoring the system parameters from indirect information represented by the eigenvalues of the deviation curvature tensor was formulated. For a double pendulum with damping, the Jacobi stability conditions were justified in terms of its free parameters. Solution to the inverse problem of restoring the pendulum parameters was obtained using the optimization approach. When minimizing the regularized criterion function, a new hybrid global optimization algorithm was applied. Numerical example is provided.

On the Jacobi Stability of Two SIR Epidemic Patterns with Demography

In the present work, two SIR patterns with demography will be considered: the classical pattern and a modified pattern with a linear coefficient of the infection transmission. By reformulating of each first-order differential systems as a system with two second-order differential equations, we will examine the nonlinear dynamics of the system from the Jacobi stability perspective through the Kosambi–Cartan–Chern (KCC) geometric theory. The intrinsic geometric properties of the systems will be studied by determining the associated geometric objects, i.e., the zero-connection curvature tensor, the nonlinear connection, the Berwald connection, and the five KCC invariants: the external force εi—the first invariant; the deviation curvature tensor Pji—the second invariant; the torsion tensor Pjki—the third invariant; the Riemann–Christoffel curvature tensor Pjkli—the fourth invariant; the Douglas tensor Djkli—the fifth invariant. In order to obtain necessary and sufficient conditions for the Jacobi stability near each equilibrium point, the deviation curvature tensor will be determined at each equilibrium point. Furthermore, we will compare the Jacobi stability with the classical linear stability, inclusive by diagrams related to the values of parameters of the system.

About the Jacobi Stability of a Generalized Hopf–Langford System through the Kosambi–Cartan–Chern Geometric Theory

In this work, we will consider an autonomous three-dimensional quadratic system of first-order ordinary differential equations, with five parameters and with symmetry relative to the z-axis, which generalize the Hopf–Langford system. By reformulating the system as a system of two second-order ordinary differential equations and using the Kosambi–Cartan–Chern (KCC) geometric theory, we will investigate this system from the perspective of Jacobi stability. We will compute the five invariants of KCC theory which determine the own geometrical properties of this system, especially the deviation curvature tensor. Additionally, we will search for necessary and sufficient conditions on the five parameters of the system in order to reach the Jacobi stability around each equilibrium point.

Jacobi and Lyapunov Stability Analysis of Circular Geodesics around a Spherically Symmetric Dilaton Black Hole

We analyze the stability of the geodesic curves in the geometry of the Gibbons–Maeda–Garfinkle–Horowitz–Strominger black hole, describing the space time of a charged black hole in the low energy limit of the string theory. The stability analysis is performed by using both the linear (Lyapunov) stability method, as well as the notion of Jacobi stability, based on the Kosambi–Cartan–Chern theory. Brief reviews of the two stability methods are also presented. After obtaining the geodesic equations in spherical symmetry, we reformulate them as a two-dimensional dynamic system. The Jacobi stability analysis of the geodesic equations is performed by considering the important geometric invariants that can be used for the description of this system (the nonlinear and the Berwald connections), as well as the deviation curvature tensor, respectively. The characteristic values of the deviation curvature tensor are specifically calculated, as given by the second derivative of effective potential of the geodesic motion. The Lyapunov stability analysis leads to the same results. Hence, we can conclude that, in the particular case of the geodesic motion on circular orbits in the Gibbons–Maeda–Garfinkle–Horowitz–Strominger, the Lyapunov and the Jacobi stability analysis gives equivalent results.

Dynamics and Jacobi stability of the controlled 3D Hindmarsh-Rose neuron model

Dynamics and Jacobi stability of the controlled 3D Hindmarsh-Rose neuron model

When Shimizu–Morioka model meets Jacobi stability analysis: Detecting chaos

This paper is concerned with the Jacobi stability of the Shimizu–Morioka model by using the KCC-theory. First, by associating the nonlinear connection and Berwald connection, five geometrical invariants of the dynamical model are obtained. Furthermore, the Jacobi stability of the Shimizu–Morioka model at equilibrium is studied in terms of the eigenvalues of the deviation curvature tensor. It shows that the three equilibria are always Jacobi unstable. Finally, the dynamical behavior of the components of the deviation vector is discussed, which geometrically characterizes the chaotic behavior of studied model near the origin. It proved the onset of chaos in the Shimizu–Morioka model.

A Study of the Jacobi Stability of the Rosenzweig–MacArthur Predator–Prey System through the KCC Geometric Theory

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

Analyzing the Jacobi Stability of Lü’s Circuit System

By reformulating the circuit system of Lü as a set of two second order differential equations, we investigate the nonlinear dynamics of Lü’s circuit system from the Jacobi stability point of view, using Kosambi–Cartan–Chern geometric theory. We will determine the five KCC invariants, which express the intrinsic geometric properties of the system, including the deviation curvature tensor. Finally, we will obtain necessary and sufficient conditions on the parameters of the system to have the Jacobi stability near the equilibrium points.

Qualitative geometric analysis of traveling wave solutions of the modified equal width Burgers equation

This paper devotes to the qualitative geometric analysis of the traveling wave solutions of MEW‐Burgers wave equation. Firstly, MEW‐Burgers equation is transformed into an equivalent planar dynamical system by using traveling wave transformation. Then the global structure of the planar system is presented, and solitary waves, kink waves (anti‐kink waves), and periodic waves are found. Secondly, Jacobi stability for the planar system is studied based on KCC theory, and Jacobi stability of any point on the trajectory of system is dicussed. The dynamical behavior of the deviation vector near the equilibrium points is analyzed, and the numerical simulation is consistent with the theoretical analysis. Lyapunov stability and Jacobi stability of the equilibrium points are also compared and analyzed. The obtaining results show that Lyapunov stability of the equilibrium points of the system is not exactly consistent with Jacobi stability. Finally, the planar system with periodic disturbance is transformed into a six dimensional nonlinear system, and the periodic, quasi‐periodic, and chaotic dynamical behaviors of the system are numerically simulated.

Novel Pehlivan–Uyarŏglu Chaotic System Variants and their CFOA Based Realization

In this paper, four variants of Pehlivan–Uyarŏglu chaotic system (PUCS) have been proposed. The properties of the proposed PUCSs are examined through numerical simulations and the parameter values are obtained by observing the bifurcation diagrams for state variables. Further, the convergence/divergence of nearby orbits is investigated by noticing the evolution of Lyapunov exponents with time. It is found that the values of Lyapunov exponents are negative, zero and positive for all proposed variants thus confirming the chaoticity of the proposals. The strangeness of the proposed variants is also studied. The stability of PUCS and its proposed variants is examined using Jacobi stability analysis. A current feedback operational amplifier (CFOA) based circuit is put forward that can realize the existing PUCS and its proposed variants, by simply adjusting the component values. The proposed realization is compact (23% saving in overall component count) in comparison to its operational amplifier (OpAmp) based counterpart. The behavior of the proposed variants in time domain, frequency domain and phase space have been examined through simulations in LTspice design environment. Furthermore, the feasibility of the proposed variants is also discussed through presenting the electronic circuit implementation of two of the variants and the results obtained are in good agreement with the LTspice simulations. Monte Carlo (MC) simulations are also included to show the robustness of the proposed circuit against parameter variations.

Stability of circular geodesics in equatorial plane of Kerr spacetime

We analyze the stability of circular geodesics for timelike as well as null geodesics of the Kerr BH spacetime with rotation parameter on the equatorial plane by Lyapunov stability analysis. Also, we verify the results of stability by presenting the phase portrait for both timelike and null geodesics. Further, by reviewing the Kosambi-Cartan-Chern (KCC) theory, we analyze the Jacobi stability for Kerr spacetime and present a comparative study of the methods used for stability analysis of geodesics.

Jacobi stability analysis and impulsive control of a 5D self-exciting homopolar disc dynamo

<p style='text-indent:20px;'>In this paper, we make a thorough inquiry about the Jacobi stability of 5D self-exciting homopolar disc dynamo system on the basis of differential geometric methods namely Kosambi-Cartan-Chern theory. The Jacobi stability of the equilibria under specific parameter values are discussed through the characteristic value of the matrix of second KCC invariants. Periodic orbit is proved to be Jacobi unstable. Then we make use of the deviation vector to analyze the trajectories behaviors in the neighborhood of the equilibria. Instability exponent is applicable for predicting the onset of chaos quantitatively. In addition, we also consider impulsive control problem and suppress hidden attractor effectively in the 5D self-exciting homopolar disc dynamo.</p>

Вычислительная диагностика неустойчивых по Якоби динамических систем с использованием гибридных алгоритмов глобальной оптимизации

The study focuses on the problems of restoration and analysis of free parameters of dynamical systems from indirect, approximately given information. In the context of the Kosambi --- Cartan --- Chern theory, a geometric description of the time-evolution of the system is introduced. Five geometric invariants are determined for the system under study. The eigenvalues of the second invariant estimate the Jacobi stability of the system. Such a study is of interest in practical applications, where it is required to identify the regions in which both Lyapunov stability and Jacobi stability occur simultaneously. The inverse problem of computational diagnostics of the system is formulated for approximately given eigenvalues of the second invariant. The solution to the regularized inverse problem is determined using an optimization approach. Scalar criterion functions are assumed to be continuous, multidimensional, locally Lipschitzian, and not necessarily everywhere differentiable. When searching for global solutions, we used new hybrid algorithms that integrate stochastic algorithms for scanning the space of variables and a deterministic local minimization procedure. The numerical scanning procedure is implemented with the use of two modified versions: quasi-opposition-based and rotation-based learning mechanisms. In the phase of local search, two-parameter smoothing approximations of criterion functions are introduced. Examples of solving problems of computational diagnostics of Jacobi unstable dynamical systems are given: the Lorentz system and a controllable elliptical pendulum