Narrow Results

The Jacobi stability of the normal form of typical bifurcations in one-dimensional dynamical systems is analyzed by introducing the concept of the production process (time-like potential) to KCC theory. This KCC theory approach shows that the geometric invariants of the system characterize the nonequilibrium dynamics of the bifurcations. For example, the deviation curvature that is one of the geometric invariants shows that the well-known two hysteresis jumps in subcritical pitchfork bifurcations differ qualitatively from each other. In the nonequilibrium region, the deviation curvature in the saddle-node and the transcritical bifurcations are a function of the bifurcation parameter alone; thus, the Jacobi stability does not depend on time. However, the deviation curvature in a pitchfork bifurcation is a function of the time-like potential, so the Jacobi stability does depend on time. This time dependence can be described by the Douglas tensor, which is a useful geometric invariant to consider how the higher-order term in the bifurcation system affects the stability structure.

In this paper, a 4D Lorenz-type multistable hyperchaotic system with a curve of equilibria is investigated by using differential geometry method, i.e. with KCC-theory. Due to the deviation curvature tensor and its eigenvalues, the curve of equilibria of this hyperchaotic system is proved analytically to be Jacobi unstable under a certain parameter condition, and a periodic orbit of this system is proved numerically to be also Jacobi unstable. Furthermore, the dynamics of contravariant vector field near the curve of equilibria and the periodic orbit are studied, respectively, and their results comply absolutely with the above analysis of Jacobi stability.

In this paper, the research of the Jacobi stability of the Chen system is performed by using the KCC-theory. By associating a nonlinear connection and a Berwald connection, five geometrical invariants of the Chen system are obtained. The Jacobi stability of the Chen system at equilibrium points and a periodic orbit is investigated in terms of the eigenvalues of the deviation curvature tensor. The obtained results show that the origin is always Jacobi unstable, while the Jacobi stability of the other two nonzero equilibrium points depends on the values of the parameters. And a periodic orbit of the Chen system is proved to be also Jacobi unstable. Furthermore, Jacobi stability regions of the Chen system and the Lorenz system are compared. Finally, the dynamical behavior of the components of the deviation vector near the equilibrium points is also discussed.

This paper considers the stability of a one-dimensional system during a catastrophic shift described by the Hill function. Because the shifting process goes through a nonequilibrium region, we applied the theory of Kosambi, Cartan, and Chern (KCC) to analyze the stability of this region based on the differential geometrical invariants of the system. Our results show that the Douglas tensor, one of the invariants in the KCC theory, affects the robustness of the trajectory during a catastrophic shift. In this analysis, the forward and backward shifts can have different Jacobi stability structures in the nonequilibrium region. Moreover, the bifurcation curve of the catastrophic shift can be interpreted geometrically, as the solution curve where the nonlinear connection and the deviation curvature become zero. KCC analysis also shows that even if the catastrophic pattern itself is similar, the stability structure in the nonequilibrium region is different in some cases, from the viewpoint of the Douglas tensor.

The purpose of the present paper is to study the stability of a prey–predator model using KCC theory. The KCC theory is based on the assumption that the second-order dynamical system and geodesics equation, in associated Finsler space, are topologically equivalent. The stability (Jacobi stability) based on KCC theory and linear stability of the model are discussed in detail. Further, the effect of parameters on stability and the presence of chaos in the model are investigated. The critical values of bifurcation parameters are found and their effects on the model are investigated. The numerical examples of particular interest are compared to the results of Jacobi stability and linear stability and it is found that Jacobi stability on the basis of KCC theory is global than the linear stability.

In this paper, the Jacobi stability of a two-degree-of-freedom mechanical system is studied by the innovative application of KCC-theory, namely differential geometric methods. We discuss the Jacobi stability of two equilibria and a periodic orbit by constructing geometric invariants. Both the regions of Jacobi stability and Lyapunov stability are presented to show the difference. We draw the phase portraits of the deviation vector near two equilibria under specific parameter values and initial conditions, and point out the sensitivity of deviation vector to initial conditions. In addition, the corresponding instability exponent and curvature are applicable for predicting the onset of chaos, which help us to detect chaotic behaviors quantitatively.

Little is known about bifurcations in two-dimensional (2D) differential systems from the viewpoint of Kosambi–Cartan–Chern (KCC) theory. Based on the KCC geometric invariants, three types of static bifurcations in 2D differential systems, i.e. saddle-node bifurcation, transcritical bifurcation, and pitchfork bifurcation, are discussed in this paper. The dynamics far from fixed points of the systems generating bifurcations are characterized by the deviation curvature and nonlinear connection. In the nonequilibrium region, the nonlinear stability of systems is not simple but involves alternation between stability and instability, even though systems are invariably Jacobi-unstable. The results also indicate that the dynamics in the nonequilibrium region are node-like for three typical static bifurcations.

This study applies the Kosambi–Cartan–Chern (KCC) theory to the Brusselator model to derive differential geometric quantities related to bifurcation phenomena. Based on these geometric quantities, the KCC stability of the Brusselator model is analyzed in linear and nonlinear cases to determine the extent to which nonequilibrium affects bifurcation and stability. The geometric quantities of the Brusselator model have a constant value in the linear case, and are functions of spatial variables with parameter dependence in the nonlinear case. Therefore, the KCC stability of the nonlinear case shows various distribution patterns, depending on the distance from the equilibrium point (EQP), as follows: in the regions near or far enough from the EQP, the distribution of KCC stability is uniform and regular; and in the intermediate nonequilibrium region, the distribution varies and shows complex patterns with parameter dependence. These results indicate that stability in the intermediate nonequilibrium region plays an important role in the dynamic complex patterns in the Brusselator model.

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.

We perform the study of the stability of the Lorenz system by using the Jacobi stability analysis, or the Kosambi–Cartan–Chern (KCC) theory. The Lorenz model plays an important role for understanding hydrodynamic instabilities and the nature of the turbulence, also representing a nontrivial testing object for studying nonlinear effects. The KCC theory represents a powerful mathematical method for the analysis of dynamical systems. In this approach, we describe the evolution of the Lorenz system in geometric terms, by considering it as a geodesic in a Finsler space. By associating a nonlinear connection and a Berwald type connection, five geometrical invariants are obtained, with the second invariant giving the Jacobi stability of the system. The Jacobi (in)stability is a natural generalization of the (in)stability of the geodesic flow on a differentiable manifold endowed with a metric (Riemannian or Finslerian) to the non-metric setting. In order to apply the KCC theory, we reformulate the Lorenz system as a set of two second-order nonlinear differential equations. The geometric invariants associated to this system (nonlinear and Berwald connections), and the deviation curvature tensor, as well as its eigenvalues, are explicitly obtained. The Jacobi stability of the equilibrium points of the Lorenz system is studied, and the condition of the stability of the equilibrium points is obtained. Finally, we consider the time evolution of the components of the deviation vector near the equilibrium points.

Geometric structures of dynamical systems are investigated based on a differential geometric method (Jacobi stability of KCC-theory). This study focuses on differences of Jacobi stability of two-dimensional second-order differential equation from that of one-dimensional second-order differential equation. One of different properties from a one-dimensional case is the Jacobi unstable condition given by eigenvalues of deviation curvature with different signs. Then, this geometric theory is applied to an overhead crane system as a two-dimensional dynamical system. It is shown a relationship between the Hopf bifurcation of linearized overhead crane and the Jacobi stability. Especially, the Jacobi stable trajectory is found for stable and unstable spirals of the two-dimensional linearized system. In case of the linearized overhead crane system, the Jacobi stable spiral approaches to the equilibrium point faster than the Jacobi unstable spiral. This means that the Jacobi stability is related to the resilience of deviated trajectory in the transient state. Moreover, for the nonlinear overhead crane system, the Jacobi stability for limit cycle changes stable and unstable over time.

In this paper, we study the Jacobi stability on the nonlinear double pendulum by the Kosambi–Cartan–Chern (KCC) theory. We assume that the mass and length of rods of two kinds of pendulums are equal, respectively. Moreover, we consider the case that initial angles of the double pendulum are equal. Under these conditions, we obtain the boundary between Jacobi stable and unstable trajectories for initial angles. It is shown that the condition of Jacobi stable or unstable depends only on deflection angles of the nonlinear double pendulum. Then, we discuss relationships between Jacobi stability, physical parameters and other concepts of stability such as Lyapunov stability and chaos. We suggest that the ratio of length of rods and the mass ratio of pendulums of the double pendulum do not affect the Jacobi stability. It is suggested that the equilibrium points in the Jacobi stable region and in the Jacobi unstable region are Lyapunov stable and Lyapunov unstable, respectively, and that the motions in the Jacobi unstable region are related to the onset of chaotic behavior.

Stability analysis of dynamical system is very useful and is able to classify the role of stable and unstable equilibrium points. In this work, Naiver–Stokes system has been studied by using KCC theory. The Jacobi stability and dynamics of the deviation vector near equilibrium points have been also studied. Further, the effect of bifurcation parameter on stability of Navier–Stokes system has been observed and found the limiting conditions for bifurcation. Numerical examples of particular interest have been taken to compare the results of Jacobi stability and linear stability. It is observed that Jacobi stability on the basis of KCC theory is more efficient than the linear stability.

Little seems to be known about the study of the chaotic system with only Lyapunov stable equilibria from the perspective of differential geometry. Therefore, this paper presents Jacobi analysis of an unusual three-dimensional (3D) autonomous chaotic system. Under certain parameter conditions, this system has positive Lyapunov exponents and only two linear stable equilibrium points, which means that chaotic attractor and Lyapunov stable equilibria coexist. The dynamical behavior of the deviation vector near the whole trajectories (including all equilibrium points) is analyzed in detail. The results show that the value of the deviation curvature tensor at equilibrium points is only related to parameters; the two equilibrium points of the system are Jacobi stable if the parameters satisfy certain conditions. Particularly, for a specific set of parameters, the linear stable equilibrium points of the system are always Jacobi unstable. A periodic orbit that is Lyapunov stable is also proven to be always Jacobi unstable. Next, Jacobi-stable regions of the Lorenz system, the Chen system and the system under study are compared for specific parameters. It can be found that although these three chaotic systems are very similar, their regions of Jacobi stable parameters are much different. Finally, by comparing Jacobi stability with Lyapunov stability, the obtained results demonstrate that the Jacobi stable parameter region is basically symmetric with the Lyapunov stable parameter region.

In this paper, Jacobi stability of a segmented disc dynamo system is geometrically investigated from viewpoint of Kosambi–Cartan–Chern (KCC) theory in Finsler geometry. First, the geometric objects associated to the reformulated system are explicitly obtained. Second, the Jacobi stability of equilibria and a periodic orbit are discussed in the light of deviation curvature tensor. It is shown that all the equilibria are always Jacobi unstable for any parameters, a Lyapunov stable periodic orbit falls into both Jacobi stable regions and Jacobi unstable regions. The considered system is not robust to small perturbations of the equilibria, and some fragments of the periodic orbit are included in fragile region, indicating that the system is extremely sensitive to internal parameters and environment. Finally, the dynamics of the deviation vector and its curvature near all the equilibria are presented to interpret the onset of chaos in the dynamo system. In a physical sense, magnetic fluxes and angular velocity can show irregular oscillations under some certain cases, these oscillations may reveal the irregularity of magnetic field’s evolution and reversals.

In this study, we discuss a relationship between the behavior of nonlinear dynamical systems and geometry of a system of second-order differential equations based on the Jacobi stability analysis. We consider how a maximal Lyapunov exponent is related to the geometric quantities. As a result of a theoretical investigation, the maximal Lyapunov exponent can be represented by a nonlinear connection and a deviation curvature. Thus, this means that the Jacobi stability given by the sign of the deviation curvature affects the change of the maximal Lyapunov exponent. Additionally, for an equation of nonlinear pendulum, we numerically confirm the theoretical results. We observe that a change of the maximal Lyapunov exponent is related to a change of an average deviation curvature. These results indicate that the deviation curvature and Jacobi stability are essential for considering the change of maximal Lyapunov exponent.

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.

The paper investigates the KCC (structural) stability of the second-order extension of the brain-stimulated Parkinson tremor dynamical system. After a brief presentation of the known linear stability results of the model, are determined and studied the KCC-invariants. It is emphasized that the spectral properties of the second KCC-invariant provide practical information for the behavior of the investigated model. The semispray of the second-order differential system associated to the basic SODE is evidentiated.