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

Abstract

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.

1. Introduction

The main aim of this paper is to study the Jacobi stability of a generalized Hopf–Langford system by following the geometric approach of the Kosambi–Cartan–Chern (KCC) theory. In order to obtain the Jacobi stability requirements, we compute the five invariant tensors of the KCC theory: the fist invariant ${\epsilon }^i$—the external force; the second invariant $P_j^i$—the deviation curvature tensor; the third KCC invariant $P_{jk}^i$—the torsion tensor, the fourth invariant $P_{jkl}^i$—the Riemann–Christoffel curvature tensor, and the fifth KCC invariants $D_{jkl}^i$—the Douglas tensor. Additionally, we determine the zero-connection curvature tensor $Z_j^i$ and the Berwald connection $G_{jl}^i$. These tensors show us the intrinsic geometric characteristics of the system and they can be symmetric, skew-symmetric, or neither, but only the deviation curvature tensor sets this geometrical Jacobi stability around each equilibrium point.

The original Hopf–Langford system represents a deterministic mathematical model for the phenomenon of turbulence in fluid dynamics [1,2]. We will focus on the generalized Hopf–Langford system with five real parameters—a, b, c, d, e:

$$\left\{\begin{array}{ccc}\hfill \dot{x}& =& ax+by+xz\hfill \\ \hfill \dot{y}& =& cx+dy+yz\hfill \\ \hfill \dot{z}& =& ez-\left(x^2+y^2+z^2\right)\hfill \end{array}\right.$$

This generalized system was introduced by Yang, Q. and Yang, T. in 2018 in [3]. After the publication of the famous paper of E. Hopf [1] in 1948, in order to model the phenomenon of turbulence in fluid dynamics, Langford presented and analyzed a three-dimensional quadratic system in 1981, in [2], by simplifying the equations introduced by Hopf in [1]. In the last decades, many works have been published related to the study of Lyapunov (or linear) stability for the Hopf–Langford system or for different generalized Hopf–Langford systems [4,5,6,7,8]. Recently, some perturbations of the Hopf–Langford system that do not modify reflection function were studied in [9] and admissible perturbations of a generalized system Hopf–Langford were introduced and studied in [10]. Particularly, various bifurcations of the invariant torus and some knotted periodic orbits for a generalized Hopf–Langford system were studied in 2021 [11].

The aim of this work is to conduct a study of another type of stability for the generalized Hopf–Langford system, the so-called Jacobi stability. This Jacobi stability represents a geometrical approach and is a natural extension of the stability for the geodesic flow on a smooth manifold endowed with a Riemann or Finsler metric to a manifold with no metric [12,13,14,15,16,17,18]. This Jacobi stability emphasizes the sturdiness of a dynamical system associated by a system of second-order differential equations (SODE), where this sturdiness represents the level of the lack of sensitivity and conformation to the changing of the dynamical system internal characteristics and to the external influences. The study of the Jacobi stability of dynamical systems by using the Kosambi–Cartan–Chern (KCC) theory [19,20,21] has been a topic of great interest in the last decade [14,15,22,23,24,25,26,27]. More precisely, crucial results about the behaviour and the nonlinear dynamics, including the chaos of the dynamical system, are obtained by using the geometric properties of the five invariants defined by the system of second-order differential equations (SODE), which is obtained from the system of first-order differential equations [28,29].

Practically, the Kosambi–Cartan–Chern theory analyzes the deviation of close integral curves and indicates how to establish the allowed perturbation around each equilibrium point of the system. This geometric theory deals with the investigation of the variation equations (or the so-called Jacobi field equations) associated with the geometry defined on the smooth manifold. At the beginning, this geometric stability, called Jacobi stability, was studied by Antonelli, Ingarden and Matsumoto in [12,14,15], but only for the geodesics of a Riemann manifold or Finsler manifold. By using the KCC-covariant derivative, a differential system in variations was obtained and then the second invariant was discovered, called the deviation curvature tensor. The sign of the real parts of the eigenvalues of this tensor is crucial to decide the stability from the Jacobi point of view, both for a geodesic and, more general, for the dynamical system associated with a system of second-order differential equations (SODE). This approach is very important because, in differential geometry, every SODE (also called semi-spray) can define a nonlinear connection on the manifold and conversely, every nonlinear connection defines a semi-spray. Therefore, every SODE can introduce a geometry on the phase space by the corresponding geometric objects, i.e., the tensors of connections, curvatures and torsion [16,30,31,32].

The origins of the KCC geometric theory come from the years 1933–1939, with the papers of Kosambi [19], Cartan [20] and Chern [21], and this explains the abbreviation KCC (Kosambi–Cartan–Chern). Sixty years later, this theory was rediscovered and developed by Antonelli, Ingarden and Matsumoto in [12,14,15] and then many applications in engineering, physics, chemistry and biology were presented in [22,33,34,35,36,37]. Furthermore, recent approaches and new perspectives of KCC theory in black hole theories can be found in [38,39,40]. Very important and useful is the paper [13] (2012), where Boehmer, Harko and Sabău studied the Jacobi stability and the relationship with classical stability, together with a lot of applications in gravitation, cosmology and astrophysics.

In Section 2, we will provide a concise presentation of the generalized Hopf–Langford system and of his equilibrium points. In Section 3, in order to search the Jacobi stability of generalized Hopf–Langford system, a comprehensive review of the framework of the geometric KCC theory was presented, by following [25,26,27]. We define the five invariants of the geometric theory and we present the basics of the Jacobi stability. In Section 4, we reformulate the generalized Hopf–Langford system as a system of second-order differential equations and we calculate the five geometric invariants. Next, in Section 5, the outcomes related to the Jacobi stability of the generalized Hopf–Langford system near the equilibrium points are presented. Finally, in Section 6, the deviation equations around each equilibrium point and the expression of the curvature of the deviation vector are determined. The sum over crossed repeated indices is understood.

2. The Generalized Hopf–Langford System

By simplifying the equations proposed by Hopf [1] in order to model the phenomenon of turbulence in fluid dynamics, Langford presented and analyzed a three-dimensional first-order quadratic system of differential equations, which was noted in [2] and called the Langford system

$$\left\{\begin{array}{ccc}\hfill \dot{x}& =& \left(\rho -1\right)x-y+xz\hfill \\ \hfill \dot{y}& =& x+\left(\rho -1\right)y+yz\hfill \\ \hfill \dot{z}& =& \rho z-\left(x^2+y^2+z^2\right)\hfill \end{array}\right.$$

(1)

where $\rho$ is a real parameter of the system and the overdot denotes differentiation with respect to the time.

In papers [4,5], S.G. Nikolov, B. Bozhkov and V.Y. Belozyorov proved the existence of a chaotic attractor for the system (1). In [6], S.G. Nikolov and V.M. Vassilev introduced and studied the following generalized Hopf–Langford system with three real parameters:

$$\left\{\begin{array}{ccc}\hfill \dot{x}& =& ax-by+xz\hfill \\ \hfill \dot{y}& =& bx+ay+yz\hfill \\ \hfill \dot{z}& =& cz-\left(x^2+y^2+z^2\right)\hfill \end{array}\right.$$

(2)

It was shown that the system is equivalent to the nonlinear force-free Duffing equation and in some cases, the solutions of the system were expressed in explicit analytical form in terms of elementary and Jacobi elliptic functions. Recently, in 2021, S.G. Nikolov and V.M. Vassilev performed a complete assessing of the nonlinear dynamical behavior of a Hopf–Langford type by using the system’s energy as a mode transformation [7].

In [3], Q. Yang and T. Yang introduced and studied the following, more generalized Hopf–Langford system with five real parameters:

$$\left\{\begin{array}{ccc}\hfill \dot{x}& =& ax+by+xz\hfill \\ \hfill \dot{y}& =& cx+dy+yz\hfill \\ \hfill \dot{z}& =& ez-\left(x^2+y^2+z^2\right)\hfill \end{array}\right.$$

(3)

It was shown that the generalized Hopf–Langford system features the coexistence of two periodic orbits and the coexistence of a periodic orbit and an invariant torus [3].

Let us observe that the generalized Hopf–Langford system (3) does not changes via the transformation $(x,y,z)\to (-x,-y,z)$, i.e., the integral curves of the system are symmetric relative to the z-axis.

Next, we will focus on this generalized Hopf–Langford system (3). As well as previous Hopf–Langford-type systems (1) and (2), this system (3) has two equilibrium points $O(0,0,0)$, $G(0,0,e)$. For the system (3), the Jacobi matrix at a point $(x,y,z)$ has the form:

$$A=\left(\begin{array}{ccc}a+z& b& x\\ c& d+z& y\\ -2x& -2y& e-2z\end{array}\right)$$

If we denote by $\Delta ={(a+d)}^2-4\left(ad-bc\right)={(a-d)}^2+4bc$, then we have:

For $O(0,0,0)$ we have the Jacobi matrix $A=\left(\begin{array}{ccc}a& b& 0\\ c& d& 0\\ 0& 0& e\end{array}\right)$, with eigenvalues ${\lambda }_1=e$, ${\lambda }_{2,3}=$$\frac{1}{2}\left(a+d\pm \sqrt{\Delta }\right)$.

Let us remark that ${\lambda }_2{\lambda }_3=ad-bc$, ${\lambda }_2+{\lambda }_3=a+d$.

For $G(0,0,e)$, we have the Jacobi matrix $A=\left(\begin{array}{ccc}a+e& b& 0\\ c& d+e& 0\\ 0& 0& -e\end{array}\right)$, with eigenvalues ${\lambda }_1=e$, ${\lambda }_{2,3}=\frac{1}{2}\left(a+d+2e\pm \sqrt{{\Delta }_1}\right)$, where ${\Delta }_1={(a+d+2e)}^2-4\left((a+e)(d+e)-bc\right)$$={(a-d)}^2+4bc=\Delta$, and ${\lambda }_2{\lambda }_3=(a+e)(d+e)-bc=e^2+(a+d)e+ad-bc$, ${\lambda }_2+{\lambda }_3=a+d+2e$.

As there are a lot of parameters which involve in the analysis of the system, we can point out the fact that it is quite difficult to determine the dynamics of the system in the neighborhood of the equilibrium points. Nevertheless, a complete study of the classical linear (Lyapunov) stability close to the equilibrium points O and G was conducted by Q. Yang and T. Yang in [3]. Therefore, in the next sections, our purpose is to study the Jacobi stability.

3. Kosambi–Cartan–Chern Geometric Theory and Jacobi Stability

In order to understand the basics of Jacobi stability, it is mandatory to present in this section the basic elements of the Kosambi–Cartan–Chern (KCC) geometric theory [12,14,15,19,20,21,22,23,25,26,27].

The KCC geometric theory is a modern geometric method of the study of the dynamics of the modelled systems. The basic idea is to associate to the system of second-order differential equations (SODE), which defines the dynamical system, the next geometrical objects: a nonlinear connection, a zero-connection curvature tensor, a Berwald connection and another five tensors (called the invariants): ${\epsilon }^i$–the external force, $P_j^i$–the deviation curvature tensor, $P_{jk}^i$–the torsion tensor, $P_{jkl}^i$–the Riemann–Christoffel curvature tensor and $D_{jkl}^i$–the Douglas curvature tensor. However, fortunately, only the second invariant (the deviation curvature tensor) determines the Jacobi stability of the system.

Let M be a real, $C^{\infty }$–manifold with dimension n and let us denote by $TM$ the tangent bundle. In applications, $M={\mathbf{R}}^n$ or $M\subset {\mathbf{R}}^n$ is an open set. Let $u=(x,y)\in TM$, where $x=\left(x^1,\dots ,x^n\right)$ and $y=\left(y^1,\dots ,y^n\right)$, with $y^i=\frac{d{x}^i}{dt}$, $i=1,\dots ,n$.

Let us consider the following second-order system of differential equations written in normalized form [12]

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

(4)

where $G^i(x,y)$ are $C^{\infty }$–functions locally defined in a coordinates chart on $TM$, i.e., an open set around of a point $(x_0,y_0)\in TM$ with initial conditions. The system (4) is inspired from the Euler–Lagrange equations of classical dynamics [12,30]

$$\left\{\begin{array}{ccc}\hfill \frac{d}{dt}\frac{\partial L}{\partial y^i}-\frac{\partial L}{\partial x^i}& =& F^i\hfill \\ \hfill y^i& =& \frac{d{x}^i}{dt}\hfill \end{array}\right.,i=1,\dots ,n.$$

(5)

where $L(x,y)$ is a regular Lagrange function on $TM$ and $F^i$ are the so-called external forces.

Usually, the system (4) does not have any geometrical meaning because “accelerations” $\frac{d^2{x}^i}{d{t}^2}$ or “forces” $G^i(x^j,y^j)$ are not tensors of type $(0,1)$ via the local charts change on $TM$

$$\left\{\begin{array}{ccc}\hfill {\tilde{x}}^i& =& {\tilde{x}}^i(x^1,\dots ,x^n)\hfill \\ \hfill {\tilde{y}}^i& =& \frac{\partial {\tilde{x}}^i}{\partial x^j}{y}^j\hfill \end{array}\right.,i=1,\dots ,n.$$

(6)

According to [12,30], we say that the system of second-order differential equations (SODE) (4) has a geometrical meaning (and it is said to be a semi-spray), if the transformations of the coefficients $G^i(x^j,y^j)$ via the local coordinates change (6) are made by the following relations

$$2{\tilde{G}}^i=2G^j\frac{\partial {\tilde{x}}^i}{\partial x^j}-\frac{\partial {\tilde{y}}^i}{\partial x^j}{y}^j.$$

(7)

The fundamental idea of the Kosambi–Cartan–Chern (KCC) geometric approach is that it is possible to change the system of second-order differential Equation (4) into an equivalent system (i.e., with the same solutions) if and only if the corresponding coefficients $G^i(x^j,y^j)$ are transformed via the local coordinates change (6) by the relations (7). This means that $G^i(x^j,y^j)$ has a geometrical meaning or is a $(0,1)$-type tensor (i.e., is a semi-spray). In conclusion, it is possible to show that this semi-spray determines five tensor fields, which are called geometrical (or differential) invariants of the KCC geometric theory [14,15]. In order to obtain the five invariants of the system (4) via the local coordinates transformation (6), it is mandatory to use the KCC-covariant derivative of a vector field $\xi ={\xi }^i\frac{\partial }{\partial x^i}$, which is defined locally, in an open set of $TM$, following the works in [14,19,20,21]

$$\frac{D{\xi }^i}{dt}=\frac{d{\xi }^i}{dt}+N_j^i{\xi }^j,$$

(8)

where $N_j^i=\frac{\partial G^i}{\partial y^j}$ are the components of a nonlinear connection N on the tangent bundle $TM$ defined by the SODE (or semi-spray) (4).

For ${\xi }^i=y^i$, it follows that

$$\frac{D{y}^i}{dt}=-2G^i+N_j^i{y}^j=-{\epsilon }^i.$$

(9)

The vector field ${\epsilon }^i$ is said to be the first invariant of KCC geometric theory. This first invariant is interpreted, from the physical point of view, as an external force [14]. Furthermore, ${\epsilon }^i$ has a geometrical meaning because, relative to the local charts change (6), the following equality holds

$${\tilde{\epsilon }}^i=\frac{\partial {\tilde{x}}^i}{\partial x^j}{\epsilon }^j.$$

If the coefficients $G^i$ are functions homogeneous of degree 2, relative to $y^i$, this means that $\frac{\partial G^i}{\partial y^j}{y}^j=2G^i$, for all $i=1,\dots ,n$, then all components of the first invariant vanish, i.e., ${\epsilon }^i=0$, for all $i=1,\dots ,n$. Therefore, the first invariant is null if and only if the semi-spray becomes a spray. This is valid for the geodesic spray defined by a Riemannian metric or a Finslerian metric [12,30].

From the dynamical point of view, the basic target of Kosambi–Cartan–Chern geometric approach is to deepen the knowledge about the behaviour of the system (4), relative to a slight deviation of the integral curves upon a certain integral curve. More precisely, a study of the dynamics of the system in variations will be conducted, and to this end, it must vary the integral curve $x^i\left(t\right)$ of (4) in a neighborhood, using the following formula:

$${\tilde{x}}^i\left(t\right)=x^i\left(t\right)+\eta {\xi }^i\left(t\right)$$

(10)

where $\left|\eta \right|$ is a small parameter and ${\xi }^i\left(t\right)$ are the coordinates of a vector field defined along the integral curves $x^i\left(t\right)$. After substituting (10) into (4) and using the limit as $\eta \to 0$, the next system of second-order differential equations is obtained [12,14,15]:

$$\frac{d^2{\xi }^i}{d{t}^2}+2N_j^i\frac{d{\xi }^j}{dt}+2\frac{\partial G^i}{\partial x^j}{\xi }^j=0$$

(11)

Following the covariant derivative from (8), the system of second-order differential Equation (11) can be rewritten in the covariant form [12,14,15]:

$$\frac{D^2{\xi }^i}{d{t}^2}=P_j^i{\xi }^j$$

(12)

where the $(1,1)$-type tensor $P_j^i$ has the components

$$P_j^i=-2\frac{\partial G^i}{\partial x^j}-2G^l{G}_{jl}^i+y^l\frac{\partial N_j^i}{\partial x^l}+N_l^i{N}_j^l$$

(13)

where

$$G_{jl}^i=\frac{\partial N_j^i}{\partial y^l}$$

(14)

represent the components of the so-called Berwald connection defined by the nonlinear connection N [12,30].

Let us remark that if all components of nonlinear connection and of the Berwald connection vanish, then the deviation curvature tensor from (13) has the coefficients $P_j^i=-2\frac{\partial G^i}{\partial x^j}$. Then, by following [35], we will define the so-called zero-connection curvature tensor Z by

$$Z_j^i=2\frac{\partial G^i}{\partial x^j}.$$

(15)

Let us point out that, for a system of dimension 2, the zero-connection curvature tensor Z is the Gaussian curvature K of the potential surface $V\left(x^i\right)=0$, where ${\dot{x}}^i=f^i\left(x^j\right)=-\frac{\partial V}{\partial x^i}\left(x^j\right)$. Moreover, if the potential surface is minimal, then $P=-K$.

This tensor $\left(P_j^i\right)$ is called the deviation curvature tensor and is the second invariant of KCC theory. Equation (11) is known as the deviation equations or even Jacobi equations. Equation (12) is also called the Jacobi equation. In terms of Riemannian geometry or Finslerian geometry, if the system of second-order differential equations (SODE) represents the geodesic motion, then the previous system of equations is just the Jacobi field equations associated with the defined geometry.

In the geometric KCC theory, the third, fourth and fifth invariant of the system of second-order differential equations (SODE) (4) are also defined. These last invariants are given by

$$\begin{array}{c}{P}_{jk}^i=\frac{1}{3}\left(\frac{\partial P_j^i}{\partial y^k}-\frac{\partial P_k^i}{\partial y^j}\right),\\ \\ P_{jkl}^i=\frac{\partial P_{jk}^i}{\partial y^l},D_{jkl}^i=\frac{\partial G_{jk}^i}{\partial y^l}.\end{array}$$

(16)

From the differential geometry perspective, the third invariant, $P_{jk}^i$, is viewed as a torsion tensor, and the fourth and fifth invariants, $P_{jkl}^i$ and $D_{jkl}^i$, represent the Riemann–Christoffel curvature tensor, and, respectively, the Douglas tensor.

In accordance with [12,14,15,23,30], let us highlight that all these five tensors always exist.

In the framework of KCC theory, the five invariants are the fundamental mathematical quantities which define the intrinsic geometrical properties of the system of second-order differential equations [12,20,30].

Next, we present a very important result of KCC theory, what is Antonelli’s contribution [14]:

Theorem 1.

The systems of second-order differential equations of type (4),

$$\frac{d^2{x}^i}{d{t}^2}+2G^i(x^j,y^j)=0,y^j=\frac{d{x}^j}{dt}$$

and

$$\frac{d^2{\tilde{x}}^i}{d{t}^2}+2{\tilde{G}}^i({\tilde{x}}^j,{\tilde{y}}^j)=0,{\tilde{y}}^j=\frac{d{\tilde{x}}^j}{dt},$$

can be transformed one into another by locally changing the coordinates (6) if and only if the five invariants, ${\epsilon }^i$, $P_j^i$, $P_{jk}^i$, $P_{jkl}^i$ and $D_{jkl}^i$, are tensors equivalent to the tensors ${\tilde{\epsilon }}^i$, ${\tilde{P}}_j^i$, ${\tilde{P}}_{jk}^i$, ${\tilde{P}}_{jkl}^i$, and, respectively, ${\tilde{D}}_{jkl}^i$.

In particular, there exist local coordinates $(x^1,\dots ,x^n)$ on the base manifold M, with $G^i=0$, for every i, if and only if all five invariant tensors are null. In this situation, the integral curves of the dynamical systems are precisely straight lines.

The name of Jacobi stability in the framework of this theory comes from the fact that when (4) is the system of second-order differential equations for a geodesic in Riemann geometry or Finsler geometry, then (12) is exactly the Jacobi field equations for the geodesic deviation. The Jacobi field Equation (12) of the Finsler (or Riemann) manifold $(M,F)$ can be put in the scalar form [17]

$$\frac{d^{2}v}{d{s}^2}+K·v=0.$$

(17)

Here, ${\xi }^i=v\left(s\right){\eta }^i$ is the Jacobi field on the geodesic curve $\gamma :x^i=x^i\left(s\right)$ and ${\eta }^i$ is the unit normal vector field on the geodesic curve $\gamma$ and K is the flag curvature of Finslerian space $(M,F)$. The sign of the flag curvature K influences the geodesic rays. More precisely, if $K>0$, then the geodesics curves merge (i.e., are Jacobi stable), and if $K<0$, then the geodesics curves disperse (i.e., are Jacobi unstable). Therefore, following the equivalence between (12) and (17), it happens that a positive (correspondingly, a negative) flag curvature is equivalent to negative (correspondingly, positive) eigenvalues of the deviation curvature tensor $P_j^i$. Then, the well-known result makes sense [13]:

Theorem 2.

The orbits (or trajectories) of the system (4) are Jacobi stable if and only if the eigenvalues or the real parts of the eigenvalues of the deviation tensor $P_j^i$ are strictly negative everywhere. Otherwise, they are Jacobi unstable.

Following [13,18,22,23], next, we present a rigorous definition of the Jacobi stability for a geodesic on a manifold with an Euclid, Riemann or Finsler structure or, more generally, for an orbit (or trajectory) $x^i=x^i\left(s\right)$ of the dynamical system corresponding to (4):

Definition 1.

An orbit (or a trajectory) $x^i=x^i\left(s\right)$ of (4) is called Jacobi stable if for every $\epsilon >0$, there is $\delta \left(\epsilon \right)>0$ so that $\parallel {\tilde{x}}^i\left(s\right)-x^i\left(s\right)\parallel <\epsilon$ holds for all $s\ge s_0$ and for all orbits ${\tilde{x}}^i={\tilde{x}}^i\left(s\right)$ with $\parallel {\tilde{x}}^i\left(s_0\right)-x^i\left(s_0\right)\parallel <\delta \left(\epsilon \right)$ and $\parallel \frac{d{\tilde{x}}^i}{ds}\left(s_0\right)-\frac{d{x}^i}{ds}\left(s_0\right)\parallel <\delta \left(\epsilon \right)$.

In the KCC theory, we will study the orbits of (4) as trajectories in a Euclidean space ${\mathbf{R}}^n$ with the norm $\parallel ·\parallel$ induced by the canonical scalar product $<·,·>$ on ${\mathbf{R}}^n$. Additionally, we will suppose that the deviation vector $\xi$ from (12) meets the initial conditions $\xi \left(s_0\right)=O$ and $\dot{\xi }\left(s_0\right)=W\ne O$, where $O\in {\mathbf{R}}^n$ is the zero vector. If we choose $s_0=0$ and $\parallel W\parallel =1$, then for $s\searrow 0$, the orbits of (4) merge if and only if the real parts of all eigenvalues of $P_j^i\left(0\right)$ are strictly negative or they disperse if and only if at least one of the real parts of the eigenvalues of $P_j^i\left(0\right)$ is positive.

This type of stability highlights the focus behavior (near to $s_0=0$) of the orbits of (4) relative to (10) that verifies the conditions $\parallel {\tilde{x}}^i\left(0\right)-x^i\left(0\right)\parallel =0$ and $\parallel \frac{d{\tilde{x}}^i}{ds}\left(0\right)-\frac{d{x}^i}{ds}\left(0\right)\parallel \ne 0$.

Let us note that the system of second-order differential equations (SODE) or semi-spray (4) is stable from the Jacobi point of view if and only if the variational system (11) (or the system in the covariant form (12)) is stable from the classical (or linear) point of view. In conclusion, the Jacobi stability approach is focused on the study of linear stability of every orbits in a domain, but without taking into account the velocity. This geometric theory, even when considered near an equilibrium point, provides information on the behavior of the orbits in an open domain around this equilibrium point.

4. SODE Formulation of the Generalized Hopf–Langford System

If we suppose that $b\ne 0$, then starting with the generalized Hopf–Langford system, (3)

$$\left\{\begin{array}{ccc}\hfill \dot{x}& =& ax+by+xz\hfill \\ \hfill \dot{y}& =& cx+dy+yz\hfill \\ \hfill \dot{z}& =& ez-\left(x^2+y^2+z^2\right)\hfill \end{array}\right.$$

by substituting

$$y=\frac{1}{b}\left(\dot{x}-ax-xz\right)$$

from the first equation of (3) and introducing into the second equation of (3), iwe obtain the following:

$$\ddot{x}=\left(a+d+2z\right)\dot{x}+x\dot{z}+\left[bc-\left(a+z\right)\left(d+z\right)\right]x,$$

because $\dot{y}=\frac{1}{b}\left(\ddot{x}-a\dot{x}-\dot{x}z-x\dot{z}\right)$.

From the third equation of (3), by substituting $y=\frac{1}{b}\left(\dot{x}-ax-xz\right)$ and by using the derivative relative to the time t of the third equation, we obtain

$$\begin{array}{ccc}\hfill \ddot{z}& =& (e-2z)\dot{z}-\frac{2}{b^2}(d+z){\left(\dot{x}\right)}^2+\frac{2}{b^2}\left[2\left(a+z\right)\left(d+z\right)-b^2-bc\right]x\dot{x}\hfill \\ & +& \frac{2}{b^2}(a+z)\left[bc-\left(a+z\right)\left(d+z\right)\right]x^2.\hfill \end{array}$$

If we consider the notations of variables in the following manner,

$$x=x^1,\dot{x}=y^1,z=x^2,\dot{z}=y^2,$$

where $y=\frac{1}{b}\left(y^1-a{x}^1-x^1{x}^2\right)$, then the last two second-order differential equations become the next system of second-order differential equations (SODE):

$$\left\{\begin{array}{ccc}{\ddot{x}}^1-\left(a+d+2x^2\right){\dot{x}}^1-x^1{\dot{x}}^2-\left[bc-\left(a+x^2\right)\left(d+x^2\right)\right]x^1\hfill & =& \hfill 0\\ {\ddot{x}}^2-(e-2x^2){\dot{x}}^2+\frac{2}{b^2}(d+x^2){\left({\dot{x}}^1\right)}^2-\frac{2}{b^2}\left[2\left(a+x^2\right)\left(d+x^2\right)-b^2-bc\right]x^1{\dot{x}}^1\hfill & & \\ -\frac{2}{b^2}(a+x^2)\left[bc-\left(a+x^2\right)\left(d+x^2\right)\right]{\left(x^1\right)}^2\hfill & =& \hfill 0\end{array}\right.$$

(18)

or, equivalently,

$$\left\{\begin{array}{ccc}\frac{d^2{x}^1}{d{t}^2}-\left(a+d+2x^2\right)y^1-x^1{y}^2-\left[bc-\left(a+x^2\right)\left(d+x^2\right)\right]x^1\hfill & =& \hfill 0\\ \frac{d^2{x}^2}{d{t}^2}-(e-2x^2)y^2+\frac{2}{b^2}(d+x^2){\left(y^1\right)}^2-\frac{2}{b^2}\left[2\left(a+x^2\right)\left(d+x^2\right)-b^2-bc\right]x^1{y}^1\hfill & & \\ -\frac{2}{b^2}(a+x^2)\left[bc-\left(a+x^2\right)\left(d+x^2\right)\right]{\left(x^1\right)}^2\hfill & =& \hfill 0\end{array}\right.$$

(19)

where $\frac{d{x}^i}{dt}=y^i$, $i=1,2$.

Then, the system (19) can be seen as a SODE from Kosambi–Cartan–Chern theory

$$\left\{\begin{array}{c}\frac{d^2{x}^1}{d{t}^2}+2G^1(x^1,x^2,y^1,y^2)=0\\ \frac{d^2{x}^2}{d{t}^2}+2G^2(x^1,x^2,y^1,y^2)=0\end{array}\right.$$

(20)

where $\frac{d{x}^i}{dt}=y^i$, $i=1,2$ and

$$\begin{array}{ccc}{G}^1(x^i,y^i)\hfill & =& -\frac{1}{2}\left\{\left(a+d+2x^2\right)y^1+x^1{y}^2+\left[bc-\left(a+x^2\right)\left(d+x^2\right)\right]x^1\right\}\hfill \\ G^2(x^i,y^i)\hfill & =& -\frac{1}{2}\left\{(e-2x^2)y^2-\frac{2}{b^2}(d+x^2){\left(y^1\right)}^2+\frac{2}{b^2}\left[2\left(a+x^2\right)\left(d+x^2\right)-b^2-bc\right]x^1{y}^1\right.\hfill \\ \hfill & & +\left.\frac{2}{b^2}(a+x^2)\left[bc-\left(a+x^2\right)\left(d+x^2\right)\right]{\left(x^1\right)}^2\right\}\hfill \end{array}$$

(21)

Next, the zero-connection $Z_j^i=2\frac{\partial G^i}{\partial x^j}$ has the coefficients:

$$\begin{array}{ccc}\hfill Z_1^1& =& -y^2-bc+\left(a+x^2\right)\left(d+x^2\right)\hfill \\ \hfill Z_2^1& =& -2y^1+\left(a+d\right)x^1+2x^1{x}^2\hfill \\ \hfill Z_1^2& =& -\frac{2}{b^2}\left[2\left(a+x^2\right)\left(d+x^2\right)-b^2-bc\right]y^1-\frac{4}{b^2}(a+x^2)\left[bc-\left(a+x^2\right)\left(d+x^2\right)\right]x^1\hfill \\ \hfill Z_2^2& =& 2y^2+\frac{2}{b^2}{\left(y^1\right)}^2-\frac{4}{b^2}\left(a+d+2x^2\right)x^1{y}^1\hfill \\ & & -\frac{2}{b^2}\left[bc-a^2-2ad-2\left(2a+d\right)x^2-3{\left(x^2\right)}^2\right]{\left(x^1\right)}^2\hfill \end{array}$$

The nonlinear connection N is defined by the coefficients $N_j^i=2\frac{\partial G^i}{\partial y^j}$:

$$\left\{\begin{array}{ccc}\hfill N_1^1& =& \frac{\partial G^1}{\partial y^1}=-\frac{1}{2}\left(a+d+2x^2\right)\hfill \\ \hfill N_2^1& =& \frac{\partial G^1}{\partial y^2}=-\frac{1}{2}{x}^1\hfill \\ \hfill N_1^2& =& \frac{\partial G^2}{\partial y^1}=\frac{2}{b^2}\left(d+x^2\right)y^1-\frac{1}{b^2}\left[2\left(a+x^2\right)\left(d+x^2\right)-b^2-bc\right]x^1\hfill \\ \hfill N_2^2& =& \frac{\partial G^2}{\partial y^2}=-\frac{1}{2}\left(e-2x^2\right)\hfill \end{array}\right.$$

(22)

It results that the Berwald connection has all coefficients $G_{jk}^i=\frac{\partial N_j^i}{\partial y^k}$ null, with a single exception, $G_{11}^2=\frac{2}{b^2}\left(d+x^2\right)$.

For the first invariant of KCC theory, ${\epsilon }^i=-\left(N_j^i{y}^j-2G^i\right)$, we obtain the components

$$\left\{\begin{array}{ccc}\hfill {\epsilon }^1& =& -\frac{1}{2}\left(a+d+2x^2\right)y^1-\frac{1}{2}{x}^1{y}^2-\left[bc-\left(a+x^2\right)\left(d+x^2\right)\right]x^1\hfill \\ \hfill {\epsilon }^2& =& -\frac{1}{2}\left(e-2x^2\right)y^2-\frac{1}{b^2}\left[2\left(a+x^2\right)\left(d+x^2\right)-b^2-bc\right]x^1{y}^1\hfill \\ \hfill & & -\frac{2}{b^2}(a+x^2)\left[bc-\left(a+x^2\right)\left(d+x^2\right)\right]{\left(x^1\right)}^2\hfill \end{array}\right.$$

(23)

Since

$$G_{jl}^i=\left\{\begin{array}{ccc}\hfill \frac{2}{b^2}\left(d+x^2\right)& ,& \mathrm{if}j=1,l=1,i=2\hfill \\ \hfill 0& ,& \mathrm{in}\mathrm{rest}\hfill \end{array}\right.$$

and in accordance with (13),

$$P_j^i=-2\frac{\partial G^i}{\partial x^j}-2G^l{G}_{jl}^i+y^l\frac{\partial N_j^i}{\partial x^l}+N_l^i{N}_j^l$$

we obtain the components of the second invariant, which means that the deviation curvature tensor associated with the generalized Hopf–Langford system (19) is

$$\begin{array}{ccc}\hfill P_1^1& =& bc-\left(a+x^2\right)\left(d+x^2\right)+\frac{1}{4}{\left(a+d+2x^2\right)}^2-\frac{1}{b^2}\left(d+x^2\right)x^1{y}^1\hfill \\ \hfill & & +\frac{1}{2b^2}\left[2\left(a+x^2\right)\left(d+x^2\right)-b^2-bc\right]{\left(x^1\right)}^2\hfill \\ \hfill P_2^1& =& -\frac{3}{2}{y}^1-\frac{3}{2}{x}^1{x}^2-\frac{3}{4}(a+d)x^1+\frac{1}{4}(e-2x^2)x^1\hfill \\ \hfill P_1^2& =& \frac{1}{b^2}\left[2\left(a+x^2\right)\left(d+x^2\right)-b^2-bc\right]\left[y^1+\left(\frac{a+d+e}{2}\right)x^1\right]\hfill \\ \hfill & & +\frac{2}{b^2}\left[bc-\left(a+x^2\right)\left(d+x^2\right)\right]\left(2a+d+3x^2\right)x^1\hfill \\ \hfill & & +\frac{2}{b^2}{y}^1{y}^2+\frac{2}{b^2}\left(d+x^2\right)\left(\frac{a+d-e}{2}+x^2\right)y^1-\frac{2}{b^2}\left(a+x^2\right)x^1{y}^2\hfill \\ \hfill P_2^2& =& -y^2-\frac{2}{b^2}{\left(y^1\right)}^2+\frac{1}{4}{\left(e-2x^2\right)}^2+\frac{1}{b^2}\left(4a+3d+7x^2\right)x^1{y}^1\hfill \\ \hfill & & +\frac{1}{b^2}\left[\frac{3bc-b^2}{2}-2a^2-3ad-\left(7a+3d\right)x^2-5{\left(x^2\right)}^2\right]{\left(x^1\right)}^2\hfill \end{array}$$

(24)

The trace and the determinant of the matrix of the deviation curvature tensor

$$P=\left(\begin{array}{cc}{P}_1^1& P_2^1\\ P_1^2& P_2^2\end{array}\right)$$

are trace $\left(P\right)=P_1^1+P_2^2$ and det $\left(P\right)=P_1^1{P}_2^2-P_1^2{P}_2^1$ and so, by using the results from the section above, we can write the following result:

Theorem 3.

All roots of the characteristic polynomial of the deviation curvature tensor P are negative or have negative real parts (which indicates Jacobi stability) if and only if

$$P_1^1+P_2^2<0\mathrm{and}{P}_1^1{P}_2^2-P_1^2{P}_2^1>0.$$

Taking into account that $P_{jk}^i=\frac{1}{3}\left(\frac{\partial P_j^i}{\partial y^k}-\frac{\partial P_k^i}{\partial y^j}\right)$, $P_{jkl}^i=\frac{\partial P_{jk}^i}{\partial y^l}$, $D_{jkl}^i=\frac{\partial G_{jk}^i}{\partial y^l}$, we will obtain the third, fourth and fifth invariants of the generalized Hopf–Langford system:

Theorem 4.

The third KCC invariant $P_{jk}^i$, called the torsion tensor, has the components

$$\left\{\begin{array}{cccc}{P}_{11}^1=0& P_{12}^1=\frac{1}{2}& P_{21}^1=-\frac{1}{2}& P_{22}^1=0\\ P_{11}^2=0& P_{12}^2=\frac{1}{b^2}\left[2y^1-\left(2a+d+3x^2\right)x^1\right]& P_{21}^2=-P_{12}^2& P_{22}^2=0\end{array}\right.$$

(25)

The fourth KCC invariant $P_{jkl}^i$, called the Riemann–Christoffel curvature tensor, has only two non-zero components out of the sixteen components, i.e.,

$$P_{121}^2=\frac{2}{b^2},P_{211}^2=-\frac{2}{b^2},P_{jkl}^i=0,inrest$$

(26)

The fifth KCC invariant $D_{jkl}^i$, called the Douglas tensor, has all sixteen components null, which means

$$D_{jkl}^i=0foralli,j,k,l$$

(27)

Let us note that none of the tensors computed in this section are not symmetric, but we can say that the third invariant and the fourth invariant of KCC theory are skew-symmetric tensors.

5. Jacobi Stability Analysis of the Generalized Hopf–Langford System

In the present section, we will compute the first invariant and the second invariant at each equilibrium point of the generalized Hopf–Langford system (3) and then, we will draw a conclusion on the Jacobi stability of the dynamical system around each equilibrium point.

Therefore, for the equilibrium points $O(0,0,0)$ and $G(0,0,e)$ of the initial generalized Hopf–Langford system (3), we consider the associated equilibrium points $O(0,0,0,0)$, and $G(0,e,0,0)$ for the second-order differential system (19).

For $O(0,0,0,0)$, we have that the first invariant ${\epsilon }^i$ has the components ${\epsilon }^1={\epsilon }^2=0$ and the second invariant has the components in the following matrix

$$P=\left(\begin{array}{cc}bc+\frac{{(a-d)}^2}{4}& 0\\ 0& \frac{e^2}{4}\end{array}\right).$$

Since $trP=bc+\frac{{(a-d)}^2}{4}+\frac{e^2}{4}$ and $detP=\left[bc+\frac{{(a-d)}^2}{4}\right]·\frac{e^2}{4}$, by using Theorem 3,we obtain the following:

Theorem 5.

The trivial equilibrium point O is Jacobi unstable for any parameter values.

For $G(0,e,0,0)$, we obtain that the first invariant ${\epsilon }^i$ has the components ${\epsilon }^1=0$, ${\epsilon }^2=0$ and the components of the second invariant are the same as for O, which means

$$P=\left(\begin{array}{cc}bc+\frac{{(a-d)}^2}{4}& 0\\ 0& \frac{e^2}{4}\end{array}\right).$$

Theorem 6.

The equilibrium point G is Jacobi unstable for any parameter values.

Remark 1.

Unfortunately, we obtained that the generalized Hopf–Langford system (3) has no Jacobi stability either for O or G, the two isolated equilibrium points! It is possible that this lack of local stability from the KCC geometric theory point of view is due to the fact that this system models the phenomenon of turbulence in dynamic fluids.

6. Dynamics of the Deviation Vector for the Generalized Hopf–Langford System

The behavior of the deviation vector with components ${\xi }^i$, $i=1,2$, near an equilibrium point is defined by the deviation equations (also called Jacobi equations) (11), or, in a covariant form, by Equation (12).

For the generalized Hopf–Langford system (3), the deviation equations become:

$$\left\{\begin{array}{c}\frac{d^2{\xi }^1}{d{t}^2}-\left(a+d+2x^2\right)\frac{d{\xi }^1}{dt}-x^1\frac{d{\xi }^2}{dt}+\hfill \\ \left[-y^2-bc+\left(a+x^2\right)\left(d+x^2\right)\right]{\xi }^1+\left[-2y^1+\left(a+d\right)x^1+2x^1{x}^2\right]{\xi }^2=0\hfill \\ \frac{d^2{\xi }^2}{d{t}^2}+\frac{2}{b^2}\left\{2\left(d+x^2\right)y^1-\left[2\left(a+x^2\right)\left(d+x^2\right)-b^2-bc\right]x^1\right\}\frac{d{\xi }^1}{dt}-\left(e-2x^2\right)\frac{d{\xi }^2}{dt}+\hfill \\ \left[-\frac{2}{b^2}\left[2\left(a+x^2\right)\left(d+x^2\right)-b^2-bc\right]y^1-\frac{4}{b^2}(a+x^2)\left[bc-\left(a+x^2\right)\left(d+x^2\right)\right]x^1\right]{\xi }^1+\hfill \\ \left[2y^2+\frac{2}{b^2}{\left(y^1\right)}^2-\frac{4}{b^2}\left(a+d+2x^2\right)x^1{y}^1-\frac{2}{b^2}\left[bc-a^2-2ad-2\left(2a+d\right)x^2-3{\left(x^2\right)}^2\right]{\left(x^1\right)}^2\right]{\xi }^2=0\hfill \end{array}\right.$$

(28)

The length of the deviation vector $\xi \left(t\right)=\left({\xi }^1\left(t\right),{\xi }^2\left(t\right)\right)$ is given by

$$∥\xi \left(t\right)∥=\sqrt{{\left({\xi }^1\left(t\right)\right)}^2+{\left({\xi }^2\left(t\right)\right)}^2}.$$

Furthermore, we are able to write the deviation equations near each equilibrium point for the SODE (19) obtained from generalized Hopf–Langford system (3). Thus, the dynamics of the deviation vector near $O(0,0,0,0)$ is described by the following SODE:

$$\left\{\begin{array}{ccc}\frac{d^2{\xi }^1}{d{t}^2}-(a+d)\frac{d{\xi }^1}{dt}+(ad-bc){\xi }^1\hfill & =& \hfill 0\\ \frac{d^2{\xi }^2}{d{t}^2}-e\frac{d{\xi }^2}{dt}\hfill & =& \hfill 0\end{array}\right.$$

(29)

The dynamics of the deviation vector near $G(0,e,0,0)$ is described by the next deviation equations:

$$\left\{\begin{array}{ccc}\frac{d^2{\xi }^1}{d{t}^2}-(a+d+2e)\frac{d{\xi }^1}{dt}+\left[(a+e)(d+e)-bc\right]{\xi }^1\hfill & =& \hfill 0\\ \frac{d^2{\xi }^2}{d{t}^2}+e\frac{d{\xi }^2}{dt}\hfill & =& \hfill 0\end{array}\right.$$

(30)

If we want to use the formula of the curvature from the differential geometry of the plane curves [33], then it means that the curvature $\kappa \left(t\right)$ of the orbit $\xi \left(t\right)=\left({\xi }^1\left(t\right),{\xi }^2\left(t\right)\right)$ associated with the system of deviation Equation (28) gives us a a quantitative representation of the dynamics related to the deviation vector and its expression is obtained by

$$\kappa \left(t\right)=\frac{{\dot{\xi }}^1\left(t\right){\ddot{\xi }}^2\left(t\right)-{\ddot{\xi }}^1\left(t\right){\dot{\xi }}^2\left(t\right)}{{\left[{\left({\dot{\xi }}^1\left(t\right)\right)}^2+{\left({\dot{\xi }}^2\left(t\right)\right)}^2\right]}^{3/2}}$$

(31)

where ${\dot{\xi }}^i\left(t\right)=\frac{d{\xi }^i}{dt}$, ${\ddot{\xi }}^i\left(t\right)=\frac{d^2{\xi }^i}{d{t}^2}$, $i=1,2$.

7. Conclusions

In this work, we conduct a study on the Jacobi stability of the generalized Hopf–Langford system through the use the geometric methods of the Kosambi–Cartan–Chern (KCC) theory. For this purpose, after the reformulation of the system of first-order differential equations into a system of second-order differential equations (SODE), the five geometrical invariants associated with this SODE (semi-spray) were determined, as well as the zero connection, the nonlinear connection, the Berwald connection. Moreover, surprisingly for a quadratic system, we obtained that the third invariant and the fourth invariant do not have all components null, and neither does the Berwald connection.

By computing the components of the deviation curvature tensor at every equilibrium point, we obtained the conclusion that there is no Jacobi stability around any equilibrium point. Furthermore, the deviation equations near each equilibrium point were written. A possible next work could be related to a numerical approach to the time evolution of the deviation vector and its associated curvature to highlight the nonlinear dynamics of the generalized Hopf–Langford system near each equilibrium point. In this sense, our next target will be to elaborate a numerical study of this generalized Hopf–Langford system for some particular values of the parameters in order to obtain useful and valuable results on the dynamics of the system and practical interpretations of this mathematical pattern for turbulence phenomenons in fluid dynamics and, especially, in the atmosphere of Earth.