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.


Introduction
The relation between dynamical systems and geometry has been widely studied.The theory developed by Kosambi [1], Cartan [2], and Chern [3] is one of the geometric theories applied to dynamical systems, and this theory is called the KCC theory.According to the KCC theory, five invariants are obtained from systems of second-order, ordinary differential equations.The first invariant has the meaning of an external force.The second invariant, the deviation curvature, is related to the stability of the perturbation to the trajectory.This invariant allows us to study Jacobi stability with respect to geodesic deviation.Unlike linear stability, Jacobi stability can be discussed in the nonequilibrium and equilibrium regions.The third invariant can be obtained from the derivative of the deviation curvature that is equivalent to the curvature of a nonlinear connection under certain circumstances.The fourth invariant is related to the deviation curvature as the Riemannian curvature.Additionally, the fifth invariant is called the Douglas tensor and is used to determine whether the original system of ordinary differential equations can be expressed in quadratic form.
In previous studies, the KCC theory has been applied to phenomena represented by ordinary differential equations to analyze the geometric structure of dynamical systems .For example, in biology, the Volterra-Hamilton system for modeling the dynamics of modular populations of a forest has been studied using Jacobi stability [18].In other studies of biology, the KCC theory was applied to a tumor growth model and the Jacobi stability of the model was investigated [23].For an oscillating system, a study regarding the Brusselator compared the Jacobi and linear stabilities [5].Moreover, a study on an overhead crane surveyed the relation between the Jacobi stability and Hopf bifurcation [6].In astrophysics, the KCC theory was applied to braneworld models and these models were analyzed from the viewpoint of the Jacobi stability [9].
The research examples presented in the previous paragraph used firstor second-order differential equations.For the previous studies regarding first-order differential equations, several approaches were considered when applying the KCC theory.Especially, in [22], the changes in the populations of organisms were discussed using the logistic equation with the Hill function, which is a first-order one-dimensional system.Then, a new second-order differential equation was created by considering the solutions of the first-order, onedimensional system as a time potential, and the catastrophic shift of the second-order differential equations was analyzed in terms of the fifth invariant.In [26], second-order differential equations of three-point vortices were obtained by differentiating both sides of the first-order differential equations.Additionally, the behavior of the point vortex was classified using the second invariant.
When [26] was studied, we noticed that the deviation curvatures of first-order one-dimensional differential equations are calculated by two methods as follows: Method 1 is only differentiating both sides of the equation.Method 2 is differentiating both sides of the equation and then substituting the original equation into the second-order system.In the study [26], the relation between the solution of the original equation and the deviation curvature was indicated using only Method 1 because the three-dimensional differential equations are complicated to investigate the difference between these methods.In cases of three-or higher-dimensional differential equations, there are multiple eigenvalues of the deviation curvature, and it is difficult to find them analytically and compare Methods 1 and 2. Instead, herein, we deal with a one-dimensional first-order differential equation for simplicity, and the purpose is to clarify the influence of the two methods on the Jacobi stability of the first-order system.Specifically, both sides of a general form of a first-order one-dimensional equation are differentiated to consider the relation between a solution of the general form and the deviation curvature as the second invariant.Furthermore, the differences in deviation curvatures between the two methods are discussed from the viewpoint of Jacobi stability in the equilibrium and nonequilibrium regions.Finally, the results are confirmed using numerical examples, the logistic equation, and its evolution equation with the Hill function.
The remainder of this article is structured as follows: In section 2, we review the KCC theory.In section 3, we calculate the deviation curvatures in the two methods and compare their properties from the viewpoint of the Jacobi stable region.In section 4, we confirm the results of section 3 using numerical examples, which are the logistic equation and its evolution equation with the Hill function.Finally, section 5 concludes this study.

Review of KCC theory and Jacobi stability
In this section, the KCC theory and Jacobi stability are reviewed based on studies [4,5,15,31,32].Notably, the Einstein convention is used throughout this section.Let M be a smooth n-dimensional manifold and (TM, π, M) be a tangent bundle.A projection π: TM → M is a mapping from TM to M. The local coordinates on M are denoted by x i , where i = 1,K,n.The local coordinates on TM are defined by ( ) x x  , where t is the time parameter and = x dx dt i i


. We consider the following system of second-order differential equations: the five invariants of the second-order system can be obtained.The first invariant E i is given by where N i j is the coefficient of the nonlinear connection: The second invariant is related to the stability of the entire trajectory of the second-order system.The trajectory , where u i is a vector field and δτ is an infinitesimal parameter.From equation (1), a time evolution equation of perturbation called a variation equation can be obtained for δτ → 0: Then, the KCC covariant derivative is defined by  By using (6), equation ( 5) can be rewritten as where P i j is the second invariant and is a deviation curvature tensor defined by The third invariant, B i jk , the fourth invariant, B i ljk , and fifth invariant, D i jkl are defined as follows: The deviation curvature tensor, P i j , gives the stability of whole trajectories via the following definition [4,5]: The trajectories of the system (1) are Jacobi stable if and only if the real parts of the eigenvalues of P i j are strictly negative everywhere, and Jacobi unstable otherwise.

Jacobi stability on the equilibrium and nonequilibrium regions
In this section, we apply the KCC theory in two ways to the following one-dimensional differential equation: Method 1 is the method that only differentiates both sides of the equation (11) like Method 2 is the method that differentiates both sides of the equation (11) and then substitutes the original equation into the equation (12) like: Furthermore, based on [21,22], we can obtain the Jacobian matrix, J, for each equation and then investigate the properties of the equilibrium point.In this case, the type of linear stability of ordinary differential equations can be classified by using J tr and J det .In the following this paper, subscripts 1 and 2 refer to Method 1 and Method 2, respectively.

Method 1: Only differentiating both sides of the equation
From the equations (4) and (9), the coefficients of the nonlinear connection, N 1 , and the Berwald connection, G 1 , can be calculated as follows: From equations (8), ( 12), (14), and (15), the deviation curvature, P 1 , can be obtained as


By substituting = x 0  in equation ( 16), the deviation curvature at the equilibrium point is expressed using Then, P 1 > 0; therefore, the equilibrium point is Jacobi unstable.Next, we consider the properties of the equilibrium point.To calculate the Jacobian matrix, equation ( 12) is rewritten by introducing a new variable, y, as simultaneous equations: Using equation (18), the Jacobian matrix, J 1 , can be calculated by Then, the trace and determinant of J 1 are obtained as follows: Therefore, from [21,22], the type of the equilibrium point of Method 1 is a nonisolated fixed point (figure 1 (a)).

Method 2:
Differentiating both sides of the equation and then substituting the original equation in the second-order system From equations (4) and (9), the coefficients of the nonlinear connection, N 2 , and the Berwald connection, G 2 , can be calculated as follows: From equations (8), ( 13), (21), and (22), the deviation curvature, P 2 , can be obtained as By substituting f (x) = 0 in equation (23), the deviation curvature at the equilibrium point is expressed as Then, P 2 > 0; therefore, the equilibrium point is Jacobi unstable.Next, we consider the properties of the equilibrium point.To calculate the Jacobian matrix, equation ( 13) is rewritten in the same way of Method 1: Using equations (25), the Jacobian matrix, J 2 , can be calculated by Then, the trace and determinant of J 2 can be obtained as follows: Therefore, from [21,22], the equilibrium point is a saddle (figure 1 (b)).
3.3.Differences in the size of the Jacobi stable region in Methods 1 and 2 for a one-dimensional dynamical system In this section, we compare the Jacobi stable regions of Methods 1 and 2. From equations ( 16) and (23), the relation between P 1 and P 2 is expressed as follows: From the expression (28), P 1 < P 2 always holds.Therefore, if there is a Jacobi stable region, the Jacobi stable region of Method 1 is larger than that of Method 2. From this result and the difference in the type of the equilibrium point, it can be seen that the difference in the type of the equilibrium point affects the size of the Jacobi stable region.Next, we assume the case of multiple equilibrium points.Expression ( 23) is rewritten as follows: As F(x) > 0, there exists c where the neighborhood of F(c) is upwardly convex.In other words, there always exists a region where the second derivative of F(x) is negative.Then, P 2 always gives the Jacobi stable region between equilibrium points.Furthermore, from P 1 < P 2 , P 1 always gives the Jacobi stable region between equilibrium points.From this result, it can be shown that the Jacobi stable region appears as a nonequilibrium region.

Jacobi stability and equilibrium or nonequilibrium region of a one-dimensional dynamical system by numerical examples
This section uses concrete examples to confirm the results of the previous section.As specific equations for the equation (11), we use the logistic equation and its evolution equation with the Hill function.In the following this paper, subscripts L and H denote the logistic equation and the Hill function, respectively.
The logistic equation is defined as follows: where R L and K L are positive parameters of intrinsic rate of natural increase and carrying capacity, respectively.The case of R L = 1 and K L = 5 is considered as a numerical example.The equilibrium points of equation (30) are given by (0, 0) and (5, 0).For Method 1, the second-order differential equation and the deviation curvature of the logistic equation, P 1L , are calculated as follows: For Method 2, the second-order differential equation and the deviation curvature of the logistic equation, P 2L , are calculated as follows: The vector fields and the Jacobi stable region obtained by Methods 1 and 2 are shown in figures 2 and 3, respectively.In these figures, the black dots are the equilibrium points, (0, 0) and (5, 0).The red and green regions in figures 2 and 3 are the Jacobi stable regions obtained from P 1L and P 2L , respectively.Figure 2 shows that the equilibrium point has a nonisolated fixed point.Figure 3 shows that the equilibrium point is a saddle.Comparing the two figures, it can be confirmed that the Jacobi stable region in figure 2 is wider than that in  figure 3, and the Jacobi stable region appears as a nonequilibrium region between the equilibrium points in both methods.
Next, we consider the logistic equation with the Hill function.According to a previous study [22], the definition of the logistic equation with the Hill function is as follows: where R H and K H are parameters.Then, the bifurcation curves of the equilibrium points of the equation (35) are given by x = 0 and As numerical examples, we consider the bifurcation diagrams of the equilibrium points in cases of K H = 5 and 10 because the difference in the parameter generates the difference in the number of equilibrium points.The bifurcation diagrams of the equilibrium points in the cases of K H = 5 and 10 are shown in figures 4 and 5, respectively.The number of equilibrium points in the case of K H = 5 is not changed (figure 4); however, in the case of K H = 10, it is changed (figure 5).
The second-order differential equation of Method 1 is as follows:


The second-order differential equation of Method 2 is as follows: For K H = 5, two deviation curvatures, P 1H and P 2H , are calculated as follows:  For K H = 10, two deviation curvatures, P 1H and P 2H , are calculated as follows: Figures 6 and 7 show the bifurcation diagrams of the equilibrium points in the cases of K H = 5 and 10, respectively.The red region is the Jacobi stable region indicated by P 1H , and the green region is the Jacobi stable region indicated by P 2H .Notably, the red region includes the green region.From figures 6 and 7, the Jacobi stable region indicated by P 1H is wider than that indicated by P 2H , and the Jacobi stable region appears as a nonequilibrium region between equilibrium points.In both cases, even if the number of equilibrium points changes, it can be confirmed that there is a Jacobi stable region between the equilibrium points.
From the above results, it can be seen that the difference in the type of equilibrium point affects the size of the Jacobi stable region, which appears as a nonequilibrium region.

Conclusions
In this study, we applied the KCC theory to a first-order one-dimensional dynamical system in two ways: Method 1: differentiating both sides of the equation and Method 2: differentiating both sides of the equation and then substituting the original equation.Additionally, we discussed Jacobi stability in the equilibrium and   nonequilibrium regions.The results show that the equilibrium point becomes Jacobi unstable in Methods 1 and 2. The equilibrium point of Method 1 becomes a nonisolated fixed point, while that of Method 2 becomes a saddle point.Furthermore, it was shown analytically that the size of the Jacobi stable region differs for Method 1 and Method 2, with Method 1 being larger.Additionally, both methods always have a Jacobi stable region in the nonequilibrium region between the equilibrium points when there are multiple equilibrium points.From the above results, it is found that differences in types of equilibrium points affect the size of the Jacobi stable region, and the Jacobi stable region appears as a nonequilibrium region where the equations cannot be linearized.

Figure 1 .
Figure 1.(a) Images of nonisolated fixed points of the type of the equilibrium point.(b) Image of the saddle of the type of equilibrium point.

Figure 2 .
Figure 2. The vector field (blue arrows) and Jacobi stable region (red area) of the logistic equation in Method 1.

Figure 3 .
Figure 3.The vector field (blue arrows) and the Jacobi stable region (green area) of the logistic equation in Method 2.

Figure 4 .
Figure 4.The bifurcation diagrams of the equilibrium points with the Hill function in the case of K H = 5.

Figure 5 .
Figure 5.The bifurcation diagrams of the equilibrium points with the Hill function in the cases of K H = 10.

Figure 6 .
Figure 6.United the bifurcation diagrams of the equilibrium points with the Hill function in cases of K H = 5 and the Jacobi stable region of both methods.

Figure 7 .
Figure 7. United the bifurcation diagrams of the equilibrium points with the Hill function in the cases of K H = 10 and the Jacobi stable region of both methods.