# Trajectories and Singular Points of Two-Dimensional Fractional-Order Autonomous Systems

Authors: Bohui Yang; Ziyang Luo; Xindong Zhang; Quan Tang; Juan Liu
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

In this paper, we study the trajectories and singular points of two-dimensional fractional-order planar autonomous linear system involving the Caputo-Fabrizio fractional derivative. By the corresponding fractional integral of the Caputo-Fabrizio fractional derivative, we obtain the analytical solutions for the fractional-order planar autonomous linear system, and then, we discuss the behavior of the trajectories for the mentioned autonomous linear system. Furthermore, we consider the existence of singular points in the trajectories. We discuss the conditions under which the singular point is stable or unstable. By determining the value range of the parameters, we obtain the theorems on the type of singular points. Finally, some examples are given to verify the analysis for the mentioned autonomous linear system.


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## Body

## 1. Introduction

A fractional differential equation is an equation which contains fractional derivatives. There are many forms of fractional derivatives, such as the Riemann-Liouville, Grünwald-Letnikov, and Caputo fractional derivatives. Podlubny summarized and studied the basic theory of fractional differential and fractional differential equation in [1]. About fractional differential equation, there are many results; Albadarneh et al. [2, 3] considered the numerical approach of the Riemann-Liouville and Caputo fractional derivative operators. Lin and Xu [4] studied the finite difference/spectral approximation of time fractional diffusion equations. Zhuang et al. [5] studied the numerical method of variable-order fractional advection diffusion equation with nonlinear source term. Nieto studied numerical solutions of fractional logistic ordinary differential equations in [6]. Liu et al. studied the RBF-FD semidiscrete numerical solution of time fractional convection diffusion equation in [7]. It has been found that the behavior of many physical systems can be properly described by using the fractional-order system theory. A fractional-order system means a system described by a fractional differential equation or fractional integral equation or a system of such equations. There are many results about autonomous systems; one can refer to [8–11]. Meanwhile, the fractional-order systems have been studied by many scholars; for example, Albadarneh et al. studied the analytical solutions of linear and nonlinear incommensurate fractional-order coupled systems in [12]. Ahmad and Harb considered autonomous chaotic systems of integer and fractional orders in [13]. Li and Chen studied the fractional-order Chen system and its control in [14]. Kingni et al. considered three-dimensional chaotic autonomous system and its fractional-order form in [15]. In [16], Caputo and Fabrizio present a new definition of fractional derivative with a smooth kernel. Losada and Nieto studied the properties of a new fractional derivatives without singular kernels in [17], and they also studied the special cases of the Caputo-Fabrizio and Atangana-Baleanu derivatives in [18]. Akman et al. studied the new discretization scheme of the Caputo-Fabrizio derivatives in [19]. Xu and Jian considered unsteady rotating electroosmotic flow with the time-fractional Caputo-Fabrizio derivative in [20]. Caputo and Fabrizio [21] focused on the applications of the Caputo-Fabrizio derivative to partial differential equation. Owolabi studied the calculation and analysis of the Caputo-Fabrizio derivatives of pseudoflat algae species model in [22]. Haq et al. studied a new approach for the qualitative study of vector born disease using the Caputo-Fabrizio derivative in [23]. Fardi and Khan proposed a finite difference spectral method for moving/nonmoving fractal transport model based on the Caputo-Fabrizio fractional derivative in [24]. Harrouche et al. considered the computational algorithm for solving uncertain pharmacokinetic models with the nonsingular nuclear Caputo-Fabrizio fractional derivative in [25]. In [26], Zhang and Li considered the exponential Euler schemes for the Caputo-Fabrizio fractional-order differential equations with multiple delays.In this paper, we consider the Caputo-Fabrizio fractional-order planar autonomous real linear system. The two-dimensional fractional-order autonomous system is of the form(1)DtαXt=AXtorDtαxt=a11xt+a12yt,Dtαyt=a21xt+a22yt,where
(2)0<α<1,Xt=xtyt,A=a11a12a21a22.The fractional derivative Dtαxt is the Caputo-Fabrizio fractional derivative with 0<α<1, defined as (one can see [16])
(3)Dtαxt=11−α∫0te−αt−s/1−αx′sds.For the Caputo-Fabrizio fractional derivative, the corresponding fractional integral of a functionxt is that
(4)Itαxt=1−αxt−x0+α∫0txsds,where x0 is the initial condition; one can refer to [17]. Thus, we have
(5)ItαDtαxt=xt+c,where c is an arbitrary constant.The main interest in studying (1) is twofold: the first one is that a large number of dynamic processes in applied sciences are governed by such systems; the second one is that the qualitative behavior of its solutions can be illustrated through the geometry in the xy-plane.The results for trajectories and singular points can be found in [27–29]. To the best of our knowledge, the only known example of fractional-order dynamical systems for trajectories and singular points can be found in [27], where the fractional derivative is in the Caputo sense. And similar results of fractional-order autonomous systems with the Caputo-Fabrizio fractional derivative have not been found. The main goal of this paper is to study the trajectories and singular points of two-dimensional fractional-order planar autonomous linear systems, where the fractional derivative is in the Caputo-Fabrizio sense. In our paper, the undefined terms and notation will follow [8, 10].


## 2. Solution of Fractional Planar Autonomous Linear Systems

In this section, we consider fractional-order planar real linear system (1) under the assumption that detA≠0. We know that system (1) has a unique singular point O0,0 if and only if detA≠0. By the analysis in [30], we know that system (1) will become
(6)DtαX~t=T−1ATX~t=BX~t,where T is a nonsingular matrix, Xt=TX~t, and X~t=ut,vt′. And B=T−1AT is one of the following normal forms:
(7)Lλ,μ=λ00μ,Mλ=λ01λ,Rη,γ=ηγ−γη.Here,λ,μ,η and γ are real numbers with μ≠0 and γ>0. The above three cases are denoted by L for left, M for middle, and R for right, respectively. If p2−4q>0, we have the real case L. If p2−4q<0, we have the complex case R. If p2−4q=0, the case L or M occurs depending on whether λ=μ has two linearly independent eigenvectors.IfdetA≠0 and a12=a21=0, system (1) will become (6) with case L. Next, we mainly consider the solution of system (1) for the case L. This implies that λ=a11 and μ=a22. Thus, system (1) will become
(8)Dtαxt=λxt,Dtαyt=μyt.Observe that forα=1, system (8) is recovered to the classical autonomous system as follows:
(9)x′t=λxt,y′t=μyt,which has the solutions xt=c1eλt and yt=c2eμt.ForA=Lλ,μ, the following theorem can be obtained.Theorem 1.
Letx0 and y0 be the initial conditions of system (8); by (4) and (5), we get the solutions of system (8) which are xt=c1eλα/1−λ+λαt and yt=c2eμα/1−μ+μαt, where c1=x0 and c2=y0.Proof.
Letxt and yt be the solutions of (8). By (4) and (5), we get that
(10)xt+c1=λItαxt=λ1−αxt−x0+λα∫0txsds,yt+c2=μItαyt=μ1−αyt−y0+μα∫0tysds.
Taking the first derivative both sides of (10), we get
(11)x′t=λ1−αx′t+λαxt,y′t=μ1−αy′t+μαyt.
By (11), for 1−λ+λα≠0 and 1−μ+μα≠0, the analytical solutions of system (8) with 0<α<1 are given by
(12)xt=c1eλαt/1−λ+λα,yt=c2eμαt/1−μ+μα,where c1=x0 and c2=y0 are real constants.IfA=Mλ, then system (1) will become as follows:
(13)Dtαxt=λxt,Dtαyt=xt+λyt.Observe that forα=1, system (13) is recovered to the classical autonomous system as follows:
(14)x′t=λxt,y′t=xt+λyt.Similar with the caseA=Lλ,μ, we can get the following theorem for A=Mλ.Theorem 2.
Letx0 and y0 be the initial conditions of system (13); then, the solutions of system (13) are xt=c1eλα/1−λ+λαt and yt=c1αt/1−λ+λα2+c2eλα/1−λ+λαt, where c1=x0 and c2=y0.IfA=Rη,γ, then system (1) will become as follows:
(15)Dtαxt=ηxt+γyt,Dtαyt=−γxt+ηyt.Observe that forα=1, system (15) is recovered to the classical autonomous system as follows:
(16)x′t=ηxt+γyt,y′t=−γxt+ηyt.Similar with the caseA=Lλ,μ, we can get the following theorem for A=Rη,γ.Theorem 3.
Letx0 and y0 be the initial conditions of system (15). Then, the solutions of system (15) are xt=c1eη1tcos−γ1t+c2 and yt=c1eη1tsin−γ1t+c2, where η1=ηα−α−α2η2+γ2/1−η+ηα2+γ−γα2 and γ1=γα/1−η+ηα2+γ−γα2, and c1,c2 satisfy that x0=c1cosc2 and y0=c1sinc2.


## 3. Singular Points in the Solution Trajectories

In this section, we discuss the trajectories and singular points of (8), (13), and (15), respectively. Since detA≠0, then O0,0 is the unique singular point of (8), (13), and (15). Further, we investigate the distribution of trajectories in the neighborhood of point O0,0. By (11), we know that x=0 and y=0 are the trajectories of (8), respectively (strictly speaking, x=0 contains three trajectories: x=0,y>0, x=0,y<0, and x=0,y=0; for convenience, we simply say that x=0 is trajectory). Similarly, we can get that x=0 is the trajectory of (13).By (12), we get that
(17)xc1μ/1−μ+μα=yc2λ/1−λ+λαory=cxμ1−λ+λα/λ1−μ+μα.where c is an arbitrary constant.The picture of all trajectories of a system is called the phase portrait of the system. Since the solutions of (8) can be determined explicitly, a complete description of its phase portrait can be given. However, as we have seen earlier, the nature of the solutions of (8) depends on the eigenvalues of the matrix A. Thus, the phase portrait of (8) depends almost entirely on the values of λ and μ. For this, there are several different cases which must be studied separately. In the following, we investigate the case L and construct the phase portraits of system (8). Since detA≠0, it follows that λμ≠0; thus, we will consider three cases as follows.Case 1.
λ=μ.
Ifλ=μ, it follows that y=cx. Therefore, the trajectories are half-lines leaving or entering singular point O0,0. By (12), if λ<0, then λα/1−λ+λα=μα/1−μ+μα<0. Thus, both xt and yt tend to zero as t⟶+∞ (singular point O0,0 is called stable node). If λ>0, however, there are two cases occur as the different selection of α. The two cases are λα/1−λ+λα=μα/1−μ+μα<0 and 0<λα/1−λ+λα=μα/1−μ+μα. If λα/1−λ+λα=μα/1−μ+μα<0, then the result is the same as λ<0. If 0<λα/1−λ+λα=μα/1−μ+μα, then all solutions tend to +∞ as t⟶+∞ (singular point O0,0 is called unstable node). The phase portraits of this case can be found in Example 1.In the following cases, we assume thatλ≠μ and consider Case 2 and Case 3.Case 2.
λμ>0.
In this case, we will consider two subcases, that isλ<0 and μ<0 or λ>0 and μ>0.
Subcase 1.λ<0 and μ<0
We may assume thatμ<λ<0 (otherwise, we can exchange x and y). Thus, we get that μ1−λ+λα/λ1−μ+μα>1. Otherwise, μ1−λ+λα/λ1−μ+μα≤1; this implies that μ≥λ, a contradiction. By (17), the trajectories are a family of parabolas which tangent to the x-axis at the origin, except x=0 and y=0. And both xt and yt tend to zero as t⟶+∞ (singular point O0,0 is called stable node).
Subcase 2.λ>0 and μ>0
We may assume thatλ>μ>0 (otherwise, we can exchange x and y). In this subcase, one of the following holds.
(i)
μ1−λ+λα/λ1−μ+μα>1. This implies that 1−λ+λα<0 and 1−μ+μα<0(ii)
0<μ1−λ+λα/λ1−μ+μα<1. This implies that 1−λ+λα>0 and 1−μ+μα>0(iii)
μ1−λ+λα/λ1−μ+μα<0. This implies that 1−λ+λα<0 and 1−μ+μα>0For (i) of Subcase 2 of Case2, by (17), the trajectories are a family of parabolas which tangent to the x-axis at the origin (stable node), except x=0 and y=0. For (ii) of Subcase 2 of Case 2, by (17), the trajectories are a family of parabolas which tangent to the y-axis at the origin (unstable node), except x=0 and y=0. For (iii) of Subcase 2 of Case 2, by (17), except x=0 and y=0, the trajectories are a family of hyperbolas which take x=0 and y=0 as asymptotes. In this case, xt,yt will stay away from 0,0 as t⟶+∞, and the origin is called saddle point which is unstable.The phase portraits of Case2 can be found in Example 2.Case 3.
λμ<0.
In this case, we will consider two subcases, that is,λ>0>μ or λ<0<μ.
Subcase 1.λ>0>μ
In this subcase, one of the following holds.(i)
λα/1−λ+λα<0 and μα/1−μ+μα<0(ii)
λα/1−λ+λα>0 and μα/1−μ+μα<0
For (i) of Subcase 1 of Case3, by (17), the trajectories are a family of parabolas which tangent to the y-axis at the origin (stable node), except x=0 and y=0. For (ii) of Subcase 1 of Case 3, by (17), except x=0 and y=0, the trajectories are a family of hyperbolas which take x=0 and y=0 as asymptotes. In this case, xt,yt will stay away from 0,0 as t⟶−∞, and the origin is called saddle point which is unstable. The phase portraits of this case can be found in Example 3.
Subcase 2.λ<0<μ
In this subcase, one of the following holds.(i)
μα/1−μ+μα<0 and λα/1−λ+λα<0(ii)
μα/1−μ+μα>0 and λα/1−λ+λα<0For Subcase 2 of Case3, the analysis is similar to Subcase 1 of Case 3. For (i) of Subcase 2 of Case 3, by (17), the trajectories are a family of parabolas which tangent to the y-axis at the origin (stable node), except x=0 and y=0. For (ii) of Subcase 2 of Case 3, by (17), except x=0 and y=0, the trajectories are a family of hyperbolas which take x=0 and y=0 as asymptotes. In this case, xt,yt will stay away from 0,0 as t⟶+∞, and the origin is called saddle point which is unstable. The phase portraits of this case can be found in Example 4.We summarize the above analysis in the following theorem forA=Lλ,μ.Theorem 4.
For the fractional-order planar autonomous linear system (1), let λ and μ be the eigenvalues of the matrix A with a12=a21=0. Then system (1) will become system (8). Then, the behavior of its trajectories near the singular point O0,0 is as follows.
(i-1) Stable node, ifλ and μ are equal and negative
(i-2) Stable node, ifλ and μ are equal and positive, and λα/1−λ+λα=μα/1−μ+μα<0 for some α
(i-3) Unstable node, ifλ and μ are equal and positive, and λα/1−λ+λα=μα/1−μ+μα>0 for some α
(ii-1) Stable node, ifλ and μ are distinct and negative
(ii-2) Stable node, ifλ and μ are distinct and positive, and μ1−λ+λα/λ1−μ+μα>1 for some α
(ii-3) Unstable node, ifλ and μ are distinct and positive, and 0<μ1−λ+λα/λ1−μ+μα<1 for some α
(ii-4) Saddle point (unstable), ifλ and μ are distinct and positive, and μ1−λ+λα/λ1−μ+μα<0 for some α
(iii-1) Stable node, ifλ and μ are of opposite sign, and μ1−λ+λα/λ1−μ+μα>0 for some α
(iii-2) Saddle point (unstable), ifλ and μ are of opposite sign, and μ1−λ+λα/λ1−μ+μα<0 for some αIn the following, we will consider system (13) and system (15), respectively. For system (13), by Theorem 2, we get that
(18)y=C1lnx+C2x,where C1=1/λ1−λ+λα and C2=c2/c1−lnc1/λ1−λ+λα. By (18), we know that
(19)limx⟶0y=0andlimx⟶0y′x=∞.Therefore, when the trajectory approaches the origin, the limit position of its tangent is they-axis. By Theorem 2, if λ<0, then λα/1−λ+λα<0. Thus, as t⟶+∞, all trajectories will tend to singular point (singular point O0,0 is called stable node). If λ>0, however, there are two cases occur as the different selection of α. If λα/1−λ+λα<0, then the result is the same as λ<0. If λα/1−λ+λα>0, then all trajectories will be far away from the singular point as t⟶+∞ (singular point O0,0 is called unstable node). The phase portraits of this case can be found in Example 5.For system (15), by Theorem 3, we get that
(20)ρ=Ce−η1θ/γ1,where C=c1ec2η1/γ1, η1=ηα−α−α2η2+γ2/1−η1−α2+γ21−α2 and γ1=γα/1−η1−α2+γ21−α2. Since γ>0, we have γ1=γα/1−η1−α2+γ21−α2>0. If η1=0, by Theorem 3 or (20), we have
(21)x2+y2=c12orρ=c1.Thus, the trajectories are closed curves, and the phase portrait of system (15) can been found in Example 6 with η=γ=2 and α=3/4 (see Figure 1(a)). In this case, the singular point O0,0 is stable but not asymptotically stable and is called a center.Figure 1
The phase portraits of system (15) at different α.
(a)
η=γ=2 with α=3/4(b)
η=−2 and γ=2(c)
η=γ=2 with α=1/4(d)
η=γ=2 with α=4/5Ifη1<0, then the effect of the factor eη1t in Theorem 3 is to change the simple closed curves into the spirals. In this case, the singular point O0,0 is asymptotically stable and is called a stable focus. Since η1=ηα−α−α2η2+γ2/1−η1−α2+γ21−α2, we know that if η<0, then η1<0; this case can be found in Example 6 with η=−2 and γ=2 (see Figure 1(b) ). However, if η≥0, we also can get η1<0 for some values of α, such as η=γ=2 and α=1/4, which will be shown in Example 6 (see Figure 1(c)).Ifη1>0, then all trajectories of system (15) spiral away from the origin O0,0 as t⟶+∞ and are illustrated in figure of Example 6. This case only happens when η≥0 for some values of α, such as η=γ=2 and α=4/5, which will be shown in Example 6 (see Figure 1(d)). In this case, the singular point O0,0 is unstable and is named an unstable focus.We summarize the above analysis in the following theorem forA=Mλ and A=Rη,γ.Theorem 5.
For the fractional-order autonomous linear system (1), let Mλ (or Rη,γ) be the normal form of matrix A. Then, the behavior of its trajectories near the singular point O0,0 is as follows.
(i-1) Ifλ<0, then λα/1−λ+λα<0 and O0,0 is stable node
(i-2) Ifλ>0 and λα/1−λ+λα<0 for some α, then O0,0 is stable node
(i-3) Ifλ>0 and λα/1−λ+λα>0 for some α, then O0,0 is unstable node
(ii-1) Forη≥0, if η1=0 at some α, then O0,0 is stable center
(ii-2) Ifη<0, then η1<0 and O0,0 is stable focus
(ii-3) Forη≥0, if η1>0 at some α, then O0,0 is stable focus
(ii-4) Forη≥0, if η1<0 at some α, then O0,0 is unstable focus


## 4. The Illustrative Examples

In fact, for the classical autonomous systems (9), (14), and (16), the phase portraits can be found in [8, 10]. If the arrow is pointing to the singular point, then the singular point O0,0 is a stable point; otherwise, it is an unstable point. Next, we make a detailed analysis of A=Lλ,μ; the results will be shown in Examples 1-4. For A=Mλ and A=Rη,γ, we can get similar results as in Example 5 and Example 6, respectively.ForA=Lλ,μ, there are three cases as analyzed in Section 3. For case λ=μ, comparing our results with those in [8, 10], we can find that the results of system (8) and system (9) are consistent for the case λ=μ<0. However, the results of the case λ=μ>0 are inconsistent. For λ=μ>0 of classical autonomous system (9), the results can be found as the same one in Example 1 Figure 2(b). For λ=μ>0 of system (8), if λα/1−λ+λα=μα/1−μ+μα<0, then the result is the same as λ<0 in Example 1 Figure 2(a), and if λα/1−λ+λα=μα/1−μ+μα>0, then the result can be found in Example 1 Figure 2(b).Figure 2
The phase portraits of system (8) for λ=μ at different α.
(a)
λ=μ=−1 or λ=μ=2 with α=1/4(b)
λ=μ=2 with α=3/4For caseλ≠μ and λμ>0, the results of system (8) and system (9) are inconsistent. Precisely speaking, for case λμ>0, the trajectories of classical autonomous system (9) are a family of parabolas; they all can be found in [8, 10]. However, for case λμ>0, the trajectories of system (8) may be a family of parabolas or hyperbolas as the different selection of α. As the results in Examples 2, if μ1−λ+λα/λ1−μ+μα>0 for some α, then the results are shown in Figures 3(a)–3(c); the trajectories of system (8) are a family of parabolas. If μ1−λ+λα/λ1−μ+μα<0 for some α, then the result is shown in Figure 3(d); the trajectories of system (8) are a family of hyperbolas.Figure 3
The phase portraits of system (8) for λ≠μ and λμ>0 at different α.
(a)
λ=−1 and μ=−2 for 0<α<1(b)
λ=4 and μ=3/2 with α=1/5(c)
λ=2 and μ=1 with α=7/8(d)
λ=10 and μ=1/2 with α=1/8Similarly, for caseλμ<0, the results of system (8) and system (9) are also inconsistent. Precisely speaking, for case λμ<0, the trajectories of classical autonomous system (9) are a family of hyperbolas; they all can be found in [8, 10]. However, for case λμ<0, the trajectories of system (8) may be a family of parabolas or hyperbolas as the different selection of α; the results can be found in Examples 3 and 4. If μ1−λ+λα/λ1−μ+μα>0 for some α, then the results are shown in Figures 4(a) and 5(a); the trajectories of system (8) are a family of parabolas. If μ1−λ+λα/λ1−μ+μα<0 for some α, then the results are shown in Figures 4(b) and 5(b); the trajectories of system (8) are a family of hyperbolas.Example 1.
In this example, we consider the case forλ=μ≠0. For λ=μ=−1 or λ=μ=2 with α=1/4, the results can be found in Figure 2(a), which verify (i-1) and (i-2) in Theorem 4. The results of λ=μ=2 with α=3/4 can be found in Figure 2(b), which verify (i-3) in Theorem 4. From Figure 2, it can be seen that our results of Case 1 are consistent with the analysis in Section 3.Example 2.
The second example is for Case2 in Section 3. For Subcase 1 of Case 2, if λ=−1 and μ=−2, then the results can be found in Figure 3(a), which verify (ii-1) in Theorem 4. If λ=4 and μ=3/2 with α=1/5, then Figure 3(b) is for Subcase 2 (i) of Case 2, which verify (ii-2) in Theorem 4. If λ=2 and μ=1 with α=7/8, then Figure 3(c) is for Subcase 2 (ii) of Case 2, which verify (ii-3) in Theorem 4. If λ=10 and μ=1/2 with α=1/8, then Figure 3(d) is for Subcase 2 (iii) of Case 2, which verify (ii-4) in Theorem 4. From Figure 3, it can be seen that our results of Case 2 are consistent with the analysis in Section 3.Example 3.
The third example is for Subcase 1 of Case3 in Section 3. If λ=−2 and μ=−1 with α=1/8, then Figure 4(a) is for Subcase 1 (i) of Case 3, which verify (iii-1) in Theorem 4. If λ=1 and μ=−1 with α=7/8, then Figure 4(b) is for Subcase 1 (ii) of Case 3, which verify (iii-2) in Theorem 4. From Figure 4, it can be seen that our results of Subcase 1 of Case 3 are consistent with the analysis in Section 3.Example 4.
The fourth example is for Subcase 2 of Case3 in Section 3. If λ=−1 and μ=2 with α=1/4, then Figure 5(a) is for Subcase 2 (i) of Case 3, which verify (iii-1) in Theorem 4. If λ=−1 and μ=2 with α=3/4, then Figure 5(b) is for Subcase 2 (ii) of Case 3, which verify (iii-2) in Theorem 4. From Figure 5, it can be seen that our results of Subcase 2 of Case 3 are consistent with the analysis in Section 3.Example 5.
In this example, we consider the phase portraits for system (13). For λ=−1, the result can be found in Figure 6(a), which verify (i-1) in Theorem 5. For λ=2 with α=1/4, the result can be found in Figure 6(b), which verify (i-2) in Theorem 5. The result of λ=2 with α=3/4 can be found in Figure 6(c), which verify (i-3) in Theorem 5. From Figure 6, it can be found that our results are consistent with the analysis in Section 3.Example 6.
This example is for system (15). If η=γ=2 with α=3/4, then the results can be found in Figure 1(a), which verify (ii-1) in Theorem 5. If η=−2 and γ=2, then the results can be found in Figure 1(b), which verify (ii-2) in Theorem 5. If η=γ=2 with α=1/4, then the results can be found in Figure 1(c), which verify (ii-3) in Theorem 5. If η=γ=2 with α=4/5, then the results can be found in Figure 1(d), which verify (ii-4) in Theorem 5. From Figure 1, it can be seen that our results are consistent with the analysis in Section 3.Figure 4
The phase portraits of system (8) for λ>0>μ at different α.
(a)
λ=2 and μ=−1 with α=1/8(b)
λ=1 and μ=−1 with α=7/8Figure 5
The phase portraits of system (8) for λ<0<μ at different α.
(a)
λ=−1 and μ=2 with α=1/4(b)
λ=−1 and μ=2 with α=3/4Figure 6
The phase portraits of system (13) at different α.
(a)
λ=−1(b)
λ=2 with α=1/4(c)
λ=2 with α=3/4


## 5. Conclusion

In this paper, we study the trajectories and singular points for two-dimensional fractional-order planar autonomous linear systems involving the Caputo-Fabrizio fractional derivative. This problem is a natural generalization of the classical autonomous systems. Compared with the classical integer-order autonomous systems, trajectories of the solutions for fractional-order autonomous systems have more abundant physical phenomena. The results of our paper will further enrich the related contents of fractional-order autonomous systems. In the future, fractional-order complex autonomous systems and fractional-order autonomous nonlinear systems can also be studied.



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