Equivalent system for a multiple-rational-order fractional differential system

1. Introduction

In 1695, fractional calculus was born, with a question about the meaning of a derivative of the order one-half. Although fractional calculus is a mathematical topic with more than a 300 year old history, the development of fractional calculus was a bit slow at the early stage and was mainly focused on the pure mathematical field. The earliest systematic studies were made in the nineteenth century. With the development of fractional calculus theory, it has been found only in recent years that the behaviours of many systems can be described by using fractional differential systems, such as viscoelastic systems, dielectric polarization, electrode–electrolyte polarization, electromagnetic waves, power-law phenomena in fluid and complex networks, allometric scaling laws in biology and ecology, coloured noise, boundary-layer effects in ducts, quantitative finance, quantum evolution of complex systems, fractional kinetics, etc.

The extensive applications of fractional differential systems in various fields of science and engineering have greatly accelerated their advance in theoretical analysis and numerical calculation, especially in stability analysis, fractional dynamics and numerical computation, etc. Over the past few decades, since the work of Matignon [1], the stability analysis of fractional differential systems has become more and more interesting and important. Matignon's stability analysis is devoted to a linear fractional differential system with a Caputo derivative whose order lies in (0,1]. Recently, Qian et al. [2] investigated the stability of fractional differential systems with Riemann–Liouville derivatives whose order belongs to (0,1). For nonlinear fractional differential systems, the stability analysis is much more difficult and only a few studies are available, including the continuous dependence of the solution on the initial conditions [3,4] and the stability in the sense of Lyapunov [5–7]. In [5], the definition of Mittag–Leffler stability was first defined, and the corresponding theoretical theorems were also derived. The generalized Mittag–Leffler stability was studied in [6]. All of the above literature deals with same-order fractional differential systems. On the other hand, the stability analysis of multiple-order fractional differential systems has been also discussed. For the multiple-rational-order (MRO) case with MRO in (0,1), we can refer to [8]. A survey on stability analysis of fractional differential systems has recently been presented, where multiple-order systems are also mentioned [9].

It is often inconvenient to study the MRO system directly. However, we change it into an equivalent system with the same derivative order. There are some studies in this respect. For more details, see [8,10–17] and references therein, where [8,16] mainly focused on the stability of solutions.

The rest of this paper is organized as follows. In §2, some definitions and properties are introduced. In §3, the equivalence and stability analysis of the MRO fractional differential system with a Caputo derivative are studied. The equivalence and stability analysis of the MRO fractional differential system with a Riemann–Liouville derivative, together with illustrative examples, are given in the following section. The conclusions are given in the final section.

2. Preliminaries and definitions

Let us denote by the set of real numbers, by denote the set of positive real numbers and by denote the set of positive integer numbers.

In this section, we will recall the main definitions and properties of the relevant fractional derivative operators. Among several definitions of the fractional derivatives, the Caputo derivative and the Riemann–Liouville derivative are often used in applied mathematics and engineering [1,8,18,19]. Throughout this paper, we always assume the existence of the fractional integral and fractional derivatives of a given function, together the composite operations, as usual. Detailed discussions of such existence can be found in [18,20,21].

Definition 2.1

The αth-order Riemann–Liouville integral of function x(t) is defined as follows:

where α>0 and Γ(⋅) is the Euler Gamma function. In some situations, we use instead of for α>0.

Definition 2.2

The αth-order Riemann–Liouville derivative of function x(t) is defined as follows:

where .

Definition 2.3

The αth-order Caputo derivative of function x(t) is defined as follows:

where .

Unlike classical differentiation and integration, fractional differentiation and integration cannot commute. Neither the Caputo-derivative nor the Riemann–Liouville-derivative operator satisfies the semigroup property. In the following, we just list some properties of the fractional calculus where the calculations involved are meaningful [18,19,21,22].

Property 2.4

The fractional integral operator satisfies the semigroup property, i.e.

where α,β>0.

Property 2.5

The compositions of Riemann–Liouville derivative operators and are as follows:

and where n−1≤α<n, m−1≤β<m and .

Property 2.6

The compositions of Riemann–Liouville derivative operator and fractional integral operator are as follows:

and where and β>0.

Properties 2.4–2.6 can be found in [19]. From [23], we can also conclude the following result on the fractional integral.

Remark 2.7

If x(t)∈C⁰[0,T] for T>0 and α>0, then

i.e.

Definition 2.8

Y_α, the convolution kernel of order α>0 for the fractional integral, is defined as follows:

where

Remark 2.9

	— According to definitions 2.1 and 2.8, the αth-order Riemann–Liouville integral of a continuous, even , causal function x(t) can be written as 2.10
	— Convolution property: Yα⋆Yβ=Yα+β for α>0 and β>0.

Definition 2.10

Y_−α, the causal distribution or the generalized function in the sense of Schwartz [24,25], is defined as follows:

where δ is the Dirac distribution, which is the neutral element of convolution.

Definition 2.11

The generalized fractional derivative with order α of a casual function or distribution x(t) (abstract fractional differential operator) is defined as

Remark 2.12

	— Convolution property: Yα⋆Yβ=Yα+β holds for any real numbers α,β.
	— Sequential property: for any real numbers α,β.
	— , where x(t) is usually a causal function or a distribution.
	— For , , 2.13
	— For , , 2.14

The above properties in remark 2.12 can be found in [22]. The generalized fractional derivative is often used in abstract analysis, see [1,22] for more details.

3. Analysis of a multiple-rational-order fractional differential system with a Caputo derivative

In this section, we investigate the equivalent system with the same order of the following system of fractional differential equations:

with the initial-value conditions where the time variable t≥0, , the vector fields , are continuous. All α_i,i=1,2,…,n, are rational numbers satisfying . For all α_i lying in (0,1), the reader can refer to [8,12,13,15] for more information. We always assume that system (3.1) with the initial-value conditions (3.2) has a solution for some b>0.

(a) Equivalent system

In this subsection, we derive the equivalent system of system (3.1) together with the initial-value conditions (3.2).

It follows from system (3.1) that there exist such that α_i=p_i/q_i, where p_i and q_i are two co-prime numbers, i=1,2,…,n. Let M be the lower common multiple of the denominators q_i, i=1,2,…,n. Let us take γ=1/M and N=M(α₁+α₂+⋯+α_n), then one can obtain the following equivalence result.

Theorem 3.1

System (3.1) with the initial-value conditions (3.2) is equivalent to the N-dimensional system of fractional differential equations with derivative order γ,

with the initial-value conditionswhere i=1,2,…,n, that is,

	— whenever [x11(t),x12(t),…,x1α1M(t),x21(t),x22(t),…,x2α2M(t),…,xn1(t),xn2(t),…,xnαnM(t)]Tis a solution to system (3.3) equipped with the initial-value conditions (3.4), [x11(t),x21(t),…,xn1(t)]T∈Cm1[0,b]×Cm2[0,b]×⋯×Cmn[0,b], then [x11(t),x21(t),…,xn1(t)]Tsolves system (3.1) and satisfies its corresponding initial-value conditions (3.2);
	— whenever [x11(t),x21(t),…,xn1(t)]T∈Cm1[0,b]×Cm2[0,b]×⋯×Cmn[0,b] is a solution to system (3.1) equipped with the initial-value conditions (3.2), thensatisfies system (3.3) and its initial-value conditions (3.4).

Proof.

(1) Suppose that the vector [x₁₁(t),x₁₂(t),…,x_1α1M(t),x₂₁(t),x₂₂(t),…,x_2α2M(t),…,x_n1(t),x_n2(t),…,x_nαnM(t)]^T is a solution to system (3.3) with the initial-value conditions (3.4), then the following relations hold:

From remark 2.12, system (3.3) and the initial-value conditions (3.4), we have

Similar to the above derivation, one can obtain

Also note that

Therefore, the first part of this theorem is completed.

(2) Suppose that [x₁₁(t),x₂₁(t),…,x_n1(t)]^T is a solution to system (3.1) with the initial-value conditions (3.2). Then, we have

i.e.

Taking into account remark 2.12 yields

Similarly,

It follows from the above reasoning that Then, applying repeatedly remark 2.12 and the above initial-value conditions leads to

Until now, we affirm that solves

and satisfies the corresponding part of the initial-value conditions (3.4).

Proceeding with the same procedure yields that the vector solves system (3.3) and satisfies the initial-value conditions (3.4).

The proof is now completed. □

Now, we study the equivalent system with the same order of the following MRO fractional differential equation:

with the initial-value conditions where , function is continuous and b_i,i=1,2,…,n, are constant numbers. The orders α_i,i=1,2,…,n, are rational numbers such that and α_n>α_n−1>⋯>α₁. In the same way, it is supposed that the initial-value problem (3.7)–(3.8) has a solution x(t)∈C[0,b] for some b>0.

Similarly, there exist such that α_i=p_i/q_i, where (p_i,q_i)=1. Let M be the lower common multiple of the denominators q_i,i=1,2,…,n, and take γ=1/M,N=α_nM. Then, the equivalent system of (3.7)–(3.8) is given in the following corollary, which is somewhat different from that discussed in [15]. Since we show interest in stability analysis, we prefer to study an MRO system like (3.7)–(3.8).

Corollary 3.2

Equation (3.7) with the initial-value conditions (3.8) is equivalent to the N-dimensional system of fractional differential equations

with the initial-value conditionswhere x₀(t)=x(t), that is,

	— whenever [x(t),x1(t),…,xM(t),…,x2M(t),…,xαnM−1(t)]T,x∈Cmn[0,b] is a solution to system (3.9), equipped with the initial-value conditions (3.10), then x(t) solves equation (3.7) and satisfies its corresponding initial-value conditions (3.8);
	— wheneverx(t),x∈Cmn[0,b] is a solution to equation (3.7) with the initial-value conditions (3.8), thensatisfies system (3.9) and its initial-value conditions (3.10).

Corollary 3.2 still holds for . From the above corollary, there is a strong connection between the ordinary differential equation (ODE) and the fractional ODE. For example,

and is equivalent to with the initial-value conditions for .

From the above example, for a given function x(t) whose first-order derivative exists, we can find another way to numerically compute its arbitrary order α=m/n∈(0,1) by constructing an equation and its equivalent system. For any irrational number β∈(0,1), the function can be numerically approximated according to the fact that an arbitrary irrational number can be approached by a rational number series to arbitrary accuracy.

(b) Stability analysis

In this subsection, we always presume that the solution to a given system can be extended to . In the following, we study the stability of the zero solution to the autonomous system:

with the initial-value conditions where , t and α_i (i=1,2,…,n) are the same as those in theorem 3.1, , are continuous, is a domain that contains the origin .

Next, we give the definition of the stability of the Caputo-type differential equation as (3.1) [1,2,9].

Definition 3.3

The autonomous system (3.11) is said to be

	— stable if and only if
	— asymptotically stable if and only if

where .

By using theorem 3.1, one obtains the following stability result, which can be regarded as a direct application of theorem 3.1.

Theorem 3.4

Assume that g_i satisfy g_i(0)=0,i=1,2,…,n, and the initial-value problem (3.11)–(3.12) has a unique solution. Then, the zero solution to system (3.11) is asymptotically stable if, where λ is the solution to the characteristic equation

γ=1/M is the same as that of theorem 3.1, E is the identity matrix with order, and A is the Jacobian matrix at the zero point of the equivalent system of (3.11)

In particular, if system (3.11) is a linear system, i.e.where the n×n matrix B=(b_ij), then

	— the zero solution to system (3.11) is asymptotically stable if and only if any solution to equation (3.13) satisfies
	— the zero solution to system (3.11) is stable if and only if either it is asymptotically stable (i.e.orand those critical solutions to equation (3.13) that satisfyhave the same algebraic and geometric multiplicities, and the zero solution to equation (3.13) has the same algebraic and geometric multiplicities if there exists the zero solution. Here,

Proof.

Based on theorem 3.1, the MRO fractional differential system (3.11) and (3.12) can be changed into a higher-dimensional fractional differential system with the same order lying in (0,1). Then, combining with [1], theorem 2, [2], remark 3.4.(b) and the linearization method of stability analysis for fractional differential equations [26–33], one can obtain the conclusions. □

Remark 3.5

By applying the properties of the determinant, equation (3.13) is equivalent to the following equation:

where is an n×n matrix.

Likewise, we consider the stability of the following autonomous MRO fractional differential equation:

with the initial-value conditions where x,m_n,b_i,α_i, (i=1,2,…,n) are the same as those in corollary 3.2, is continuous, is a domain that contains the origin x(t)=0. One can then derive the following result.

Corollary 3.6

Suppose that f(y(t)) is a real-valued continuous function such that f(0)=b_n, and equation (3.15) with the initial-value conditions (3.16) has a unique solution. Then, the zero solution to equation (3.15) is asymptotically stable ifwhere λ is the solution to the characteristic equation

γ=1/M is the same as that of theorem 3.1, E is the identity matrix with order N=α_nM, and A is the Jacobian at the zero point of the equivalent system with the same rational order of (3.15) as follows:

In particular, if equation (3.15) is a linear equation and b_n=0, i.e. f(x(t))=b₀x(t), where the constant numberthen

	— the zero solution to equation (3.15) is asymptotically stable if and only if any solution λ to equation (3.17) satisfies
	— the zero solution to equation (3.15) is stable if and only if either it is asymptotically stable (i.e.orand those critical solutions to equation (3.17) that satisfyhave the same algebraic and geometric multiplicities, and the zero solution to equation (3.17) has the same algebraic and geometric multiplicities if there exists a zero solution. Here, 3.19

Remark 3.7

By applying the properties of the determinant, equation (3.17) is equivalent to the following equation:

(c) Several examples

In the sequel, we will give several concrete examples to illustrate theorem 3.1 and corollary 3.2.

Example 3.8

We consider the following MRO system of fractional differential equations:

with the initial-value conditions

By using theorem 3.1, system (3.21), together with (3.22), is equivalent to a 17-dimensional system that reads

with the initial-value conditions

Next, we will give another example to illustrate how to obtain the equivalent system with the same order of an MRO fractional differential equation.

Example 3.9

We study the MRO fractional differential equation

with the initial-value conditions

Applying corollary 3.2, one has that system (3.25) with the corresponding initial-value conditions (3.26) is equivalent to the following system in :

with the initial-value conditions

At last, we consider an interesting model in vibration mechanics.

Example 3.10

Consider the famous Bagley–Torvik equation [34]

with the initial-value conditions

This model was originally established by Bagley and Torvik. They considered the motion of a half-space Newtonian viscous fluid induced by a prescribed transverse motion of a rigid plate on the surface. Their aim was to demonstrate that the resulting shear stress at any point in the fluid can be characterized directly in terms of a fractional derivative of the fluid velocity profile. In the above model, we assume that the mass of the plate, which is immersed in the Newtonian fluid with density ρ and viscosity μ, is a unit. This thin rigid plate is connected by a massless spring of stiffness K to a fixed point outside the fluid. f(t) relates to the force, the constant coefficient a depends upon the area of the plate, the fluid density ρ, and viscosity μ, and b relies on the stiffness K of the spring outside the Newtonian fluid.

It is easy to know that equation (3.29) with initial conditions (3.30) is equivalent to

with the initial-value conditions by utilizing corollary 3.2.

Next, we consider the stability of the Bagley–Torvik equation without the external forcing term, i.e. f(t)=0. That is to say, we consider the stability of the following system with the same order:

Here, , and If the zero solution to the characteristic equation of (3.33) satisfies the condition of corollary 3.6, then the stability problem will be settled. The characteristic equation can be written as We obtain the solutions to the characteristic equation satisfying λ³(λ+a)+b=0.

If b=0 but a≠0 (i.e. the thin rigid plate is immersed in the fluid, but is free from the spring), we can see that the characteristic equation has a zero solution, whose algebraic multiplicity is not equal to the geometric multiplicity. So, in this situation, the zero solution of equation (3.29) with the force f(t)=0, b=0, a≠0 is unstable. Such a theoretical result fits well the real situation.

4. Analysis of a multiple-rational-order fractional differential system with a Riemann–Liouville derivative

For simplicity, we first study the MRO with fractional order lying in (0,1) as

with the initial-value conditions where the time variable t>0, , . All α_i, i=1,2,…,n, are rational numbers satisfying 0<α_i<1. Also, we assume that the initial-value problem (4.1)–(4.2) has a unique solution .

(a) Equivalent system

For the rational number α_i∈(0,1), i=1,2,…,n, we note that there exist such that α_i=p_i/q_i, where p_i and q_i are two co-prime numbers, i=1,2,…,n. Let M be the lower common multiple of the denominators q_i, i=1,2,…,n. Let us take γ=1/M and N=M(α₁+α₂+⋯+α_n), then one can obtain the following equivalence result.

Theorem 4.1

System (4.1) with the initial-value conditions (4.2) is equivalent to the N-dimensional system (4.3) of fractional differential equations with order γ,

subject to the initial-value conditionswhere i=1,2,…,n, that is,

	— whenever [x11(t),x12(t),…,x1α1M(t),x21(t),x22(t),…,x2α2M(t),…,xn1(t),xn2(t),…,xnαnM(t)]Twith [x11(t),x21(t),…,xn1(t)]T∈(C1[0,b])n is a solution to system (4.3), equipped with the initial-value conditions (4.4), then [x11(t),x21(t),…,xn1(t)]Tsolves system (4.1) and satisfies its corresponding initial-value conditions (4.2);
	— whenever [x11(t),x21(t),…,xn1(t)]T∈(C1[0,b])n i=1,2,…,n, is a solution to system (4.1) with the initial-value conditions (4.2), then the vectorsatisfies system (4.3) and its initial-value conditions (4.4).

Proof.

(1) Suppose that [x₁₁(t),x₁₂(t),…,x_1α1M(t),x₂₁(t),x₂₂(t),…,x_2α2M(t),…,x_n1(t),x_n2(t),…,x_nαnM(t)]^T is a solution to system (4.3) with the initial-value conditions (4.4), then the following relations hold:

First, using repeatedly the composition formula of the fractional integral operator and the Riemann–Liouville derivative operator (2.8) and initial-value conditions (4.4), we have

i.e. In the same manner, we have So, the initial-value conditions (4.2) are valid.

Second, using repeatedly the composition formula of fractional integral operator and the Riemann–Liouville derivative operator (2.5) and the initial-value conditions (4.4) yields

Similar to the above derivation, one can obtain

Therefore, the vector [x₁₁(t),x₂₁(t),…,x_n1(t)]^T solves system (4.1) and satisfies its corresponding initial-value conditions (4.2), and the first part of this theorem is completed.

(2) Suppose that [x₁₁(t),x₂₁(t),…,x_n1(t)]^T, x_i1(t)∈C_1−αi[0,b], i=1,2,…, n, is a solution of system (4.1) with the initial-value conditions (4.2), then we have

i.e.

For i=1,2,…,n, it follows from the initial-value conditions (4.2) that

In fact, from (2.4), we have Taking into account , in other words, there exists δ>0 such that the function is bounded on the interval [0,δ]. Then, we arrive at with the conclusion of remark 2.7.

Next, using repeatedly (2.5) and (4.8), we obtain

where i=1,2,…,n. In addition, for i=1,2,…,n and k=1,2,…,α_iM−2, that is to say, and

So, the vector satisfies system (4.3) and its initial-value conditions (4.4). The proof is completed. □

Next, we extend theorem 4.1 to the more general MRO system of the following form:

with the initial-value conditions where i=1,2,…,n, the time variable t>0, , . All α_i, i=1,2,…,n, are rational numbers satisfying .

With almost the similar reasoning as theorem 4.1, we obtain the following theorem.

Theorem 4.2

System (4.9) with the initial-value condition (4.10) is equivalent to the N-dimensional system of equations with derivative order γ

subject to the initial-value conditionswhere α_i=p_i/q_i, p_i and q_i are two co-prime numbers, i=1,2,…,n. M is the lower common multiple of the denominators q_i, i=1,2,…,n, and γ=1/M, N=M(α₁+α₂+…+α_n). That is

	— whenever [x11(t),x12(t),…,x1α1M(t),x21(t),x22(t),…,x2α2M(t),…,xn1(t),xn2(t),…,xnαnM(t)]Twith [x11(t),x21(t),…,xn1(t)]T∈Cm1[0,b]×Cm2[0,b]×⋯×Cmn[0,b] is a solution to system (4.11), equipped with the initial-value conditions (4.12), then [x11(t),x21(t),…,xn1(t)]Tsolves system (4.9) and satisfies its corresponding initial-value conditions (4.10);
	— whenever [x11(t),x21(t),…,xn1(t)]T∈Cm1[0,b]×Cm2[0,b]×⋯×Cmn[0,b] is a solution to system (4.9) with the initial-value conditions (4.10), then the vector satisfies system (4.11) and its initial-value conditions (4.12).

Now, we study the equivalent system with the same order of the following MRO fractional differential equation:

with the initial-value condition where , function , and a_i, i=1,2,…,n, are constant numbers. The orders α_i, i=1,2,…,n, are rational numbers such that 0<α_i<1 and α_n>α_n−1>⋯>α₁. Here, we assume that the initial-value problem (4.13)–(4.14) has a solution x(t)∈L^α_n(0,b) for some b>0.

Similarly, there exist such that α_i=p_i/q_i, where (p_i,q_i)=1. Let M be the lower common multiple of the denominators q_i, i=1,2,…,n, and take γ=1/M,N=α_nM.

Corollary 4.3

Equation (4.13) with the initial-value conditions (4.14) is equivalent to the N-dimensional system of fractional differential equations,

with the initial-value conditionswherex₀(t)=x(t), that is,

	— whenever [x(t),x1(t),…,xαnM−1(t)]T,x∈C1[0,b] x(t)∈C1−αn[0,b], for some b>0, is a solution to system (4.15), equipped with the initial-value conditions (4.16), then x(t) solves equation (4.13) and satisfies its corresponding initial-value conditions (4.14);
	— wheneverx(t)∈C1[0,b] is a solution to equation (4.13) with the initial-value conditions (4.14), thensatisfies system (4.15) and its initial-value conditions (4.16).

Remark 4.4

In suitable conditions, the solutions to Riemann–Liouville-type fractional differential equations can be extended to  [35].

(b) Stability analysis

In the following, we study the stability of the zero solution of the linear MRO fractional differential system that is widely used in control processing:

with the initial-value conditions where , t and α_i (i=1,2,…,n) are the same as those in theorem 4.1, and .

In the following, we introduce the stability definition of system (4.17) [2,9].

Definition 4.5

The linear fractional differential system (4.17) is said to be

	— stable if and only if , there exist ε>0 and δ>0 such that for t≥δ;
	— asymptotically stable if and only if system (4.17) is stable and .

When α₁=α₂=⋯=α_n=α, the stability of system (4.17) has been studied in [2], the corresponding conclusion is as follows. Since system (4.17) is a linear one with a constant coefficient matrix, we can obtain the necessary and sufficient condition of the stability of the solution to this system.

Lemma 4.6

The linear fractional differential system (4.17) equipped with the initial-value conditions (4.18), where α₁=α₂=…=α_n=α, 0<α<1, is

	— asymptotically stable if and only if all the non-zero eigenvalues of A satisfyor A has k-multiple zero eigenvalues corresponding to a Jordan block diag(J1,J2,…,Ji), where Jlis a Jordan canonical form with order, and nlα<1,1≤l≤i.
	— stable if and only if either it is asymptotically stable, or those critical eigenvalues that satisfyhave the same algebraic and geometric multiplicities, or A has k-multiple zero eigenvalues corresponding to a Jordan block matrix diag(J1,J2,…,Ji), whereJlis a Jordan canonical form with orderandnlα≤1,1≤l≤i.

By using theorem 4.1 and lemma 4.6, one has the following stability result.

Theorem 4.7

If the solution of system (4.17) with the initial-value conditions (4.18) satisfies theorem 3.1, then system (4.17) is

	— asymptotically stable if and only if all the non-zero eigenvalues ofsatisfyorhas k-multiple zero eigenvalues corresponding to a Jordan block diag(J1,J2,…,Ji), where Jl is a Jordan canonical form with orderand nlγ<1,1≤l≤i;
	— stable if and only if either it is asymptotically stable, or those critical eigenvalues that satisfyhave the same algebraic and geometric multiplicities, orhas k-multiple zero eigenvalues corresponding to a Jordan block matrix diag(J1,J2,…,Ji), where Jl is a Jordan canonical form with orderand nlγ≤1,1≤l≤i,

where γ=1/M is the same as that of theorem 4.1, denotes the eigenvalues of matrix.

andwhereE_ii are the identity matrices with orders α_iM−1 and O_ij are (α_iM−1)×(α_jM−1) zero matrices, i,j=1,2,…,n.

Proof.

Based on theorem 4.1, the MRO fractional differential system (4.17) and (4.18) can be changed into a higher-dimensional fractional differential system with the same order γ lying in (0,1),

with the initial-value condition where the vector X(t)=[x₁₁(t),x₁₂(t),…,x_1α1M(t),x₂₁(t),x₂₂(t),…,x_2α2M(t),…,x_n1(t),x_n2(t),…,x_nαnM(t)]^T and X₀=[0,0,…,x₁₀,0,0,…,x₂₀,…,0,0,…,x_n0]^T.

Then, according to lemma 4.6, one can obtain conclusions. □

Remark 4.8

By applying the properties of the determinant, the eigenvalues of matrix in theorem 4.7, i.e. the zero solutions λ of

satisfy the following equation: where E is the identity matrix with order .

In the following, we study the stability of the linear nonautonomous differential system associated with system (4.9)

subject to the initial-value conditions (4.10), where i=1,2,…,n, the time variable t>0, , , are continuous functions, j=1,2,…,n. All α_i, i=1,2,…,n, are rational numbers satisfying .

From theorem 4.1, we know that system (4.19) equipped with the initial-value conditions (4.10) is equivalent to the N-dimensional differential system with the same order γ,

with the initial-value condition where γ and N are the same as those of theorem 4.2, the vector X(t)=[x₁₁(t),x₁₂(t),…,x_1α1M(t),x₂₁(t),x₂₂(t),…,x_2α2M(t),…,x_n1(t),x_n2(t),…,x_nαnM(t)]^T, X₀=[0,0,…,x₁₀,0,0,…,x₂₀,…,0,0,…,x_n0]^T. A=(A_ij), B(t)=(B_ij(t)), where E_ii are the identity matrices with orders α_iM−1 and O_ij are (α_iM−1)×(α_jM−1) zero matrices.

Equation (4.20) is a linear system but with a variable coefficient matrix, so we only obtain the sufficient condition of the stability of its solution.

Theorem 4.9

Suppose that the matrix A satisfies |spec(A)|≠0,, the critical eigenvalues that satisfyhave the same algebraic and geometric multiplicities, andis bounded. Then, the zero solution of (4.19) is stable, where spec(A) denotes the eigenvalues of matrix A.

Likewise, we consider the stability of the following autonomous MRO fractional differential equation:

with the initial-value conditions where x,a_i,α_i, i=1,2,…,n, are the same as those in corollary 4.3.

It follows from corollary 4.3 that equation (4.22) with the initial-value conditions (4.23) is equivalent to the following system:

with the initial-value conditions i.e. with the initial-value condition where and N=α_nM.

One can derive the following result.

Corollary 4.10

The zero solution to equation (4.22) is

	— asymptotically stable if and only ifwhere λ is the solution of the characteristic equation 4.25 γ=1/M is the same as that of corollary 4.3, E is an identity matrix with orderN=αnM; or equation (4.25) hask-multiple zero eigenvalues corresponding to a Jordan block diag(J1,J2,…,Ji), whereJlis a Jordan canonical form with orderandnlγ<1, 1≤l≤i;
	— stable if and only if eitherorand those critical solutions of equation (4.25) that satisfyhave the same algebraic and geometric multiplicities, or equation (4.25) hask-multiple zeros corresponding to a Jordan block matrix diag(J1,J2,…,Ji), whereJlis a Jordan canonical form with orderandnlγ≤1,1≤l≤i.

Proof.

This corollary can be proved in the same manner as that in the proof of theorem 4.7, so is omitted here. □

Remark 4.11

By applying the properties of the determinant, equation (4.25) is equivalent to the following equation:

(c) Several examples

In this subsection, we will give several numerical simulations to illustrate the main results derived in this section.

Example 4.12

Consider the following fractional differential system:

with the initial-value conditions

It is obvious that , and it follows from theorem 4.1 that system (4.27) equipped with the initial-value conditions (4.28) is equivalent to a five-dimensional system that reads

with the initial-value conditions

Now, taking a₁₁=−2, a₁₂=0.2, a₂₁=0 and a₂₂=−1.3, we obtain

Using a simple calculation yields the eigenvalues λ_k (k=1,2,3,4,5) of , which satisfy , so system (4.27) with the initial-value condition (4.28) is asymptotically stable from theorem 4.7. At the same time, we give a figure to demonstrate this, see figure 1.

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Example 4.13

Consider the following MRO fractional differential equation:

with the initial-value condition

In the same way, based on corollary 4.3, we see that and N=2 from equation (4.31). Furthermore, equation (4.31) with the initial-value condition (4.32) is equivalent to the following system in :

with the initial-value conditions

Next, we take a₁=0.002, a₂=0.05 and investigate the stability of equation (4.31) with the initial-value condition (4.32). According to corollary 4.10, it is needed to compute the eigenvalues of the coefficient matrix of system (4.33). The coefficient matrix of system (4.33) can be written as

and it is easy to obtain the eigenvalues of B, We can see that λ₁ and λ₂ satisfy , k=1,2. Therefore, the zero solution to equation (4.31) is asymptotically stable. In figure 2, we numerically simulate the above stability result in the light of different initial values.

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Example 4.14

For simplicity, we consider the following linear nonautonomous fractional differential system:

with the initial-value conditions

Next, we determine the stability of the zero solution to system (4.35). According to theorem 4.2, we can change it into the following equivalent system in R¹⁷:

with the initial-value conditions By tedious calculation, the matrix B(t) and the eigenvalues λ_k (k=1,2,…,17) of system matrix A in system (4.35) satisfy the conditions of theorem 4.9, so the zero solution of system (4.35) is stable. After numerical simulations, we also find that its zero solution is stable, see figure 3, which coincides with the theoretical analysis.

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5. Conclusion

In this paper, we study Caputo-type and Riemann–Liouville-type MRO fractional differential systems. By using the properties of the fractional calculus, we can change the original systems in Caputo and Riemann–Liouville senses into their respective equivalent ones. Through these systems, we can conveniently study the stability of the equilibria to the original systems. Various examples are also displayed, which support the theoretical results.

Footnotes

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