First, we illustrate the definitions of the fractionalorder integral, Caputo fractional derivative, and MittagLeffler function; see [9].
Definition 2.1
The fractional integral of order \(q\in R^{+}\) of the function \(g(t)\), \(t>0\), is defined by
$$ I^{q}g(t)= \int _{0}^{t}\frac{(ts)^{q1}}{\varGamma (q)}g(s)\,ds. $$
(1)
Definition 2.2
The Caputo fractional derivative of order \(q>0\) of \(g(t)\), \(t>0\), is defined by
$$ D_{\ast }^{q} g(t)=I^{nq}D^{n}g(t), $$
(2)
where \(D=d/dt\) and \(n1< q\leq n\), \(n\in \mathbb{N} \).
For properties of fractional derivatives and integrals, see [9].
Definition 2.3
The MittagLeffler function of parameter \(q>0\) is defined as
$$ E_{q}(z)=\underset{j=0}{\overset{\infty }{\sum }} \frac{z^{j}}{\varGamma (qj+1)}. $$
(3)
Let \(q\in (0,1]\) and consider the system
$$\begin{aligned}& D_{\ast }^{q}y_{1}(t)=g_{1}(y_{1},y_{2},y_{3}), \\& D_{\ast }^{q}y_{2}(t)=g_{2}(y_{1},y_{2},y_{3}), \end{aligned}$$
(4)
$$\begin{aligned}& D_{\ast }^{q}y_{3}(t)=g_{3}(y_{1},y_{2},y_{3}), \\& y_{1}(0)=y_{o1},\qquad y_{2}(0)=y_{o2}, \quad \text{and}\quad y_{3}(0)=y_{o3}. \end{aligned}$$
(5)
Definition 2.4
The constant \((y_{1}^{\mathrm{eq}},y_{2}^{\mathrm{eq}},y_{3} ^{\mathrm{eq}})\) can only be an equilibrium point of the fractional dynamic model (4) if and only if
$$ g_{i}\bigl(y_{1}^{\mathrm{eq}},y_{2}^{\mathrm{eq}},y_{3}^{\mathrm{eq}} \bigr)=0, \quad i=1,2,3. $$
Theorem 2.1
The equilibrium points of system (4) are locally asymptotically stable (LAS) if all eigenvalues
\(r_{i}\)
of the Jacobian matrix evaluated at the equilibrium points
\begin{array}{c}B=\left[\begin{array}{ccc}{b}_{11}& {b}_{12}& {b}_{13}\\ {b}_{21}& {b}_{22}& {b}_{23}\\ {b}_{31}& {b}_{32}& {b}_{33}\end{array}\right],\hfill \\ {b}_{ij}=\frac{\partial {g}_{i}}{\partial {y}_{j}}{}_{\mathrm{eq}},\phantom{\rule{1em}{0ex}}i,j=1,2,3,\hfill \end{array}
satisfy
$$ \bigl\vert \arg (r_{i}) \bigr\vert >\frac{q\pi }{2}, \quad i=1,2,3. $$
(6)
Proof
Refer to [10,11,12,13]. □
Figure 1 shows the condition of the stability of the fractionalorder model with order α.
We use the following polynomial to obtain an equation for the eigenvalues of the equilibrium point \((y_{1}^{\mathrm{eq}},y_{2}^{\mathrm{eq}},y_{3} ^{\mathrm{eq}})\):
$$ p(r)=r^{3}+b_{1}r^{2}+b_{2}r+b_{3}=0, $$
(7)
and its discriminant \(D(P)\) is given by
D(P)=\left\begin{array}{ccccc}1& {b}_{1}& {b}_{2}& {b}_{3}& 0\\ 0& 1& {b}_{1}& {b}_{2}& {b}_{3}\\ 3& 2{b}_{1}& {b}_{2}& 0& 0\\ 0& 3& 2{b}_{1}& {b}_{2}& 0\\ 0& 0& 3& 2{b}_{1}& {b}_{2}\end{array}\right=18{b}_{1}{b}_{2}{b}_{3}+{({b}_{1}{b}_{2})}^{2}4{b}_{3}{b}_{1}^{3}4{b}_{2}^{3}27{b}_{3}^{2}.
(8)
Proposition 2.1
([10])
The equilibrium point
\((y_{1}^{\mathrm{eq}},y_{2}^{\mathrm{eq}},y_{3}^{\mathrm{eq}})\)
of system (4) is locally asymptotically stable if one of the following conditions is satisfied:

(i)
\(D(P)>0\), \(b_{1}>0\), \(b_{3}>0\), and
\(b_{1}b_{2}b_{3}>0\).

(ii)
\(D(P)<0\), \(b_{1}\geq 0\), \(b_{2}\geq 0\), \(b_{3}>0\), and
\(q<2/3\).

(iii)
\(D(P)<0\), \(b_{1}>0\), \(b_{2}>0\), \(b_{1}b_{2}b_{3}=0\), and for all
\(q\in (0,1)\).

(iv)
The imperative condition is
\(b_{3}>0\).
Now, consider the following autonomous system:
$$ D_{\ast }^{q}y(t)=g(y),\quad q\in (0,1). $$
(9)
The following lemmas help us to prove the globally asymptotical stability (GAS) of equilibrium points.
Lemma 2.2
(See [14])
Suppose
D
is a bounded closed set. Every solution of (9) starts from a point in
D
and remains in
D
all the time. If
\(\exists V(y):D\longrightarrow \mathbb{R} \)
with continuous first partial derivatives satisfies following condition:
$$ D_{\ast }^{q}V \vert _{\text{(9)}}\leq 0. $$
Let
\(E=\{ D_{\ast }^{q}V \vert _{\text{(9)}}=0,y\in D\}\), and let
M
be the largest invariant set of
E. Then every solution
\(y(t)\)
originating in
D
tends to
M
as
\(t\rightarrow \infty \). Particularly, if
\(M=\{0\}\), then
\(y\rightarrow 0\)
as
\(t\rightarrow \infty \).
Lemma 2.3
(See [15])
Let
\(y(t)\in \mathbb{R} ^{+}\)
be a continuous and differentiable function. Then, for any time instant
\(t\geq t_{0}\),
$$ D_{\ast }^{q} \biggl[ y(t)y^{\ast }y^{\ast }\ln \frac{y(t)}{y^{ \ast }} \biggr] \leq \biggl( 1\frac{y^{\ast }}{y(t)} \biggr) D_{ \ast }^{q}y(t), \quad y^{\ast }\in \mathbb{R} ^{+},\forall q\in (0,1). $$
(10)