1 Introduction

Fractional calculus is a generalization of the standard integer calculus, extending the derivative and integral operators to non-integer orders [1]. In recent years, fractional calculus experienced a rapid development, and its applications became common in mechanics, physics, chemistry, and other disciplines. Fractional-order (FO) models can describe many real-world systems better than their integer-order counterparts [2], namely when phenomena like long-range correlations, memory, and heredity effects are present.

State-space models of FO systems can be divided into two categories, namely commensurate and incommensurate. In general, if the state variables are of the same order, the system is called commensurate, otherwise it is denoted incommensurate. Multi-order systems are usually incommensurate, while commensurate systems are just special cases of the incommensurate ones. Studying the stability problem of incommensurate FO systems is crucial, but also demanding.

Stability analysis is one of the most important issues when studying dynamical systems [3,4,5]. The stability analysis of FO systems, meaning those described by FO differential equations, is often more difficult than that of integer-order systems, since fractional derivatives are non-local and have weakly singular kernels [6, 7]. In the past decade, many results regarding the stability of commensurate FO systems were derived, including for linear [8], nonlinear [9], and delayed systems [10]. However, for incommensurate FO systems, fewer studies were reported in the literature, since their stability analysis is very demanding. The stability analysis of two-term FO differential equations was investigated in various papers [11,12,13,14,15]. The necessary and sufficient stability and instability conditions for linear FO differential equations with three Caputo derivatives were considered in the work [16]. Robust stability of FO systems described in pseudo-state space with incommensurate FO was addressed in the references [17, 18].

We should note that all references mentioned previously, dealing with the stability of FO incommensurate systems, focus mainly on linear system [11,12,13,14,15,16,17,18]. Indeed, to the best of the authors knowledge, the stability of multidimensional FO nonlinear systems was not explored so far. In this paper, by using the Mittag–Leffler function, the Laplace transform, and the generalized Gronwall inequality, an asymptotic condition for FO nonlinear systems with two different derivatives is proposed.

The structure of this paper is as follows. Section 2 gives some preliminaries on fractional calculus and basic definitions. Section 3 is dedicated to proving the main results of the paper. Section 4 presents numerical examples for illustrating the theoretical findings. Finally, Sect. 5 lists the conclusions of the study.

2 Preliminaries and basic tools

Definition 1

[19] The Riemann–Liouville fractional integral of order \(\alpha \in R^{+}\) of function x(t) is

$$\begin{aligned} D_{t_{0}, t}^{-\alpha } x(t)=\frac{1}{\Gamma (\alpha )} \int _{t_{0}}^{t}(t-\tau )^{\alpha -1} x(\tau ) \textrm{d}\tau , \end{aligned}$$

where \(\Gamma (\cdot )\) is the gamma function, \(\Gamma (\tau )=\int _{0}^{\infty } t^{\tau -1}\textrm{e}^{-t} \text {d}t\).

Definition 2

[19] The Caputo fractional integral of order \(\alpha \in R^{+}\) of function x(t) is

$$\begin{aligned} \begin{aligned} { }^{C} D_{t_{0}, t}^{-\alpha } x(t)= \frac{1}{\Gamma (\alpha )} \int _{t_{0}}^{t}(t-\tau )^{\alpha -1} x(\tau ) \textrm{d} \tau . \end{aligned} \end{aligned}$$

Definition 3

[19] The Caputo fractional derivative of order \(\alpha \in R^{+}\) of function x(t) is

$$\begin{aligned} \begin{aligned} { }^{C} D_{t_{0}, t}^{\alpha } x(t)= \frac{1}{\Gamma (n-\alpha )} \int _{t_{0}}^{t}(t-\tau )^{n-\alpha -1} x^{(n)}(\tau ) \textrm{d} \tau , n-1 \le \alpha <n \in Z^{+}, \end{aligned} \end{aligned}$$

where \(n=\lceil \alpha \rceil \), with \(\lceil \alpha \rceil \) being the smallest integer greater than or equal to \(\alpha \).

Proposition 1

[19] The Laplace transform of the Caputo fractional derivative is

$$\begin{aligned} L\left\{ { }^{C} D_{t}^{\alpha } x(t); s\right\} =s^{\alpha } X(s)-\sum _{k=0}^{n-1} s^{\alpha -k-1} x^{(k)}(0)\quad , n-1<\alpha \leqslant n, \end{aligned}$$

where X(s) represents the Laplace transform of function x(s).

Definition 4

[19] The multi-variant Mittag–Leffler function is defined as follows:

$$\begin{aligned} E_{\left( a_{1}, \ldots , a_{n}\right) , b}\left( z_{1}, \ldots , z_{n}\right) =\sum _{\begin{array}{c} k=0 \\ l_{1} \geqslant 0, \ldots l_{n} \geqslant 0 \end{array}}^{\infty } \sum _{\begin{array}{c} l_{1}+\cdots +l_{1}=k \end{array}} \frac{k !}{l_{1} ! \times \cdots \times l_{n} !} \frac{\prod _{i=1}^{n} z_{i}^{l_{i}}}{\Gamma \left( b+\sum _{i=1}^{n} a_{i} l_{i}\right) }, \end{aligned}$$

where \(b>0\), \(a_{i}>0\), \(|z_{i} |<\infty \), \(i=1, \ldots ,n\).

When \(n=1\), the Mittag–Leffler function with one parameter is obtained as follows:

$$\begin{aligned} E_{a_{1}, b}\left( z_{1}\right) =\sum _{k=0}^{\infty } \frac{z_{1}^{k}}{\Gamma \left( b+k a_{1}\right) } \quad a_{1}, b>0, |z_{1} |<\infty . \end{aligned}$$

Moreover, the Laplace transform of the Mittag–Leffler function is

$$\begin{aligned} \begin{aligned} L\left\{ t^{\alpha k+\beta -1} E_{\alpha , \beta }^{(k)}\left( \pm a t^{\alpha }\right) ; s\right\}&=\int _{0}^{\infty } \textrm{e}^{-st} t^{\alpha k+\beta -1} E_{\alpha , \beta }^{(k)}\left( \pm a t^{\alpha }\right) \textrm{d}t \\&=\frac{k ! s^{\alpha -\beta }}{\left( s^{\alpha } \mp a\right) ^{k+1}}, {\text {Re}}\{s\}> |a |^{\frac{1}{\alpha }}. \end{aligned} \end{aligned}$$

Lemma 1

[20] If \(\alpha \ge 1\), then for \(\beta =1, 2, \alpha \), we have

$$\begin{aligned} E_{\alpha , \beta }\left( A t^{\alpha }\right) \le \Vert \textrm{e}^{A t^{\alpha }} \Vert . \end{aligned}$$

Lemma 2

[21] Suppose that \(\alpha > 0\), a(t) is a nonnegative function locally integrable on \( 0\le t<T\) (some \( T \le +\infty \)), and g(t) is a nonnegative, nondecreasing continuous function defined on \(0\le t<T\), with \(g(t)\le M\) (constant). Also, suppose that u(t) is nonnegative and locally integrable on \(0\le t<T\), with

$$\begin{aligned} u(t)\le a(t)+g(t)\int _{0}^{t}(t-s)^{\alpha -1} u(s)\textrm{d}s \end{aligned}$$

on this interval. Then,

$$\begin{aligned} u(t) \le a(t)+\int _{0}^{t}\left[ \sum _{n=1}^{\infty } \frac{(g(t) \Gamma (\alpha ))^{n}}{\Gamma (n\alpha )}(t-s)^{n \alpha -1} a(s)\right] \textrm{d}s. \end{aligned}$$

Moreover, if a(t) is a nondecreasing function on [0, T), then

$$\begin{aligned} u(t) \le a(t) E_{\alpha , 1}\left( g(t) \Gamma (\alpha ) t^{\alpha }\right) . \end{aligned}$$

Lemma 3

[19] If \(0<\alpha <2\), \(\beta \) is an arbitrary real number, \(\mu \) satisfies \(\pi \alpha /2<\mu <\min \{\pi ,\pi \alpha \}\), and \(C_1\), \(C_2\) are real constants, then

$$\begin{aligned} \begin{aligned} |E_{\alpha , \beta }(z) |&\le C_{1}(1+|z |)^{(1-\beta ) / \alpha } \exp \left( {\text {Re}}\left\{ z^{1 / \alpha }\right\} \right) +\frac{C_{2}}{1+ |z |}, \end{aligned} \end{aligned}$$

where \(|\arg (z) |\le \mu , |z |\ge 0\).

3 Main results

The FO nonlinear system with two different derivatives is considered:

$$\begin{aligned} A{ }^{C} D_{t}^{p} x(t)+B{ }^{C} D_{t}^{q} x(t)=f(x(t)), \end{aligned}$$
(1)

where \(x(t)=\left( x_{1}(t), x_{2}(t), \ldots , x_{n}(t)\right) ^{T} \in R^{n}\) denotes the state vector, \(A, B \in R^{n \times n}\) are constant matrices, \(f(x(t))\in R^n\) represents a nonlinear vector, and satisfies \(f(0)=0\), and the Lipschitz condition with Lipschitz constant L, that is, \(\Vert f(x_1(t))-f(x_2(t))\Vert \le L\Vert x_1(t)-x_2(t)\Vert \), and the FO p, q belong to \(0<p<1<1+p \le q<2\).

The system (1) can be used to describe a standard heat diffusion process, the standard voltage equations for impedance at the drive of lossy transmission lines, as well as other phenomena useful in practical engineering.

Theorem 1

The system (1) has asymptotically stable behavior if \(Re\{\lambda (C)\}<0\) and \(w=-\max \{\text {Re}\{\lambda (C)\}>(L\Vert B^{-1}\Vert \Gamma (q))^{\frac{1}{q}}\), where \(C=B^{-1}A\).

Proof

Taking the Laplace transform on (1), one has

$$\begin{aligned} As^{p} X(s)-As^{p-1} x(0)+Bs^{q}X(s)-Bs^{q-1}x(0)-Bs^{q-2} x^{\prime }(0)=F(x(s)), \end{aligned}$$

where \(F(x(s))=L(f(x(t)))\). Thus, we can obtain

$$\begin{aligned} X(s)= & {} \frac{A s^{p-1} x(0)+Bs^{q-1} x(0)+Bs^{q-2} x^{\prime }(0)+F(x(s))}{As^{p}+Bs^{q}} \nonumber \\= & {} \frac{As^{p-1} x(0)}{As^{p}+Bs^{q}}+\frac{Bs^{q-1} x(0)}{As^{p}+Bs^{q}}+\frac{Bs^{q-2} x^{\prime }(0)}{As^{p}+Bs^{q}}+\frac{F(x(s))}{As^{p}+Bs^{q}}. \end{aligned}$$
(2)

Applying the inverse Laplace transform on (2), it yields

$$\begin{aligned} L^{-1}\bigg \{\frac{As^{p-1}x(0)}{As^{p}+Bs^{q}}; s\bigg \}= & {} B^{-1}Ax(0) t^{q-p} E_{q-p,q-p+1} \left( -B^{-1}At^{q-p}\right) ,\nonumber \\ L^{-1}\bigg \{\frac{F(x(s))}{As^{p}+Bs^{q}}; s\bigg \}= & {} f(x(t)) * \left[ B^{-1}t^{q-1}E_{q-p,q}\left( -B^{-1}At^{q-p}\right) \right] ,\nonumber \\ L^{-1}\bigg \{\frac{Bs^{q-1}x(0)}{As^{p}+Bs^{q}}; s\bigg \}= & {} x(0) E_{q-p,1} \left( -B^{-1}At^{q-p}\right) ,\nonumber \\ L^{-1}\bigg \{\frac{Bs^{q-2}x^{\prime }(0)}{As^{p}+Bs^{q}}; s\bigg \}= & {} x^{\prime }(0) t E_{q-p,2} \left( -B^{-1}At^{q-p}\right) . \end{aligned}$$
(3)

In view of (3), the solution of (2) can be obtained as follows:

$$\begin{aligned} x(t)= & {} B^{-1}Ax(0) t^{q-p} E_{q-p, q-p+1}\left( -B^{-1}At^{q-p}\right) \nonumber \\{} & {} +x(0) E_{q-p, 1}\left( -B^{-1}At^{q-p}\right) +x^{\prime }(0) t E_{q-p,2} \left( -B^{-1}At^{q-p}\right) \nonumber \\{} & {} +\int _{0}^{t}B^{-1}(t-\tau )^{q-1} E_{q-p, q}\left( -B^{-1}At^{q-p}\right) f(\tau )\text {d}\tau . \end{aligned}$$
(4)

It follows from Lemma 1 that

$$\begin{aligned} E_{q-p,q-p+1}\left( -B^{-1}At^{q - p}\right)\le & {} \Vert \textrm{e}^{-B^{-1}At^{q-p}}\Vert ,\\ E_{q-p,2}\left( -B^{-1}At^{q - p}\right)\le & {} \Vert \textrm{e}^{-B^{-1}At^{q-p}}\Vert , \\ E_{q-p,1}\left( -B^{-1}At^{q - p}\right)\le & {} \Vert \textrm{e}^{-B^{-1}At^{q-p}}\Vert ,\\ E_{q-p,q}\left( -B^{-1}At^{q - p}\right)\le & {} \Vert \textrm{e}^{-B^{-1}At^{q-p}}\Vert . \end{aligned}$$

Combining (4) and (5) and denoting \(C=B^{-1}A \), one obtain

$$\begin{aligned} \Vert x(t)\Vert\le & {} \Vert C\textrm{e}^{-Ct^{q-p}}\Vert \Vert x(0)\Vert t^{q - p}+\Vert \textrm{e}^{-Ct^{q-p}}\Vert \Vert x(0)\Vert +\Vert \textrm{e}^{-Ct^{q -p}}\Vert \Vert x^{\prime }(0)\Vert t\nonumber \\{} & {} +\Vert B^{-1}\Vert \int _{0}^{t}(t-\imath )^{q-1}\Vert \textrm{e}^{-C(t-\tau )^{q-p}}\Vert \cdot \Vert f(\tau )\Vert \textrm{d} \tau . \end{aligned}$$
(5)

Since C is a stability matrix, one has

$$\begin{aligned} \Vert \textrm{e}^{Ct^{q -p}}\Vert \le \textrm{e}^{-wt^{q -p}}. \end{aligned}$$
(6)

Substituting (6) into (5), it yields

$$\begin{aligned} \Vert x(t)\Vert\le & {} \Vert C\Vert \Vert \textrm{e}^{-wt^{q-p}}\Vert \Vert x(0)\Vert t^{q - p}+\Vert \textrm{e}^{-wt^{q-p}}\Vert \Vert x(0)\Vert +\Vert \textrm{e}^{-wt^{q -p}}\Vert \Vert x^{\prime }(0)\Vert t\nonumber \\{} & {} +\Vert B^{-1}\Vert \int _{0}^{t}(t-\imath )^{q-1} \Vert \textrm{e}^{-w(t-\tau )^{q-p}}\Vert \cdot \Vert f(\tau )\Vert \textrm{d} \tau . \end{aligned}$$
(7)

Multiplying both sides of (7) by \(\textrm{e}^{wt^{q-p}}\), one has

$$\begin{aligned} \begin{aligned} \textrm{e}^{wt^{q-p}}\Vert x(t)\Vert&\le \Vert C\Vert \Vert x(0)\Vert t^{q-p}+\Vert x(0)\Vert +t \Vert x^{\prime }(0)\Vert \\&\quad +\Vert B^{-1}\Vert \int _{0}^{t}(t-\tau )^{q-1}\cdot \textrm{e}^{w\tau ^{q-p}} \cdot \Vert f(\tau )\Vert \text {d}\tau . \end{aligned} \end{aligned}$$
(8)

Let us denote \(u(t)=\textrm{e}^{wt^{q-p}}\Vert x(t)\Vert \), \(g(t)=L\Vert B^{-1}\Vert \), and \(a(t)=t^{q-p}\Vert x(0)\Vert +\Vert x(0)\Vert +t \Vert x^{\prime }(0)\Vert \). Then, (8) can be rewritten as follows:

$$\begin{aligned} u(t)\le a(t)+L\Vert B^{-1}\Vert \cdot \int _{0}^{t}(t-\tau )^{q-1}u(\tau )\text {d}\tau . \end{aligned}$$
(9)

From Lemma 2, it yields

$$\begin{aligned} \textrm{e}^{wt^{q-p}}\Vert x(t)\Vert \le a(t)\cdot E_{q,1}(L\Vert B^{-1}\Vert \cdot \Gamma (q)t^q). \end{aligned}$$
(10)

It follows from Lemma 3 that there exist two real constants \(C_1, C_2 > 0 \) such that

$$\begin{aligned} \textrm{e}^{wt^{q-p}}\Vert x(t) \Vert \le a(t) \cdot C_{1} \cdot \textrm{e}^{\left[ L\Vert B^{-1}\Vert \cdot \Gamma (q) \cdot t^{q}\right] ^{\frac{1}{q}}} +\frac{C_{2} \cdot a(t)}{1+L\Vert B^{-1}\Vert \cdot \Gamma (q) \cdot t^{q}}. \end{aligned}$$
(11)

Multiplying both sides of (11) by \(\textrm{e}^{-wt^{q-p}}\), one has

$$\begin{aligned} \begin{aligned} \Vert x(t)\Vert&\le a(t) \cdot C_{1} \cdot \textrm{e}^{\left[ L\Vert B^{-1}\Vert \cdot \Gamma (q) \cdot t^{q}\right] ^{\frac{1}{q}}-wt^{q-p}}\\&\quad +\frac{C_{2} \cdot a(t)}{\left[ 1+L\Vert B^{-1}\Vert \cdot \Gamma (q) \cdot t^{q}\right] \cdot \textrm{e}^{wt^{q-p}}}. \end{aligned} \end{aligned}$$
(12)

According to the condition in Theorem 1, when \(t\rightarrow \infty \), \(\Vert x(t)\Vert \rightarrow 0 \), which implies that system (1) has asymptotically stable behavior. \(\square \)

Remark 1

Note that references [11,12,13,14,15,16,17,18] focus on the stability condition, or region, of incommensurate FO systems expressed using state-space equations. Even though the nonlinear term in (1) turns into a linear one, the model considered in those works is different from (1).

Remark 2

Theorem 1 is also valid for FO systems with irrational order.

4 Numerical examples

Example 1

Let us consider the nonlinear FO system (1) with parameters \(p=\frac{\sqrt{3}}{3}\), \(q=\frac{\sqrt{3}}{3}+1\), \(A=B=\left[ \begin{array}{cc} 1 &{} 0 \\ 0 &{} -1 \\ \end{array}\right] \) and \(f(x)=\left[ \begin{array}{c} \sin (x_1) \sin (x_2) \\ \sin ^2(x_2) \end{array}\right] .\) By using simple calculations, one can verify that \(w=-\max \{{\text {Re}} \{\lambda ({C})\}\}=1>0.9296=(1\times \Gamma (\frac{\sqrt{3}}{3}+1)^{\frac{1}{\frac{\sqrt{3}}{3}+1 } }\). Therefore, the condition of Theorem 1 is satisfied and the system is stable.

Figure 1 depicts the time response of the system, when the initial states are chosen as \(x_{0}=[0; 0.1]\). Clearly, we verify that the system is stable.

Fig. 1
figure 1

Time response of the selected system

Full size image

Example 2

(The fractional Bagley–Torvik equation) The fractional Bagley-Torvik equation can be used in many scenarios, such as forced damping vibrations. Let us consider the equation:

$$\begin{aligned} A{ }^{C} D_{t}^{p} x(t)+B{ }^{C} D_{t}^{q} x(t)+x(t)= f(x(t)). \end{aligned}$$

We choose \(p=0.5\) and \(q=1.6\), which satisfy \(0<p<1<1+p \le q<2\). Herein, let \(A=\left[ \begin{array}{cc} 1 &{} 0 \\ 0 &{} -1 \\ \end{array}\right] \), \(B=\left[ \begin{array}{cc} -1 &{} 0 \\ 0 &{} 1 \\ \end{array}\right] \) and \(f(x(t))=\left[ \begin{array}{c} \sin (x_1(t)) \sin (x_2(t))+x_1(t) \\ \sin ^2(x_2(t))+x_2(t) \end{array}\right] \). The system of Example 2 is equivalent to (1). As such, by using simple calculations, one can verify that \(\omega = -\max \{Re\{\lambda (C)\}\} = 1>0.9320=(1\times 1 \times \Gamma (1.6))^{\frac{1}{1.6} } \). Thus, the condition of Theorem 1 is satisfied and the system is stable.

Figure 2 depicts the time response of the system, when the initial states are chosen as \(x_0= [0; 0.1]\). Clearly, we verify that the system is stable.

Fig. 2
figure 2

Time response of the Bagley–Torvik system

Full size image

5 Conclusion

We investigated the asymptotic behavior of systems described by nonlinear differential equations with two fractional derivatives, p, q, verifying \(0<p<1<1+p \le q<2\). The Mittag–Leffler function, the Laplace transform and properties of fractional calculus were adopted to derive a sufficient asymptotic stability condition. Numerical examples illustrated the effectiveness of the theoretical condition. Possible generalizations to the case of multi-term FO delayed differential equations will be addressed in future work.