Stability Analysis for a Class of Caputo Fractional Time-Varying Systems with Nonlinear Dynamics

Abstract

This paper investigates mainly stability problem of equilibrium points for a class of Caputo fractional time-varying systems with nonlinear dynamics. By employing Gronwall-Bellman’s inequality, Laplace transform and estimates of Mittag-Leffler functions, when the fractional-order belongs to the interval (0, 2), several stability criterions for fractional time-varying system described by Caputo’s definition are presented. Besides, some problems about the stability of fractional time-varying systems in existing literatures are pointed out. Finally, an example and corresponding numerical simulations are presented to show the validity and feasibility of the proposed stability criterions.

Appendix

Let us modify the system (12) as the following system, where $\alpha =0.95$.

$$\begin{aligned} \left\{ \begin{aligned}&\frac{d^{\alpha }x_1(t)}{dt^{\alpha }}=-x_{1}(t)+10x_{2}(t)+0.1x_1(t)x_2(t),\\&\frac{d^{\alpha }x_2(t)}{dt^{\alpha }}=(-1+0.9\cos (5t))x_{1}(t)+0.95x_{2}(t), \end{aligned} \right. \end{aligned}$$

(13)

where $\frac{d^{\alpha }}{dt^{\alpha }}$ stands for Caputo fractional derivative operator, $x(t)=[x_{1}(t),x_{2}(t)]^T$ is the pseudo-state vector of the system. Subsequently, one gets

By the equations $\det (\lambda I-\bar{A})=0$ and , one can calculate the corresponding eigenvalues as:

$$\begin{aligned} {\lambda _{1,2}} = \frac{{ - 0.05 \pm 0.89303i}}{2} \; \mathrm {and} \; {\tilde{\lambda } _{1,2}} = \frac{{ - 0.05 \pm 8.49691i}}{2}. \end{aligned}$$

Then, $|{\arg ({\lambda _i}(\bar{A}))}|=1.6267$ and , which implies that . Consequently, it is easy to calculate that $\Vert {\bar{A}}\Vert =10.0952$, , , $\alpha \Vert {\bar{A}}\Vert = 9.5904$, , and ${\mathop {\lim }\limits }_{x \rightarrow 0} \frac{{\left\| {f(x(t))} \right\| }}{{\left\| {x(t)} \right\| }} = {\mathop {\lim }\limits }_{x \rightarrow 0} {\frac{0.1\sqrt{{{({x_{1}}{x_{2}})}^{2}}}}{\sqrt{x_{1}^{2} + x_{2}^{2}}}} \le 0.05 {\mathop {\lim }\limits }_{x \rightarrow 0} \sqrt{x_{1}^{2} + x_{2}^{2}} = 0$.

Obviously, the system (13) satisfies the three conditions which are shown in the literature [10]. However, the state of system (13) cannot converge to zero as time t tends to infinity. In the following, Adams-Bashforth-Moulton predictor-corrector algorithm [14] is employed to the numerical calculation of fractional system. When the initial condition $(x_1(0), x_2(0))=(0.1, 0.1)$, the system (13) can generate subharmonic, harmonic or almost-periodic oscillation and even yield a chaotic trajectory. Its chaotic figure is shown in Fig. 3, while Fig. 4 depicts the evolution of the trajectories $x_1(t)$ and $x_2(t)$ of system (13). Therefore, the proposed results of paper [10] are not correct. Since the matrix A(t) of system may depend on time t, we cannot select the lower and upper boundaries of the time matrix A(t) to determine the stability of the zero solution of the time-varying system with integer-order or fractional-order differential operator. Meanwhile, the stability criterions of our presented paper is not used to check the convergence and divergence of the system (13) as well; it will be our future topic of research.

Fig. 3

Chaotic behavior of system

Fig. 4

Evolution of the state $x_{1}(t)$ and $x_{2}(t)$ of the system