Stability Theory Of Fractional Systems Research Articles

Conditions for the local and global asymptotic stability of the time–fractional Degn–Harrison system

Abstract Fractional calculus has been shown to improve the dynamics of differential system models and provide a better understanding of their dynamics. This paper considers the time–fractional version of the Degn–Harrison reaction–diffusion model. Sufficient conditions are established for the local and global asymptotic stability of the model by means of invariant rectangles, the fundamental stability theory of fractional systems, the linearization method, and the direct Lyapunov method. Numerical simulation results are used to illustrate the theoretical results.

Observer-based output consensus of leader-following fractional-order heterogeneous nonlinear multi-agent systems

ABSTRACT In this paper, the observer-based output consensus of leader-following fractional-order heterogeneous nonlinear multi-agent systems with potentially different state dimensions and different dynamics are investigated. Considering the agents' states may be not available in practice, the distributed state observer is designed for each agent. Then corresponding distributed control protocols based a proposed virtual exosystem are proposed to convert the output consensus problem into the stabilisation problem. By employing matrix theory, algebraic graph theory and the stability theory of fractional system, the state consensus of the virtual exosystem and the output consensus of heterogenous fractional nonlinear MAS is achieved. Moreover, the sufficient conditions are also obtained. Simulation examples are presented to verify the effectiveness of the main results.

Pinning Synchronization of Fractional General Complex Dynamical Networks with Time Delay

In this paper, pinning synchronization of fractional general complex dynamical networks with time delay is investigated. Based on the stability theory of fractional systems and the inequality of the fractional derivative, some pinning synchronization criteria are abtained for both directed and undirected complex networks. Moreover, an adaptive pinning controller is designed to achieve synchronization of fractional general complex networks with time delay. Finally, some numerical examples are given to verify the proposed theoretical analysis.

Mittag–Leffler Stability and Global Asymptotically $$\omega $$ ω -Periodicity of Fractional-Order BAM Neural Networks with Time-Varying Delays

This article studies the Mittag–Leffler stability and global asymptotical $$\omega $$ -periodicity for a class of fractional-order bidirectional associative memory (BAM) neural networks with time-varying delays by using Laplace transform, stability theory of fractional systems and some integration technique. Firstly, some sufficient conditions are given to ensure the boundedness and global Mittaag-Leffler stability of fractional-order BAM neural networks with time-varying delays. Next, S-asymptotical $$\omega $$ -periodicity and global asymptotical $$\omega $$ -periodicity of fractional-order BAM neural networks with time-varying delays are also explored. Finally, some numerical examples and simulation are performed to show the effectiveness of theoretical results.

Dynamic analysis of a 5D fractional-order hyperchaotic system

In this paper, the fractional-order 5D hyperchaotic system is proposed based on the hyperchaotic Lorenz system. Fractional-order chaotic systems are often three- or four-dimensional. There are few results about high dimension fractional-order systems. For this 5D hyperchaotic system, the stability of equilibrium points is analyzed by means of the stability theory of fractional systems. Then the fractional bifurcation is investigated. It is found that the system admits bifurcations with varying fractional-order and parameters, respectively. Under different bifurcation parameters, some conditions ensuring the bifurcations are presented. Finally, numerical simulations are presented to confirm the given analytical results.