Stability analysis of linear fractional differential system with multiple time delays

By Weihua Deng, Changpin Li, Jinhu Lu. 2007

915 citations recorded by the source corpus.

In this paper, we study the stability of n-dimensional linear fractional differential equation with time delays, where the delay matrix is defined in (R+)n×n. By using the Laplace transform, we introduce a characteristic equation for the above system with multiple time delays. We discover that if all roots of the characteristic equation have negative parts, then the equilibrium of the above linear system with fractional order is Lyapunov globally asymptotical stable if the equilibrium exist that is almost the same as that of classical differential equations. As its an application, we apply our theorem to the delayed system in one spatial dimension studied by Chen and Moore [Nonlinear Dynamics29, 2002, 191] and determine the asymptotically stable region of the system. We also deal with synchronization between the coupled Duffing oscillators with time delays by the linear feedback control method and the aid of our theorem, where the domain of the control-synchronization parameters is determined.

Subfields: Computer Networks and Communications, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/e1ef4845321934d594cbda48f98a14f98aebd45c

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In Computer Networks and Communications

1. Dumitru Băleanu
2. W. Zheng
3. Fahd Jarad
4. Jinde Cao
5. Thabet Maraaba
6. Liping Chen
7. R. Wu
8. Song Liang

In Statistical and Nonlinear Physics

1. Thabet Abdeljawad
2. Fahd Jarad
3. Dumitru Băleanu
4. Jinde Cao
5. Thabet Maraaba
6. Liping Chen
7. R. Wu
8. W. Zheng

Citations in the corpus

• Fractional-order delayed predator–prey systems with Holling type-II functional response (2015, methodology): “In section, we extend the analysis to study the global stability conditions [10,22] for the fractional-order delay differential system.”
• LAPLACE TRANSFORM OF FRACTIONAL ORDER DIFFERENTIAL EQUATIONS (2015, methodology): “In [5], Deng et al studied the stability of n-dimensional linear fractional differential equation with time delays by Laplace transform method.”
• STABILITY RESULTS AND EXISTENCE THEOREMS FOR NONLINEAR DELAY-FRACTIONAL DIFFERENTIAL EQUATIONS WITH $ \varphi^*_P $-OPERATOR (2020, background): “[9] proved stability results for a system of fractional DEs with delays and provided applications.”
• Stability of Fractional-Order Systems with Rational Orders (2008, neutral): “characteristic roots of the system (17) have negative real parts [ 15 ].”
• Fractional-order PD control at Hopf bifurcations in a fractional-order congestion control system (2017, background): “Corollary 2 ([50]) Given αi = α ∈ (0, 1].”
• Analysis of the compact difference scheme for the semilinear fractional partial differential equation with time delay (2017, background): “[11] studied stability of linear fractional differential system with multiple time delays and the Lyapunov global asymptotic stability of n-dimensional linear fractional differential equation with time delay.[12] Zhou et al.”
• Fractional variational optimal control problems with delayed arguments (2010, background): “In [ 24 ], the stability of linear fractional differential equations was analyzed.”
• On the existence and the uniqueness theorem for fractional differential equations with bounded delay within Caputo derivatives (2008, methodology): “This leads us to the idea to replace the classical potential by a non-local one and implicitly to use the combination of fractional and delay [17, 18 ] techniques.”