Stability analysis of linear fractional differential system with multiple time delays

Abstract

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 Dynamics 29, 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.

Chaos synchronization in fractional differential systems

“…Especially, based on the stability theory of delayed fractional differential systems, CS of delayed fractional chaotic systems by one-way coupling was investigated by Deng et al [76], who simulated CS of the coupled Duffing oscillators.…”

“…Consider the PC driveresponse configuration with the drive system given by the fractional Chen system (with subscript m) For the error dynamical system of systems (3.20) and (3.21), by applying the stability theorem of multi-rational-order fractional differential systems [76], CS is achieved for the parameters…”

Adaptive feedback control and synchronization of non-identical chaotic fractional order systems

“…Our aim is to determine the controller u(t) for the global synchronization of non-identical fractional order Rössler systems (38) and (39). For this purpose, we design the controller u(t) as follows,…”

“…and so, systems (38) and (39) are synchronized if k 1 , k 2 and k 3 satisfy the law, Therefore, under the controller,…”

Uniform Euler approximation of solutions of fractional-order delayed cellular neural network on bounded intervals

“…Various results on fractional-order neural networks have been obtained as a result of development of fractional calculus [5][6][7][8]. The discrete time fractional-order artificial neural networks were presented in [9,10].…”