Synchronization of fractional-order hyper-chaotic systems based on a new adaptive sliding mode control

A novel continuous time-varying sliding mode controller for robustly synchronizing non-identical fractional-order chaotic systems precisely at any arbitrary pre-specified time

“…the above fractional differential equation is equivalent to the following Volterra integral equation: 10 Trajectories of (51) and (52) after applying the proposed controller in [29] to (52) when k i = 2, λ i = 30, μ i = 5 and ρ i = 0.25 for i = 1, 2, 3…”

“…To show the difference between synchronization at a pre-specified time and finite-time synchronization, the (51) and (52) after applying the proposed controller in [29] to (52) when k i = 2, λ i = 30, μ i = 5 and ρ i = 0.25 for i = 1, 2, 3 performance of the proposed controller is compared with the controller which has been presented in [29] for finite-time synchronization of fractional chaotic systems. To have a comparison under the same circumstances, the adaptive laws are not considered and the controller proposed in [29] is applied to the fractional chaotic systems (51) and (52) with the same initial conditions as the previous simulations.…”

“…To have a comparison under the same circumstances, the adaptive laws are not considered and the controller proposed in [29] is applied to the fractional chaotic systems (51) and (52) with the same initial conditions as the previous simulations. Since the synchronization time has been set at t = 1 in the third sim- Fig.…”

“…Since the synchronization time has been set at t = 1 in the third sim- Fig. 12 Control signals produced by proposed controller in [29] for synchronization purpose when the control parameters have been set as k i = 2, λ i = 30, μ i = 5 and ρ i = 0.25 for i = 1, 2, 3 ulation, we choose the parameters of the controller in [29] as k i = 2, λ i = 30, μ i = 5 and ρ i = 0.25 for i = 1, 2, 3 in order to ensure that the synchronization time is less than one, i.e. t synch < 1.…”

“…Gao et al in [28] have utilized a modified sliding mode controller for the finite-time synchronization of typical threedimensional fractional chaotic systems. Mohadeszadeh and Delavari in [29] have proposed a novel adaptive sliding mode approach for the synchronization of fractional chaotic and hyper-chaotic systems with unknown parameters and model uncertainties and have proved its finite-time stability, as well. Even though synchronization in a finite time has been accomplished in these works, they could not specify exactly when the two fractional chaotic systems would be synchronized with each other.…”