Synchronization of uncertain fractional-order chaotic systems with time delay based on adaptive neural network control

Abstract

Time delay frequently appears in many phenomena of real life and the presence of time delay in a chaotic system leads to its complexity. It is of great practical significance to study the synchronization control of fractional-order chaotic systems with time delay. This is because it is closer to the real life and its dynamical behavior is more complex. However, the chaotic system is usually uncertain or unknown, and may also be affected by external disturbances, which cannot make the ideal model accurately describe the actual system. Moreover, in most of existing researches, they are difficult to realize the synchronization control of fractional-order time delay chaotic systems with unknown terms. In this paper, for the synchronization problems of the different structural fractional-order time delay chaotic systems with completely unknown nonlinear uncertain terms and external disturbances, based on Lyapunov stability theory, an adaptive radial basis function (RBF) neural network controller, which is accompanied by integer-order adaptive laws of parameters, is established. The controller combines RBF neural network and adaptive control technology, the RBF neural network is employed to approximate the unknown nonlinear functions, and the adaptive laws are used to adjust corresponding parameters of the controller. The system stability is analyzed by constructing a quadratic Lyapunov function. This method not only avoids the fractional derivative of the quadratic Lyapunov function, but also ensures that the adaptive laws are integer-order. Based on Barbalat lemma, it is proved that the synchronization error tends to zero asymptotically. In the numerical simulation, the uncertain fractional-order Liu chaotic system with time delay is chosen as the driving system, and the uncertain fractional-order Chen chaotic system with time delay is used as the response system. The simulation results show that the controller can realize the synchronization control of the different structural fractional-order chaotic systems with time delay, and has the advantages of fast response speed, good control effect, and strong anti-interference ability. From the perspective of long-term application, the synchronization of different structures has greater research significance and more development prospect than self synchronization. Therefore, the results of this study have great theoretical significance, and have a great application value in the field of secure communication.

Adaptive fuzzy synchronization of uncertain fractional-order chaotic systems with different structures and time-delays

“…Remark 4 According to Eq. ( 41) and inequalities (49), we see that θ θ θ (t) 2 ≤ 2V (t) ≤ 2ε. Then we see that θi (t) can be arbitrarily small eventually.…”

“…Remark 1 Comparing with the integer-order parameter adaptation in Ref. [49] that has the same aim as this study, the integer-order adaptive laws in the paper have been improved, adding a term (for example, the adaptive law (24) adds a term -ξ i ξi ε * i (t)). We can easily learn from the integer-order adaptive laws in this paper: ε * i (t) = -ξ i ξi < 0, and ε * i (t) has a maximal value on t ∈ [0, +∞).…”

“…Remark 3 Noting that besides the form of controller (22) being different from controller's in Ref. [49], it adds a feedback gain variable l i (t) that is automatically adjustable with the change of 0 D μ-1 t e i (t). And it strengthens the connection between U U U(t) and e e e(t) and makes the flexibility of U U U(t) be more obvious.…”