Dynamic Analysis Of Fractional-order Neural Networks With Two Types Of Activation Functions

Fractional-order neural networks are important biological networks.Due to their broad application prospects in many aspects such as the research on parameter estimation in s-tatistical theory,the description of quantum motion in physics and network security com-munications,many scholars are increasingly paying close attention to fractional-order neu-ral networks all over the world.Compared with classical integer-order neural networks,fractional-order neural networks model can simulate the behavior of human brain accurately,and describe memory property of neuron and dependence of the historical data effectively.In other words,fractional-order neural networks not only accurately describe the character-istics of the system,but also extend the capability of neural network.This paper focuses on a class of fractional-order neural networks with two types of activation functions(i.e.,con-tinuous or discontinuous activation function),including global projective synchronization and finite-time stabilization of fractional-order neural networks with continuous activation functions,and global Mittag-Leffler synchronization,global dissipativity and finite-time sta-bilization of fractional-order neural networks with discontinuous activation functions.The corresponding contents and innovations are listed as follows.1.The global asymptotical projective synchronization of fractional-order neural net-works with continuous activation functions is investigated.We design a new fractional-order integral sliding mode controller to deal with this problem.By using sliding mode control theory,fractional Lyapunov direct methods and the properties of fractional calculus,some new criteria on global projective synchronization of nonidentical fractional-order neural net-works are obtained.Compared with the previous synchronization results of fractional-order neural networks,our system is more general,more practical,and provide convenience for its practical applications.2.The global Mittag-Leffler synchronization for a class of fractional-order neural net-works with discontinuous activation functions is discussed.Under the framework of Fil?ippov solution,the existence of global solution is given basing on a singular Gronwall in-equality.By using the nonsmooth analysis and control theory,some sufficient criteria for the global Mittag-Leffler synchronization of such system are derived.These criteria can be easily verified,and our results not only cover corresponding ones of integer-order neural networks,but also enrich and enhance the earlier ones of fractional-order neural networks with continuous activation functions.3.The global dissipativity of a class of fractional-order neural networks with time delays and discontinuous activation functions is analyzed.Based on nonsmooth analysis,differential inclusion theory,and comparison theorem and stability theorem for a class of fractional-order systems with time delays,some new sufficient conditions are derived to ensure dissipativity of solutions.Theoretical proof and numerical simulations demonstrate the validity and superiority of the obtained results.4.The robust finite-time stabilization for a class of fractional-order neural networks with two types of activation functions(i.e.,discontinuous or continuous activation function)under uncertainty is investigated.The existence of global solution under the framework of Filippov for such discontinuous system is guaranteed by limiting discontinuous activation functions.In order to handle the existence of equilibrium point of fractional-order neural networks with discontinuous activation functions,the common Banach contraction fixed point theorem has not been used.So we choose Kakutani's fixed point theorem to solve this problem.Based on nonsmooth analysis,differential inclusion theory and fractional Lyapunov stability theory,several new sufficient conditions are given to ensure finite-time stabilization via a novel discontinuous controller.The obtained results improve corre-sponding ones of integer-order neural networks with discontinuous or continuous activation functions.Due to the operation and efficiency of practical engineering practice,finite-time stability control(stabilization)is of great practical significance.