Multiple Asymptotic Periodicity Analysis Of Fractional-Order Neural Networks With Time Delays

In recent years,many valuable results have been obtained about multistability and multiperiodicity of neural networks.However,these results are mostly based on the traditional integer-order neural network models.Due to its own special properties,fractional calculus can more accurately describe physical processes with memory properties and hereditary properties.Unlike traditional integer-order neural networks,a non-autonomous fractional-order neural network cannot generate exactly nonconstant periodic solution.However,it is found that the solution trajectory can have an approximately periodic state in numerical experiments.At present,people mainly focuses on the multiple almost periodicity and multiple asymptotic periodicity of fractional-order neural networks.So far,the results on the multiple asymptotic periodicity of fractional-order neural networks are rarely seen,and these results mainly focus on the neural network models with bounded activation function.However,in actual engineering,neural networks with unsaturating piecewise linear activation functions have important application value in reducing computational costs,and due to the unboundedness of functions,neural networks with such activation function have more complex dynamic behavior.This dissertation discussed the multiple asymptotic periodicity of fractional-order Hopfield neural networks with and without time delays,where the activation function is unsaturating piecewise linear function.In the first chapter,the research significance and current research status are discussed,which concern the multistability and multiperiodicity of traditional integer-order neural networks and the multistability and multiple asymptotical periodicity of fractional-order neural networks.The main contents and the main contributions of this dissertation are also expounded according to the abovementioned analysis.In the second chapter,the dynamic behavior of the S-asymptotically ?-periodic solution is discussed for fractional-order Hopfield neural networks with unsaturating piecewise linear activation function.By analyzing the geometric characteristics of unsaturating piecewise linear activation function,applying fractional calculus theories,Mittag-Leffler function properties,and Arzela-Ascoli theorem,the sufficient conditions are obtained for the global asymptotic periodicity and multiple asymptotic periodicity of the neural networks.Finally,two examples with their computer simulations are presented to verify the correctness and validity of the theory.In the third chapter,we have studied the global asymptotic periodicity and multiple asymptotic periodicity of fractional-order delayed Hopfield neural network with unsaturating piecewise linear activation function.By applying the theory of fractional calculus,fractional Leibniz rule,Lyapunov function method,and combining with the geometric characteristics of the unsaturating piecewise linear activation function,the theoretical criteria related to the fractional order ? and the time delay range ? are given.Finally,two examples with their computer simulations are presented to verify the correctness and validity of the theory.In the last chapter,the research work of this dissertation is summarized,and the prospect for the future work is made.