Research On Periodic Solution Problems Of Two Neural Networks With Discontinuous Activations

Neural networks are widely studied by scholars since its numerous applications in engineering problems.Its dynamical behaviors restrict the applications of neural networks,so,research the dynamical behaviors can promote the corresponding applications.As an important dynamical behavior of neural networks,periodicity has much attention and existence in practice,such as heartbeat and memorization in biological activities.Periodicity is also an important factor in describing differential equations,and has been studied by many scholars.In this paper,the existence and stability of the periodic solutions of two neural networks are studied.The existence and stability of the periodic solution of Cohen-Grossberg neural networks with discontinuous activations functions and mixed delays are studied in this paper.Currently,the related conclusion are based on the assumptions that activation functions satisfy Lipschitz conditions.The considered activations in this paper only satisfy non-decreasing which weakened above conditions.Base on the differential inclusion theory,Leary-Schauder alternative theorem and Lyapunov functional theory,some new sufficient conditions are derived to ensure the existence and global exponential stability of the periodic solution.Finally,an application and some numerical examples illustrate the feasibility and superiority of the obtained results.Inertial neural networks have extensive applications in the field of secure communication and image encryption process and so on.In practice,the inertia terms are also critical factors which will cause complex branches and chaos.For the periodic solution of second-order inertial neural networks with time delays,the existence and global exponential stability are investigated in this paper.Generally,it is difficult to deal with the inertial terms and to select the Lyapunov functional.To solve these problems,by a suitable variable transformation,we transform the second-order inertial neural networks into first-order neural networks with better properties.Under some assumptions,the existence and global exponential stability of the periodic solution for the transformed systems are proved,then the existence and global exponential stability of the periodic solution for the original systems can be derived.Effectiveness of the theoretical results are verified by some numerical examples.