Dynamical behaviors for discontinuous and delayed neural networks in the framework of Filippov differential inclusions.
This paper is concerned with the periodic dynamics of a class of delayed neural networks with discontinuous neural activation functions. Under the Filippov framework, the cone expansion and compression fixed point theorems of set-valued maps are successfully employed to derive the existence of the ω-periodic positive solution. However, before the discussion of the periodicity, there still remains a fundamental issue about viability to be solved due to the presence of general mixed time-delays involving both time-varying delays and distributed delays. This difficulty can be overcome by a transformation and the continuation theorem. Then, for the discontinuous and delayed neural network system with time-periodic coefficients, the uniqueness and global exponential stability of the periodic state solution are proved by using non-smooth analysis theory with generalized Lyapunov approach. Furthermore, the global convergence in measure of the periodic output is also investigated. The obtained results are a very good extension and improvement of previous works on discontinuous dynamical neuron systems with a broad range of neuron activations dropping the assumption of boundedness or monotonicity. Finally, numerical simulations are provided to illustrate the effectiveness of the theoretical results.
College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, PR China; Hunan Women's University, Changsha, Hunan 410002, PR China. Electronic address: lhhuang@hnu.edu.cn.
Lihong
L
Huang
Zuowei
Z
Cai
Lingling
L
Zhang
Lian
L
Duan
D000465:Algorithms; D008959:Models, Neurological; D016571:Neural Networks, Computer; D009474:Neurons / Q000502:physiology; D017711:Nonlinear Dynamics; D010507:Periodicity
Delayed neural networks; Differential inclusions; Discontinuous neuron activation; Global convergence; Krasnoselskii’s fixed point theorem of set-valued maps; Positive periodic solution