Global searching ability of chaotic neural networks

This paper aims to theoretically prove that both transiently chaotic neural networks (TCNN´s) and discrete-time recurrent neural networks (DRNN´s) have a global attracting set which ensures that the neural networks carry out a global search. A significant property of TCNN´s and DRNN´s is that their attracting sets are generated by a bounded fixed point, which is the unique repeller when absolute values of the self-feedback connection weights in TCNN and the difference time in DRNN are sufficiently large. We provide sufficient conditions under which the neural networks have a trapping region where the global unstable set of the fixed point actually evolves into a global attracting set. We also prove the coexistence of an attracting set and a transversal homoclinic orbit in the same region, which may result in complicated chaotic dynamics. For combinatorial optimization with neural networks, this paper shows that TCNN´s and DRNN´s do have global searching ability and their attracting set encloses not only local minima, but also global minima for the commonly used objective functions. To demonstrate the theoretical results of this paper, several numerical simulations are provided as illustrative examples