Complete periodic behaviours of real and complex bang bang dynamical systems

Discontinuous dynamical systems arise in major engineering problems involving discontinuous control functions such as the Heaviside function H. Yet complete analyses of these models are extremely difficult and to our knowledge, even the prototype mathematical model defined over the set of integers has not been studied. In this paper, we give a complete analysis of its periodic behaviours. In particular, we show that any solution is periodic and can be determined by two of its consecutive terms, that there are solutions with arbitrary large periods and that there are exactly four types of periodic solutions. As applications, we can then give exact steady-state solutions to neural networks with the above equation as its steady-state equation, and we can also provide the exact periodic behaviours of the complex dynamical system . It is hoped that the techniques used in this paper can be applied to many other discontinuous dynamical systems in the future.