On the importance of the coexistence of time and state-dependent switching

Abstract

Switching systems have many applications in various fields of physics and engineering. Almost all the switching systems contain either the state-dependent switching or the time-dependent switching. However systems with the coexistence of time and state-dependent switching are very rare in the literature, as we are not aware of the importance and applications of such systems. In this short communication, we focus on the importance of such switched systems, where state and time-dependent switching coexist. In this article, we answer an important open question related to the bifurcation theory of dynamical systems. We show that all kind of periodic transitions are possible due to the border collision bifurcation in this type of switching systems. We have also derived the explicit mathematical expressions of the periodic attractors, existing before and after the border collision bifurcation. Besides, we present some shreds of evidence to show that the complexity of the human behaviour and human intelligence can sometimes be modelled nicely by combination of the state-dependent and the time-dependent switching.