This paper proposes a method for calculation of local bifurcation points in discrete-time dynamical systems with piecewise nonlinear characteristics (PNDDS). First, an -dimensional PNDDS, which has two piecewise nonlinear maps, is shown and its variational equation is derived. Next, a calculation method for the local bifurcation points that utilizes the conditional equation for the periodic solution and the characteristic equation is proposed. It is essential to calculate the derivatives of the map with an initial value and with a bifurcation parameter to obtain the bifurcation points continuously in the parameter space. The above calculation process is a key component of the proposed method, and is explained in detail. Finally, we apply the proposed method to a two-dimensional PNDDS and calculate the local bifurcation points in order to confirm the validity of the proposed method.

In this study, we analyze the characteristics of an interrupted electric circuit. In particular, we focus on a special situation where the switching action of the circuit is delayed because of a time lag in the response to the switching signal. This situation is observed in switching circuits driven by a high-frequency switching signal. However, the fundamental characteristics of this type of circuit have not yet been clarified. To address this shortfall, we assume that a time lag of the response to the switching signal occurs in simple interrupted electric circuits, and investigate how this time lag affects circuit characteristics. First, we show the model of a circuit whose switching action is the same as that of a current-mode-controlled dc/dc converter. Here by using logic circuits, we impose an artificial time lag on the response to the switching signal. Next, we define a sampled data model (i.e., a return map) that we analyze in detail. Based on the return map, we derive one- and two-parameter bifurcation diagrams. Finally, we compare the bifurcation diagrams constructed with time lag to those constructed without time lag. The results clearly show that time lag is responsible for a new structure in the return map that does not occur in circuits with ideal switching. This new return map structure is a key to understanding the essential characteristics of circuits with time lag. Furthermore, the mathematical results are verified experimentally.