Chaotic phenomena in four circuits with an ideal diode due to the change of the oscillation frequency

Four simple chaos-generating circuits that include a diode are proposed for which both rigorous analyses and physical explanations of the generation of chaos are given. The idealized case in which the diode in the circuit operates as a simple on-off switching element is considered. In this case, the circuit equation is represented as an equation in which two linear second-order ordinary differential equations that have considerably different oscillation frequency are connected to each other by the switching operation of the diode. It is clarified that this effect causes a stretching and folding mechanism. Moreover, it is possible to derive a Poincare map strictly as a one-dimensional map that is similar to a logistic map, and it explains the generation of chaos