The twist-and-flip circuit contains only three circuit elements: two linear capacitors connected across the ports of a gyrator characterized by a nonlinear gyration conductance function g(v/sub 1/, v/sub 2/). When driven by a square-wave voltage source of amplitude a and frequency omega , the resulting circuit is described by a system of two nonautonomous state equations. For almost any choice of nonlinear g(v/sub 1/, v/sub 2/)>0, and over a very wide region of the a- omega parameter plane, the twist-and-flip circuit is imbued with the full repertoire of complicated chaotic dynamics typical of those predicted by the classic KAM theorem from Hamiltonian dynamics. The significance of the twist-and-flip circuit is that its associated nonautonomous state equations have an explicit Poincare map, called the twist-and-flip map, thereby making it possible to analyze and understand the intricate dynamics of the system, including its many fractal manifestations. The focus is on the many fractals associated with the twist-and-flip circuit.>