FRACTALS IN THE TWIST-AND-FLIP CIRCUIT

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(upsilon(1), upsilon(2)). When driven by a square-wave voltage source of amplitude ''alpha'' and frequency ''omega,'' the resulting circuit is described by a system of two nonautonomous state equations. For almost any choice of nonlinear g(upsilon(1), upsilon(2)) > 0, and over a very wide region of the alpha - 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, and widely observed numerically from plasma and accelerator dynamics, as well as from celestial mechanics. 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. Although the properties of the twist-and-flip circuit are best understood from an in-depth analysis of its many elegant mathematical properties (published elsewhere), this paper will focus on the many fractals associated with the twist-and-flip circuit, in keeping with the theme of this special issue. Behind the masks of the colorful fractals, however, lies an immensely rich variety of chaotic phenomena, whose unique mathematical tractability is responsible for this circuit's potential application as a paradigm for ''nonautonomous'' chaos.