A singular bifurcation into instant chaos in a piecewise-linear circuit

Strange bifurcation route to chaos is found in a piecewise-linear second-order nonautonomous differential equation derived from a simple electronic circuit. When a limit cycle loses its stability, the attractor changes directly to chaos (instant chaos) without undergoing a period doubling bifurcation or an intermittency. The width of the attractor´s band is continuous at the bifurcation point, and the chaotic band grows larger continuously as the system parameter is varied. We call this bifurcation a singular bifurcation into instant chaos. The purpose of this paper is to show that the singular phenomenon arises from the piecewise-linearity of the system. To analyze this phenomenon in detail, the degenerate approach is applied. In this simplified case, the Poincare map is derived rigorously as a one-dimensional mapping. By analyzing it, we prove with computer assistance that the Liapunov exponent jumps discontinuously from minus to plus at the bifurcation point