A 555 Timer IC Chaotic Circuit: Chaos in a Piecewise Linear System With Stable but No Unstable Equilibria

Abstract

The 555 timer IC is a well-known integrated circuit having been widely used as a pulse generator. We propose a circuit containing a single 555 IC, an LED, an inductor, two capacitors, and three resistors that exhibits chaos. The IC serves as a hysteretic switch causing the system to be alternately attracted to each of two stable equilibria. Ours appears to be the first chaotic circuit governed by piecewise-linear equations that have stable but no unstable equilibria. Also unique is the sensitive dependence on initial conditions resulting only from a square-root map. This arises from the grazing impact of two voltages. The circuit is also unusual in exhibiting periodic orbits with a Farey tree structure and a transition to chaos via a period-adding cascade. This and other bifurcations seen experimentally are in agreement with those of the governing three-dimensional ODEs with hysteretic conditions. Inclusion of three extra resistors in the circuit lowers the characteristic frequency so that the bifurcations can be detected via the LED without the need of an oscilloscope. The circuit is therefore also suitable for experimentation by electronics hobbyists with limited resources.