Discontinuity-Induced Bifurcations and Chaos in a Linear Ring Oscillator with a Piecewise Linear Reverse Coupling

The bifurcations of periodic solutions and the generation of chaos in a ring of three unidirectionally coupled linear elements with a single reverse coupling through a piecewise linear function are considered. A discontinuous and a continuous piecewise linear function are employed for the reverse coupling. A chaotic attractor is generated immediately through a Hopf-like boundary equilibrium bifurcation of a focus in both cases. A chaotic attractor is also generated directly through a grazing bifurcation in the case of the discontinuous function, which is replaced with a cascade of period-doubling bifurcations in the case of the continuous function. A chaotic oscillation with the same form is also observed in an experiment on an analog circuit constructed with operational amplifiers. In a smooth version of the system, a ring of three unidirectionally coupled sigmoid neurons with a reverse coupling, the Hopf-like boundary equilibrium bifurcation is replaced with a period-doubling cascade following after the Hopf bifurcation.