Seven Limit Cycles in a Periodically Forced Three-Zone Equation

Abstract

We construct a seven-cycle example for a periodically forced scalar equation with a continuous three-zone piecewise-linear nonlinearity. Symbolic integration and linear elimination reduce the four-flight matching conditions to two equations in the residence times. The competing exterior visits produce an additional turn of the critical response and lead to the rational parameters $(A,B,μ)=(9,3/2,181363/100000)$. At these parameters, finite Taylor estimates give three sign changes of the return displacement. Scalar order, analyticity along transverse itineraries, and half-period reflection then establish at least seven distinct isolated periodic orbits on the phase cylinder.