Continuous-Time Limit of a Shift Map Yielding an Exactly Solvable Chaotic Differential Equation

A chaotic nonlinear differential equation possessing an exact analytic solution is derived from a continuous-time limit of an iterated shift map. The resulting equation consists of an unstable harmonic oscillator forced by a nonlinear piecewise-constant feedback. The continuous time solution exhibits a shift dynamic at integer return times, and the waveform is described completely in terms of the symbols of the contained shift map. Significantly, this exactly solvable nonlinear system provides the first example of a chaotic ordinary differential equation possessing an exact symbolic dynamics.