Iterated dynamical maps offer an ideal setting to investigate quantum dynamical bifurcations and are well adapted to few-qubit quantum computer realisations. We show that a single trapped ion, subject to periodic impulsive forces, exhibits a rich str
ucture of dynamical bifurcations derived from the Jahn-Teller Hamiltonian flow model. We show that the entanglement between the oscillator and electronic degrees of freedom reflects the underlying dynamical bifurcation in a Floquet eigenstate.
