  We present a theoretical study of the spin dynamics of a single electron
confined in a quantum dot. Spin dynamics is induced by the interplay of
electrical driving and the presence of a spatially disordered magnetic field,
the latter being transverse to a homogeneous magnetic field. We focus on the
case of strong driving, i.e., when the oscillation amplitude $A$ of the
electron's wave packet is comparable to the quantum dot length $L$. We show
that electrically driven spin resonance can be induced in this system by
subharmonic driving, i.e., if the excitation frequency is an integer fraction
(1/2, 1/3, etc) of the Larmor frequency. At strong driving we find that (i) the
Rabi frequencies at the subharmonic resonances are comparable to the Rabi
frequency at the fundamental resonance, and (ii) at each subharmonic resonance,
the Rabi frequency can be maximized by setting the drive strength to an
optimal, finite value. Our simple model is applied to describe electrical
control of a spin-valley qubit in a weakly disordered carbon nanotube.
