  Chemically active, or autophoretic, particles that isotropically emit or
absorb solute molecules undergo spontaneous self-propulsion when their activity
is increased beyond a critical P\'{e}clet number ($Pe$). Here, we conduct
numerical computations, using a spectral-element based method, of a rigid,
spherical autophoretic particle in unsteady rectilinear translation. The
particle can be freely suspended (or `unforced') or subject to an external
force field (or `forced'). The motion of an unforced particle progresses
through four regimes as $Pe$ is increased: quiescent, steady, stirring, and
chaos. The particle is stationary in the quiescent regime, and the solute
profile is isotropic about the particle. At $Pe=4$ the fore-aft symmetry in the
solute profile is broken, resulting in its steady self-propulsion. Our
computations indicate that the self-propulsion speed scales linearly with
$Pe-4$ near the onset of self-propulsion, as has been predicted in previous
studies. A further increase in $Pe$ gives rise to the stirring regime at
$Pe\approx27$, where the fluid undergoes recirculation, while the particle
remains essentially stationary. As $Pe$ is increased even further, the particle
dynamics are marked by chaotic oscillations at $Pe\approx55$ and higher, which
we characterize in terms of the mean square displacement and velocity
autocorrelation of the particle. Our results for an autophoretic particle under
a weak external force are in good agreement with recent asymptotic predictions
(Saha, Yariv, and Schnitzer, J. Fluid Mech., vol. 916, A47, 2021).
Additionally, we demonstrate that the strength and temporal scheduling of the
external force may be tuned to modulate the chaotic dynamics at large $Pe$.
