  We perform an analytical study of the stability of the background solution of
the model in which an inflaton, through an axionic coupling to a $U(1)$ gauge
field, causes an amplification of the gauge field modes that strongly backreact
on its dynamics. To this goal, we study the evolution of the gauge field modes
coupled to the inflaton zero mode, treating perturbatively the deviation of the
inflaton velocity from its mean-field value. As long as the system is in the
strong backreaction regime we find that the inflaton velocity performs
oscillations of increasing amplitude about the value it would have in the
approximation of constant velocity, confirming an instability that has been
observed in numerical studies.
