  The stability and dynamics of almost strong zero and $\pi$ modes in weakly
non-integrable Floquet spin chains are investigated. Such modes can also be
viewed as localized Majorana modes at the edge of a topological superconductor.
Perturbation theory in the strength of integrability-breaking interaction $J_z$
is employed to estimate the decay rates of these modes, and compared to decay
rates obtained from exact diagonalization. The structure of the perturbation
theory and thus the lifetime of the modes is governed by the conservation of
quasi-energy modulo $2 \pi/T$, where $T$ is the period of the Floquet system.
If the quasi-energies of minimally $4 n-1$ quasi-particles adds up to zero (or
$\pi/T$ for a $\pi$ mode), the lifetime is proportional to $1/J_z^{2 n}$. Thus
the lifetime is sensitively controlled by the width of the single-particle
Floquet bands. For regimes where the decay rates are quadratic in $J_z$, an
analytic expression for the decay rate in terms of an infinite temperature
autocorrelation function of the integrable model is derived, and shown to agree
well with exact diagonalization.
