  We study the quench dynamics of a topological $p$-wave superfluid with two
competing order parameters, $\Delta_\pm(t)$. When the system is prepared in the
$p+ip$ ground state and the interaction strength is quenched, only
$\Delta_+(t)$ is nonzero. However, we show that fluctuations in the initial
conditions result in the growth of $\Delta_-(t)$ and chaotic oscillations of
both order parameters. We term this behavior phase III'. In addition, there are
two other types of late time dynamics -- phase I where both order parameters
decay to zero and phase II where $\Delta_+(t)$ asymptotes to a nonzero constant
while $\Delta_-(t)$ oscillates near zero. Although the model is nonintegrable,
we are able to map out the exact phase boundaries in parameter space.
Interestingly, we find phase III' is unstable with respect to breaking the time
reversal symmetry of the interaction. When one of the order parameters is
favored in the Hamiltonian, the other one rapidly vanishes and the previously
chaotic phase III' is replaced by the Floquet topological phase III that is
seen in the integrable chiral $p$-wave model.
