  Recently, effects of nonlinearity on topologically nontrivial systems have
attracted attention and the stability of topologically protected edge states
has been studied for a quantum walk with nonlinear effects, which is akin to
time-periodically driven systems (Floquet systems). In the previous work, it
has been found that the edge states can be stable attractors or unstable
repellers depending on their intrinsic topological property, while the
stability is not affected by the strength of nonlinearity. In the present work,
we find additional bifurcations at which edge states change from stable
attractors to unstable repellers with increasing the strength of nonlinearity
in nonlinear quantum walks, for the first time. The new bifurcations are unique
to Floquet systems, since we take dynamical properties of Floquet systems into
consideration by directly applying the time-evolution operator of the quantum
walks to the linear stability analysis. Our results shed new light on nonlinear
effects on topological edge states in Floquet systems.
