Delayed q-deformed logistic map

•The interplay between complexity brought by q-deformation and order by delay is studied in logistic map.•Delay suppresses chaos for Tsallis and Qu-group type of deformations with stabilization of period-1 and -2, respectively.•Increasing delay maintains steady state in Tsallis type while chaotic state is recovered in Qu-group type of deformation. The delay logistic map with two types of q-deformations: Tsallis and Quantum-group type are studied. The stability of the logistic map and its bifurcation scheme is analyzed as a function of the deformation and delay parameters. Chaos is suppressed in a certain region of deformation and delay parameter space. By introducing delay, the steady state in one type of deformation is maintained while chaotic behavior is recovered in another type.