  The coupled electron-nuclear spin system in an InGaAs semiconductor as
testbed of nonlinear dynamics can develop auto-oscillations, resembling
time-crystalline behavior, when continuously excited by a circularly polarized
laser. We expose this system to deviations from continuous driving by periodic
modulation of the excitation polarization, revealing a plethora of nonlinear
phenomena that depend on modulation frequency and depth. We find ranges in
which the system's oscillations are entrained with the modulation frequency.
The width of these ranges depends on the polarization modulation depth,
resulting in an Arnold tongue. Outside the tongue, the system shows a variety
of fractional subharmonic responses connected through bifurcation jets when
varying the modulation frequency. Here, each branch in the frequency spectrum
forms a devil's staircase. When an entrainment range is approached by going
through an increasing order of bifurcations, chaotic behavior emerges. These
findings can be described by an advanced model of the periodically pumped
electron-nuclear spin system. We discuss the connection of the obtained results
to different phases of time matter.
