Transition from order to chaos in reduced quantum dynamics

We study a damped kicked top dynamics of a large number of qubits ($N \rightarrow \infty$) and focus on an evolution of a reduced single-qubit subsystem. Each subsystem is subjected to the amplitude damping channel controlled by the damping constant $r\in [0,1]$, which plays the role of the single control parameter. In the parameter range for which the classical dynamics is chaotic, while varying $r$ we find the universal period-doubling behavior characteristic to one-dimensional maps: period-two dynamics starts at $r_1 \approx 0.3181$, while the next bifurcation occurs at $r_2 \approx 0.5387$. In parallel with period-four oscillations observed for $r \leq r_3 \approx 0.5672$, we identify a secondary bifurcation diagram around $r\approx 0.544$, responsible for a small-scale chaotic dynamics inside the attractor. The doubling of the principal bifurcation tree continues until $r \leq r_{\infty} \sim 0.578$, which marks the onset of the full scale chaos interrupted by the windows of the oscillatory dynamics corresponding to the Sharkovsky order.