It is shown that the efficiency of the excitation of Bloch oscillations depends not only on the carrier frequency, but also on the frequency of the amplitude oscillations of the driving force. In the nonlinear regime a period doubling bifurcation occurs and the chain of these bifurcations results in chaotic Bloch oscillations of the wave envelope bouncing in the system.

We study the response of an optical system with the Kerr nonlinearity demonstrating Bloch oscillations to a periodic train of coherent pulses. It has been found out that the intensity of the field excited in the system by pulses resonantly depends on the train period. It is demonstrated numerically and analytically that the response of the system is stronger when the period of the driving pulses is commensurate with the period of the Bloch oscillations. Moreover, large enough pulses are capable to induce the instabilities which eventually lead to onset of chaotic Bloch oscillations of the wave-function envelope bouncing both in time and space. The analysis reveals that these instabilities are associated with period-doubling bifurcations. A cascade of such bifurcations with increase of the pulses' amplitude triggers the chaotic behaviour.