All-solid-state passively Q-switched lasers can exhibit nonlinear behaviors such as period-doubling, injection locking, and chaos under specific conditions, offering new applications in fields like secure communication and random number generation. As a result, the nonlinear dynamics of laser systems has become increasingly important. Pump modulation is a typical method for controlling the nonlinear dynamical states of solid-state lasers. This study investigates the nonlinear dynamical characteristics of an all-solid-state passively Q-switched Nd:YAG/Cr:YAG laser under pump modulation by solving the four-level rate equation system using the Runge-Kutta method. The results demonstrate that by adjusting key parameters including modulation frequency, modulation amplitude, and unmodulated pump rate, the laser system can exhibit rich dynamical states, including period-one, period-two, multi-period, and chaotic pulsation. By analyzing the bifurcation diagram, the evolution pattern of output laser pulse characteristics with parameter changes are revealed． The system mainly enters chaos through period-doubling and quasi-periodic routes, while exhibiting a unique phenomenon where the pulse peak and pulse frequency follow synchronized evolutionary paths but with opposite trends in intensity variation, indicating dynamic coupling effects between frequency and intensity domains ． By constructing the nonlinear dynamical distributions within a three-dimensional pump modulation parameter space, this study systematically investigates the combined effects of modulation frequency, modulation amplitude, and unmodulated pump rate on the evolution of the laser's nonlinear dynamics. The results show that at lower unmodulated pump rates, the system cannot be driven into nonlinear states even when the modulation amplitude and frequency are relatively large. In contrast, under higher unmodulated pump rates, appropriate tuning of the modulation amplitude and frequency can enable the system to transition from periodic states to chaotic behavior. This research not only clarifies the modulation mechanisms of pump parameters on the laser's nonlinear dynamics, but also provides theoretical guidance for optimizing laser output performance and designing high-performance chaotic lasers, holding significant value for promoting applications of Q-switched lasers in precision measurement and secure communication fields．