<p>Understanding the generative mechanisms and evolutionary laws of bursting behaviours, as well as developing effective strategies for encouraging or suppressing bursting, is an important topic in nonlinear dynamics. In the present work, we pay close attention to the impact of the slow-changing amplitude-modulated excitation (AME) on the dynamic behaviours in a hybrid van der Pol–Rayleigh oscillator. Typically, two symmetric pulse-shaped explosion (PSE)-type bursting behaviours, i.e., symmetric relaxation oscillation and symmetric Hopf/Hopf-Hopf/Hopf bursting, can be exhibited in the unmodulated system. We demonstrate that the AME can significantly alter the symmetry and bifurcation structures of the system. Consequently, the AME can induce the generation of new asymmetric bursting behaviours. Specifically, the symmetric relaxation oscillation can be modulated into two different dynamic behaviours, namely asymmetric relaxation oscillation and asymmetric Hopf/Hopf bursting and the symmetric Hopf/Hopf–Hopf/Hopf bursting are modulated into two different bursting patterns called asymmetric Hopf/Hopf bursting and asymmetric Hopf/Hopf–Hopf/Hopf bursting. We explore the mechanism of the bursting behaviours of the amplitude-modulated system by the slow–fast dynamic analysis. The results indicate that the AME is a valuable method for encouraging or suppressing the generation of bursting behaviours and it has specific theoretical and engineering value for controlling the system’s dynamic behaviours.</p>