Abstract
Amplitude modulation, a recently focused mode of modulation in fast–slow dynamics, plays a crucial role in governing the behaviors of relaxation oscillations. This modulation mode involves three distinct factors: modulation index, modulation frequency and modulation phase. Previous studies have primarily concentrated on the influence of the first two factors on relaxation oscillations, whereas this paper emphasizes the effects of modulation phase on relaxation oscillations by taking the forced Duffing system as an example. Through numerical simulations, it has been ascertained that the introduction of modulation phase results in a shift in relaxation oscillations. Significantly, as modulation phase increases to some critical value, relaxation oscillations will disappear and transit into a conventional oscillatory pattern, which persists in a specific parameter interval. Moreover, as modulation phase continues to increase and reaches the other critical value, relaxation oscillations are generated again. We show that the introduction of modulation phase may lead to a qualitative alteration in the bifurcation behavior of the vector field. The variation in modulation phase can result in the disappearance of fold bifurcation points, which is associated with the disappearance of relaxation oscillations. Furthermore, the alteration in modulation phase can also induces the sudden generation of fold bifurcation points, offering an explanation for the transition from conventional oscillatory patterns to relaxation oscillations. Our research supplements the third factor i.e. modulation phase that influences amplitude modulation in fast–slow dynamics and reveals its effects on relaxation oscillations. In particular, a three-parameter bifurcation analysis is employed in this paper to illustrate the universality of modulation index selection in normal modulation, thereby providing a reference for future investigations concerning cases of overmodulation.
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