This study delves into the exploration of wave propagation in spatially homogeneous systems governed by a Klein-Gordon–type equation with a periodically time-varying cutoff frequency. Through a combination of analytical calculations and numerical simulations, intriguing and distinctive features in the dispersion diagram of these systems are uncovered. Notably, the examined configurations demonstrate some remarkable transitions as the modulation frequency increases. These transitions encompass a transformation from a frequency gap to a wave-number (q) gap around q=0, with the transition point corresponding to a gapless Dirac dispersion with an exceptional point of degeneracy. Subsequently, the q gap undergoes a bifurcation into two symmetric gaps at positive and negative wave numbers. At this second transition point, the dispersion diagram takes the form of an imaginary Dirac dispersion relation and exhibits an isolated exceptional point at the center of the q=0 gap. These findings contribute to a deeper understanding of wave dynamics in periodically modulated media, uncovering tunable phenomena.