A discontinuous and noninvertible circle map is introduced to describe modulated relaxation oscillators. The boundaries of the parameter regions are calculated where images of the jumping gap completely cover the noninvertible region so that the dynamics of the system are abruptly changed. We show that a kind of structure called a ‘‘hole’’ may appear due to the interaction between these two competing characteristics and may lead to some dynamic features. The influence of the holes on the boundaries of Arnold tongues is also discussed.

• Received 7 December 1994

DOI:https://doi.org/10.1103/PhysRevE.52.453