  This paper analyzes the asymptotic behavior of inter-event times in planar
linear systems, under event-triggered control with a general class of
scale-invariant event triggering rules. In this setting, the inter-event time
is a function of the angle of the state at an event. This viewpoint allows us
to analyze the inter-event times by studying the fixed points of the angle map,
which represents the evolution of the angle of the state from one event to the
next. We provide a sufficient condition for the convergence or non-convergence
of inter-event times to a steady state value under a scale-invariant
event-triggering rule. Following up on this, we further analyze the inter-event
time behavior in the special case of threshold based event-triggering rule and
we provide various conditions for convergence or non-convergence of inter-event
times to a constant. We also analyze the asymptotic average inter-event time as
a function of the angle of the initial state of the system. With the help of
ergodic theory, we provide a sufficient condition for the asymptotic average
inter-event time to be a constant for all non-zero initial states of the
system. Then, we consider a special case where the angle map is an
orientation-preserving homeomorphism. Using rotation theory, we comment on the
asymptotic behavior of the inter-event times, including on whether the
inter-event times converge to a periodic sequence. We illustrate the proposed
results through numerical simulations.
