  Perturbation theory is an important tool in the analysis of oscillators and
their response to external stimuli. It is predicated on the assumption that the
perturbations in question are "sufficiently weak", an assumption that is not
always valid when perturbative methods are applied. In this paper, we identify
a number of concrete dynamical scenarios in which a standard perturbative
technique, based on the infinitesimal phase response curve (PRC), is shown to
give different predictions than the full model. Shear-induced chaos, i.e.,
chaotic behavior that results from the amplification of small perturbations by
underlying shear, is missed entirely by the PRC. We show also that the presence
of "sticky" phase-space structures tend to cause perturbative techniques to
overestimate the frequencies and regularity of the oscillations. The phenomena
we describe can all be observed in a simple 2D neuron model, which we choose
for illustration as the PRC is widely used in mathematical neuroscience.
