  The traditional phase sensitivity function (PSF) has manifested its efficacy
in investigating synchronization behaviors for limit-cycle oscillators.
However, some subtle details may be ignored when the phase value is accumulated
in space or the perturbation is space-dependent. In this paper, we compared
spatial PSF with the traditional PSF for oscillators close to the saddle-node
homoclinic (SNH) bifurcation, also known as saddle-node on invariant circle
(SNIC) bifurcation. It is found that the spatial phase sensitivity function
could reveal the phase accumulation feature on the limit cycle. Moreover, it is
proved that for any two-dimensional smooth dynamical system, type II phase
response curve is the only possible type. Finally, the synchronization
distributions of uncoupled SNH oscillator driven by common and independent
noises are studied, which shows the space-dependent coupling function of common
noise could have significant influences on the synchronization behavior.
