  Gravitational lensing provides a unique and powerful probe of the mass
distributions of distant galaxies. Four-image lens systems with fold and cusp
configurations have two or three bright images near a critical point. Within
the framework of singularity theory, we derive analytic relations that are
satisfied for a light source that lies a small but finite distance from the
astroid caustic of a four-image lens. Using a perturbative expansion of the
image positions, we show that the time delay between the close pair of images
in a fold lens scales with the cube of the image separation, with a constant of
proportionality that depends on a particular third derivative of the lens
potential. We also apply our formalism to cusp lenses, where we develop
perturbative expressions for the image positions, magnifications and time
delays of the images in a cusp triplet. Some of these results were derived
previously for a source asymptotically close to a cusp point, but using a
simplified form of the lens equation whose validity may be in doubt for sources
that lie at astrophysically relevant distances from the caustic. Along with the
work of Keeton et al. (2005), this paper demonstrates that perturbation theory
plays an important role in theoretical lensing studies.
