  We analyse the spatial inhomogeneities ('spatial clustering') in the
distribution of particles accelerated by a force that changes randomly in space
and time. To quantify spatial clustering, the phase-space dynamics of the
particles must be projected to configuration space. Folds of a smooth
phase-space manifold give rise to catastrophes ('caustics') in this projection.
When the inertial particle dynamics is damped by friction, however, the
phase-space manifold converges towards a fractal attractor. It is believed that
caustics increase spatial clustering also in this case, but a quantitative
theory is missing. We solve this problem by determining how projection affects
the distribution of finite-time Lyapunov exponents. Applying our method in one
spatial dimension we find that caustics arising from the projection of a
dynamical fractal attractor ('fractal catastrophes') make a distinct and
universal contribution to the distribution of spatial finite-time Lyapunov
exponents. Our results explain a projection formula for the spatial fractal
correlation dimension, and how a fluctuation relation for the distribution of
finite-time Lyapunov exponents for white-in-time Gaussian force fields breaks
upon projection. We explore the implications of our results for heavy particles
in turbulence, and for wave propagation in random media.
