# Fractal catastrophes

## TL;DR

This paper develops a quantitative theory for how fractal attractors and caustics influence spatial clustering and Lyapunov exponents of particles in random force fields, with applications to turbulence and wave propagation.

## Contribution

It introduces a method to analyze the effect of projection on the distribution of finite-time Lyapunov exponents for fractal attractors, revealing universal contributions of fractal catastrophes.

## Key findings

- Caustics from fractal attractors significantly affect spatial Lyapunov exponent distributions.
- The study explains the breakdown of a fluctuation relation upon projection.
- Results have implications for turbulence particle dynamics and wave propagation in media.

## Abstract

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.