# Lyapunov exponents for the random product of two shears

## TL;DR

This paper provides accurate bounds on Lyapunov exponents for random shear matrix products, aiding in understanding stability and chaos in systems like fluid stirring, with implications for numerical method validation.

## Contribution

It introduces invariant cone-based bounds for Lyapunov exponents of random shear matrices, improving accuracy over existing bounds and assisting in numerical exponent estimation.

## Key findings

- Bounds are highly accurate and improve with increasing shear.
- Invariant cone method effectively estimates Lyapunov exponents.
- Useful for testing numerical methods for exponent computation.

## Abstract

We give lower and upper bounds on both the Lyapunov exponent and generalised Lyapunov exponents for the random product of positive and negative shear matrices. These types of random products arise in applications such as fluid stirring devices. The bounds, obtained by considering invariant cones in tangent space, give excellent accuracy compared to standard and general bounds, and are increasingly accurate with increasing shear. Bounds on generalised exponents are useful for testing numerical methods, since these exponents are difficult to compute in practice.