In this thesis we study a number of systems with varying degrees of hyperbolicity, including uniform and non-uniform hyperbolicity, and discuss the calculation of Lyapunov exponents in these cases. We derive and construct explicit, elementary bounds on the Lyapunov exponents of a collection of systems, which are collectively formed via the composition of shear mappings, upon the 2-torus. These bounds utilise the existence of invariant cones in tangent space to restrict the range of vectors considered in the calculations.
The bounds, with appropriate modifications, are (primarily) used to bound the Lyapunov exponents of two types of system in which their explicit calculation is not possible: a random dynamical system formed by choosing at random a hyperbolic toral automorphism, formed via shear composition, at each iterate, and the linked twist map, a deterministic system which has been used to model various physical phenomena in fluid mixing.
Following the derivation of the bounds, we discuss ways in which their accuracy can be improved. These improvements largely focus on finding a way to narrow the invariant cones used in the bounds, by considering possible preceding matrices within the orbit. We also investigate the practicality of the bounds, and how they compare to other bounds and methods of estimation for Lyapunov exponents.
