Computing unstable periodic orbits (UPOs) for systems governed by ordinary differential equations (ODEs) is a fundamental problem in the study of nonlinear dynamical systems that exhibit chaotic dynamics. Success of any existing method to compute UPOs relies on the availability of very good initial guesses for both the UPO and its time period. This thesis presents a computational framework for computing UPOs that are extremal, in the sense that they optimise the infinite-time average of a certain observable. Constituting this framework are two novel techniques.

The first is a method to localise extremal UPOs for polynomial ODE systems that does not rely on numerical integration. The UPO search procedure relies on polynomial optimisation to construct nonnegative polynomials whose sublevel sets approximately localise parts of the extremal periodic orbit. Points inside the relevant sublevel sets can then be computed efficiently through direct nonlinear optimisation. Such points provide good initial conditions for UPO computations with existing algorithms.

The second technique involves the addition of a control term to the original polynomial ODE system to reduce the instability of the extremal UPO, and, in some cases, to provably stabilise it. This control methodology produces a family of controlled systems parametrised by a control amplitude, to which existing UPO-finding algorithms are often more easily applied.

The practical potential of these techniques is demonstrated by applying them to find extremal UPOs for a nine-dimensional model of sinusoidally forced shear flow, an extended version of the Lorenz system, and two different three-dimensional chaotic ODE systems. Extensions of the framework to non-polynomial and Hamiltonian ODE systems are also discussed.