### Abstract

The remarkable universality observed in period-doubling cascades for families of unimodal maps with integer degree critical point has been studied extensively. Feigenbaum offered an explanation in terms of a renormalisation operator. In this thesis we begin by examining one-parameter families of unimodal maps with quadratic critical point. Using rigorous computer-assisted methods we calculate tight bounds on the renormalisation for point relevant for period doubling, by using a contraction mapping argument in a space of analytic functions, also leading to rigorous bounds on the universal scaling constant, α.Bounds on the spectrum of the derivative of the renormalisation operator at the fixed point are used to prove hyperbolicity in the function space and to provide bounds on the eigenvalues. We explore analytical solutions to the associated eigenproblem and the relationship between the eigenfunctions corresponding to different choices of normalisation-preserving scale factor in the definition of the renormalisation operator.

We use a novel application of the contraction mapping method to provide rigorous bounds on the eigenvalue-eigenfunction pair for the universal constant δ controlling the asymptotic rate of accumulation of period-doublings, and also to provide rigorous bounds on a particular eigenvalue-eigenfunction pair for the operator corresponding to the universal scaling of added uncorrelated noise. We also explore analytical solutions to the noise eigenproblem.

We extend this rigorous work to cover families with quartic and cubic critical points and apply the method numerically for higher integer degrees.

Further, by casting the attractor at the accumulation of period-doublings as the limit set of an iterated function system we use a rigorous method to bound its Hausdorff dimension in the case of quadratic, quartic and cubic critical points.

We explore universality in two-variable, unidirectionally-coupled systems using the corresponding renormalisation operator via numerical calculations, and take steps towards a rigorous, computer-assisted proof of existence of a fixed point and rigorous bounds on associated constants. Using the knowledge gained and methods developed for the single variable system, we suggest a direction for future work in this area.

Date of Award | Sept 2021 |
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Original language | English |

Supervisor | Andrew Burbanks (Supervisor) & Andrew Harold Osbaldestin (Supervisor) |