  Topological methods can provide a way of proposing new metrics and methods of
scrutinising data, that otherwise may be overlooked. In this work, a method of
quantifying the shape of data, via a topic called topological data analysis
will be introduced. The main tool within topological data analysis (TDA) is
persistent homology. Persistent homology is a method of quantifying the shape
of data over a range of length scales. The required background and a method of
computing persistent homology is briefly discussed in this work. Ideas from
topological data analysis are then used for nonlinear dynamics to analyse some
common attractors, by calculating their embedding dimension, and then to assess
their general topologies. A method will also be proposed, that uses topological
data analysis to determine the optimal delay for a time-delay embedding. TDA
will also be applied to a Z24 Bridge case study in structural health
monitoring, where it will be used to scrutinise different data partitions,
classified by the conditions at which the data were collected. A metric, from
topological data analysis, is used to compare data between the partitions. The
results presented demonstrate that the presence of damage alters the manifold
shape more significantly than the effects present from temperature.
