  In this article we study Chen's flow of curves from theoreical and numerical
perspectives. We investigate two settings: that of closed immersed
$\omega$-circles, and immersed lines satisfying a cocompactness condition. In
each of the settings our goal is to find geometric conditions that allow us to
understand the global behaviour of the flow: for the cocompact case, the
condition is straightforward and the argument is largely standard. For the
closed case however, the argument is quite complex. The flow shrinks every
initial curve to a point if it does not become singular beforehand, and we must
identify a condition to ensure this behaviour as well as identify the point in
order to perform the requisite rescaling. We are able to successfully conduct a
full analysis of the rescaling under a curvature condition. The analysis
resembles the case of the mean curvature flow more than other fourth-order
curvature flow such as the elastic flow or the curve diffusion flow, despite
the lack of maximum and comparison principles. Our work is informed by a
numerical study of the flow, and we include a section that explains the
algorithms used and gives some further simulations.
