This is a three-part talk where I’ll cover piecewise-linear maps, mechanical systems with impacts, and neurons. Piecewise-linear maps provide a useful tool to explore robust chaos in dynamical systems. One such map is the two-dimensional border-collision normal form which exhibits bifurcation structures that are still not fully understood. In this talk, I will first show how renormalization can be utilized to better understand these. I will also verify Devaney’s definition of chaos, and how robust chaos extends to maps with higher dimensions. In the second part of my talk, I will discuss simple impacting systems that occur in engineering problems. Impacts are onset by grazing bifurcations, and these generate complex dynamics including chaos because the corresponding Poincare map has a highly nonlinear and destabilising square-root term. The main goal of this talk is to determine why leading-order approximations to such maps, such as the Nordmark map, are often only effective in extremely small parameter ranges. This can be caused by a resonance effect, resulting in nearby period-doubling and saddle-node bifurcations. To numerically continue the curves of these bifurcations, we found it necessary to develop a new tool that allows us to use root-finding methods such as Newton's without the method falling off the side of the square root. Finally, in the last part of my talk, I will explore a two-dimensional Chialvo neuron map and how different bifurcation scenarios occur on perturbing the map with electromagnetic flux.