Serial interconnections of 1-contracting and 2-contracting systems

The flow of contracting systems contracts 1dimensional polygons (i.e. lines) at an exponential rate. One reason for the usefulness of contracting systems is that many interconnections of contracting sub-systems yield an overall contracting system. A recent generalization of contracting systems is called k-contracting systems, where k ∈{1,...,n}. The flow of such systems contracts k-dimensional polygons at an exponential rate, and in particular they reduce to contracting systems when k = 1. Here, we analyze serial interconnections of 1-contracting and 2-contracting systems. We provide conditions guaranteeing that such interconnections have a well-ordered asymptotic behaviour, and demonstrate the theoretical results using several examples.