Chaos in Two-Party Democracies

In two-party democracies (e.g. US and UK) the two parties often alternate at the government almost periodically. Here, we present a very simple continuous-time model with three state variables (social welfare and size of the lobbies associated with the two parties) that explains this tendency to cyclic behavior. However, the analysis of the model shows that when the lobbies are unbalanced, much more complex behaviors, including chaos, can emerge. The bifurcation structure of the system is interesting: it contains a countable number of codimension-2 points (associated with pseudo-equilibria of a particular Filippov system) which are the roots of Arnold tongues delimited by two border collision bifurcations, and in each one of these tongues the sequence of the parties at the government is a particular periodic sequence.