The analysis using the game and statistical models has uncovered a hitherto unknown, powerful phenomenon at the heart of what happens before, during and after a congressional election. The phenomenon is a bifurcation pattern, comparable to that in Figure 3, that is qualitatively well modeled using the normal form equations for a dynamical system subject to Hopf bifurcation. The success of the hypotheses predicting particular changes between 1984 and 1986 in the stability of the dynamics for different types of PACs provides strong evidence that the dynamic patterns recovered by the 4DH model are substantively real. Because they strongly support the hypotheses, the recovered dynamics tend to verify a central result that the game model implies about the effect of variation in challenger quality. And through the argument used to motivate the hypotheses, the recovered dynamics connect to core facts about the American political process, in particular the midterm loss phenomenon and the partisan biases of different types of PACs' allocations of financial contributions. That the recovered dynamics are in these profound and surprising ways substantively meaningful is the best kind of evidence of the reality of the mathematical phenomenon--the bifurcation--that is the primary connection between the game and statistical models. The evidence that a bifurcation that includes features of both the Hopf and the saddle connection bifurcations is a nonlinear phenomenon inherent in the politics of congressional elections is therefore strong.
The apparent existence of the bifurcation has many substantive and methodological implications. Here I consider a few that seem to me to be among the most important.