Ill-funded challengers are uncompetitive in U.S. congressional elections (Jacobson 1980; Jacobson 1985; Abramowitz 1991; Krasno 1994), but it is not clear how incumbents produce such challengers. Better challengers are more likely to enter a race when they are more likely to win (Bond, Covington and Fleisher 1985; Banks and Kiewiet 1989; Jacobson 1989; Jacobson 1990a). But why do potential challengers take their chances of winning as given? Large incumbent ``war chests'' of campaign funds in particular can deter quality challengers (Epstein and Zemsky 1995; Box-Steffensmeier 1996). But it is not clear why incumbents are able to accumulate such war chests: why would financial contributors want to make the election uncompetitive? Several interesting theoretical arguments suggest that district service such as casework and ``pork barrel'' spending ought to benefit incumbents (Austen-Smith 1987; Baron 1989a, 1989b, 1994; Hinich and Munger 1989; Snyder 1990; Morton and Cameron 1992), but the effects have been remarkably difficult to identify in empirical work. There is good evidence that local federal expenditure varies in response both to incumbents' involvement in ``policy subsystems'' (Stein and Bickers 1995) and to the proportion of Democratic voters in each district (Levitt and Snyder 1995), but evidence that district service affects votes has been hard to come by (Feldman and Jondrow 1984; Cain, Ferejohn and Fiorina 1987; Fiorina 1989; Stein and Bickers 1994; Levitt and Snyder 1997), and evidence regarding effects of district service on campaign contributions has been mixed (Kau and Rubin 1982; McAdams and Johannes 1987; Grier and Munger 1986; Snyder 1990; Endersby and Munger 1992; McCarty and Rothenberg 1996).
I argue that the lack of simple and reliable empirical relationships among campaign contributions, district service, challenger quality and election outcomes reflects a nonlinearity inherent in the strategic interactions of political parties, candidates and contributors. I use a two-stage game model in which the second stage is a realization of a system of ordinary differential equations. The type of district service and the quality of challenger that, respectively, the incumbent and the opposing party are most likely to choose in the first stage of the game induce a particular kind of nonlinearity in the second stage dynamics. For service type and challenger quality values near the values that have the highest probability of occurring in the perfect Nash equilibrium solution of the game, the dynamics exhibit Hopf and saddle connection bifurcations (Guckenheimer and Holmes 1983, 150, 290). I show how the nonlinearity of the dynamics may be the reason for the complicated and contradictory empirical relationships that have been reported in the literature. I develop a nonlinear statistical model based on the normal form equations that local bifurcation theory specifies for Hopf bifurcation. I estimate the model using cross-sectional data from the 1984 and 1986 election periods. I use statistical tests to examine how well the dynamics the model recovers match predictions from the game.