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Political candidates often shift their positions opportunistically in hopesof capturing more votes. When there are only two candidates, the best strategyfor each of them is often to move towards the other. This eventually results intwo centrists with coalescing views. However, the strategy of moving towardsthe other candidate ceases to be optimal when enough voters abstain instead ofvoting for a centrist who does not represent their views. These observations,formalized in various ways, have been made many times. Our own formalization isbased on differential equations. The surprise and main result derived fromthese equations is that the final candidate positions can jump discontinuouslyas the voters' loyalty towards their candidate wanes. The underlyingmathematical mechanism is a blue sky bifurcation.

A blue sky bifurcation in the dynamics of political candidates