When a heat transfer coefficient is varied in a lumped-parameter model of a nuclear reactor, the model can undergo period-doubling pitchfork bifurcations leading to aperiodic behavior. Until aperiodicity commences, the model behaves in the universal manner predicted by Feigenbaum, and the Poincaré map for the excess neutron population behaves as a typical one-dimensional map with a quadratic maximum. In the aperiodic region, though, this Poincaré map displays a hysteresis-like folding. At all times, the model's dynamic evolution remains bounded.