Insects often undergo regular outbreaks in population density but identifying the causal mechanism for such outbreaks in any particular species has proven difficult. Here, we show that outbreak cycles in the tea tortrix Adoxophyes honmai can be explained by temperature-driven changes in system stability. Wavelet analysis of a 51-year time series spanning more than 200 outbreaks reveals a threshold in outbreak amplitude each spring when temperature exceeds 15 degrees C and a secession of outbreaks each fall as temperature decreases. This is in close agreement with our independently parameterized mathematical model that predicts the system crosses a Hopf bifurcation from stability to sustained cycles as temperature increases. These results suggest that temperature can alter system stability and provide an explanation for generation cycles in multivoltine insects.