| Summary: | Early warning signals (EWS) have been tested on many real-life systems to early detect bifurcations. In particular, the study of EWS for the homoclinic bifurcation is currently scarce and underdeveloped. Out of the several EWS, the effectiveness of the lag-1 autocorrelation and variance is tested in this thesis on theoretical systems exhibiting the saddle-node, hopf and homoclinic bifurcations. Our results show that their effectiveness varies across the board and neither of the two is a better EWS. In some cases, the behaviour of one EWS indicates an incoming bifurcation but the behaviour of the other indicates otherwise, and vice versa. In other cases, they falsely detect bifurcation when there is none. The limited success of the EWS on the homoclinic bifurcation motivated the development of two specific novel detection techniques, which are the minimum speed and curvature of the limit cycles near the saddle-point. Based on theoretical calculations, the minimum speed is expected to decrease towards zero as the system slows down near the saddle-point. On the other hand, the curvature is expected to increase towards infinity because a kink eventually forms at the saddle-point when homoclinic bifurcation is reached. Our results indeed show that as homoclinic bifurcation is approaching, the minimum speed and curvature tend towards zero and very large values.
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