Studying the performance of critical slowing down indicators in a biological system with a period-doubling route to chaos

Abstract

This paper aims to investigate critical slowing down indicators in different situations where the system’s parameters change. Variation of the bifurcation parameter is important since it allows finding bifurcation points. A system’s parameters can vary through different functions. In this paper, five cases of bifurcation parameter variation are considered in a biological model with a period-doubling route to chaos. The first case is a slow and small stepwise variation of the bifurcation parameter. The second case is a cyclic, state-dependent variation of the bifurcation parameter. In the third case, a small cyclic variation is combined with a sizeable stochastic resonance. The fourth case involves variations by a large noise, and finally, in the fifth case, significant stepwise changes in the parameter are studied. To identify the conditions under which critical slowing down occurs, an improved version of four well-known critical slowing down indicators (autocorrelation at lag-1, variance, kurtosis, and skewness) are used. The results show that when bifurcations are caused by a sudden change in a parameter or state, critical slowing down cannot be observed before the bifurcation points. However, in cases with slowly varying parameters, critical slowing down can be detected before the bifurcation points. Thus critical slowing down indicators can predict these bifurcation points. In other words, in three cases, the system approaches bifurcation points slowly. In other cases, the bifurcations occur suddenly because of a significant shift in the parameter or state. Thus critical slowing down indicators cannot predict those bifurcation points. However, critical slowing down indicators can predict the bifurcation points in other cases.