Application to Oscillating Systems

Oscillating reactions are of central importance in chemical nonlinear dynamics and for the explanation of many biological phenomena (e.g. cell cycles, circadian rhythm). The sensitivity functions of oscillating reactions are continuously increasing if the corresponding periodic time sensitivity is not zero, i.e. if the time period of the oscillation depends on the value of any of the parameters. A truly periodic sensitivity function can be calculated from the raw sensitivity function and a linearly increasing term (Edelson and Thomas 1981 Barter 1983 Zak et al. 2005 Lu and Yue 2010)  

Here is the cleaned out (periodic) sensitivity matrix, S is the original (inflating) sensitivity matrix, t is time, t is the time period of the oscillation, f is the RHS of the ODE and is the vector of period sensitivities. The elements of vector are the sensitivities of the periodic time t with respect to the parameters  

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