Quenching of chemical oscillations: General theory

Abstract

We develop the theoretical basis for quenching analysis of small-amplitude chemical oscillations near a Hopf bifurcation in a concentration space of arbitrary dimension. (Quenching of a limit-cycle oscillation is carried out experimentally by instantaneously changing the concentrations of chemical species in a definite phase of the oscillation.) It is shown that quenching by addition of a species determines the reciprocal of the corresponding component of an eigenvector of the transposed Jacobi matrix. The set of all independent quenchings determines the tangent space of the stable manifold at the associated saddle focus even if only one species is monitored. Reconstruction of the oscillations from quenching data in an n-dimensional concentration space requires additional information if n>3. We show that if n−2 chemical species are monitored the oscillatory parts of the remaining two (unknown) concentrations can be exactly calculated from the result of n quenchings. In this case one additional quenching by dilution suffices to determine also the two unknown average concentrations. Reconstruction from m quenchings with m−2 species monitored may still be possible with m less than the dimension; it will be exact if the species tested by quenching are the only ones oscillating, and an accurate approximation if the remaining n−m species oscillate with sufficiently small amplitudes. Experimental quenching data can be used for quantitative tests of models of an oscillatory system, and we show how quenching data can be calculated from models.