Sensitivity analysis of limit cycles with application to the Brusselator

Sensitivity analysis, by which it is possible to determine the dependence of the solution of a system of differential equations to variations in the parameters, is applied to systems which have a limit cycle solution in some region of parameter space. The resulting expressions for the sensitivity coefficients, which are the gradients of the limit cycle solution in parameter space, are analyzed by a Fourier series approach; the sensitivity coefficients are found to contain information on the sensitivity of the period and other features of the limit cycle. The intimate relationship between Lyapounov stability analysis and sensitivity analysis is discussed. The results of our general derivation are applied to two limit cycle oscillators: (1) an exactly soluble two-species oscillator and (2) the Brusselator.