Fast computation of limit cycles in an industrial application

A model problem is described that requires the study of a system of the form ̇v(1) = εFp (v(t). t) which depends on a set of parameters P, and where ε ≪ 1. The problem comes from an industrial application where it is a kernel of an optimization procedure. The optimization depends on computing the limit cycle, and the problem needs to be solved repeatedly. Short computation time is therefore essential. The naive approach is to integrate the equation forward in time, starting from an arbitrary initial condition, until the transients disappear and the limit cycle is approximated within a given tolerance. This approach is too slow and thus impractical in the context of the optimization procedure. The problem involves two types of asymptotic considerations: long-time asymptotics and small-parameter asymptotics. Here a simple approach is demonstrated, based on implementing the averaging method. This reduces the solution time to the point that the optimization procedure becomes feasible.