Computation of Limit Cycles and Their Isochrons: Fast Algorithms and Their Convergence

In this paper we develop efficient algorithms to compute limit cycles and their isochrones (i.e. the sets of points with the same asymptotic phase) for planar vector fields. We formulate a functional equation for the parameterization of the invariant cycle and its isochrones and we show that it can be solved by means of a Newton method. Using the right transformations, we can solve the equation of the Newton step efficiently. The algorithms are efficient in the sense that if we discretize in N points, a Newton step requires O(N) storage and O(N log(N)) operations (in Fourier discretization) or O(N) operations in other discretizations. We prove convergence of the algorithms and present a validation theorem in an a-posteriori format. That is, we show that if there is an approximate solution of the invariance equation that satisfies some some mild non-degeneracy conditions, then, there is a true solution nearby. Thus, our main theorem can be used to validate numerically computed solutions. The theorem also shows that the isochrones are analytic and depend analytically on the base point. Moreover, it establishes smooth dependence of the solutions on parameters and provides efficient algorithms to compute perturbative expansions with respect to external parameters. We include a discussion on the numerical implementation of the algorithms as well as numerical results for a representative example.