AN ADAPTIVE NEWTON-PICARD ALGORITHM WITH SUBSPACE ITERATION FOR COMPUTING PERIODIC-SOLUTIONS

This paper is concerned with the efficient computation of periodic orb its in large-scale dynamical systems that arise after spatial discreti zation of partial differential equations (PDEs). A hybrid Newton-Picar d scheme based on the shooting method is derived, which in its simples t form is the recursive projection method (RPM) of Shroff and Keller [ SIAM J. Numer. Anal., 30 (1993), pp. 1099-1120] and is used to compute and determine the stability of both stable and unstable periodic orbi ts. The number of time integrations needed to obtain a solution is sho wn to be determined only by the system's dynamics. This contrasts with traditional approaches based on Newton's method, for which the number of time integrations grows with the order of the spatial discretizati on. Two test examples are given to show the performance of the methods and to illustrate various theoretical points.