Computation of periodic solution bifurcations in ODEs using bordered systems

Abstract

We consider numerical methods for the computation and continuation of the three generic secondary periodic solution bifurcations in autonomous ordinary differentialequations (ODEs), namely the fold, the period-doubling (or flip) bifurcation, and the torus (or Neimark-Sacker) bifur- cation. In the fold and flip cases we append one scalar equation to the standard periodic boundary value problem (BVP) that defines the periodic solution; in the torus case four scalar equations are appended. Evaluation of these scalar equations and their derivatives requires the solution of lin- ear BVPs, whose sparsity structure (after discretization) is identical to that of the linearization of the periodic BVP. Therefore the calculations can be done using existing numerical linear algebra techniques, such as those implemented in the software auto and colsys.