Deflation of Periodic Orbits in Large-Scale Systems: Algorithm and Parallel Implementation

We consider a nonlinear autonomous large-scale dynamical system with a parameter. Such systems usually stem from the discretization of initial boundary value problems for nonlinear partial differential equations. We are interested in finding all periodic orbits of such a system for a particular value of the parameter. We employ the deflation method together with the Poincare section method to solve the problem. The deflation operator is constructed in the hyperplane used for the Poincare return map and is iterated by the Newton-Raphson method. Two approaches are used for parallelization: the solution of the dynamical system, which is symmetrically parallel, and the deflation process, which is parallel and relies on the master-slave approach. The master controls the process of deflation and exchanges information between slaves. We show that the process can be successfully applied to the discrete system obtained from the Galerkin projection of the Navier-Stokes equations.