Convergence analysis of a numerical method to solve the adjoint linearized periodic orbit equations

The equation adjoint to the linearization of the periodic orbit equations in a dynamical system is fundamental in the study of sensitivity issues for periodic orbits, e.g. in the synchronization of networks of weakly coupled oscillators. It is also fundamental in the computation of normal form coefficients for bifurcations of limit cycles. Numerically, the adjoint equations can be solved in a variety of ways. In the case where the periodic orbit equations are solved as a boundary value problem by collocation at Gauss points, a recent method allows one to compute the solution to the adjoint equations as a byproduct of Newton's method applied to solve the boundary value problem. This method is practically cost-free since it requires only the solution to an already factorized linear system. Moreover, it provides the solution to the adjoint equations in exactly the form needed in the applications. So far, the method has not been analyzed carefully and no rigorous convergence results have been proved. We prove that the method is equivalent to a collocation method for the adjoint equations so that convergence of order h m + 1 holds at all points and of order h 2 m at the points of the coarse mesh; here h is the maximum length of the mesh intervals and m is the degree of the approximating piecewise polynomials. We support this by extensive numerical tests.