Nonlocal singularities on families of periodic solutions to ordinary differential equations

We consider degenerate solutions on families of periodic solutions to ordinary differential equations. Degeneracy is understood as an arbitrary property of a solution that isolates this solution from generic cases. This can be either a bifurcation on a family or some topological peculiarity of the family, which causes a failure of a numerical algorithm applicable to generic cases. We suggest a means to compute these singular solutions with application of variational equations of higher order and with the same accuracy as ordinary solutions. The method is based on a symbolic recursive differentiation of an ODE with respect to initial values and parameters.