# Rank deficient Jacobian in discretized periodic solutions to autonomous ODE

I'm trying to numerically find periodic solutions to different systems of autonomous nonlinear ordinary differential equations. I decided to use a finite difference scheme and solve the resulting system of equations iteratively, which works reasonably well using a sufficiently good initial guess.

Now most of the systems considered depend on an additional parameter $\lambda$, with a solution in the limit $\lambda\to 0$ easily found, so I would like to use some sort of numerical continuation method to generate a family of solutions for different values of $\lambda$. All literature I found on the subject, e.g. http://dx.doi.org/10.1137/1.9780898719154, assumes the Jacobian to have full rank, which it does not in my case: There is a zero eigenvalue corresponding to a shift in the parametrization of the solution to the original system of ODEs.

The ODEs typically look like $$\ddot{x} = - \sqrt{\dot{x}^2+\dot{y}^2}\ f(x, y, \lambda)\ \dot{y},\\ \ddot{y} = \sqrt{\dot{x}^2+\dot{y}^2}\ f(x, y,\lambda)\ \dot{x},$$ with the constraints $x(0) = x(1)$, $y(0) = y(1)$.

Is there either

• a good way to somehow fix the parametrization of the solution to remove this degree of freedom or
• a numerical continuation method that works with a rank deficient Jacobian?