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?

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.