Calculating eigenvalues and eigenfunctions
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

In the previous example, we have found stationary solutions, but we could not obtain the stability of these solutions. To explicitly calculate the eigenvalues, we can use the method :py:meth:`~pyoomph.generic.problem.Problem.solve_eigenproblem` of the problem class, which requires the number of eigenvalues we want to calculate. Here, there is just one degree of freedom, namely :math:`x`, so we should only aim for a single eigenvalue. The method will return a list of eigenvalues (complex numbers) and a list of the corresponding eigenvectors. Of course, using :py:meth:`~pyoomph.generic.problem.Problem.solve_eigenproblem` is only meaningful if the problem is currently on a stationary solution.

The rest is more or less the same as the previous code, i.e. first importing the problem and equation from the previous example :download:`bifurcation_transient_transcritical.py` and then using :py:meth:`~pyoomph.generic.problem.Problem.solve_eigenproblem` after finding the stationary solutions via the :py:meth:`~pyoomph.generic.problem.Problem.solve` call.

.. literalinclude:: bifurcation_eigenvalues_transcritical.py
   :language: python
   :start-at: from bifurcation_transient_transcritical import * #Import the equation and problem from the previous script
   :end-at: print("Thus, this solution is "+("unstable" if eigen_vals[0].real>0 else "stable"))

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