Spectral signature of the pitchfork bifurcation: Liouville equation approach

The time evolution of probability densities of one-dimensional nonlinear vector fields is studied using a Liouville equation approach. It is shown that the Liouville operator admits a discrete spectrum of eigenvalues of decaying type if the vector field is far from bifurcation. The associated right and left eigenvectors are explicitly constructed for simple models and shown to be distributions rather than regular functions. On the other hand, the spectrum of the Liouville operator may become continuous at the bifurcation point, a phenomenon illustrated explicitly in the paper in the case of the pitchfork bifurcation. The relationship between the spectral decompositions of the Liouville and of the Fokker-Planck equations is discussed. In particular, the spectral decompositions constructed here for the Liouville equation are obtained as the noiseless limit of the well known spectral decompositions of the Fokker-Planck equation of the associated stochastic process.