Liouvillian dynamics of the Hopf bifurcation.

Two-dimensional vector fields undergoing a Hopf bifurcation are studied in a Liouville-equation approach. The Liouville equation rules the time evolution of statistical ensembles of trajectories issued from random initial conditions, but evolving under the deterministic dynamics. The time evolution of the probability densities of such statistical ensembles can be decomposed in terms of the spectrum of the resonances (i.e., the relaxation rates) of the Liouvillian operator or the related Frobenius-Perron operator. The spectral decomposition of the Liouvillian operator is explicitly constructed before, at, and after the Hopf bifurcation. Because of the emergence of time oscillations near the Hopf bifurcation, the resonance spectrum turns out to be complex and defined by both relaxation rates and oscillation frequencies. The resonance spectrum is discrete far from the bifurcation and becomes continuous at the bifurcation. This continuous spectrum is caused by the critical slowing down of the oscillations occurring at the Hopf bifurcation and it leads to power-law relaxation as 1/square root of [t] of the probability densities and statistical averages at long times t-->infinity. Moreover, degeneracy in the resonance spectrum is shown to yield a Jordan-block structure in the spectral decomposition.