Liouvillian dynamics of the Hopf bifurcation - art. no. 056232

Two-dimensional vector fields undergoing a Hopf bifurcation are studied in a Liouville-equation approach, The Liouville equation rules the time evolut ion of statistical ensembles of trajectories issued from random initial con ditions. but evolving under the deterministic dynamics. The time evolution of the probability densities of such statistical ensembles can be decompose d in terms of the spectrum of the resonances (i.e., the relaxation rates) o f the Liouvillian operator or the related Frobenius-Perron operator. The sp ectral decomposition of the Liouvillian operator is explicitly constructed before, at, and after the Hopf bifurcation. Because of the emergence of tim e oscillations near the Hopf bifurcation, the resonance spectrum turns out to be complex and defined by both relaxation rates and oscillation frequenc ies. The resonance spectrum is discrete far from the bifurcation and become s 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/roott of the probability densiti es and statistical averages at long times t-->infinity. Moreover. degenerac y in the resonance spectrum is shown to yield a Jordan-block structure in t he spectral decomposition.