Gaspard P, Tasaki S
Center for Nonlinear Phenomena and Complex Systems, Faculté des Sciences, Université Libre de Bruxelles, Campus Plaine, Code Postal 231, B-1050 Brussels, Belgium.
Phys Rev E Stat Nonlin Soft Matter Phys. 2001 Nov;64(5 Pt 2):056232. doi: 10.1103/PhysRevE.64.056232. Epub 2001 Oct 30.
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.
采用刘维尔方程方法研究了经历霍普夫分岔的二维向量场。刘维尔方程支配着从随机初始条件出发、但在确定性动力学下演化的轨迹统计系综的时间演化。这种统计系综概率密度的时间演化可以根据刘维尔算子或相关弗罗贝尼乌斯 - 佩龙算子的共振谱（即弛豫率）进行分解。在霍普夫分岔之前、之时和之后，明确构建了刘维尔算子的谱分解。由于在霍普夫分岔附近出现时间振荡，共振谱结果是复数的，并且由弛豫率和振荡频率共同定义。共振谱在远离分岔处是离散的，在分岔处变为连续的。这种连续谱是由霍普夫分岔处发生的振荡的临界减慢引起的，并且它导致长时间(t \to \infty)时概率密度和统计平均值的幂律弛豫，形式为(1 / \sqrt{t})。此外，共振谱中的简并性在谱分解中产生了若尔当块结构。
