Extended Phase Diagram of the Lorenz Model
Extended Phase Diagram of the Lorenz Model
复制标题
洛伦兹模型的扩展相图
DOI：
10.1142/s021812740701883x
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发表时间：
2005
期刊：
影响因子：
--
通讯作者：
S. Grossmann
中科院分区：
文献类型：
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作者：
H. Dullin;Sven Schmidt;P. Richter;S. Grossmann
研究了三变量非线性Lorenz方程的各种吸引解作为归一化Rayleigh数r和Prandtl数σ的函数对参数的依赖性.以前的工作，无论是固定的σ和所有的r或沿着σ <$r和，推广到整个（r，σ）参数平面。一个洋葱状的周期图案被发现，这是由于对称和非对称周期轨道的交替稳定。通过考虑大r和σ的非平凡极限并因此在上述情况之间插值来解释这种周期性图案。数学分析使用之前工作中介绍的艾里函数，但我们不集中于洛伦兹映射，而是分析全相空间中的轨迹。艾里函数的周期性允许解析地计算（r，σ）平面中的周期洋葱结构。以前的观察序列的分叉得到证实，并报告了更多的细节，关于它们的对称性。
The parameter dependence of the various attractive solutions of the three variable nonlinear Lorenz equations is studied as a function of r, the normalized Rayleigh number, and of σ, the Prandtl number. Previous work, either for fixed σ and all r or along σ ∝ r and , is extended to the entire (r, σ) parameter plane. An onion-like periodic pattern is found which is due to the alternating stability of symmetric and nonsymmetric periodic orbits. This periodic pattern is explained by considering non-trivial limits of large r and σ and thus interpolating between the above mentioned cases. The mathematical analysis uses Airy functions as introduced in previous work, but instead of concentrating on the Lorenz map we analyze the trajectories in full phase space. The periodicity of the Airy function allows to calculate analytically the periodic onion structure in the (r, σ)-plane. Previous observations about sequences of bifurcations are confirmed, and more details regarding their symmetry are reported.