Reick Christian H
Alfred-Wegener-Institute for Polar and Marine Research, Columbusstrasse, D-27568 Bremerhaven, Germany.
Phys Rev E Stat Nonlin Soft Matter Phys. 2002 Sep;66(3 Pt 2A):036103. doi: 10.1103/PhysRevE.66.036103. Epub 2002 Sep 6.
The present numerical study provides strong evidence that at standard parameters the response of the Lorenz system to small perturbations of the control parameter r is linear. This evidence is obtained not only directly by determining the response in the observable A(x)=z, but also indirectly by validating various implications of the assumption of a linear response, like a quadratic response at twice the perturbation frequency, a vanishing response in A(x)=x, the Kramers-Kronig relations, and relations between different response functions. Since the Lorenz system is nonhyperbolic, the present results indicate that in contrast to a recent speculation the large system limit (thermodynamic limit) need not be invoked to obtain a linear response for chaotic systems of this type.
目前的数值研究提供了有力证据，表明在标准参数下，洛伦兹系统对控制参数r的小扰动的响应是线性的。这一证据不仅通过直接确定可观测量A(x)=z中的响应而获得，还通过验证线性响应假设的各种推论而间接获得，比如在两倍扰动频率下的二次响应、A(x)=x中的零响应、克莱默斯-克勒尼希关系以及不同响应函数之间的关系。由于洛伦兹系统是非双曲的，目前的结果表明，与最近的一种推测相反，对于这类混沌系统，无需引入大系统极限（热力学极限）来获得线性响应。
