Nishikawa Jun, Gohara Kazutoshi
Department of Applied Physics, Hokkaido University, Sapporo, Hokkaido, Japan.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Mar;77(3 Pt 2):036210. doi: 10.1103/PhysRevE.77.036210. Epub 2008 Mar 18.
We studied an anomaly in fractal dimensions measured from the attractors of dynamical systems driven by stochastically switched inputs. We calculated the dimensions for different switching time lengths in two-dimensional linear dynamical systems, and found that changes in the dimensions due to the switching time length had a singular point when the system matrix had two different real eigenvalues. Using partial dimensions along each eigenvector, we explicitly derived a generalized dimension D(q) and a multifractal spectrum f(alpha) to explain this anomalous property. The results from numerical calculations agreed well with those from analytical equations. We found that this anomaly is caused by linear independence, inhomogeneity of eigenvalues, and overlapping conditions. The mechanism for the anomaly could be identified for various inhomogeneous systems including nonlinear ones, and this reminded us of anomalies in some physical values observed in critical phenomena.
我们研究了由随机切换输入驱动的动力系统吸引子所测得的分形维数中的一种异常现象。我们计算了二维线性动力系统中不同切换时间长度下的维数，发现当系统矩阵有两个不同的实特征值时，由于切换时间长度导致的维数变化存在一个奇点。利用沿每个特征向量的部分维数，我们明确推导了广义维数D(q)和多重分形谱f(α)来解释这种异常特性。数值计算结果与解析方程的结果吻合得很好。我们发现这种异常是由线性独立性、特征值的不均匀性和重叠条件引起的。这种异常现象的机制可以在包括非线性系统在内的各种不均匀系统中得到识别，这使我们想起了在临界现象中观察到的一些物理值的异常。
