Anomaly of fractal dimensions observed in stochastically switched systems
Anomaly of fractal dimensions observed in stochastically switched systems
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DOI：
10.1103/physreve.77.036210
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发表时间：
2008-03-01
影响因子：
2.4
通讯作者：
Gohara, Kazutoshi
中科院分区：
文献类型：
--
作者：
Nishikawa, Jun;Gohara, Kazutoshi
我们研究了由随机切换输入驱动的动力系统吸引子测量的分形维数异常。计算了二维线性动力系统中不同开关时间长度的维数，发现当系统矩阵具有两个不同的实特征值时，开关时间长度引起的维数变化存在一个奇点。利用沿每个特征向量的偏维数，我们明确地推导出广义维数D-q和多重分形谱f（alpha）来解释这种反常性质。数值计算结果与解析方程结果吻合较好。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。在包括非线性系统在内的各种非均匀系统中，异常机制都可以被识别出来，这让我们想起了在临界现象中观测到的一些物理值的异常。
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.