Anomaly of fractal dimensions observed in stochastically switched systems

概要

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We studied an anomaly in fractal dimensions measured from the attractors of dynamical systems driven bystochastically switched inputs. We calculated the dimensions for different switching time lengths in twodimensionallinear dynamical systems, and found that changes in the dimensions due to the switching timelength had a singular point when the system matrix had two different real eigenvalues. Using partial dimensionsalong each eigenvector, we explicitly derived a generalized dimension Dq and a multifractal spectrumf to explain this anomalous property. The results from numerical calculations agreed well with those fromanalytical 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 inhomogeneoussystems including nonlinear ones, and this reminded us of anomalies in some physical values observedin critical phenomena.