Anomaly of fractal dimensions observed in stochastically switched systems

Abstract

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.