Anomaly of fractal dimensions observed in stochastically switched systems

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 $f\left(\alpha \right)$ 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.

Figure 1

Correlation dimension $D_2$ to switching time length $T_l$ of inputs for matrix (I) with homogeneous eigenvalues. Crosses $(+)$ indicate dimensions computed from fractal set $C$ on Poincaré section $\Sigma$, while the dotted curve indicates the theoretical curve of Eq. (16). (a), (b), (c), (d), and (e) in the top panels indicate point sets of fractal $C$ in input time lengths of 0.5, 1.3, 1.8, 3.0, and 5.0.

Figure 2

Correlation dimension $D_2$ to switching time length $T_l$ of inputs for matrix (I) with heterogeneous eigenvalues. Curves $A$ and $B$ indicate theoretical curves discussed in Sec. 3. Curve $A$ satisfies Eq. (23), and curve $B$ satisfies Eq. (28).

Figure 3

Correlation dimension $D_2$ to switching time length $T_l$ of inputs for matrix (II).

Figure 4

Correlation dimension $D_2$ to switching time length $T_l$ of inputs for matrix (III).

Figure 5

Multiple scaling properties of fractal set $C$. The figure shows the relationship of logarithm of correlation integral $C\left(r\right)$ to logarithm of scale $r$. The base of logarithm is 10 in this figure. Calculated $D_2$ is 0.90 in the region from $-2\le r<0$, 0.79 in $-6\le r<-2$, and 0.73 in $-10\le r<-6$.