Method for measuring unstable dimension variability from time series

Many of the results in the theory of dynamical systems rely on the assumption of hyperbolicity. One of the possible violations of this condition is the presence of unstable dimension variability (UDV), i.e., the existence in a chaotic attractor of sets of unstable periodic orbits, each with a different number of expanding directions. It has been shown that the presence of UDV poses severe limitations to the length of time for which a numerically generated orbit can be assumed to lie close to a true trajectory of such systems (the shadowing time). In this work we propose a method to detect the presence of UDV in real systems from time series measurements. Variations in the number of expanding directions are detected by determining the local topological dimension of the unstable space for points along a trajectory on the attractor. We show for a physical system of coupled electronic oscillators that with this method it is possible to decompose attractors into subsets with different unstable dimension and from this gain insight into the times a typical trajectory spends in each region.

Figure 1

The distribution of points around a chosen trajectory point. Expanding directions, $\leftarrow \to$, and the neutral direction of flow (shaded arrow) show a spread of points, whereas contracting directions, $\to \leftarrow$, appear almost flat. Thus the dimension of the containing region is strongly related to the number of unstable dimensions.

Figure 2

The kicked double rotor.

Figure 3

Distributions of local mean distances, $\overline{\mu }$, between points in the embedded attractor for the kicked double-rotor system. The distribution shows two peaks around the values corresponding to two-dimensional and three-dimensional distributions of points, in agreement with other results on this system confirming the existence of UDV.

Figure 4

Circuit based on the van der Pol oscillator. Units are coupled via two unidirectional links from point $V$ of one oscillator to $g$ of the other and vice versa.

Figure 5

Bifurcation diagram for the system as a function of the dimensionless parameter $\beta$. For $\beta \lesssim 3.72$ there is an almost-symmetric attractor which is not shown. The value $x$ is the first nondimensional embedded coordinate, which is exactly equal in magnitude to the measured voltage response of the system.

Figure 6

Time series in a region of parameter space showing clear intermittency between the three sets.

Figure 7

Evolution of the three largest Lyapunov exponents of the system of coupled oscillators for varying $\beta$.

Figure 8

Histograms of numerically generated short-time Lyapunov exponents. The second and third exponents have been summed to give the peaks shown. The spread of values seen in (b) across positive and negative values is an indication of UDV.

Figure 9

Two-dimensional projection of periodically sampled orbits of the numerically separated invariant regions. The small sets were found to have one positive Lyapunov exponent and the large set to have two.

Figure 10

Distributions of local mean distances, $\overline{\mu }$, between points in the embedded attractor. Before the crisis the values of $\overline{\mu }$ are located around one peak close to the value corresponding to a two-dimensional distribution of points (a). After the crisis, and for nearby values of $\beta$, the distribution shows two peaks around the values corresponding to two-dimensional and three-dimensional distributions of points, (b) and (c). For larger values of $\beta$ the distribution of points is mainly three-dimensional, (d). The values of $\beta$ for (a), (b), (c), and (d) are 3.69, 3.73, 3.78, and 3.87, respectively.

Figure 11

Singular values, ${\sigma }_i$, at points on a trajectory in the embedded attractor. Although there exist regions where the dimensionality of the attractor is different, the actual dimensionality is difficult to ascertain.

Figure 12

Histograms of experimentally obtained short-time Lyapunov exponents. The peak shown in (a) shows the second STLE of the precrisis attractor, whereas the second and third exponents of the hyperchaotic attractor have been summed to give the distribution shown in (b).

Figure 13

Projection of the invariant regions within the attractor, reconstructed from experimental data, and separated using information from the application of the method of Sec. 2.