Comparison of tests for embeddings

It is possible to compare results for the classical tests for embeddings of chaotic data with the results of a recently proposed test. The classical tests, which depend on real numbers (fractal dimensions, Lyapunov exponents) averaged over an attractor, are compared with a topological test that depends on integers. The comparison can only be done for mappings into three dimensions. We find that the classical tests fail to predict when a mapping is an embedding and when it is not. We point out the reasons for this failure, which are not restricted to three dimensions.

Figure 1

Fraction of false near neighbors as a function of time delay $\tau$ for embedding dimensions $N=2,3,4$.

Figure 2

(Color online) Fraction of false near neighbors under the projections to the $(X,Y,Z)$ (dark) and $(X,Y,U)$ (light) subspaces.

Figure 3

Logarithm of the correlation integral as a function of the logarithm of the distance $ϵ$ for the three-dimensional projection $(X,Y,Z,U)\to (X,Y,U)$ of the dynamo model for $\beta =4.0$.

Figure 4

Correlation dimension estimate, as a function of time delay, for a three-dimensional differential-delay mapping of $x\left(t\right)$ from the fluid model.

Figure 5

(Color online) Estimates of the fractal dimension $D_2$ (top curve) and the LLE (bottom curve) for the $(X,Y,Z)$ projection of the dynamo model.

Figure 6

(Color online) Estimates of the fractal dimension $D_2$ (top curve) and the LLE (bottom curve) for the $(X,Y,U)$ projection of the dynamo model.

Figure 7

Estimate of the logarithm of the divergence as a function of the number of forward time steps for the projection $(X,Y,Z,U)\to (X,Y,U)$ of the dynamo model at $\beta =4.0$. The largest Lyapunov exponent is to be estimated from the slope of the “locally linear” region. This curve was computed simultaneously with the curve shown in Fig. 3, which has a single linear region.

Figure 8

Estimate of the logarithm of the divergence as a function of the number of forward time steps for the fluid data. The largest Lyapunov exponent is to be estimated from the slope of the “locally linear” region.

Figure 9

Estimate of the LLE for differential-delay mapping of the scalar time series $x\left(t\right)$ from the fluid model as a function of delay $\tau$.

Figure 10

Projection of the lowest period orbit onto the $x\text{−}\dot{x}$ plane is independent of the time delay $\tau$ in the mixed differential-delay mapping. There are four crossing points in this projection.

Figure 11

The difference between the coordinates $x_3$ of the two segments that cross, at each of the four crossing points, is shown, plotted as a function of the time delay $\tau$. When this difference is zero the orbit has a self-intersection and the mapping fails to be an embedding in a region around the intersection point.

Figure 12

Minimum distance between two orbits as a function of $\beta$ in the chaotic attractor in ${\mathbb{R}}^4$ and under two projections into three-dimensional spaces. Parameter values: $(R,\sigma ,\nu ,\Lambda )=(74.667,10,8∕3,3.2)$.

Figure 13

(Color online) Estimates of LLE (bottom curve) and $D_2$ over 1001 projections into $(X,Y,Z)$ with $\beta$ fixed at 4.0.

Figure 14

Comparison of the suitably normalized statistical distribution of the LLE and $D_2$ estimates with the Gaussian normal distribution.

Figure 15

Cardboard model of the self-intersection of the projection of the chaotic attractor into the $(X,Y,Z)$ subspace for $0.6<\beta <5.4$.