Grid-based partitioning for comparing attractors

Refinements to the boundary transformation vector representation of attractor shape deformation to enhance system parameter identification

A new method of quantifying parameter changes in chaotic systems using estimates of how the boundaries of Poincare sections deform was recently developed. Refinements that improve the number and quality of the boundary transformation vectors produced by this method are proposed and analyzed here. Collectively, these refinements offer the ability to better match closely spaced linear segments of Poincare sections typical of fractal geometry, better handle boundary gaps, and more uniformly sample the boundary, resulting in additional data. The refinements are tested using Poincare sections constructed in three ways for five different dynamical systems and are shown to enhance results in all cases.

Boundary transformation representation of attractor shape deformation

Detecting parameter changes in chaotic systems depends on characterizing the deformation of the strange attractor. Here, we present a new method for comparing the geometry of two attractors by examining their boundaries in 2D via shape context analysis. Poincaré sections for each attractor are sampled along their outer limits, and a boundary transformation is computed that warps one set of points into the other. This boundary transformation is a rich descriptor of the attractor deformation and approximately proportional to a system parameter change in specific regions. Both simulated and experimental data with various levels of noise are used to demonstrate the effectiveness of this method.

Testing dynamical system variables for reconstruction

Analyzing data from dynamical systems often begins with creating a reconstruction of the trajectory based on one or more variables, but not all variables are suitable for reconstructing the trajectory. The concept of nonlinear observability has been investigated as a way to determine if a dynamical system can be reconstructed from one signal or a combination of signals [L. A. Aguirre, IEEE Trans. Educ. 38, 33 (1995); C. Letellier, L. A. Aguirre, and J. Maquet, Phys. Rev. E 71, 066213 (2005); L. A. Aguirre, S. B. Bastos, M. A. Alves, and C. Letellier, Chaos 18, 013123 (2008); L. A. Aguirre and C. Letellier, Phys. Rev. E 83, 066209 (2011); and E. Bianco-Martinez, M. S. Baptista, and C. Letellier, Phys. Rev. E 91, 062912 (2015)]; however, nonlinear observability can be difficult to calculate for a high dimensional system. In this work, I compare the results from nonlinear observability to a continuity statistic that indicates the likelihood that there is a continuous function between two sets of multidimensional points-in this case, two different reconstructions of the same attractor from different signals are simultaneously measured. Without a metric against which to test the ability to reconstruct a system, the predictions of nonlinear observability and continuity are ambiguous. As an additional test on how well different signals can predict the ability to reconstruct a dynamical system, I use the fitting error from training a reservoir computer.

Dimension from covariance matrices

We describe a method to estimate embedding dimension from a time series. This method includes an estimate of the probability that the dimension estimate is valid. Such validity estimates are not common in algorithms for calculating the properties of dynamical systems. The algorithm described here compares the eigenvalues of covariance matrices created from an embedded signal to the eigenvalues for a covariance matrix of a Gaussian random process with the same dimension and number of points. A statistical test gives the probability that the eigenvalues for the embedded signal did not come from the Gaussian random process.

Chaotic sequences for noisy environments

There have been many attempts to apply chaotic signals to communications or radar, but one obstacle has been that there is no effective way to recover chaotic signals from noise larger than the signal. In this work, we create "pseudo-chaotic" signals by concatenating dictionary sequences generated from a chaotic attractor. Because the number of dictionary sequences is finite, these pseudo-chaotic signals are not actually chaotic, but they can still contain some of the desirable properties of chaos. Using dictionary sequences allows the pseudo-chaotic signal to be recovered from noise using a correlation detector and a Viterbi decoder, so the signal can be recovered from noise or interference that is larger than the signal itself.