Geometry of a Relativistic Quantum Chaos: New approach to dynamics of quantum systems in electromagnetic field and some applications

In the paper we present development of a new relativistic chaos-geometric and quantum-dynamic approach to solve problems of complete modelling relativistic chaotic dynamics of atoms in an electromagnetic field. As usually the approach consistently includes a number of new relativistic quantum models and a number of new or improved methods of analysis (correlation integral, fractal analysis, algorithms, average mutual information, false nearest neighbors, Lyapunov exponents, surrogate data, non-linear prediction, spectral methods, etc.) For the first time we present the corresponding atomic ionization quantitative data for some atoms in a microwave external field.

Fractal Dimension of Seismicity at Cangar, Arjuno-Welirang Complex, East Java

This study aimed to determine the value of fractal dimensions in the Cangar region, Arjuno-Welirang Volcano Hosted Geothermal System (VHG) based on seismic time series data. The determination of fractal dimension values is done using the delay embedding theorem method. The process starts from determining the value of the delay time by using the autocorrelation function and also determining the value of embedding dimensions using the False Nearest Neighbors (FNN) method. These two parameters are used to reconstruct the attractor diagram. Quantization of the attractor diagram provides a correlation dimension curve that is directly related to the fractal dimension value. The fractal dimension value in the study area tends to be high, ranging from 5.43 to 6.29. This high value is associated with the amount of energy needed by hot water to make their way out to the surface through rock with very little permeability. The emergence of fractures due to the continuous heat discharge process in Cangar hot water and the gas release process triggers the appearance of seismic signals.

Reliable Estimation of Minimum Embedding Dimension Through Statistical Analysis of Nearest Neighbors

False nearest neighbors (FNN) is one of the essential methods used in estimating the minimally sufficient embedding dimension in delay-coordinate embedding of deterministic time series. Its use for stochastic and noisy deterministic time series is problematic and erroneously indicates a finite embedding dimension. Various modifications to the original method have been proposed to mitigate this problem, but those are still not reliable for noisy time series. Here, nearest-neighbor statistics are studied for uncorrelated random time series and contrasted with the corresponding deterministic and stochastic statistics. New composite FNN metrics are constructed and their performance is evaluated for deterministic, correlates stochastic, and white random time series. In addition, noise-contaminated deterministic data analysis shows that these composite FNN metrics are robust to noise. All FNN results are also contrasted with surrogate data analysis to show their robustness. The new metrics clearly identify random time series as not having a finite embedding dimension and provide information about the deterministic part of correlated stochastic processes. These metrics can also be used to differentiate between chaotic and random time series.

Statistical Characterization of Nearest Neighbors to Reliably Estimate Minimum Embedding Dimension

False nearest neighbors (FNN) is one of the essential methods used in estimating the minimally sufficient embedding dimension in delay coordinate embedding of deterministic time series. Its use for stochastic and noisy deterministic time series is problematic and erroneously indicates a finite embedding dimension. Various modifications to the original method have been proposed to mitigate this problem, but those are still not reliable for noisy time series. Nearest neighbor statistics are studied for uncorrelated random time series and contrasted with the deterministic statistics. A new FNN metric is constructed and its performance is evaluated for deterministic, stochastic, and random time series. The results are also contrasted with surrogate data analysis and show that the new metric is robust to noise. It also clearly identifies random time series as not having a finite embedding dimension and provides information about the deterministic part of stochastic processes. The new metric can also be used for differentiating between chaotic and random time series.