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Throughout the DOE complex and the INL, there is a wide range of highly nonlinear dynamical systems which need to be mathematically modeled as well as qualitatively understood. Such an understanding is the first step in controlling such processes. These complex systems include industrial processes (Welding), fuel processing (coal desulfurization), bio-remediation (TCE – degradation), transport through the vadose zone (water and contaminate flow), etc. We at the INL are investigating and developing new tools in the fields of chaotic mathematics and complexity theory in order to accomplish these goals. Many of these tools have their footholds in the traditional tools of the field such as fractal dimension, surrogate data, symbol string analysis, mutual information theory, false nearest neighbors, Lyapunov Exponents, and attractor reconstruction. We have current activities in the fields of lacunarity measurements, integrate and fire dynamics, the search for dynamics within generic signals and Time-Based Clustering methods.

We are in the process of developing a new statistical measurement of texture, known as lacunarity. This statistical measure is akin to fractal dimension. The term is derived from the Latin "lacuna" which means gap. Mandelbrot theorized that a texture could be described using three measures: fractal dimension, lacunarity, and connectivity. While fractal dimension is well understood, the same can not be said of the latter two measures. The importance of being able to describe generic textures quantitatively must not be under estimated. The applications are wide ranging from biofilm growth modeling to target detection and tracking problems. In order to develop this new measure of lacunarity, we are investigating the underlining principles behind lacunarity. We have developed a new lacunarity measurement for ramified data sets and implement its estimation algorithm. We are currently developing a measurement for dense sets. Once a means for quantifying lacunarity is achieve, it can be used to model biofilm growth over time. Furthermore, the time series measurements of lacunarity can be used via time series analysis of biological systems to help infer if they are stochastically or deterministically driven.

Whitney/Taken/Sauer embedding theorems are key to the concept of phase space reconstruction. Based on these methods new predictive models can be developed. Moreover, new controllers can also be developed to stabilize unmodeled chaotic processes using the reconstructed phase space. Here at the INL we are studying the effects of integrate and fire dynamic systems in light of Sauer’s latest integrate and fire embedding theorem. We are exploring the usefulness of current chaos tools to two dynamic systems, which exhibit both continuous state dynamics as well as integrate and fire dynamics. These systems are the welding process, and water flow through fractured basalt. Within the welding process one can measure directly the voltage and current states of the process as well as the integrated dynamics via droplet detachments. Likewise, in the water flow system one can measure the constant in flux of water to the system as well as the integrated droplets of the out flow.

It is important for the engineer to take time steps short enough so that information is not lost. This is the under-sampling problem or Nyquist rate; however, it is not the whole story. Symbol train analysis researchers, must be careful to avoid making the sampling interval too small. The symbol to symbol dynamics can be overwhelmed by minute changes. Stated in a more figurative manner: One looses the concept of a forest by looking at only one tree at a time.

One method for choosing an optimal time step for the symbol train creation process uses Mutual Information Theory. One drawback to both Mutual Information Theory and a standard sampling algorithm is the inability of either to adapt to the changing dynamics within a sampled signal. In short, Mutual Information Theory gives only an average best sampling time.

Here at the INL we are developing a new sampling algorithm that adapts to the changing dynamics within a signal of interest. This new algorithm also contains the special case of the traditional Mutual Information Theory solution but makes it possible for much more advanced localized dynamic effects to be interwoven within the symbol trains of interest. Furthermore, more advanced phase space reconstructions can be attempted using this method. This new method is called Time-Based Clustering (TBC). In addition, TBC gives rise to the concept of Mutual Information Space. In the original form of TBC only single embeddments have been taken into account; however, the Time-Based Clustering concept is expandable to include numerous embeddments of interest. This is the focus area of current and future research programs.

Additional Reading:

• Lacunarity definition for ramified data sets based on optimal cover — 618kB PDF
• Suboptimal Minimum Cluster Volume Cover-Based Method for Measuring Fractal Dimension — 913kB PDF
• Image characterization fact sheet — 2.1MB PDF
• Biofilm characterization fact sheet — 2.1MB PDF
• Chaos of welding, technical paper, "Is there evidence of determinism in droplet detachment within the gas metal arc welding process?," International Conference on Trends in Welding, April 2002. — 484kB PDF