Entropy rate estimates from mutual information.

We show how to estimate the Kolmogorov-Sinai entropy rate for chaotic systems using the mutual information function, easily obtainable from experimental time series. We state the conditions under which the relationship is exact, and explore the usefulness of the approach for both maps and flows. We also explore refinements of the method, and study its convergence properties as a function of time series length.

Geometry of repeated measurements in chaotic systems.

We use joint probability matrices for measurements at different times to describe chaotic systems. By coarse graining the range of the measured variable into uniformly sized bins we can generate matrices that contain both topological and metric information about the systems being studied. Armed with this tool we examine two extreme families of chaotic systems. In the case of one-dimensional piecewise linear maps, we can construct transfer matrices that depend on the map and partition used, and which allow us to generate the respective joint probability matrices for all times as well as the exact time evolution of the mutual information function. We find that the mutual information decays linearly or exponentially depending on whether the second-largest eigenvalue of the transfer matrix is zero or not. In the case of three-dimensional, continuous-time chaotic systems we generate the joint probability matrices directly from numerical data. We show that these matrices directly provide attractor reconstructions with information about the attractor's probability measure.

Advances in Atom Probe Specimen Fabrication from Planar Multilayer Thin Film Structures.

A sample preparation method has been developed whereby sharp needle-shaped specimens for atom probe analysis are fabricated from multilayer thin films deposited onto silicon substrates. The specimens are fabricated in an orientation such that atom probe composition profiles across the layer interfaces can be determined with atomic-layer spatial resolution, i.e., the layer normals are parallel to the needle axis. The method uses standard silicon etching techniques and focused ion-beam milling. The feasibility and utility of this technique are shown through its application to a NiFe/CoFe/Cu/CoFe-based thin film structure.