The Bifurcation Interpreter: A Step Towards the Automatic Analysis of Dynamical Systems

The Bifurcation Interpreter is a computer program that autonomously explores the steady-state orbits of one-parameter families of periodically- driven oscillators. To report its findings, the Interpreter generates schematic diagrams and English text descriptions similar to those appearing in the science and engineering research literature. Given a system of equations as input, the Interpreter uses symbolic algebra to automatically generate numerical procedures that simulate the system. The Interpreter incorporates knowledge about dynamical systems theory, which it uses to guide the simulations, to interpret the results, and to minimize the effects of numerical error.

Stable Mixing of Complete and Incomplete Information

An increasing number of parameter estimation tasks involve the use of at least two information sources, one complete but limited, the other abundant but incomplete. Standard algorithms such as EM (or em) used in this context are unfortunately not stable in the sense that they can lead to a dramatic loss of accuracy with the inclusion of incomplete observations. We provide a more controlled solution to this problem through differential equations that govern the evolution of locally optimal solutions (fixed points) as a function of the source weighting. This approach permits us to explicitly identify any critical (bifurcation) points leading to choices unsupported by the available complete data. The approach readily applies to any graphical model in O(n^3) time where n is the number of parameters. We use the naive Bayes model to illustrate these ideas and demonstrate the effectiveness of our approach in the context of text classification problems.

Transport of Sub-micron Aerosols in Bifurcations

The convective-diffusive transport of sub-micron aerosols in an oscillatory laminar flow within a 2-D single bifurcation is studied, using order-of-magnitude analysis and numerical simulation using a commercial software (FEMLAB®). Based on the similarity between momentum and mass transfer equations, various transient mass transport regimes are classified and scaled according to Strouhal and beta numbers. Results show that the mass transfer rate is highest at the carinal ridge and there is a phase-shift in diffusive transport time if the beta number is greater than one. It is also shown that diffusive mass transfer becomes independent of the oscillating outer flow if the Strouhal number is greater than one.