Estimation of the Maximal Lyapunov Exponent in Thermoacoustic Engines

Nonlinear dynamics and chaos theory have given insights into the behavior of a wide variety of systems. Nonlinearities in acoustic systems can result in intriguing behavior, such as period doubling and structured chaos. Historically, mathematical modeling for thermoacoustic engines has been in the analytic realm. In this work we visit an alternative visual realm to study their behavior. The possible presence of chaos was tested by looking for a key element of chaos: a positive Lyapunov exponent. Twenty-one data sets of discrete pressure data gathered from thermoacoustic prime movers were analyzed with two programs. One estimated the maximal Lyapunov exponent, and another displayed phase-space plots. For each data set, a positive Lyapunov exponent was found and corresponding phase-space plots qualitatively confirmed orbital divergence. These results strongly suggest that chaotic behavior is obtainable in thermoacoustic systems. A careful study of the experimental time series data suggests an intuitive quantitative model consisting of discrete sigmoid modulated growth. We chose two simple component functions, sine modulations and Verhulst logistic growth. Using this method, experimental data can be faithfully projected to an event horizon at about 15 iterations.