DETERMINING LYAPUNOV EXPONENTS FOR CHAOTIC WATERWHEEL DATA
The chaotic waterwheel is a simple mechanical system that can exhibit chaotic motion. In fact the equations of motion are a special case of the famous Lorenz equations, the study of which marked the beginning of the modern study of chaos in the 1960s. The experimental wheel can exhibit three broad classes of motion: uniform rotation in one direction, simple oscillatory motion, and motion in which the reversals of direction are chaotic instead of periodic. Chaotic systems are often characterized by recording their behavior while changing only one parameter. In our laboratory, this parameter is usually the strength of the velocity-dependent braking imposed on the wheel. The experimental data show significant differences between the wheel’s behavior at large values of the imposed braking compared to numerical simulation of the idealized equations of motion. The idealized equations predict simple periodic motion at high brake strength, while experimental data appear to show chaotic behavior. In this talk, we will present the results of calculations of a definitive measure of the degree of chaos (called the maximal Lyapunov exponent, or MLE) for the experimental data. These results will be compared to other measures of chaos in order to determine whether the experimental results really indicate chaotic motion rather than multi-periodic or noisy periodic motion. Experimental data will also be presented that indicate period-3 motion at lower brake strength, another curious departure from the predictions of the idealized equations of motion.
Miller, Mitch, "DETERMINING LYAPUNOV EXPONENTS FOR CHAOTIC WATERWHEEL DATA" (2019). University Research Symposium. 328.