PEER has just published Report No. 2016/07 titled “Hybrid Simulation Theory for a Classical Nonlinear Dynamical System.” It was authored by Paul L. Drazin and Sanjay Govindjee.
Hybrid simulation is an experimental and computational technique that allows one to study the time evolution of a system by physically testing a subset of it while the remainder is represented by a numerical model that is attached to the physical portion via sensors and actuators. The technique allows the study of large or complicated mechanical systems while only requiring a subset of the complete system to be present in the laboratory. This results in vast cost savings as well as the ability to study systems that simply cannot be tested due to scale. However, the errors that arise from splitting the system in two requires careful attention if a valid simulation is to be guaranteed. To date, efforts to understand the theoretical limitations of hybrid simulation have been restricted to linear dynamical systems. The research reported herein considers the behavior of hybrid simulation when applied to nonlinear dynamical system. The model problem focuses on the damped, harmonically-driven nonlinear pendulum. This system offers complex nonlinear characteristics, in particular periodic and chaotic motions. We are able to demonstrate that the application of hybrid simulation to nonlinear systems requires careful understanding of what one expects from such an experiment. In particular, when system response is chaotic, we advocate using multiple metrics to characterize the difference between two chaotic systems via Lyapunov exponents and Lyapunov dimensions, as well as correlation exponents. When system response is periodic we advocate using L2 norms. Further, we demonstrate that hybrid simulation can falsely predict chaotic or periodic response when the true system has the opposite characteristic. In certain cases, control system parameters can mitigate this issue.