Qualitative Analysis of Invariant Tori in a Dynamical System

Invariant tori are examples of invariant manifolds in dynamical systems. Usual tools in dynamical systems such as analysis and numerical simulations alone are often not sufficient to understand the complicated mechanisms that cause changes in these manifolds. Computer-graphical visualization is a natural and powerful addition to these tools used for the qualitative study of dynamical systems, especially for the study of invariant manifolds. The dynamics of two linearly coupled oscillators is the focus of this case study. With little or no coupling between the oscillators, an invariant torus is present but it breaks down for strong coupling. Visualization has been employed to gain a qualitative understanding of this breakdown process. The visualization has allowed key features of the tori to be recognized, and it has proven to be indispensable in developing and testing hypotheses about the tori.