Fractal classifications of trajectories in a nonlinear mass-spring system

Nonlinear continuous dynamical systems can exhibit extremely complicated behavior. We investigate a physical system consisting of a single mass anchored to two fixed points in the plane by two ideal linear springs. Though the spring forces used in the system are linear, their interactions in two dimensions create a nonlinear system, which forms the basis for more sophisticated dynamical systems used in physics and computer animation.This paper uses a numerical solution to investigate the dynamics of the trajectories in this system, and to quantify specific characteristics. We find a complicated space of trajectories which gives rise to a class of intricate fractal shapes in the system's four-dimensional phase space. We investigate the fractal properties of these shapes that help quantify and classify features of the system's trajectories, and enable visualization of the structure of its four-dimensional phase space.