A phase space is an abstract mathematical concept that describes the dynamics of a physical system. More concretely, each dimension of the space represents an independent state variable of the system, such as position or velocity. In the phase space of the buckling beam, the x-axis represents position and the y-axis represents velocity. The center represents when the beam is not bent, while the two symmetrically positioned "eyes" represent the two buckled states (buckled left or buckled right).

Poincare introduced the geometric theory of modern dynamical systems, revolutionizing the study of nonlinear phenomena. Important dynamical behaviors of physical systems are described by qualitative geometric features, such as singular points, curves, regions, or volumes in phase space.

In the buckling beam phase space, we see a number of sample "trajectories" showing how the beam's state changes over time. For example, if the state is just to the northeast of the center, then stepping through time will follow the path that spirals around and eventually ends up in the right "eye". Along this spiral, the position and velocity repeatedly increase and then decrease. This represents the oscillation of the beam back and forth around the final resting bent position.

Understanding the geometric structure of a phase space can lead to powerful algorithms to control the underlying physical system. The buckling beam can be controlled by determining the current state of the beam and choosing a control action that moves the state along "flow pipes" through the phase space to a desired state.

Various publications by our group discuss computational models for phase spaces and phase-space based control algorithms in more detail.