The structural analysis of phase space trajectories

Multidimensional phase space trajectories can be difficult to understand due to their complexity and the typically high dimensionality of the space. This paper proposes a global perspective to the problem having two aspects: first, the trajectory is viewed geometrically and analyzed structurally and second, the detailed trajectory information is compacted into a small number of averaged but global quantities. Several basic structural parameters such as the center of mass, principal moments of inertia, eigenvalues and eigenvectors of the momenta ellipsoid, average speed, etc. are defined to characterize trajectories. Additionally, the sensitivity coefficients of these newly defined quantities are examined. As illustrative examples, the Lotka oscillator and the H₂O₂ oxidation system are discussed. Finally, some interesting asymptotic properties of oscillatory systems are presented as a result of this work.