Characterization of a spring pendulum phase-space trajectories

We study the geometrical properties of phase-space trajectories (or orbits) of a spring pendulum as functions of the energy. Poincaré maps are used to describe the properties of the system. The points in the Poincaré maps of regular orbits (non-chaotic) cluster around separated segments or in chains of islands. Looking at how segments are formed, we conclude that the orbits are closely related to torus knots. Examining the toroidal and poloidal turns of the orbits, we introduce the definition of a rational parameter Ω, which is closely related to the concept of frequency used in the analysis of dynamical systems. Algorithms were developed to calculate Ω, and we found that this parameter naturally describes the orbits in terms of Farey sequences; also, calculations show that orbits with the same Ω have similar dynamics. Orbits corresponding to chains of islands are identified with cable knots that can be characterized using two parameters analogous to Ω. In some cases, non-trivial cable knots were found. With the analysis presented in this study, it is shown that Ω follows predictable distributions in the (z,Ω) space.