Visualization of orbits and pattern evocation for the double spherical pendulum

Abstract

This paper explores pattern evocation and the visualization of orbits of the double spherical pendulum. Pattern evocation is a phenomenon where patterns emerge when the ow of a dynamical system is viewed in a frame that rotates relative to the inertial frame. The paper begins with a summary of the theory on pattern evocation for mechanical systems with symmetry. The result of this theory is that if the motion in the reduced space is periodic (respectively, quasiperiodic or almost periodic), then when viewed in a suitably chosen rotating frame with constant velocity, the motion in the unreduced space is periodic (respectively, quasiperiodic or almost periodic). The motion of the system viewed in this rotating frame may have a particular pattern or symmetry. Examples of this theory are demonstrated for the double spherical pendulum. A dierential-algebraic model is created for the double spherical pendulum and is integrated with the simulation package MEXX as well as a custom energy-momentum integrator.