Detecting Symmetry in the Chaotic and Quasi-periodic Motions of Three Coupled Droplet Oscillators

Symmetry detectives offer an automated method for classifying the symmetries of solutions to dynamical systems. In this paper, symmetry detectives are applied to conservative motions of coupled-droplet oscillators. Previous application of detectives has been for the determination of symmetries of attractors as well as the detection of symmetry-changing bifurcations. We analyze the trajectories of a fourth-order $S_3$ symmetric model of three coupled liquid droplets, where motions are assumed frictionless. Since there is no dissipation in the model, there are no asymptotically stable attractors, only centers. Solutions away from equilibrium are the focus. In particular, we examine trajectories starting with no initial velocity. Detection of symmetry is achieved by mapping a trajectory into an appropriate representation space where distances to fixed-point subspaces of subgroups are computed. Results of the symmetry-detective approach are contrasted to the more conventional computation of the largest Lyapunov exponent as a signal of chaotic or quasi-periodic dynamics. Both methods can be applied to a grid of initial conditions in an automated fashion. Our results reveal the strong correlation between symmetries and nonlinear dynamics. [PUBLICATION ABSTRACT]