Stable Divergence Angles of a Magnetic Dipole Spiral Array

Abstract

An analytical model is introduced for the experiment of Douady and Couder [Phys. Rev. Lett.68, 2098 (1992), where phyllotactic patterns appear as a dynamical result of the interaction between magnetic dipoles. The difference equation for the divergence angle (i.e. the angle between successive radial vectors) is obtained by solving the equations of motion with a second nearest neighbor (SNN) approximation. A one-dimensional map analysis as well as a comprehensive analytical proof shows that the divergence angle always converges to a single attractor regardless of the initial conditions. This attractor is approximately the Fibonacci angle(~ 138°) within variations due to a growth factor μ of the pattern. The system is proved to be stable with the SNN approximation. Further analysis with a third nearest neighbor approximation (TNN) shows extra linearly stable attractors may appear around the Lucas angle (~ 99.5°).