  The normal modes for three to seven particle two-dimensional (2D) dust
clusters in a complex plasma are investigated using an N-body simulation. The
ion wakefield downstream of each particle is shown to induce coupling between
horizontal and vertical modes. The rules of mode coupling are investigated by
classifying the mode eigenvectors employing the Bessel and trigonometric
functions indexed by order integers (m, n). It is shown that coupling only
occurs between two modes with the same m and that horizontal modes having a
higher shear contribution exhibit weaker coupling. Three types of resonances
are shown to occur when two coupled modes have the same frequency. Discrete
instabilities caused by both the first and third type of resonances are
verified and instabilities caused by the third type of resonance are found to
induce melting. The melting procedure is observed to go through a two-step
process with the solid-liquid transition closely obeying the Lindemann
criterion.
