Eigenmodes in a conformally mapped domain

Using Trefethen's Chebyshev spectral methods, this program computes and plots the Laplace eigenmodes in a conformally mapped domain. The domain is the one-parameter family of Neumann ovals, defined by the parameter beta. Details of the conformal mapping are given in the readme.pdf file.
A simple GUI is created which allows the user to vary the shape of the Neumann oval via a slider and automatically plot the new eigenmodes. The Neumann oval varies from a circle (beta=0) to two separate circles (beta=1). However, in this program the maximum beta=0.7 as beta>0.7 gives `choppy' eigenmodes due to the poor coverage of the of the highly eccentric Neumann oval by the Chebyshev grid. Better coverage is achieved by increasing M (and to a lesser extent N), but in the interests of achieving a responsive GUI, I have not done this.

An interesting phenomenon occurs between beta=0.56 and beta=0.63 where the second mode goes from an `up-down' bimodal structure to a trimodal structure.

Note that I am merely a novice with MATLAB GUIs so please let me know if I have done anything obviously wrong in that regard.