Torus-ODE

Repo for MATLAB material somehow related to the motion of vortices on curved surfaces. Some related resources:

1. Repo link

2. Research advisor's paper [external link]

vortex_simulation.m

Driver code for vortex simulation; choose how many vortices you'd like to see the dynamics of here, as well as their starting locations (isothermal coordinates).

jacobitheta1.m and Djacobitheta1.m

Crude MATLAB function files to compute the first several terms (double precision is usually reached before a truncation of 15 or so terms) of the first Jacobi theta function and its derivative.

vortex_velocity.m and complexpotential.m

MATLAB functions that computes the physical velocity of a set of vortices and their corresponding complex potential function, respectively, using a relation involving Jacobi theta functions is used (see Fetter et. al. for more details).

hamiltonian.m

This function computes the total energy of a system of vortices; it also returns classical, quantum and surface curvature contributions.

plotwrapped.m

A MATLAB function to plot data wrapped on the surface of a torus nicely (based on MATLAB's wrapToPi command).

ManyOrbitsOneTorus.m

A MATLAB function to plot multiple trajectories simultaneously as a parameter is varied, integrated using vortex_velocity.m.

fixed_points.m

A script that takes an initial guess (seed) of a fixed point zstar of the torus ODE's and iteratively searches for a solution to the corresponding nonlinear least squares problem (using the Levenberg-Marquardt algorithm option).