kevinhikaruevans / chaos-stuff

Numerically checking out chaos in BECs

chaos-stuff

This is some stuff related to my undergrad thesis in physics at Washington State University. It's not my best work and I don't think it has any applications, outside of being a somewhat-interesting artifact of mathematics and physics.

In this thesis, we arbitrarily calculate the Lyapunov exponent of BECs modeled by the Gross-Pitaevskii equation. The exponent is found using the $L^2$-norm of the many-body wavefunction. It seems that the GPE exhibits chaotic motion for positive coupling constants $g$, but I have no idea if this is real (or just some ✨funky maths✨). It is interesting that it is linear between $\lambda$ and $g$ for $g \in \mathbb{R}^+$ (although I do not understand why!).

Abstract

The Gross-Pitaevskii equation (GPE) is a nonlinear Schrödinger equation used in modeling Bose-Einstein condensates (BECs). Using Lyapunov exponents, chaos was characterized in the GPE in the one-dimensional case. The GPE was simulated using Python using finite-difference methods in space and an adaptive Runge-Kutta solver was used to evolve the equation in time. When an initial state of the GPE with positive coupling constant is perturbed, positive Lyapunov exponents were found. There was a proportionality found between positive Lyapunov exponents and the nonlinear coupling constant of the GPE, $g$. Further research can be done to analyze this proportionality and to extend this research to higher dimensions.

Contents

• My undergraduate thesis (pdf)
• GPE IPython notebook (scratch-work)
• Lorenz IPython notebook (scratch-work)

Side notes

• I appreciate having the chance of conducting research as an undergrad, but did not find this topic interesting (nor motivating).
• I dislike the LaTeX template we were using.
• This was conducted before I had my classes in QM.