Undergraduate thesis: Transition to Chaos

Quick summary: Thesis here, slides here, related notes here.

This post serves as a collection of information and work related to my undergraduate thesis, which I wrote in summer 2010 under the supervision of Professor Peter H. Richter at the Physics department of the University of Bremen, Germany. A brief summary aimed at a general audience can be found below in this post.

The title of my thesis is: Melnikov’s Method and the transition to chaotic behaviour in Cardan-mounted Euler tops. It is written in German. The thesis is available here and the presentation slides here. As a way of learning about Melnikov’s method, I wrote these notes prior to the thesis. The Cardan-mounted Euler top to which the method is applied is depicted in the picture below.

The Cardan-mounted Euler top to which Melnikov’s Method is applied in my thesis. The reference coordinate system is (x,y,z), the current system is (1,2,3).

Animations and source code

I wrote a little Matlab program which visualises the motion of the Cardan-mounted Euler top. It is hosted on Matlab File Exchange.  Below, you can see two videos, one showing regular (oscillating) motion and the other showing irregular (chaotic) motion.

A brief summary

Here is a brief summary aimed at a general audience:

Euler’s top is a rigid body (with arbitrary shape, say, a potato) suspended from its centre of gravity (CoG). Its motion is regular and well-known. In most cases, however, the CoG cannot be reached by a regular suspension method like a thread or rod (one would have to reach inside the potato, i.e. pierce a hole through it). But the problem can be solved using a more sophisticated suspension mechanism: a ‘Cardan mounting’ (e.g. helicopters or gyro compasses are similar to Cardan-mounted rigid bodies).

But as the dynamic properties of the Cardan-mounted Euler top are different from those of the normal Euler top, the resulting motion is much more complex: the spectrum of possible motion ranges from regular (i.e. harmonic, predictable) to chaotic (i.e. unpredictable) in a multitude of varying degrees. My thesis provides a description of the transition from regular to chaotic motion, using methods from nonlinear dynamics and Melnikov’s Method from perturbation theory.