Have you ever used a compass to find your way through unfamiliar territory? Did you know that some thirty thousand years ago the compass needle would have pointed in the opposite direction? Yes, indeed, there have been times in the history of the Earth during which the north and south poles of the magnetic field were swapped! There is a complex set of mathematical equations – called partial differential equations, to be precise – that is thought to describe how the magnetic field is actually generated. The mathematical object that scientists believe is “responsible” for the reversal of the poles is called a heteroclinic cycle. Technically speaking, this cycle consists of several steady states (where the magnetic field does not change over time) and the connecting times, when the north and south pole of the magnetic field swap. This kind of stop-and-go dynamics is encountered in many other scientific applications as well, such as modeling the evolution of animal species competing with each other, or communication networks within the human brain.
This variety of applications is why I think it is highly interesting to study this phenomenon from an abstract, mathematical point of view. In particular, I study the stability properties of these heteroclinic cycles. The concept of stability is crucial for distinguishing pure mathematics from what we can actually “see” in real life. For example, think of a ball rolling along a hilly surface. We would never see the ball come to a stop right on the top of a hill – even though math says that is theoretically possible. Even the smallest movement will cause it to move away from the hilltop, and never return. This is a classic example of an “unstable state.” In my research, I am working on finding mathematical conditions for a heteroclinic cycle to be stable rather than unstable, so that we may understand better how the parts influence the systems in which we discover them.
So, next time you’re navigating through the woods with a compass, don’t despair when you get lost – there is a slight chance that it is just the Earth’s magnetic poles swapping positions that keeps you from finding your way…
– Alex, University of Hamburg, Germany