Hill Equation

By Terry Shang, The University of Sydney

This summer I looked at investigating a particular second order differential equation called a Hill equation. The study of differential equations is the study of change. There is a “rule”, the differential equation, which governs how a quantity will change. Our job is simply to let time run and observe how this quantity evolves.

The Hill equation is a simple looking thing: an operator consisting of the second derivative summed with a periodic function acts on a variable to produce that variable multiplied by a constant. The exploration of this equation however has revealed many intricate patterns and curious behaviours.

In general for any random periodic function we can’t write out an explicit solution to the differential equation. To understand the future behaviour of these variables therefore it is not enough to simply calculate their function and evaluate it at a later time. Indeed what is required is that we tease out information about the solution.

Aiding us in our endeavours is Floquet theory which simplifies the study of these periodic differential equations. It allows us to consider only the evolution of the system through a period; intuitively this makes sense, given that the “rule” which governs the change of the variable repeats itself we need only look at one repetition to begin to understand the variable.

We are interested in solutions to the Hill equation which do not go to infinity as time goes by. Applying the idea of Floquet theory we are interested in situations where once the variable goes through repetition its size does not change: any change either negative or positive, however small will result in the variable reaching infinity in either positive or negative time. For certain values of the constant within the differential equation, these solutions appear. We call the constant and eigenvalue in this case.

Thus far the study of Hill equations has been limited to real periodic functions. This is very neat since the operator is “self-adjoint” which means that the eigenvalues of this system are only real. Thus the subject of Hill equation is relatively simple: real periodic functions, real eigenvalues.

This project looks to extend the theory of Hill equation to complex periodic functions. Now the eigenvalues are no longer necessarily real and where they appear in the complex plane is hard to pin down. While it is just as hard to exactly find where these eigenvalues appear as it is to explicitly solve these equations, we’d like to narrow down and establish some patterns for where these solutions appear.

The answers to these questions have implications beyond just this differential equation. There is a connection between the eigenvalues of these equations and the behaviour of fluids. This simple second order differential equation is a rabbit-hole to a world of curious behaviour and intricate patterns. It’s been a pleasure trying to unpack this puzzle.

 

Terry Shang was one of the recipients of a 2016/17 AMSI Vacation Research Scholarship.

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