Math 401 Project Ideas

Here are some topics in Differential Equations (or dynamical systems) that may be interesting to study for projects. I've given the very briefest descriptions. If you want to know more about a given topic, I can tell you lots more about any of them.

The Lorenz Attractor: One of the first systems of differential equations known to have chaos, it also has an attractor with a characteristic butterfly shape (seen on the cover of your textbook)
Laplace Transforms: We won't discuss them in this class, but this is an important technique for solving linear ODEs of any order with forcing. Very, very common in engineering applications
Transfer Functions: Related to Laplace transforms, Transfer Functions are a compact description of the dynamical properties of a system. Particularly useful for vibrating mechanical systems and signal processing.
RLC Circuits: A classic application of second-order linear differential equations, the current in a circuit consisting of resistors, capacitors and inductors obeys a nice DE.
Chaotic Maps: The Shift Map: A very simple map that can be proven to be chaotic, lots of other chaotic maps can be analyzed by relating them to this one.
Chaotic Maps: ax(1-x): This map has a bifurcation to chaos at a certain parameter value and generates some of the famous pictures depicting chaos.
Chaotic Maps: The Cat Map: A two-dimensional map, this one has been made famous by investigating its effects on a drawing of a cat.
Lead in the body: A great application of compartment models, this one appears in your textbook in section 6.1
Pumping a playground swing: A mathematical model of just how to get the playground swing to go higher and higher.
Planar Systems - Hilbert's 16th Problem: David Hilbert proposed in 1900 the following problem: How many limit cycles can a planar system with polynomial euqations of degree n have? This problem is still not completely solved.
Stable and Unstable Manifold: These geometric objects are extremely important to the dynamics of nonlinear dynamical systems.
Chaotic Tangles: What happens when stable and unstable manifolds cross? They get tangled up.
Chaotic Advection: Stirring fluids can lead to chaotic motion of the parcels in the fluid.
The 3 body problem: In gravitational dynamics, two bodies move in circles, but three bodies can do all kinds of crazy things.
Escape in Finite Time: In a gravitational system of 5 point masses, it is possible to eject one or more of them to infinite distances in finite time. Huh.
Lyapunov Exponent Calculations: Lyapunov Exponents are a way of measuring chaos. How are they computed in practice?
Quantum Chaos: I'm not really sure what this is, but it sounds cool, doesn't it?
Numerical Methods: Solving differential equations on a computer is very useful. There's also lots of different ways to do it: Euler, Runge-Kutta, BDF, etc.
Numerical Methods: One of the important features of a numerical method for solving differential equations is it's stability properties: what kind of equations will it work on?
Control Theory: Another engineering application, control theor studies how external inputs to a system of differential equations can be controlled to give a desired response.