Up to this point in our course, you have been learning methods to solve differential equations. These are incredibly useful skills for you to have. At the same time, you might have noted that most of the equations you know how to solve are linear. Most scientific phenomena of interest, however, are actually nonlinear, and nonlinear equations cannot often be solved. That is to say, it's not that you can't solve them because you haven't learned the correct technique, but rather, the equations themselves do not have solutions that can be written down in closed form. There's an entirely different class of tools we can use to study such systems, namely the tools of dynamical systems theory. There are so many examples of crucial nonlinear systems in biology, physics, chemistry, economics, and more, that it is difficult to choose merely one to show you. So I will instead show you a somewhat less pivotal but very cool example of a nonlinear system. This demonstrates one of the amazing things nonlinear systems can do, namely, synchronize. Perhaps this is the takeaway message?
By the end of this lesson, you should be able to:
- Explain what a dynamical system is.
- Classify whether a differential equation is a PDE or ODE, is linear or nonlinear, and is autonomous or non autonomous
- Take a given dynamical system and write it in standard form.
- Determine the phase space of a dynamical system and represent it graphically.
- Explain what a trajectory is.
- Distinguish between two different representations of of dynamical systems: trajectories in phase space and plots of solutions.
- Compare ODE solution approach with a geometric dynamical systems approach.
- Analyze a 1-d dynamical system (find fixed points, draw a vector field, determine local/global in/stability, and draw qualitatively correct solution curves).
- Compare and contrast exponential and logistic growth.
- Analyze population growth problems.
Before class, please:
- Read Strogatz, Chapter 1 and Sections 2.0 - 2.3.
- Watch this pencast (first six bullet points above) and this pencast (last four bullet points above).
- Check yourself before you wreck yourself.
In class, we will:
After class, please:
- Do these problems.