Today we're gonna talk about chemical cycles and chaos. This overall idea fits into one of our main interests in this course which is, what properties and processes are easy to obtain through physical dynamics alone? And what we'll describe here today is how simple systems of chemical reactions can lead to cycling behavior and can also lead to chaotic dynamics. So before we can go into the details of those chemical reactions, and the mathematical analysis of them, I first want to define a few different things. So the first is a limit cycle. What a limit cycle is, is imagine you have two parameters, X and Y, coupled together through, let's say a set of differential equations. And a limit cycle is a case where the only way to define a steady-state solution to this system of differential equations, is through a closed curve, rather than a single point. So we can imagine that the steady-state of some system goes to a paired value of X and Y. In the case of a limit cycle, it goes to the set of paired values X and Y that lie along one curve and so will continually, over time, cycle through all these different values. Now, these limit cycles are also related to chaotic dynamics and we'll show how some of these limit cycles become chaotic dynamics in our simple set of equations. And here is a tracing of the dynamics through now X, Y, and Z coordinates projected in this 2-D space where you can see that the dynamics trace out a large variety of combinations of X, Y, and Z values. They're sort of contained in this ruffly understandable picture, but yet the state space that is being explored by X, Y, and Z is very large, and we'll talk about that today.