Secrets to Success

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Some problems with a colleague have me thinking back to the list I gave my students at the beginning of the semester:

  • Show up on time for class.
  • Bring a smile or a kind word for someone in our class every single day.
  • Stay in the classroom for the entire class period.
  • Participate.
  • Use class time productively for class work.
  • Be professional.

Is that everything required for success? Of course not. But if you are struggling with the list, you are probably also struggling to achieve success. Strange how that goes hand-in-hand.


What are we learning in class these days? We are working on Project 1, which is all about the Lorenz attractor. We are going to use Matlab to solve the system of differential equations in the Lorenz attractor, and then make graphs that show clearly that we see the butterfly effect.

Merriam-Webster definition of BUTTERFLY EFFECT:

a property of chaotic systems (as the atmosphere) by which small changes in initial conditions can lead to large-scale and unpredictable variation in the future state of the system

The Lorenz attractor is an example of a chaotic system

Merriam-Webster definition of CHAOS:

2b : the inherent unpredictability in the behavior of a complex natural system (as the atmosphere, boiling water, or the beating heart)

I skipped definition 1 and 2a, since 2b is the one of concern to us. The Lorenz attractor is unpredictable, but it is deterministic there is no element of chance involved; it does what it does.

One of the first struggles that students have is to figure out what system is the the Lorenz attractor a mathematical model for? Students often answer that it is a mathematical model for the butterfly effect, because that is what we will demonstrate in the project we do for our class. Lorenz’s equations actually model convection in the atmosphere where you have a fluid (such as air) heated from below and cooled from above. We expect that heat rises, so the air molecules rise when heated, then they fall when cooled. This forms a circulatory pattern, which is, unsurprisingly, what we see when we solve the differential equations for the Lorenz attractor.

One question I posed to my students is whether or not the behavior of

$$\frac{dx}{dt} = x \qquad \hbox{with}\ x(0) = 1 \qquad \hbox{or}\ x(0)= 1.001$$
is an example of the butterfly effect and chaos. Notice that solutions are of the form
$$x(t) = Ae^t$$ with \(A = 1\) or \(A=1.001\) respectively. This is clearly not chaotic, we can predict what \(x(t)\) will equal at any time. But equally clearly, it is very sensitive to the initial conditions; small changes in the initial condition lead to large changes at \(x(10)\) or \(x(15)\). Is this an example of the butterfly effect?

This example is not chaos. This is a demonstration of sensitive dependence on initial conditions. But since it is not in a chaotic system, it is not a demonstration of the butterfly effect. Now contrast that picture with what you see for the Lorenz attractor.

I hope running through these questions help them to figure out (mathematical) chaos, and just how cool the Lorenz attractor is.


I am using MathJax to make the mathematics on this page!