Chapter 12

Visualizing chaos, and sensitive dependence on initial conditions & parameters.

Lorenz attractors.

The system of ODE's we care about is the following Lorenz system, with constants $a > 1, b > 0, c > 0$:

\begin{align} x' & = a(y-x) \\ y' & = bx - y - xz \\ z' & = xy -cz. \end{align}

Lorenz focused on $(a, b, c) = (10, 28, 8/3)$, for reasons within the theory of dynamical systems. We also focus on this case here. Then the system of ODE's becomes

\begin{align} x' & = 10(y-x) \\ y' & = 28x - y - xz \\ z' & = xy -\dfrac{8}{3}z. \end{align}

The figure below shows the trajectory of a Lorenz attractor with initial value $(x_0,y_0,z_0)=(1,1,1)$ from time $t =0$ to time $t=20$. You can also play with the figure yourself; move your mouse to the figure and you will see various viewing options on the top right corner of the figure.

Here is a colorful demonstration of 50 Lorenz attractors with nearby yet distinct initial positions. This animation is essentially taken from here. You can see that their trajectories are eventually totally unrelated.

Now let's try it yourself. Choose an initial value near $(1,1,1)$ and hit the “Run the attractors!” button. Then you will see (the $xy$-projection of) the trajactories of two Lorenz attractors, one with intial value $(x_0,y_0,z_0)=(1,1,1)$ and the other with the intial value of your choice. For clarity, only the trajectories in the previous five seconds are kept showing. You will see that initially the two trajectories are very close to each other (so you probably will only be able to see one), but after a while they become unrelated. You may need to be patient (or increase the speed) to see the divergent behavior from the two nearby initial points.