Chapter 1

Visualizing the Chen attractor.

Manipulating one Chen attractor trajectory.

As we will learn later in this course, some ODE systems exhibit “chaos” (i.e., extreme sensitivity to initial conditions). One example is given by the following system of equations, which describes the velocity of a point. This system is called the Chen attractor, with parameters $a,b,c$. You can read more about it here.

\begin{align} x' & = a(y-x) \\ y' & = (c-a)x - xz + cy\\ z' & = xy -bz. \end{align}

Here we will use $(a, b, c) = (40, 3, 28)$. Then the system of ODE's becomes

\begin{align} x' & = 40(y-x) \\ y' & = -12x - xz + 28y\\ z' & = xy -3z. \end{align}

Notice that this system is non-linear. The figure below shows the trajectory of a Chen attractor with initial value $(x_0,y_0,z_0)=(-0.1,0.5,-0.6)$ from time $t =0$ to time $t=20$. You can see that it looks like a double scroll. 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.

Sensitivity to the starting point.

Now let's try it yourself. Choose an initial value near $(-0.1,0.5,-0.6)$ and hit the “Run the attractors!” button. Then you will see (the $xy$-projection of) the trajactories of two Chen attractors, one with intial value $(x_0,y_0,z_0)=(-0.1,0.5,-0.6)$ and the other with the intial value of your choice. For clarity, only the trajectories in the previous two 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.