Classwork 3

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Directions: Add work to this notebook to solve the problems below.

1. The Lorenz Equations

Consider the vector field $$F(x,y,z) = \big(\sigma(y-x), rx - y -xz, -bz + xy\big).$$ The system of differential equations $$F\big(x(t),y(t),z(t)\big) = \big(x'(t), y'(t), z'(t)\big)$$ is called the Lorenz equations.

The equation is inspired by the study of turbulence in the earth's atmosphere, and reasonable parameters for this are given by $\sigma=10$ and $b=\frac{8}{3}$. The parameter $r$ varies with atmospheric conditions (and is related to the differences in temperature between top and bottom layers of the atmosphere), with larger values of $r$ then bringing more chaotic turbulence.

A reference for this is

• Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems, 9th edition, section 9.8.

Part 1. Plot the vector field $F$ for values $\sigma=10$ and $b=\frac{8}{3}$ and $r \in \{1/2, 21, 28\}$ over the box $(x,y,z) \in [-20, 20]^3$.

Use Sage's plot_vector_field3d. Unfortunately, there is a bug where the vectors are scaled so the longest vector has length one, which doesn't make sense in $40 \times 40 \times 40$ cube. You can fix this by conjugating your vector field by the map $(x,y,z) \mapsto \frac{1}{10} (x,y,z)$. Thus, you should plot the vector field $\frac{1}{10} F(10x,10y,10z)$ over the box $[-2,2]^3$.

Part 2. Draw pictures of flow lines for $\sigma=10$ and $b=\frac{8}{3}$ and $r \in \{1/2, 21, 28\}$. Draw pictures of the trajectory starting at $(x_0,y_0,z_0)=(5,5,5)$ in the $(t,x)$-plane, in the $(x,y)$-plane, and in $(x,y,z)$-space. Your can use the time interval $[0, 20]$.

Part 3. Observe that the flow lines exhibit sensitive dependence on initial condititions when $r=28$ and the other parameters are as above. Observe that starting at $(5,5,5)$ and another point $(x_0', y_0', z_0')$ that is within distance $1/100$ away, that the trajectories separate over the interval $[0, 20]$. You can see this by looking at the $(t,x)$-plane for instance. You can just demonstrate this in an example.