Project 6.8.1. Project—The Lorenz Equations.
In 1963, an MIT professor, Edward N. Lorenz, published a paper on his research in meteorology. Using differential equations, Lorenz had developed a simplified system to model certain weather-related phenomena. When he analyzed the system, however, he found that the trajectories of the solutions were incredibly convoluted and effectively unpredictable for certain parameters. The Lorenz equations can be written as
\begin{align*}
\frac{dx}{dt} & = -\sigma x + \sigma y\\
\frac{dy}{dt} & = \rho x - y - xz\\
\frac{dz}{dt} & = -\beta z + xy,
\end{align*}
where \(\sigma\text{,}\) \(\rho\text{,}\) and \(\beta\) are constants. The Existence and Uniqueness Theorem for systems of differential equations guarantees a unique solution for each set of initial conditions,
\begin{align*}
x(0) & = x_0\\
y(0) & = y_0\\
z(0) & = z_0.
\end{align*}
However, Lorenz discovered that the trajectories of the solutions were incredibly convoluted and effectively unpredictable for certain parameters. For certain values of \(\sigma\text{,}\) \(\rho\text{,}\) and \(\beta\text{,}\) the trajectories are extremely sensitive to initial conditions. Since real data always has some inherent uncertainty, initial values are never precisely known, and we may have little success modeling real world phenomena. In addition, solutions can stay in a bounded region of the three dimensional version of the phase plane and wind through the region along an incredibly convoluted path. There is much more freedom to move around in three dimensions than there is in two.
(a)
Lorenz noticed that the system behaved strangely, when he let \(\sigma =10\text{,}\) \(\rho = 28\text{,}\) and \(\beta = 8/3\text{.}\) Thus, our system
\begin{align*}
\frac{dx}{dt} & = -10 x + 10 y\\
\frac{dy}{dt} & = 28 x - y - xz\\
\frac{dz}{dt} & = -\frac{8}{3} z + xy,
\end{align*}
defines a vector field in \({\mathbb R}^3\text{,}\)
\begin{equation*}
{\mathbf F}(x, y, z)
=
\left(
-10 x + 10 y,
28 x - y - xz,
-\frac{8}{3} z + xy
\right),
\end{equation*}
and the equilibrium solutions occur exactly when this vector field is zero. That is, \((x,y,z)\) is an equilibrium solution if
\begin{equation*}
{\mathbf F}(x, y, z)
=
\left(
-10 x + 10 y,
28 x - y - xz,
-\frac{8}{3} z + xy
\right)
=
(0, 0, 0).
\end{equation*}
Find all of the equilibrium solutions for this system.
(b)
If we wish to understand the nature of the equilibrium solutions of the Lorenz system, it makes sense to linearize the system. Compute the Jacobian matrix of the system and determine the nature of each of the equilibrium solutions that you found in Task 6.8.1.a.
(c)
Consider the Lorenz system
\begin{align*}
x' & = -a x + ay\\
y' & = rx -y -xz\\
z' & = -bz + xy.
\end{align*}
In this exercise, we shall show that the solutions of the system are bounded.
(i)
If
\begin{equation*}
V(x, y, z) = rx^2 +ay^2 + a(z - 2r)^2,
\end{equation*}
Show that
\begin{equation*}
\dot{V} = -2a [rx^2 + y^2 + b(z - r)^2 - br^2].
\end{equation*}
(ii)
It is true that \((z - r)^2 \geq (z - 2r)^2 - r^2\text{.}\) Assuming this, show that
\begin{equation*}
rx^2 + y^2 + b(z - r)^2 \geq mV - br^2,
\end{equation*}
where \(m\) is the smallest of the three numbers 1, \(1/a\text{,}\) and \(b/2a\text{.}\)
(iii)
Use parts Task 6.8.1.c.i and Task 6.8.1.c.ii to show that
\begin{equation*}
\dot{V} \leq -2maV + 4abr^2.
\end{equation*}
(iv)
Use part Task 6.8.1.c.iii to show that \(\dot{V} \leq -2abr^2 \lt 0\) everywhere outside of the ellipsoid
\begin{equation*}
R = \left\{ (x, y, z) | V(x, y, z) \leq 3br^2/m \}\right\}.
\end{equation*}
(v)
Use part Task 6.8.1.c.iv to show that the ellipsoid is invariant, and that every solution curve ends up inside of \(R\text{.}\)
