High dimensional and Chaotic flows

Lorenz equations

In the early 1960s E.N. Lorenz published some studies on a highly reduced and simplified model of atmospheric convection. Early simplifications had approximated using the 12- and 7-dimensional models; Lorenz further isolated what seemed to be the most important three variables, resulting in this system:

$x' = \sigma (y - x)$

$y' = x(\rho - z) - y$

$z' = x y - \beta z$

The classic values (Lorenz 1963) are $\sigma = 10$ , $\beta = 8/3$ , and $\rho = 28$ . The $\rho$ parameter is related to a ratio of the Reynolds number of the flow to a critical value, so that at $\rho = 1$ it is expected that some instability becomes important. The parameter $\sigma$ is related to the Prandtl number of the original hydrodynamics.

The fixed points of the flow are easy to compute: the origin is always a fixed point, and if $\rho>1$ two new equilibria appear at

$x = y = \pm \sqrt{\beta(\rho-1)}, \ \ \ \ z = \rho-1.$

These new equilibria correspond to a steady pattern of counter-rotating vortex tubes in the original hydrodynamic scenario.

At the origin, the Jacobian matrix has eigenvalues $\lambda_1 = -\beta$ and

$\lambda_{2,3} = \frac{-1 - \sigma \pm \sqrt{(1+\sigma)^2 - 4 \sigma (1-\rho)}}{2}$

For $\rho < 1$ then all three of the origin's eigenvalues are real and negative, so it is a sink. In fact it is a global sink. For $\rho > 1$ , one of the eigenvalues becomes positive.

The characteristic equation for the other fixed points is

$\lambda^3 + (\sigma + \beta + 1)\lambda^2 + (\rho + \sigma) \beta \lambda + 2 \sigma \beta (\rho - 1) = 0$

Since all of the coefficients of $\lambda$ in the above equation are positive (we assume that all three parameters are positive), there is always a real negative eigenvalue for those fixed points. For the classic values of $\sigma = 10$ , $\beta = 8/3$ , there are three negative eigenvalues for $\rho < 1.34561\ldots$ .

A general feature of the Lorenz flow for all positive parameter values is that it shrinks any small volume element at a constant rate since the divergence is:

$\nabla \cdot F = \frac{\partial x'}{\partial x} + \frac{\partial y'}{\partial y} + \frac{\partial z'}{\partial z} = - \sigma - 1 - b < 0$

Another general feature is that the flow is bounded. If we use the linear change of variable $u = z - \rho - \sigma$ , then the function $L = \frac{1}{2}(x^2 + y^2 + u^2)$ has a time derivative

$\frac{d L}{d t} = - \sigma x^2 - y^2 - b(u + \frac{\rho + \sigma}{2})^2 + \beta (\rho + \sigma)^2/4$

which is negative for large enough $x$ , $y$ , and $u$ . So for any given parameters the flow will eventually reside in a sphere.

These two facts combined mean that the invariant set for the flow is bounded and must be zero volume.

In the classic case, the Lorenz system has sensitive dependence on initial conditions. In the plot below, the x-coordinates of two trajectories with very close initial conditions are shown; once they diverge, they behave quite independently. This is was one of the striking observations made by Lorenz (with the help of Margaret Hamilton on the computations; she would later show extraordinary ability in leading the design of the Apollo mission flight computer).

x coordinates of two Lorenz equation solutions — Two Lorenz equation solutions (x coordinate)

Lorenz noticed a pattern in the $z$ values; here are two plots of $z$ values starting from close initial conditions:

z coordinates of two Lorenz equation solutions — Two Lorenz equation solutions (z coordinate)

If the locations of extreme $z$ values are picked out, we can make a plot of the $i$ th maximum $z$ value versus the $(i+1)$ th maximum $z$ value, which results in a map much like the tent map:

Lorenz return map — Z-maxima return map; the red line is the identity function

The Rikitake model

In 1958, Tsuneji Rikitake formulated a simple model of the Earth's magnetic core to explain the oscillations in the polarity of the magnetic field.

The equations for his model are:

$x' = -\mu x + y z$

$y' = -\mu y + (z-a)x$

$z' = 1 - xy$

where $a$ and $\mu$ are positive constants.

Rössler attractor

Rössler attractor (1976)

$x' = - y - z$

$y' = x + a y$

$z' = b + z(x-c)$

Rössler looked at $a = 0.2$ , $b = 0.2$ , $c = 5.7$ , shown in the animation below.

Chua's circuit: All knots and links:

Another remarkable 3D system is Chua's circuit, which is extremely close to being linear but possesses almost inconceivably complicated periodic orbits.

Despite linear on almost all of $R^3$ , it has been proven that for

$\phi(x) = \frac{2x}{7} - \frac{3}{14} \left ( |x+1| - |x-1| \right )$

there is an open set of $\beta \in [6.5, 10.5]$ for which

$x' = 7(y - \phi(x))$

$y' = x - y + z$

$z' = - \beta y$

has representatives of all knots and links.

The Halvorsen Attractor

While the Rössler system is one of the simplest with a strange attractor, it can also be nice to have a system with more symmetry. One three-dimensional system with cyclic symmetry that can have a strange attractor is the Halvorsen system, which with parameters $A$ , $B$ , and $C$ looks like

$x' = - A x - B y - B z - C y^2$

$y' = - A y - B z - B x - C z^2$

$z' = - A z - B x - B y - C x^2$

Some trajectories near the attractor for $A=1.5$ , $B=-4$ , $C=1$ are shown below:

Exercises

Describe and/or sketch the trajectories of the system

$x_1' = -x_1 + x_2$

$x_2' = -x_1 - x_2$

$x_3' = x_3$
Compute the Picard iterates $\phi_1$ , $\phi_2$ , and $\phi_3$ if $\phi_0 = \left ( \begin{array}{c} x(0) \\ y(0) \\ z(0) \end{array} \right ) = \left ( \begin{array}{c} 1 \\ 1 \\ 1 \end{array} \right )$ for the system in the previous exercise. Recall that

$\phi_{i+1} = \left ( \begin{array}{c} x(0) \\ y(0) \\ z(0) \end{array} \right ) + \int_0^t F(\phi_i(s)) \ ds$
Find a Lyapunov function for the system

$x_1' = -2 x_1 + x_2 - x_1^3$

$x_2' = - x_1 - 3 x_2$

$x_3' = - x_3$