Research in Scientific Computing in Undergraduate Education

Edward Lorenz’s equations and the Lorenz attractor

Edward Lorenz (born in New England – West Hartford, Connecticut in 1917, and died in April 2008 in Cambridge, Massachusetts, aged 90) set up a simplified model of convection rolls arising in the equations of the atmosphere, in 1963.

Edward Lorenz

His simplified model involves 3 linked differential equations of 3 variables. Specifically, the equations are:

\frac{dx}{dt} = \sigma (y - x)

\frac{dy}{dt} = x (\rho - z) - y

\frac{dz}{dt} = xy - \beta z

In these equations, all parameters \sigma,\rho, \beta are positive.

The parameters \sigma and \rho   are related to physical properties of a fluid in motion: kinematic viscosity and heat flow (the Prandtl number and Rayleigh number, respectively). In these equations it is common to take \sigma = 10, \beta = 8/3.

Different values of \rho give qualitatively different solution curves for this system of equations. For example, for \rho = 28 the solution is chaotic, meaning that nearby points diverge approximately exponentially rapidly (so prediction with time is difficult to impossible). The solution curves for these values form an object in 3-space that is known as the Lorentz attractor:

The Lorenz attractor

For other values of \rho the system has solutions that are periodic, but knotted in space.  A classic example of a knotted solution occurs when \rho = 99.96

The study of knotted periodic solutions to the Lorenz equations was initiated by Birman and Williams in 1983: J. S. Birman and R. F. Williams, Knotted Periodic Orbits in Dynamical Systems-I: Lorenz’s Equations, Topology 22, No. 1, 47-82, 1983. A further study was conducted by R. Bedient in 1985: R. Bedient, Classifying 3-trip Lorenz knots, Topology and its Applications 20 (1985), 89-96. Recently (2008) Birman and Kofman have carried out a detailed study of which torus knots are embedded in the Lorenz attractor (lorenz_knots_birman_kofman).

A very beautiful article on Lorenz knots is Lorenz and Modular Flows: A Visual Introduction by Etienne Ghysand Jos Leys.

A knot that is embdedded as a periodic orbit in the Lorenz attractor

A knot that is embedded as a periodic orbit in the Lorenz attractor

A different study study of periodic knotted solutions to the Lorenz equations was also undertaken by Jonas Bergman at Uppsala University when he was an undergraduate in 2004 (jonas_bergman_knots_lorenz). Bergman’s idea is to look for periodic knotted solutions to the Lorenz equations for higher Rayleigh numbers. His abstract follows:

“The properties of the periodic solutions to the Lorenz equations are investigated and summarized. By varying the Raleigh number r, a parameter in the equations, both chaotic and periodic behavior is observed. The knots corresponding to the different periodic solutions of the system are determined and classified. In particular the hyperbolic figure eight knot is searched for to collect information about when and how it appears. By finding this knot the suspicion that there exists a new route to chaos would be strengthened. Unfortunately this knot was not found. However, two other, more complicated, hyperbolic knots were found. This fact strengthens the suspicion that the figure eight knot exists as a solution to the Lorenz equations. This, in turn, could be a sign of the existence of a route to chaos other than those known today. Several torus knots were also found and classified. “

Bergman’s conclusion is:

“Since the investigation of knots in the Lorenz equations for higher parameter values is limited and no general approach is known, one needs to proceed either by constructing templates for higher parameter values or to numerically investigate the Lorenz equations to be able to determine what knots are embedded in the equations. “

This is a very nice piece of undergraduate research, that opens new avenues of investigation.

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