Lorenz system

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A sample solution in the Lorenz attractor when ρ = 28, σ = 10, and β = 8/3

The Lorenz system is a system of ordinary differential equations (the Lorenz equations) first studied by Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the Lorenz attractor is a set of chaotic solutions of the Lorenz system which, when plotted, resemble a butterfly or figure eight.

Overview

In 1963, Edward Lorenz developed a simplified mathematical model for atmospheric convection.[1] The model is a system of three ordinary differential equations now known as the Lorenz equations:

\begin{align} \frac{\mathrm{d}x}{\mathrm{d}t} &= \sigma (y - x), \\ \frac{\mathrm{d}y}{\mathrm{d}t} &= x (\rho - z) - y, \\ \frac{\mathrm{d}z}{\mathrm{d}t} &= x y - \beta z. \end{align}

Here $x$, $y$, and $z$ make up the system state, $t$ is time, and $\sigma$, $\rho$, $\beta$ are the system parameters. The Lorenz equations also arise in simplified models for lasers,[2] dynamos,[3] thermosyphons,[4] brushless DC motors,[5] electric circuits,[6] chemical reactions[7] and forward osmosis.[8]

From a technical standpoint, the Lorenz system is nonlinear, three-dimensional and deterministic. The Lorenz equations have been the subject of at least one book length study.[9]

Analysis

One normally assumes that the parameters $\sigma$, $\rho$, and $\beta$ are positive. Lorenz used the values $\sigma = 10$, $\beta = 8/3$ and $\rho = 28$. The system exhibits chaotic behavior for these values.[10]

If $\rho < 1$ then there is only one equilibrium point, which is at the origin. This point corresponds to no convection. All orbits converge to the origin when $\rho < 1$.[11]

A pitchfork bifurcation occurs at $\rho = 1$, and for $\rho > 1$ two additional critical points appear at

$\left( \pm\sqrt{\beta(\rho-1)}, \pm\sqrt{\beta(\rho-1)}, \rho-1 \right).$

These correspond to steady convection. This pair of equilibrium points is stable only if

$\rho < \sigma\frac{\sigma+\beta+3}{\sigma-\beta-1},$

which can hold only for positive $\rho$ if $\sigma > \beta+1$. At the critical value, both equilibrium points lose stability through a Hopf bifurcation.[12]

When $\rho = 28$, $\sigma = 10$, and $\beta = 8/3$, the Lorenz system has chaotic solutions (but not all solutions are chaotic). The set of chaotic solutions make up the Lorenz attractor, a strange attractor and a fractal with a Hausdorff dimension which is estimated to be 2.06 ± 0.01 and the correlation dimension estimated to be 2.05 ± 0.01.[13]

The Lorenz attractor is difficult to analyze, but the action of the differential equation on the attractor is described by a fairly simple geometric model. Proving that this is indeed the case is the fourteenth problem on the list of Smale's problems. This problem was the first one to be resolved, by Warwick Tucker in 2002.[14]

For other values of $\rho$, the system displays knotted periodic orbits. For example, with $\rho = 99.96$ it becomes a T(3,2) torus knot.

Example solutions of the Lorenz system for different values of ρ
ρ=14, σ=10, β=8/3 (Enlarge) ρ=13, σ=10, β=8/3 (Enlarge)
ρ=15, σ=10, β=8/3 (Enlarge) ρ=28, σ=10, β=8/3 (Enlarge)
For small values of ρ, the system is stable and evolves to one of two fixed point attractors. When ρ is larger than 24.74, the fixed points become repulsors and the trajectory is repelled by them in a very complex way.
Java animation showing evolution for different values of ρ
Sensitive dependence on the initial condition
Time t=1 (Enlarge) Time t=2 (Enlarge) Time t=3 (Enlarge)
These figures — made using ρ=28, σ = 10 and β = 8/3 — show three time segments of the 3-D evolution of 2 trajectories (one in blue, the other in yellow) in the Lorenz attractor starting at two initial points that differ only by 10-5 in the x-coordinate. Initially, the two trajectories seem coincident (only the yellow one can be seen, as it is drawn over the blue one) but, after some time, the divergence is obvious.
Java animation of the Lorenz attractor shows the continuous evolution.

Derivation of the Lorenz equations as a model of atmospheric convection

The Lorenz equations are derived from the Oberbeck-Boussinesq approximation to the equations describing fluid circulation in a shallow layer of fluid, heated uniformly from below and cooled uniformly from above.[1] This fluid circulation is known as Rayleigh-Bénard convection. The fluid is assumed to circulate in two dimensions (vertical and horizontal) with periodic rectangular boundary conditions.

The partial differential equations modeling the system's stream function and temperature are subjected to a spectral Galerkin approximation: the hydrodynamic fields are expanded in Fourier series, which are then severely truncated to a single term for the stream function and two terms for the temperature. This reduces the model equations to a set of three coupled, nonlinear ordinary differential equations. A detailed derivation may be found, for example, in nonlinear dynamics texts.[15] The Lorenz system is a reduced version of a larger system studied earlier by Barry Saltzman.[16]

Notes

1. ^ a b
2. ^
3. ^
4. ^
5. ^
6. ^
7. ^
8. ^
9. ^
10. ^ Hirsch, Smale & Devaney (2003), pp. 303­–305
11. ^ Hirsch, Smale & Devaney (2003), pp. 306+307
12. ^ Hirsch, Smale & Devaney (2003), pp. 307+308
13. ^
14. ^
15. ^ Hilborn (2000), Appendix C; Bergé, Pomeau & Vidal (1984), Appendix D
16. ^