Everything about The Lorenz Attractor totally explained
The
Lorenz attractor, named for
Edward N. Lorenz, is a 3-dimensional structure corresponding to the long-term behavior of a
chaotic flow, noted for its
butterfly shape. The map shows how the state of a
dynamical system (the three variables of a three-dimensional system) evolves over time in a complex, non-repeating pattern.
Overview
The attractor itself, and the equations from which it's derived, were introduced by
Edward Lorenz in
1963, who derived it from the simplified equations of
convection rolls arising in the equations of the
atmosphere.
From a technical standpoint, the system is
nonlinear, three-dimensional and
deterministic. In 2001 it was proven by Warwick Tucker that for a certain set of parameters the system exhibits chaotic behavior and displays what is today called a
strange attractor. The strange attractor in this case is a
fractal of
Hausdorff dimension between 2 and 3. Grassberger (1983) has estimated the Hausdorff dimension to be 2.06 ± 0.01 and the
correlation dimension to be 2.05 ± 0.01.
The system arises in
lasers,
dynamos, and specific waterwheels.
Equations
The equations that govern the Lorenz attractor are:
»
where
is called the
Prandtl number and
is called the
Rayleigh number. All
,
,
> 0, but usually
= 10,
= 8/3 and
is varied. The system exhibits chaotic behavior for
= 28 but displays knotted periodic orbits for other values of
. For example, with
it becomes a
T(3,2)
torus knot.
Butterfly effect
| Butterfly effect |
| Time t=1 |
Time t=2 |
Time t=3 |
|
|
|
| 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's drawn over the blue one) but, after some time, the divergence is obvious. |
Java animation of the Lorenz attractor shows the continuous evolution. |
Rayleigh number
| The Lorenz attractor for different values of ρ |
|
|
| ρ=14, σ=10, β=8/3 |
ρ=13, σ=10, β=8/3 |
|
|
| ρ=15, σ=10, β=8/3 |
ρ=28, σ=10, β=8/3 |
| For small values of ρ, the system is stable and evolves to one of two fixed point attractors. When ρ is larger than 24.28, the fixed points become repulsors and the trajectory is repelled by them in a very complex way, evolving without ever crossing itself. |
| Java animation showing evolution for different values of ρ |
Further Information
Get more info on 'Lorenz Attractor'.
|
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