Analysis
One normally assumes that the parameters, and are positive. Lorenz used the values, and . The system exhibits chaotic behavior for these values.
If 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 .
A saddle-node bifurcation occurs at, and for two additional critical points appear at
These correspond to steady convection. This pair of equilibrium points is stable only if
which can hold only for positive if . At the critical value, both equilibrium points lose stability through a Hopf bifurcation.
When, and, 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.
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.
For other values of, the system displays knotted periodic orbits. For example, with 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. | ||
Read more about this topic: Lorenz System
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