 A plot of the trajectory Lorenz system for values ρ=28, σ = 10, β = 8/3
 A trajectory of Lorenz's equations, rendered as a metal wire to show direction and three-dimensional structure The Lorenz attractor is a chaotic map, 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, often described as beautiful. Image File history File links Lorenz_attractor_yb. ...
Image File history File links Lorenz_attractor_yb. ...
A projection of the three-dimensional Lorenz attractor. ...
Image File history File links Download high-resolution version (1920x1440, 408 KB) Finite segment of a trajectory of Lorenzs equations, computed by numerical integration and rendered as a metal wire. ...
Image File history File links Download high-resolution version (1920x1440, 408 KB) Finite segment of a trajectory of Lorenzs equations, computed by numerical integration and rendered as a metal wire. ...
A plot of the trajectory Lorenz system for values r = 28, σ = 10, b = 8/3 In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under certain conditions exhibit a phenomenon known as chaos. ...
Families Superfamily Hesperioidea: Hesperiidae Superfamily Papilionoidea: Papilionidae Pieridae Nymphalidae Lycaenidae Riodinidae A butterfly is an insect of the order Lepidoptera, it belongs to either the Hesperioidea (the skippers) or Papilionoidea (all other butterflies) Superfamilies. ...
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. ...
Most mathematicians derive aesthetic pleasure from their work, and from mathematics in general. ...
The attractor itself, and the equations from which it is 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. Dr. Lorenz at work Edward Norton Lorenz is an American mathematician and meteorologist, and a contributor to the chaos theory and inventor of the strange attractor notion. ...
1963 (MCMLXIII) was a common year starting on Tuesday (the link is to a full 1963 calendar). ...
Convection is the internal movement of currents within fluids (i. ...
Layers of Atmosphere (NOAA) Air redirects here. ...
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. In mathematics, nonlinear systems represent systems whose behavior is not expressible as a sum of the behaviors of its descriptors. ...
In mathematics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. ...
In dynamical systems, an attractor is a set to which the system evolves after a long enough time. ...
The boundary of the Mandelbrot set is a famous example of a fractal. ...
In mathematics, the Hausdorff dimension is an extended non-negative real number (that is a number in the closed infinite interval [0, ∞]) associated to any metric space . ...
In chaos theory the correlation dimension (denoted by ν) is a measure of the dimensionality of the space occupied by a set of random points. ...
The system arises in lasers, dynamos, and specific waterwheels [1]. A laser (acronym for light amplification by stimulated emission of radiation) is an optical source that emits photons in a coherent beam. ...
An electrical generator is a device that converts mechanical energy to electrical energy, generally using electromagnetic induction. ...
The equations that govern the Lorenz attractor are:
   where σ is called the Prandtl number and ρ is called the Rayleigh number. All σ, ρ, β > 0, but usually σ = 10, β = 8/3 and ρ is varied. The system exhibits chaotic behavior for ρ = 28 but displays knotted periodic orbits for other values of ρ. For example, with ρ = 99.96 it becomes a T(3,2) torus knot. The Prandtl Number is a dimensionless number approximating the ratio of momentum diffusivity and thermal diffusivity. ...
In fluid mechanics, the Rayleigh number for a fluid is a dimensionless number associated with the heat transfer within the fluid. ...
A (3,7)-3D torus knot rendered by Apple Grapher. ...
The butterfly effect in the Lorenz attractor
| Butterfly effect | | Time t=1
(larger) | Time t=2
(larger) | Time t=3
(larger) |
 |
 |
 | | 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. | | A Java animation of the Lorenz attractor shows the continuous evolution. | Point attractors in 2D phase space. ...
Image File history File links Lorenz_caos1. ...
Image File history File links Lorenz_caos2. ...
Image File history File links Lorenz_caos3. ...
Image File history File links Lorenz_caos1-175. ...
Image File history File links Lorenz_caos2-175. ...
Image File history File links Lorenz_caos3-175. ...
Using different values for the Rayleigh number | The Lorenz attractor for different values of ρ |
 |
 | | ρ=14, σ=10, β=8/3
(larger) | ρ=13, σ=10, β=8/3
(larger) |
 |
 | | ρ=15, σ=10, β=8/3
(larger) | ρ=28, σ=10, β=8/3
(larger) | | 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, evolving without ever crossing itself. | | Java animation showing evolution for different values of ρ | Image File history File links Lorenz_Ro14_20_41_20-200px. ...
Image File history File links Lorenz_Ro13-200px. ...
Image File history File links Lorenz_Ro14_20_41_20. ...
Image File history File links Lorenz_Ro13. ...
Image File history File links Lorenz_Ro15-200px. ...
Image File history File links Lorenz_Ro28-200px. ...
Image File history File links Lorenz_Ro15. ...
Image File history File links Lorenz_Ro28. ...
See also In mathematics, a chaotic map is a map that exhibits some sort of chaotic behavior. ...
In mathematics, Takens delay embedding theorem is a result of Floris Takens on the embedding dimension of nonlinear (chaotic) systems. ...
References - Lorenz, E. N. (1963). "Deterministic nonperiodic flow". J. Atmos. Sci. 20: 130-141. DOI:10.1175/1520-0469(1963)020%3C0130:DNF%3E2.0.CO;2.
- Frøyland, J., Alfsen, K. H. (1984). "Lyapunov-exponent spectra for the Lorenz model". Phys. Rev. A 29: 2928–2931.
- Tucker, W. (2002). "A Rigorous ODE Solver and Smale's 14th Problem". Found. Comp. Math. 2: 53-117.
- Strogatz, Steven H. (1994). Nonlinear Systems and Chaos. Perseus publishing.
- Jonas Bergman, Knots in the Lorentz system, Undergraduate thesis, Uppsala University 2004.
- P. Grassberger and I. Procaccia (1983). "Measuring the strangeness of strange attractors". Physica D 9: 189-208. DOI:10.1016/0167-2789(83)90298-1.
A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ...
A digital object identifier (or DOI) is a standard for persistently identifying a piece of intellectual property on a digital network and associating it with related data, the metadata, in a structured extensible way. ...
External links |
Feeds |
Contact