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| [[File:A Trajectory Through Phase Space in a Lorenz Attractor.gif|frame|border|right|A sample solution in the Lorenz attractor when ρ = 28, σ = 10, and β = 8/3.]]
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| The '''Lorenz system''' is a system of [[ordinary differential equation]]s (the '''Lorenz equations''') first studied by [[Edward Norton Lorenz|Edward Lorenz]]. It is notable for having [[Chaos theory|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.
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| ==Overview==
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| In 1963, [[Edward Norton Lorenz|Edward Lorenz]] developed a simplified mathematical model for atmospheric convection.<ref name=lorenz>{{harvtxt|Lorenz|1963}}</ref> The model is a system of three ordinary differential equations now known as the Lorenz equations:
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| : <math> \begin{align}
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| \frac{\mathrm{d}x}{\mathrm{d}t} &= \sigma (y - x), \\
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| \frac{\mathrm{d}y}{\mathrm{d}t} &= x (\rho - z) - y, \\
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| \frac{\mathrm{d}z}{\mathrm{d}t} &= x y - \beta z.
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| \end{align} </math>
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| Here <math>x</math>, <math>y</math>, and <math>z</math> make up the system state, <math>t</math> is time, and <math>\sigma</math>, <math>\rho</math>, <math>\beta</math> are the system parameters. The Lorenz equations also arise in simplified models for [[laser]]s,<ref>{{harvtxt|Haken|1975}}</ref> [[electrical generator|dynamos]],<ref>{{harvtxt|Knobloch|1981}}</ref> thermosyphons,<ref>{{harvtxt|Gorman|Widmann|Robbins|1986}}</ref> brushless DC motors,<ref>{{harvtxt|Hemati|1994}}</ref> electric circuits,<ref>{{harvtxt|Cuomo|Oppenheim|1993}}</ref> and chemical reactions.<ref>{{harvtxt|Poland|1993}}</ref>
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| From a technical standpoint, the Lorenz system is [[nonlinearity|nonlinear]], three-dimensional and [[deterministic system (mathematics)|deterministic]]. The Lorenz equations have been the subject of at least one book length study.<ref>{{harvtxt|Sparrow|1982}}</ref>
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| ==Analysis==
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| One normally assumes that the parameters <math>\sigma</math>, <math>\rho</math>, and <math>\beta</math> are positive. Lorenz used the values <math>\sigma = 10</math>, <math>\beta = 8/3</math> and <math>\rho = 28 </math>. The system exhibits chaotic behavior for these values.<ref>{{harvtxt|Hirsch|Smale|Devaney|2003}}, pp. 303–305</ref>
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| If <math>\rho < 1</math> 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 <math>\rho < 1</math>.<ref>{{harvtxt|Hirsch|Smale|Devaney|2003}}, pp. 306+307</ref>
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| A [[pitchfork bifurcation]] occurs at <math>\rho = 1</math>, and for <math>\rho > 1 </math> two additional critical points appear at
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| :<math>\left( \pm\sqrt{\beta(\rho-1)}, \pm\sqrt{\beta(\rho-1)}, \rho-1 \right). </math>
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| These correspond to steady convection. This pair of equilibrium points is stable only if
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| :<math>\rho < \sigma\frac{\sigma+\beta+3}{\sigma-\beta-1}, </math>
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| which can hold only for positive <math>\rho</math> if <math>\sigma > \beta+1</math>. At the critical value, both equilibrium points lose stability through a [[Hopf bifurcation]].<ref>{{harvtxt|Hirsch|Smale|Devaney|2003}}, pp. 307+308</ref>
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| When <math>\rho = 28</math>, <math>\sigma = 10</math>, and <math>\beta = 8/3</math>, the Lorenz system has chaotic solutions (but not all solutions are chaotic). The set of chaotic solutions make up the Lorenz attractor, a [[Attractor#Strange attractor|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.<ref>{{harvtxt|Grassberger|Procaccia|1983}}</ref>
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| 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.<ref>{{harvtxt|Tucker|2002}}</ref>
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| For other values of <math>\rho</math>, the system displays knotted periodic orbits. For example, with <math>\rho = 99.96</math> it becomes a ''T''(3,2) [[torus knot]].
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| <blockquote>
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| {|class="wikitable" width=777px
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| ! colspan=2|Example solutions of the Lorenz system for different values of ρ
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| |align="center"|[[Image:Lorenz Ro14 20 41 20-200px.png]]
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| |align="center"|[[Image:Lorenz Ro13-200px.png]]
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| |align="center"|'''ρ=14, σ=10, β=8/3''' [[:Image:Lorenz Ro14 20 41 20.png|(Enlarge)]]
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| |align="center"|'''ρ=13, σ=10, β=8/3''' [[:Image:Lorenz Ro13.png|(Enlarge)]]
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| |align="center"|[[Image:Lorenz Ro15-200px.png]]
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| |align="center"|[[Image:Lorenz Ro28-200px.png]]
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| |align="center"|'''ρ=15, σ=10, β=8/3''' [[:Image:Lorenz Ro15.png|(Enlarge)]]
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| |align="center"|'''ρ=28, σ=10, β=8/3''' [[:Image:Lorenz Ro28.png|(Enlarge)]]
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| |-
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| |align="center" colspan=2| 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.
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| |align="center" colspan=2|[http://to-campos.planetaclix.pt/fractal/lorenz_eng.html Java animation showing evolution for different values of ρ]
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| |}
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| </blockquote>
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| <blockquote>
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| {|class="wikitable" width=777px
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| ! colspan=3| Sensitive dependence on the initial condition
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| |align="center"|'''Time t=1''' [[:Image:Lorenz caos1.png|(Enlarge)]]
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| |align="center"|'''Time t=2''' [[:Image:Lorenz caos2.png|(Enlarge)]]
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| |align="center"|'''Time t=3''' [[:Image:Lorenz caos3.png|(Enlarge)]]
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| |align="center"|[[Image:Lorenz caos1-175.png]]
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| |align="center"|[[Image:Lorenz caos2-175.png]]
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| |align="center"|[[Image:Lorenz caos3-175.png]]
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| |align="center" colspan=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<sup>-5</sup> 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.
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| |align="center" colspan=3| [http://to-campos.planetaclix.pt/fractal/lorenz_eng.html Java animation of the Lorenz attractor shows the continuous evolution.]
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| |}
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| </blockquote> | |
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| == Derivation of the Lorenz equations as a model of atmospheric convection ==
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| The Lorenz equations are derived from the [[Boussinesq approximation (buoyancy)|Oberbeck-Boussinesq approximation]] to the equations describing fluid circulation in a shallow layer of fluid, heated uniformly from below and cooled uniformly from above.<ref name=lorenz /> 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.
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| The partial differential equations modeling the system's [[stream function]] and [[temperature]] are subjected to a spectral [[Galerkin method|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.<ref>{{harvtxt|Hilborn|2000}}, Appendix C; {{harvtxt|Bergé|Pomeau|Vidal|1984}}, Appendix D</ref> The Lorenz system is a reduced version of a larger system studied earlier by Barry Saltzman.<ref>{{harvtxt|Saltzman|1962}}</ref>
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| == Gallery ==
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| <gallery>
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| File:Lorenz system r28 s10 b2-6666.png|A solution in the Lorenz attractor plotted at high resolution in the x-z plane.
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| File:Lorenz attractor.svg|A solution in the Lorenz attractor rendered as an SVG.
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| File:A Lorenz system.ogv|An animation showing trajectories of multiple solutions in a Lorenz system.
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| File:Lorenzstill-rubel.png|A solution in the Lorenz attractor rendered as a metal wire to show direction and [[Three-dimensional space|3D]] structure.
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| File:Lorenz.ogv|An animation showing the divergence of nearby solutions to the Lorenz system.
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| File:Intermittent Lorenz Attractor - Chaoscope.jpg|A visualization of the Lorenz attractor near an intermittent cycle.
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| </gallery>
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| == See also ==
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| * [[List of chaotic maps]]
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| * [[Takens' theorem]]
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| == Notes ==
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| {{reflist|2}}
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| == References ==
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| * {{cite book | last1=Bergé | first1=Pierre | last2=Pomeau | first2=Yves | last3=Vidal | first3=Christian | title=Order within Chaos: Towards a Deterministic Approach to Turbulence | publisher=[[John Wiley & Sons]] | location=New York | isbn=978-0-471-84967-4 | year=1984 | ref=harv}}
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| * {{cite journal | last1=Cuomo | first1=Kevin M. | last2=Oppenheim | first2=Alan V. | author2-link=Alan V. Oppenheim | title=Circuit implementation of synchronized chaos with applications to communications | doi=10.1103/PhysRevLett.71.65 | year=1993 | journal=[[Physical Review Letters]] | issn=0031-9007 | volume=71 | issue=1 | pages=65–68 | ref=harv|bibcode = 1993PhRvL..71...65C }}
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| * {{cite journal | last1=Gorman | first1=M. | last2=Widmann | first2=P.J. | last3=Robbins | first3=K.A. | title=Nonlinear dynamics of a convection loop: A quantitative comparison of experiment with theory | doi=10.1016/0167-2789(86)90022-9 | year=1986 | journal=Physica D | volume=19 | issue=2 | pages=255–267 | ref=harv|bibcode = 1986PhyD...19..255G }}
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| * {{cite journal | last1=Grassberger | first1=P. | last2=Procaccia | first2=I. | title=Measuring the strangeness of strange attractors | journal=Physica D | year = 1983 | volume = 9 | issue=1–2 | pages=189–208 | bibcode = 1983PhyD....9..189G | doi = 10.1016/0167-2789(83)90298-1 | ref=harv}}
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| * {{cite journal | last1=Haken | first1=H. | title=Analogy between higher instabilities in fluids and lasers | doi=10.1016/0375-9601(75)90353-9 | year=1975 | journal=[[Physics Letters A]] | volume=53 | issue=1 | pages=77–78 | ref=harv|bibcode = 1975PhLA...53...77H }}
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| * {{cite journal | last1=Hemati | first1=N. | title=Strange attractors in brushless DC motors | doi=10.1109/81.260218 | year=1994 | journal=IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications | issn=1057-7122 | volume=41 | issue=1 | pages=40–45 | ref=harv}}
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| * {{cite book | last1=Hilborn | first1=Robert C. | title=Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers | publisher=[[Oxford University Press]] | edition=second | isbn=978-0-19-850723-9 | year=2000 | ref=harv}}
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| * {{cite book | last1=Hirsch | first1=Morris W. | author1-link=Morris Hirsch | last2=Smale | first2=Stephen | author2-link=Stephen Smale | last3=Devaney | first3=Robert | title=Differential Equations, Dynamical Systems, & An Introduction to Chaos | publisher=[[Academic Press]] | location=Boston, MA | edition=Second | isbn=978-0-12-349703-1 | year=2003 | ref=harv}}
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| * {{cite journal | last1=Lorenz | first1=Edward Norton | author1-link=Edward Norton Lorenz | title=Deterministic nonperiodic flow | journal=Journal of the Atmospheric Sciences | year=1963 | volume=20 | issue=2| pages=130–141 | doi=10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 | ref=harv | bibcode=1963JAtS...20..130L}}
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| * {{cite journal | last1=Knobloch | first1=Edgar | title=Chaos in the segmented disc dynamo | doi=10.1016/0375-9601(81)90274-7 | year=1981 | journal=Physics Letters A | volume=82 | issue=9 | pages=439–440 | ref=harv |bibcode = 1981PhLA...82..439K }}
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| * {{cite journal | last1=Poland | first1=Douglas | title=Cooperative catalysis and chemical chaos: a chemical model for the Lorenz equations | doi=10.1016/0167-2789(93)90006-M | year=1993 | journal=Physica D | volume=65 | issue=1 | pages=86–99 | ref=harv|bibcode = 1993PhyD...65...86P }}
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| * {{cite journal | last1=Saltzman | first1=Barry | title=Finite Amplitude Free Convection as an Initial Value Problem—I | year=1962 | journal=Journal of the Atmospheric Sciences | volume=19 | issue=4 | pages=329–341 | ref=harv|bibcode = 1962JAtS...19..329S |doi = 10.1175/1520-0469(1962)019<0329:FAFCAA>2.0.CO;2 }}
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| * {{cite book | last1=Sparrow | first1=Colin | title=The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors | publisher=Springer | year=1982 | ref=harv}}
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| * {{cite journal | last1=Tucker | first1=Warwick | title=A Rigorous ODE Solver and Smale's 14th Problem | journal=Foundations of Computational Mathematics | volume=2 | issue=1 | year=2002 | pages=53–117 | doi=10.1007/s002080010018 | url=http://www.math.cornell.edu/~warwick/main/rodes/JFoCM.pdf | ref=harv}}
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| == External links ==
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| {{commons category|Lorenz attractors}}
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| * {{springer|title=Lorenz attractor|id=p/l060890}}
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| * {{MathWorld|urlname=LorenzAttractor|title=Lorenz attractor}}
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| * [http://demonstrations.wolfram.com/LorenzAttractor/ Lorenz attractor] by Rob Morris, [[Wolfram Demonstrations Project]].
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| * [http://planetmath.org/encyclopedia/LorenzEquation.html Lorenz equation] on planetmath.org
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| * [http://video.google.com/videoplay?docid=2875296564158834562&q=strogatz&ei=xr9OSJ_SOpeG2wKB3Iy2DA&hl=en Synchronized Chaos and Private Communications, with Kevin Cuomo]. The implementation of Lorenz attractor in an electronic circuit.
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| * [http://toxi.co.uk/lorenz/ Lorenz attractor interactive animation] (you need the Adobe Shockwave plugin)
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| * [http://amath.colorado.edu/faculty/juanga/3DAttractors.html 3D Attractors: Mac program to visualize and explore the Lorenz attractor in 3 dimensions]
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| * [http://frank.harvard.edu/~paulh/misc/lorenz.htm Lorenz Attractor implemented in analog electronic]
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| * [http://sourceforge.net/projects/lorenz/ Lorenz Attractor interactive animation] (implemented in Ada with GTK+. Sources & executable)
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| * [http://highfellow.github.com/lorenz-attractor/attractor.html Web based Lorenz Attractor] (implemented in javascript / html /css)
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| {{Chaos theory}}
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| [[Category:Chaotic maps]]
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