|
|
| Line 1: |
Line 1: |
| [[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.]]
| | Hello. Allow me introduce the author. Her title is Refugia Shryock. I utilized to be unemployed but now I am a librarian and the salary has been really satisfying. The factor she adores most is body developing and now she is attempting to earn money with it. Minnesota has usually been his house but his wife desires them to move.<br><br>my web site: [http://www.hotporn123.com/blog/154316 hotporn123.com] |
| 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.
| |
| | |
| ==Overview==
| |
| 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:
| |
| : <math> \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} </math>
| |
| | |
| 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>
| |
| | |
| 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>
| |
| | |
| ==Analysis==
| |
| 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>
| |
| | |
| 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>
| |
| | |
| A [[pitchfork bifurcation]] occurs at <math>\rho = 1</math>, and for <math>\rho > 1 </math> two additional critical points appear at
| |
| :<math>\left( \pm\sqrt{\beta(\rho-1)}, \pm\sqrt{\beta(\rho-1)}, \rho-1 \right). </math>
| |
| These correspond to steady convection. This pair of equilibrium points is stable only if
| |
| :<math>\rho < \sigma\frac{\sigma+\beta+3}{\sigma-\beta-1}, </math>
| |
| 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>
| |
| | |
| 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>
| |
| | |
| 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>
| |
| | |
| 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]].
| |
| | |
| <blockquote>
| |
| {|class="wikitable" width=777px
| |
| |-
| |
| ! colspan=2|Example solutions of the Lorenz system for different values of ρ
| |
| |-
| |
| |align="center"|[[Image:Lorenz Ro14 20 41 20-200px.png]]
| |
| |align="center"|[[Image:Lorenz Ro13-200px.png]]
| |
| |-
| |
| |align="center"|'''ρ=14, σ=10, β=8/3''' [[:Image:Lorenz Ro14 20 41 20.png|(Enlarge)]]
| |
| |align="center"|'''ρ=13, σ=10, β=8/3''' [[:Image:Lorenz Ro13.png|(Enlarge)]]
| |
| |-
| |
| |align="center"|[[Image:Lorenz Ro15-200px.png]]
| |
| |align="center"|[[Image:Lorenz Ro28-200px.png]]
| |
| |-
| |
| |align="center"|'''ρ=15, σ=10, β=8/3''' [[:Image:Lorenz Ro15.png|(Enlarge)]]
| |
| |align="center"|'''ρ=28, σ=10, β=8/3''' [[:Image:Lorenz Ro28.png|(Enlarge)]]
| |
| |-
| |
| |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.
| |
| |-
| |
| |align="center" colspan=2|[http://to-campos.planetaclix.pt/fractal/lorenz_eng.html Java animation showing evolution for different values of ρ]
| |
| |}
| |
| </blockquote> | |
| | |
| <blockquote> | |
| {|class="wikitable" width=777px
| |
| |-
| |
| ! colspan=3| Sensitive dependence on the initial condition
| |
| |-
| |
| |align="center"|'''Time t=1''' [[:Image:Lorenz caos1.png|(Enlarge)]]
| |
| |align="center"|'''Time t=2''' [[:Image:Lorenz caos2.png|(Enlarge)]]
| |
| |align="center"|'''Time t=3''' [[:Image:Lorenz caos3.png|(Enlarge)]]
| |
| |-
| |
| |align="center"|[[Image:Lorenz caos1-175.png]]
| |
| |align="center"|[[Image:Lorenz caos2-175.png]]
| |
| |align="center"|[[Image:Lorenz caos3-175.png]]
| |
| |-
| |
| |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.
| |
| |-
| |
| |align="center" colspan=3| [http://to-campos.planetaclix.pt/fractal/lorenz_eng.html Java animation of the Lorenz attractor shows the continuous evolution.]
| |
| |}
| |
| </blockquote>
| |
| | |
| == Derivation of the Lorenz equations as a model of atmospheric convection ==
| |
| 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.
| |
| | |
| 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>
| |
| | |
| == Gallery ==
| |
| <gallery>
| |
| File:Lorenz system r28 s10 b2-6666.png|A solution in the Lorenz attractor plotted at high resolution in the x-z plane.
| |
| File:Lorenz attractor.svg|A solution in the Lorenz attractor rendered as an SVG.
| |
| File:A Lorenz system.ogv|An animation showing trajectories of multiple solutions in a Lorenz system.
| |
| File:Lorenzstill-rubel.png|A solution in the Lorenz attractor rendered as a metal wire to show direction and [[Three-dimensional space|3D]] structure.
| |
| File:Lorenz.ogv|An animation showing the divergence of nearby solutions to the Lorenz system.
| |
| File:Intermittent Lorenz Attractor - Chaoscope.jpg|A visualization of the Lorenz attractor near an intermittent cycle.
| |
| </gallery>
| |
| | |
| == See also ==
| |
| * [[List of chaotic maps]]
| |
| * [[Takens' theorem]]
| |
| | |
| == Notes ==
| |
| {{reflist|2}}
| |
| | |
| == References ==
| |
| * {{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}}
| |
| * {{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 }}
| |
| * {{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 }}
| |
| * {{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}}
| |
| * {{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 }}
| |
| * {{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}}
| |
| * {{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}}
| |
| * {{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}}
| |
| * {{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}}
| |
| * {{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 }}
| |
| * {{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 }}
| |
| * {{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 }}
| |
| * {{cite book | last1=Sparrow | first1=Colin | title=The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors | publisher=Springer | year=1982 | ref=harv}}
| |
| * {{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}}
| |
| | |
| == External links ==
| |
| {{commons category|Lorenz attractors}}
| |
| * {{springer|title=Lorenz attractor|id=p/l060890}}
| |
| * {{MathWorld|urlname=LorenzAttractor|title=Lorenz attractor}}
| |
| * [http://demonstrations.wolfram.com/LorenzAttractor/ Lorenz attractor] by Rob Morris, [[Wolfram Demonstrations Project]].
| |
| * [http://planetmath.org/encyclopedia/LorenzEquation.html Lorenz equation] on planetmath.org
| |
| * [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.
| |
| * [http://toxi.co.uk/lorenz/ Lorenz attractor interactive animation] (you need the Adobe Shockwave plugin)
| |
| * [http://amath.colorado.edu/faculty/juanga/3DAttractors.html 3D Attractors: Mac program to visualize and explore the Lorenz attractor in 3 dimensions]
| |
| * [http://frank.harvard.edu/~paulh/misc/lorenz.htm Lorenz Attractor implemented in analog electronic]
| |
| * [http://sourceforge.net/projects/lorenz/ Lorenz Attractor interactive animation] (implemented in Ada with GTK+. Sources & executable)
| |
| * [http://highfellow.github.com/lorenz-attractor/attractor.html Web based Lorenz Attractor] (implemented in javascript / html /css)
| |
| | |
| {{Chaos theory}}
| |
| | |
| [[Category:Chaotic maps]]
| |