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由moyuefeng创建，尚未翻译（2021.11.07）
{{Short description|System of ordinary differential equations with chaotic solutions}}
{{Distinguish|Lorenz curve|Lorentz distribution}}
[[File:A Trajectory Through Phase Space in a Lorenz Attractor.gif|frame|right|A sample solution in the Lorenz attractor when ρ = 28, σ = 10, and β = 8/3]]
The '''Lorenz system''' is a system of [[ordinary differential equation]]s 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. In popular media the "[[butterfly effect]]" stems from the real-world implications of the Lorenz attractor, i.e. that in any physical system, in the absence of perfect knowledge of the initial conditions (even the minuscule disturbance of the air due to a butterfly flapping its wings), our ability to predict its future course will always fail. This underscores that physical systems can be completely deterministic and yet still be inherently unpredictable even in the absence of quantum effects. The shape of the Lorenz attractor itself, when plotted graphically, may also be seen to resemble a butterfly.
==Overview==
In 1963, [[Edward Norton Lorenz|Edward Lorenz]], with the help of [[Ellen Fetter]], 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), \\[6pt]
\frac{\mathrm{d}y}{\mathrm{d}t} &= x (\rho - z) - y, \\[6pt]
\frac{\mathrm{d}z}{\mathrm{d}t} &= x y - \beta z.
\end{align} </math>
The equations relate the properties of a two-dimensional fluid layer uniformly warmed from below and cooled from above. In particular, the equations describe the rate of change of three quantities with respect to time: <math>x</math> is proportional to the rate of convection, <math>y</math> to the horizontal temperature variation, and <math>z</math> to the vertical temperature variation.<ref name="Sparrow 1982">{{harvtxt|Sparrow|1982}}</ref> The constants <math>\sigma</math>, <math>\rho</math>, and <math>\beta</math> are system parameters proportional to the [[Prandtl number]], [[Rayleigh number]], and certain physical dimensions of the layer itself.<ref name="Sparrow 1982"/>
The Lorenz equations also arise in simplified models for [[laser]]s,<ref>{{harvtxt|Haken|1975}}</ref> [[electrical generator|dynamos]],<ref>{{harvtxt|Knobloch|1981}}</ref> [[thermosyphon]]s,<ref>{{harvtxt|Gorman|Widmann|Robbins|1986}}</ref> brushless [[DC motor]]s,<ref>{{harvtxt|Hemati|1994}}</ref> [[electric circuit]]s,<ref>{{harvtxt|Cuomo|Oppenheim|1993}}</ref> [[chemical reaction]]s<ref>{{harvtxt|Poland|1993}}</ref> and [[forward osmosis]].<ref>{{harvtxt|Tzenov|2014}}{{citation needed|date=June 2017<!--doesn't point anywhere-->}}</ref> The Lorenz equations are also the governing equations in Fourier space for the [[Malkus waterwheel]].<ref>{{harvtxt|Kolář|Gumbs|1992}}</ref><ref>{{harvtxt|Mishra|Sanghi|2006}}</ref> The Malkus waterwheel exhibits chaotic motion where instead of spinning in one direction at a constant speed, its rotation will speed up, slow down, stop, change directions, and oscillate back and forth between combinations of such behaviors in an unpredictable manner.
From a technical standpoint, the Lorenz system is [[nonlinearity|nonlinear]], non-periodic, three-dimensional and [[deterministic system (mathematics)|deterministic]]. The Lorenz equations have been the subject of hundreds of research articles, and at least one book-length study.<ref name="Sparrow 1982"/>
==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 (and nearby) 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, which is a global [[attractor]], 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( \sqrt{\beta(\rho-1)}, \sqrt{\beta(\rho-1)}, \rho-1 \right) </math> and <math>\left( -\sqrt{\beta(\rho-1)}, -\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 subcritical [[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). Almost all initial points will tend to an invariant set{{snd}}the Lorenz attractor{{snd}}a [[Attractor#Strange attractor|strange attractor]], a [[fractal]], and a [[Hidden attractor#Self-excited attractors|self-excited attractor]] with respect to all three equilibria. Its [[Hausdorff dimension]] is estimated from above by the [[Lyapunov dimension|Lyapunov dimension (Kaplan-Yorke dimension)]] as 2.06 ± 0.01,<ref name=Kuznetsov-2020-ND>{{Cite journal |
first1=N.V. |last1=Kuznetsov| first2=T.N. |last2=Mokaev | first3=O.A. |last3=Kuznetsova | first4=E.V. |last4=Kudryashova|
title=The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension|
journal= Nonlinear Dynamics|year=2020 |
doi=10.1007/s11071-020-05856-4| url=https://link.springer.com/article/10.1007/s11071-020-05856-4|doi-access=free }}
</ref> and the [[correlation dimension]] is estimated to be 2.05 ± 0.01.<ref>{{harvtxt|Grassberger|Procaccia|1983}}</ref>
The exact Lyapunov dimension formula of the global attractor can be found analytically under classical restrictions on the parameters:<ref>{{harvtxt|Leonov|Kuznetsov|Korzhemanova|Kusakin|2016}}</ref><ref name=Kuznetsov-2020-ND/><ref name=2020-KuznetsovR>{{cite book | first1= Nikolay | last1=Kuznetsov |
first2=Volker | last2=Reitmann | year = 2020| title = Attractor Dimension Estimates for Dynamical Systems: Theory and Computation|
publisher = Springer| location = Cham|url=https://www.springer.com/gp/book/9783030509866}}</ref>
<math> 3 - \frac{2 (\sigma + \beta + 1)}{\sigma + 1 + \sqrt{(\sigma-1)^2 + 4 \sigma \rho}}. </math>
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.<ref>{{Cite journal|title = Structural stability of Lorenz attractors|journal = Publications Mathématiques de l'Institut des Hautes Études Scientifiques|date = 1979-12-01|issn = 0073-8301|pages = 59–72|volume = 50|issue = 1|doi = 10.1007/BF02684769|first = John|last = Guckenheimer|first2 = R. F.|last2 = Williams|url = http://www.numdam.org/item/PMIHES_1979__50__59_0/}}</ref> 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 name="Tucker 2002">{{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]].
{|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.
|}
{|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 two 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.
|}
==Connection to tent map==
[[File:Lorenz_Map.png|thumb|A recreation of Lorenz's results created on [[Mathematica]]. Points above the red line correspond to the system switching lobes.|upright=1.3]]
In Figure 4 of his paper,<ref name=lorenz /> Lorenz plotted the relative maximum value in the z direction achieved by the system against the previous relative maximum in the z direction. This procedure later became known as a Lorenz map (not to be confused with a [[Poincaré plot]], which plots the intersections of a trajectory with a prescribed surface). The resulting plot has a shape very similar to the [[tent map]]. Lorenz also found that when the maximum z value is above a certain cut-off, the system will switch to the next lobe. Combining this with the chaos known to be exhibited by the tent map, he showed that the system switches between the two lobes chaotically.
==Simulations==
===MATLAB simulation===
<syntaxhighlight lang="matlab">
% Solve over time interval [0,100] with initial conditions [1,1,1]
% ''f'' is set of differential equations
% ''a'' is array containing x, y, and z variables
% ''t'' is time variable
sigma = 10;
beta = 8/3;
rho = 28;
f = @(t,a) [-sigma*a(1) + sigma*a(2); rho*a(1) - a(2) - a(1)*a(3); -beta*a(3) + a(1)*a(2)];
[t,a] = ode45(f,[0 100],[1 1 1]); % Runge-Kutta 4th/5th order ODE solver
plot3(a(:,1),a(:,2),a(:,3))
</syntaxhighlight>
=== Mathematica simulation ===
Standard way:
<syntaxhighlight lang="mathematica">
tend = 50;
eq = {x'[t] == σ (y[t] - x[t]),
y'[t] == x[t] (ρ - z[t]) - y[t],
z'[t] == x[t] y[t] - β z[t]};
init = {x[0] == 10, y[0] == 10, z[0] == 10};
pars = {σ->10, ρ->28, β->8/3};
{xs, ys, zs} =
NDSolveValue[{eq /. pars, init}, {x, y, z}, {t, 0, tend}];
ParametricPlot3D[{xs[t], ys[t], zs[t]}, {t, 0, tend}]
</syntaxhighlight>
Less verbose:
<syntaxhighlight lang="mathematica">
lorenz = NonlinearStateSpaceModel[{{σ (y - x), x (ρ - z) - y, x y - β z}, {}}, {x, y, z}, {σ, ρ, β}];
soln[t_] = StateResponse[{lorenz, {10, 10, 10}}, {10, 28, 8/3}, {t, 0, 50}];
ParametricPlot3D[soln[t], {t, 0, 50}]
</syntaxhighlight>
=== Python simulation ===
<syntaxhighlight lang="python">
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint
from mpl_toolkits.mplot3d import Axes3D
rho = 28.0
sigma = 10.0
beta = 8.0 / 3.0
def f(state, t):
x, y, z = state # Unpack the state vector
return sigma * (y - x), x * (rho - z) - y, x * y - beta * z # Derivatives
state0 = [1.0, 1.0, 1.0]
t = np.arange(0.0, 40.0, 0.01)
states = odeint(f, state0, t)
fig = plt.figure()
ax = fig.gca(projection="3d")
ax.plot(states[:, 0], states[:, 1], states[:, 2])
plt.draw()
plt.show()
</syntaxhighlight>
== Derivation of the Lorenz equations as a model for 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>
== Resolution of Smale's 14th problem ==
Smale's 14th problem says 'Do the properties of the Lorenz attractor exhibit that of a [[Attractor#Strange attractor|strange attractor]]?', it was answered affirmatively by [[Warwick Tucker]] in 2002.<ref name="Tucker 2002"/> To prove this result, Tucker used rigorous numerics methods like [[interval arithmetic]] and [[Normal form (dynamical systems)|normal forms]]. First, Tucker defined a cross section <math>\Sigma\subset \{x_3 = r - 1 \}</math> that is cut transversely by the flow trajectories. From this, one can define the first-return map <math>P</math>, which assigns to each <math>x\in\Sigma</math> the point <math>P(x)</math> where the trajectory of <math>x</math> first intersects <math>\Sigma</math>.
Then the proof is split in three main points that are proved and imply the existence of a strange attractor.<ref name="Viana 2000">{{harvtxt|Viana|2000}}</ref> The three points are:
* There exists a region <math>N\subset\Sigma</math> invariant under the first-return map, meaning <math>P(N)\subset N</math>
* The return map admits a forward invariant cone field
* Vectors inside this invariant cone field are uniformly expanded by the derivative <math>DP</math> of the return map.
To prove the first point, we notice that the cross section <math>\Sigma</math> is cut by two arcs formed by <math>P(\Sigma)</math> (see <ref name="Viana 2000"/>). Tucker covers the location of these two arcs by small rectangles <math>R_i</math>, the union of these rectangles gives <math>N</math>. Now, the goal is to prove that for all points in <math>N</math>, the flow will bring back the points in <math>\Sigma</math>, in <math>N</math>. To do that, we take a plan <math>\Sigma'</math> below <math>\Sigma</math> at a distance <math>h</math> small, then by taking the center <math>c_i</math> of <math>R_i</math> and using Euler integration method, one can estimate where the flow will bring <math>c_i</math> in <math>\Sigma'</math> which gives us a new point <math>c_i'</math>. Then, one can estimate where the points in <math>\Sigma</math> will be mapped in <math>\Sigma'</math> using Taylor expansion, this gives us a new rectangle <math>R_i'</math> centered on <math>c_i</math>. Thus we know that all points in <math>R_i</math> will be mapped in <math>R_i'</math>. The goal is to do this method recursively until the flow comes back to <math>\Sigma</math> and we obtain a rectangle <math>Rf_i</math> in <math>\Sigma</math> such that we know that <math>P(R_i)\subset Rf_i</math>. The problem is that our estimation may become imprecise after several iterations, thus what Tucker does is to split <math>R_i'</math> into smaller rectangles <math>R_{i,j}</math> and then apply the process recursively.
Another problem is that as we are applying this algorithm, the flow becomes more 'horizontal' (see <ref name="Viana 2000"/>), leading to a dramatic increase in imprecision. To prevent this, the algorithm changes the orientation of the cross sections, becoming either horizontal or vertical.
== Contributions ==
Lorenz acknowledges the contributions from [[Ellen Fetter]] in his paper, who is responsible for the numerical simulations and figures.<ref name="lorenz"/> Also, [[Margaret Hamilton (software engineer)|Margaret Hamilton]] helped in the initial, numerical computations leading up to the findings of the Lorenz model.<ref>{{harvtxt|Lorenz|1960}}</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.
File:Lorenz apparition small.gif|Two streamlines in a Lorenz system, from rho=0 to rho=28 (sigma=10, beta=8/3)
File:Lorenz(rho).gif| Animation of a Lorenz System with rho- dependence
</gallery>
== See also ==
* [[Eden's conjecture]] on the Lyapunov dimension
* [[Lorenz 96 model]]
* [[List of chaotic maps]]
* [[Takens' theorem]]
== Notes ==
{{Reflist|30em}}
== 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}}
* {{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|bibcode = 1993PhRvL..71...65C | pmid=10054374}}
* {{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|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}}
* {{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|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}}
* {{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}}
* {{cite book | last1=Hirsch | first1=Morris W. | author1-link=Morris Hirsch | last2=Smale | first2=Stephen | author2-link=Stephen Smale | last3=Devaney | first3=Robert|author3-link= Robert L. Devaney| 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}}
* {{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|bibcode = 1981PhLA...82..439K }}
* {{cite journal | last1=Kolář | first1=Miroslav | last2=Gumbs | first2=Godfrey |title=Theory for the experimental observation of chaos in a rotating waterwheel | year=1992 | journal=Physical Review A | volume=45 | issue=2 | pages=626–637 | doi=10.1103/PhysRevA.45.626 | pmid=9907027}}
* {{cite journal
| last1=Leonov | first1=G.A.
| last2=Kuznetsov | first2=N.V.
| last3=Korzhemanova | first3=N.A.
| last4=Kusakin | first4=D.V.
| title=Lyapunov dimension formula for the global attractor of the Lorenz system
| journal=Communications in Nonlinear Science and Numerical Simulation
| year = 2016 | volume = 41 | pages=84–103
| doi = 10.1016/j.cnsns.2016.04.032| arxiv=1508.07498| bibcode=2016CNSNS..41...84L}}
* {{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| bibcode=1963JAtS...20..130L| doi-access=free }}
* {{cite journal | last1=Mishra | first1=Aashwin | last2=Sanghi | first2=Sanjeev |title=A study of the asymmetric Malkus waterwheel: The biased Lorenz equations| year=2006 | journal=Chaos: An Interdisciplinary Journal of Nonlinear Science | volume=16 | issue=1 | pages=013114 | doi=10.1063/1.2154792 | pmid=16599745 | bibcode=2006Chaos..16a3114M}}
* {{cite journal | last1=Pchelintsev | first1=A.N. | title=Numerical and Physical Modeling of the Dynamics of the Lorenz System | journal=Numerical Analysis and Applications | year = 2014 | volume = 7 | issue=2 | pages=159–167 | doi = 10.1134/S1995423914020098}}
* {{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|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|bibcode = 1962JAtS...19..329S |doi = 10.1175/1520-0469(1962)019<0329:FAFCAA>2.0.CO;2 | doi-access=free }}
* {{cite book | last1=Sparrow | first1=Colin | title=The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors | publisher=Springer | year=1982}}
* {{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=https://link.springer.com/content/pdf/10.1007/s002080010018.pdf| citeseerx=10.1.1.545.3996 }}
* {{cite arXiv |last=Tzenov |first=Stephan |eprint=1406.0979v1 |title=Strange Attractors Characterizing the Osmotic Instability |class=physics.flu-dyn |year=2014 }}
* {{cite journal |last1=Viana |first1=Marcelo |title=What's new on Lorenz strange attractors? |journal=The Mathematical Intelligencer |date=2000 |volume=22 |issue=3 |pages=6–19|doi=10.1007/BF03025276 }}
* {{cite journal|last1=Lorenz|first1=Edward N.|author1-link=Edward N. Lorenz|title=The statistical prediction of solutions of dynamic equations.|journal=Symposium on Numerical Weather Prediction in Tokyo|year=1960|url=http://eaps4.mit.edu/research/Lorenz/The_Statistical_Prediction_of_Solutions_1962.pdf}}
== Further reading ==
* {{cite journal
|author1=G.A. Leonov |author2=N.V. Kuznetsov
|name-list-style=amp | year = 2015
| title = On differences and similarities in the analysis of Lorenz, Chen, and Lu systems
| journal = Applied Mathematics and Computation
| volume = 256
| pages = 334–343
| url = http://www.ee.cityu.edu.hk/~gchen/pdf/LN2015.pdf
| doi = 10.1016/j.amc.2014.12.132
| doi-access = free
}}
== 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
* [https://www.youtube.com/watch?v=J-ca_bqWp4I 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]
* [https://archive.is/20121211081109/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)
* [https://highfellow.github.com/lorenz-attractor/attractor.html Web based Lorenz Attractor] (implemented in JavaScript/HTML/CSS)
* [https://alpha.iodide.io/notebooks/34/?viewMode=report Interactive web based Lorenz Attractor] made with Iodide
{{Chaos theory}}
{{Authority control}}
[[Category:Chaotic maps]]
[[Category:Articles containing video clips]]
[[Category:Articles with example Python (programming language) code]]
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[[Category:Articles with example Julia code]]
{{Short description|System of ordinary differential equations with chaotic solutions}}
{{Distinguish|Lorenz curve|Lorentz distribution}}
[[File:A Trajectory Through Phase Space in a Lorenz Attractor.gif|frame|right|A sample solution in the Lorenz attractor when ρ = 28, σ = 10, and β = 8/3]]
The '''Lorenz system''' is a system of [[ordinary differential equation]]s 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. In popular media the "[[butterfly effect]]" stems from the real-world implications of the Lorenz attractor, i.e. that in any physical system, in the absence of perfect knowledge of the initial conditions (even the minuscule disturbance of the air due to a butterfly flapping its wings), our ability to predict its future course will always fail. This underscores that physical systems can be completely deterministic and yet still be inherently unpredictable even in the absence of quantum effects. The shape of the Lorenz attractor itself, when plotted graphically, may also be seen to resemble a butterfly.
==Overview==
In 1963, [[Edward Norton Lorenz|Edward Lorenz]], with the help of [[Ellen Fetter]], 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), \\[6pt]
\frac{\mathrm{d}y}{\mathrm{d}t} &= x (\rho - z) - y, \\[6pt]
\frac{\mathrm{d}z}{\mathrm{d}t} &= x y - \beta z.
\end{align} </math>
The equations relate the properties of a two-dimensional fluid layer uniformly warmed from below and cooled from above. In particular, the equations describe the rate of change of three quantities with respect to time: <math>x</math> is proportional to the rate of convection, <math>y</math> to the horizontal temperature variation, and <math>z</math> to the vertical temperature variation.<ref name="Sparrow 1982">{{harvtxt|Sparrow|1982}}</ref> The constants <math>\sigma</math>, <math>\rho</math>, and <math>\beta</math> are system parameters proportional to the [[Prandtl number]], [[Rayleigh number]], and certain physical dimensions of the layer itself.<ref name="Sparrow 1982"/>
The Lorenz equations also arise in simplified models for [[laser]]s,<ref>{{harvtxt|Haken|1975}}</ref> [[electrical generator|dynamos]],<ref>{{harvtxt|Knobloch|1981}}</ref> [[thermosyphon]]s,<ref>{{harvtxt|Gorman|Widmann|Robbins|1986}}</ref> brushless [[DC motor]]s,<ref>{{harvtxt|Hemati|1994}}</ref> [[electric circuit]]s,<ref>{{harvtxt|Cuomo|Oppenheim|1993}}</ref> [[chemical reaction]]s<ref>{{harvtxt|Poland|1993}}</ref> and [[forward osmosis]].<ref>{{harvtxt|Tzenov|2014}}{{citation needed|date=June 2017<!--doesn't point anywhere-->}}</ref> The Lorenz equations are also the governing equations in Fourier space for the [[Malkus waterwheel]].<ref>{{harvtxt|Kolář|Gumbs|1992}}</ref><ref>{{harvtxt|Mishra|Sanghi|2006}}</ref> The Malkus waterwheel exhibits chaotic motion where instead of spinning in one direction at a constant speed, its rotation will speed up, slow down, stop, change directions, and oscillate back and forth between combinations of such behaviors in an unpredictable manner.
From a technical standpoint, the Lorenz system is [[nonlinearity|nonlinear]], non-periodic, three-dimensional and [[deterministic system (mathematics)|deterministic]]. The Lorenz equations have been the subject of hundreds of research articles, and at least one book-length study.<ref name="Sparrow 1982"/>
==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 (and nearby) 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, which is a global [[attractor]], 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( \sqrt{\beta(\rho-1)}, \sqrt{\beta(\rho-1)}, \rho-1 \right) </math> and <math>\left( -\sqrt{\beta(\rho-1)}, -\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 subcritical [[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). Almost all initial points will tend to an invariant set{{snd}}the Lorenz attractor{{snd}}a [[Attractor#Strange attractor|strange attractor]], a [[fractal]], and a [[Hidden attractor#Self-excited attractors|self-excited attractor]] with respect to all three equilibria. Its [[Hausdorff dimension]] is estimated from above by the [[Lyapunov dimension|Lyapunov dimension (Kaplan-Yorke dimension)]] as 2.06 ± 0.01,<ref name=Kuznetsov-2020-ND>{{Cite journal |
first1=N.V. |last1=Kuznetsov| first2=T.N. |last2=Mokaev | first3=O.A. |last3=Kuznetsova | first4=E.V. |last4=Kudryashova|
title=The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension|
journal= Nonlinear Dynamics|year=2020 |
doi=10.1007/s11071-020-05856-4| url=https://link.springer.com/article/10.1007/s11071-020-05856-4|doi-access=free }}
</ref> and the [[correlation dimension]] is estimated to be 2.05 ± 0.01.<ref>{{harvtxt|Grassberger|Procaccia|1983}}</ref>
The exact Lyapunov dimension formula of the global attractor can be found analytically under classical restrictions on the parameters:<ref>{{harvtxt|Leonov|Kuznetsov|Korzhemanova|Kusakin|2016}}</ref><ref name=Kuznetsov-2020-ND/><ref name=2020-KuznetsovR>{{cite book | first1= Nikolay | last1=Kuznetsov |
first2=Volker | last2=Reitmann | year = 2020| title = Attractor Dimension Estimates for Dynamical Systems: Theory and Computation|
publisher = Springer| location = Cham|url=https://www.springer.com/gp/book/9783030509866}}</ref>
<math> 3 - \frac{2 (\sigma + \beta + 1)}{\sigma + 1 + \sqrt{(\sigma-1)^2 + 4 \sigma \rho}}. </math>
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.<ref>{{Cite journal|title = Structural stability of Lorenz attractors|journal = Publications Mathématiques de l'Institut des Hautes Études Scientifiques|date = 1979-12-01|issn = 0073-8301|pages = 59–72|volume = 50|issue = 1|doi = 10.1007/BF02684769|first = John|last = Guckenheimer|first2 = R. F.|last2 = Williams|url = http://www.numdam.org/item/PMIHES_1979__50__59_0/}}</ref> 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 name="Tucker 2002">{{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]].
{|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.
|}
{|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 two 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.
|}
==Connection to tent map==
[[File:Lorenz_Map.png|thumb|A recreation of Lorenz's results created on [[Mathematica]]. Points above the red line correspond to the system switching lobes.|upright=1.3]]
In Figure 4 of his paper,<ref name=lorenz /> Lorenz plotted the relative maximum value in the z direction achieved by the system against the previous relative maximum in the z direction. This procedure later became known as a Lorenz map (not to be confused with a [[Poincaré plot]], which plots the intersections of a trajectory with a prescribed surface). The resulting plot has a shape very similar to the [[tent map]]. Lorenz also found that when the maximum z value is above a certain cut-off, the system will switch to the next lobe. Combining this with the chaos known to be exhibited by the tent map, he showed that the system switches between the two lobes chaotically.
==Simulations==
===MATLAB simulation===
<syntaxhighlight lang="matlab">
% Solve over time interval [0,100] with initial conditions [1,1,1]
% ''f'' is set of differential equations
% ''a'' is array containing x, y, and z variables
% ''t'' is time variable
sigma = 10;
beta = 8/3;
rho = 28;
f = @(t,a) [-sigma*a(1) + sigma*a(2); rho*a(1) - a(2) - a(1)*a(3); -beta*a(3) + a(1)*a(2)];
[t,a] = ode45(f,[0 100],[1 1 1]); % Runge-Kutta 4th/5th order ODE solver
plot3(a(:,1),a(:,2),a(:,3))
</syntaxhighlight>
=== Mathematica simulation ===
Standard way:
<syntaxhighlight lang="mathematica">
tend = 50;
eq = {x'[t] == σ (y[t] - x[t]),
y'[t] == x[t] (ρ - z[t]) - y[t],
z'[t] == x[t] y[t] - β z[t]};
init = {x[0] == 10, y[0] == 10, z[0] == 10};
pars = {σ->10, ρ->28, β->8/3};
{xs, ys, zs} =
NDSolveValue[{eq /. pars, init}, {x, y, z}, {t, 0, tend}];
ParametricPlot3D[{xs[t], ys[t], zs[t]}, {t, 0, tend}]
</syntaxhighlight>
Less verbose:
<syntaxhighlight lang="mathematica">
lorenz = NonlinearStateSpaceModel[{{σ (y - x), x (ρ - z) - y, x y - β z}, {}}, {x, y, z}, {σ, ρ, β}];
soln[t_] = StateResponse[{lorenz, {10, 10, 10}}, {10, 28, 8/3}, {t, 0, 50}];
ParametricPlot3D[soln[t], {t, 0, 50}]
</syntaxhighlight>
=== Python simulation ===
<syntaxhighlight lang="python">
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint
from mpl_toolkits.mplot3d import Axes3D
rho = 28.0
sigma = 10.0
beta = 8.0 / 3.0
def f(state, t):
x, y, z = state # Unpack the state vector
return sigma * (y - x), x * (rho - z) - y, x * y - beta * z # Derivatives
state0 = [1.0, 1.0, 1.0]
t = np.arange(0.0, 40.0, 0.01)
states = odeint(f, state0, t)
fig = plt.figure()
ax = fig.gca(projection="3d")
ax.plot(states[:, 0], states[:, 1], states[:, 2])
plt.draw()
plt.show()
</syntaxhighlight>
== Derivation of the Lorenz equations as a model for 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>
== Resolution of Smale's 14th problem ==
Smale's 14th problem says 'Do the properties of the Lorenz attractor exhibit that of a [[Attractor#Strange attractor|strange attractor]]?', it was answered affirmatively by [[Warwick Tucker]] in 2002.<ref name="Tucker 2002"/> To prove this result, Tucker used rigorous numerics methods like [[interval arithmetic]] and [[Normal form (dynamical systems)|normal forms]]. First, Tucker defined a cross section <math>\Sigma\subset \{x_3 = r - 1 \}</math> that is cut transversely by the flow trajectories. From this, one can define the first-return map <math>P</math>, which assigns to each <math>x\in\Sigma</math> the point <math>P(x)</math> where the trajectory of <math>x</math> first intersects <math>\Sigma</math>.
Then the proof is split in three main points that are proved and imply the existence of a strange attractor.<ref name="Viana 2000">{{harvtxt|Viana|2000}}</ref> The three points are:
* There exists a region <math>N\subset\Sigma</math> invariant under the first-return map, meaning <math>P(N)\subset N</math>
* The return map admits a forward invariant cone field
* Vectors inside this invariant cone field are uniformly expanded by the derivative <math>DP</math> of the return map.
To prove the first point, we notice that the cross section <math>\Sigma</math> is cut by two arcs formed by <math>P(\Sigma)</math> (see <ref name="Viana 2000"/>). Tucker covers the location of these two arcs by small rectangles <math>R_i</math>, the union of these rectangles gives <math>N</math>. Now, the goal is to prove that for all points in <math>N</math>, the flow will bring back the points in <math>\Sigma</math>, in <math>N</math>. To do that, we take a plan <math>\Sigma'</math> below <math>\Sigma</math> at a distance <math>h</math> small, then by taking the center <math>c_i</math> of <math>R_i</math> and using Euler integration method, one can estimate where the flow will bring <math>c_i</math> in <math>\Sigma'</math> which gives us a new point <math>c_i'</math>. Then, one can estimate where the points in <math>\Sigma</math> will be mapped in <math>\Sigma'</math> using Taylor expansion, this gives us a new rectangle <math>R_i'</math> centered on <math>c_i</math>. Thus we know that all points in <math>R_i</math> will be mapped in <math>R_i'</math>. The goal is to do this method recursively until the flow comes back to <math>\Sigma</math> and we obtain a rectangle <math>Rf_i</math> in <math>\Sigma</math> such that we know that <math>P(R_i)\subset Rf_i</math>. The problem is that our estimation may become imprecise after several iterations, thus what Tucker does is to split <math>R_i'</math> into smaller rectangles <math>R_{i,j}</math> and then apply the process recursively.
Another problem is that as we are applying this algorithm, the flow becomes more 'horizontal' (see <ref name="Viana 2000"/>), leading to a dramatic increase in imprecision. To prevent this, the algorithm changes the orientation of the cross sections, becoming either horizontal or vertical.
== Contributions ==
Lorenz acknowledges the contributions from [[Ellen Fetter]] in his paper, who is responsible for the numerical simulations and figures.<ref name="lorenz"/> Also, [[Margaret Hamilton (software engineer)|Margaret Hamilton]] helped in the initial, numerical computations leading up to the findings of the Lorenz model.<ref>{{harvtxt|Lorenz|1960}}</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.
File:Lorenz apparition small.gif|Two streamlines in a Lorenz system, from rho=0 to rho=28 (sigma=10, beta=8/3)
File:Lorenz(rho).gif| Animation of a Lorenz System with rho- dependence
</gallery>
== See also ==
* [[Eden's conjecture]] on the Lyapunov dimension
* [[Lorenz 96 model]]
* [[List of chaotic maps]]
* [[Takens' theorem]]
== Notes ==
{{Reflist|30em}}
== 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}}
* {{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|bibcode = 1993PhRvL..71...65C | pmid=10054374}}
* {{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|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}}
* {{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|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}}
* {{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}}
* {{cite book | last1=Hirsch | first1=Morris W. | author1-link=Morris Hirsch | last2=Smale | first2=Stephen | author2-link=Stephen Smale | last3=Devaney | first3=Robert|author3-link= Robert L. Devaney| 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}}
* {{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|bibcode = 1981PhLA...82..439K }}
* {{cite journal | last1=Kolář | first1=Miroslav | last2=Gumbs | first2=Godfrey |title=Theory for the experimental observation of chaos in a rotating waterwheel | year=1992 | journal=Physical Review A | volume=45 | issue=2 | pages=626–637 | doi=10.1103/PhysRevA.45.626 | pmid=9907027}}
* {{cite journal
| last1=Leonov | first1=G.A.
| last2=Kuznetsov | first2=N.V.
| last3=Korzhemanova | first3=N.A.
| last4=Kusakin | first4=D.V.
| title=Lyapunov dimension formula for the global attractor of the Lorenz system
| journal=Communications in Nonlinear Science and Numerical Simulation
| year = 2016 | volume = 41 | pages=84–103
| doi = 10.1016/j.cnsns.2016.04.032| arxiv=1508.07498| bibcode=2016CNSNS..41...84L}}
* {{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| bibcode=1963JAtS...20..130L| doi-access=free }}
* {{cite journal | last1=Mishra | first1=Aashwin | last2=Sanghi | first2=Sanjeev |title=A study of the asymmetric Malkus waterwheel: The biased Lorenz equations| year=2006 | journal=Chaos: An Interdisciplinary Journal of Nonlinear Science | volume=16 | issue=1 | pages=013114 | doi=10.1063/1.2154792 | pmid=16599745 | bibcode=2006Chaos..16a3114M}}
* {{cite journal | last1=Pchelintsev | first1=A.N. | title=Numerical and Physical Modeling of the Dynamics of the Lorenz System | journal=Numerical Analysis and Applications | year = 2014 | volume = 7 | issue=2 | pages=159–167 | doi = 10.1134/S1995423914020098}}
* {{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|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|bibcode = 1962JAtS...19..329S |doi = 10.1175/1520-0469(1962)019<0329:FAFCAA>2.0.CO;2 | doi-access=free }}
* {{cite book | last1=Sparrow | first1=Colin | title=The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors | publisher=Springer | year=1982}}
* {{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=https://link.springer.com/content/pdf/10.1007/s002080010018.pdf| citeseerx=10.1.1.545.3996 }}
* {{cite arXiv |last=Tzenov |first=Stephan |eprint=1406.0979v1 |title=Strange Attractors Characterizing the Osmotic Instability |class=physics.flu-dyn |year=2014 }}
* {{cite journal |last1=Viana |first1=Marcelo |title=What's new on Lorenz strange attractors? |journal=The Mathematical Intelligencer |date=2000 |volume=22 |issue=3 |pages=6–19|doi=10.1007/BF03025276 }}
* {{cite journal|last1=Lorenz|first1=Edward N.|author1-link=Edward N. Lorenz|title=The statistical prediction of solutions of dynamic equations.|journal=Symposium on Numerical Weather Prediction in Tokyo|year=1960|url=http://eaps4.mit.edu/research/Lorenz/The_Statistical_Prediction_of_Solutions_1962.pdf}}
== Further reading ==
* {{cite journal
|author1=G.A. Leonov |author2=N.V. Kuznetsov
|name-list-style=amp | year = 2015
| title = On differences and similarities in the analysis of Lorenz, Chen, and Lu systems
| journal = Applied Mathematics and Computation
| volume = 256
| pages = 334–343
| url = http://www.ee.cityu.edu.hk/~gchen/pdf/LN2015.pdf
| doi = 10.1016/j.amc.2014.12.132
| doi-access = free
}}
== 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
* [https://www.youtube.com/watch?v=J-ca_bqWp4I 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]
* [https://archive.is/20121211081109/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)
* [https://highfellow.github.com/lorenz-attractor/attractor.html Web based Lorenz Attractor] (implemented in JavaScript/HTML/CSS)
* [https://alpha.iodide.io/notebooks/34/?viewMode=report Interactive web based Lorenz Attractor] made with Iodide
{{Chaos theory}}
{{Authority control}}
[[Category:Chaotic maps]]
[[Category:Articles containing video clips]]
[[Category:Articles with example Python (programming language) code]]
[[Category:Articles with example MATLAB/Octave code]]
[[Category:Articles with example Julia code]]