View source for Lorenz system

[[File:A Trajectory Through Phase Space in a Lorenz Attractor.gif|frame|right|A sample solution in the Lorenz attractor when {{math|1=''ρ'' = 28}}, {{math|1=''σ'' = 10}}, and {{math|1=''β'' = {{sfrac|8|3}}}}]]
The '''Lorenz system''' is a system of [[Ordinary differential equation|ordinary differential equation]]s first studied by mathematician and meteorologist [[Biography: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|butterfly effect]]" stems from the real-world implications of the Lorenz attractor, namely that several different initial chaotic conditions evolve in phase space in a way that never repeats, so all chaos is unpredictable. This underscores that chaotic systems can be completely deterministic and yet still be inherently unpredictable over long periods of time. Because chaos continually increases in systems, we cannot predict the future of systems well. E.g., even the small flap of a butterfly’s wings could set the world on a vastly different trajectory, such as by causing a hurricane. The shape of the Lorenz attractor itself, when plotted in [[Physics:Phase space|phase space]], may also be seen to resemble a butterfly.

==Overview==
{{See also|Chaos theory#Lorenz's pioneering contributions to chaotic modeling}}In 1963, [[Biography:Edward Norton Lorenz|Edward Lorenz]], with the help of [[Biography:Ellen Fetter|Ellen Fetter]] who was responsible for the numerical simulations and figures,<ref name="lorenz"/> and [[Biography:Margaret Hamilton (software engineer)|Margaret Hamilton]] who helped in the initial, numerical computations leading up to the findings of the Lorenz model,<ref>{{harvtxt|Lorenz|1960}}</ref>  developed a simplified mathematical model for [[Earth:Atmospheric convection|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: {{mvar|x}} is proportional to the rate of convection, {{mvar|y}} to the horizontal temperature variation, and {{mvar|z}} to the vertical temperature variation.<ref name="Sparrow 1982">{{harvtxt|Sparrow|1982}}</ref> The constants {{mvar|σ}}, {{mvar|ρ}}, and {{mvar|β}} are system parameters proportional to the [[Physics:Prandtl number|Prandtl number]], [[Rayleigh number]], and certain physical dimensions of the layer itself.<ref name="Sparrow 1982"/>

The Lorenz equations can arise in simplified models for [[Physics:Laser|laser]]s,<ref>{{harvtxt|Haken|1975}}</ref> [[Engineering:Electrical generator|dynamos]],<ref>{{harvtxt|Knobloch|1981}}</ref> thermosyphons,<ref>{{harvtxt|Gorman|Widmann|Robbins|1986}}</ref> brushless [[Engineering:DC motor|DC motor]]s,<ref>{{harvtxt|Hemati|1994}}</ref> electric circuits,<ref>{{harvtxt|Cuomo|Oppenheim|1993}}</ref> [[Chemistry:Chemical reaction|chemical reaction]]s<ref>{{harvtxt|Poland|1993}}</ref> and [[Physics:Forward osmosis|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 nonlinear, aperiodic, three-dimensional and 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 {{mvar|σ}}, {{mvar|ρ}}, and {{mvar|β}} are positive. Lorenz used the values {{math|1=''σ'' = 10}}, {{math|1=''β'' = {{sfrac|8|3}}}} and {{math|1=''ρ'' = 28}}. The system exhibits chaotic behavior for these (and nearby) values.<ref>{{harvtxt|Hirsch|Smale|Devaney|2003}}, pp. 303–305</ref>

If {{math|''ρ'' < 1}} 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|attractor]], when {{math|''ρ'' < 1}}.<ref>{{harvtxt|Hirsch|Smale|Devaney|2003}}, pp. 306+307</ref>

A [[Pitchfork bifurcation|pitchfork bifurcation]] occurs at {{math|1=''ρ'' = 1}}, and for {{math|''ρ'' > 1}} two additional critical points appear at
<math display=block>\left( \sqrt{\beta(\rho-1)}, \sqrt{\beta(\rho-1)}, \rho-1 \right) \quad\text{and}\quad \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 {{mvar|ρ}} if {{math|''σ'' > ''β'' + 1}}. 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|1=''ρ'' = 28}}, {{math|1=''σ'' = 10}}, and {{math|1=''β'' = {{sfrac|8|3}}}}, 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|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 {{val|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 |volume=102|issue=2|pages=713–732|
doi=10.1007/s11071-020-05856-4|doi-access=free }}
</ref> and the [[Correlation dimension|correlation dimension]] is estimated to be {{val|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 = 2021| 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{\left(\sigma-1\right)^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|first1 = John|last1 = Guckenheimer|first2 = R. F.|last2 = Williams|s2cid = 55218285|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 {{mvar|ρ}}, the system displays knotted periodic orbits. For example, with {{math|1=''ρ'' = 99.96}} it becomes a {{math|''T''(3,2)}} [[Torus knot|torus knot]].

{|class="wikitable" width=777px
|-
! colspan=2|Example solutions of the Lorenz system for different values of {{mvar|ρ}}
|-
|align="center"|[[Image:Lorenz Ro14 20 41 20-200px.png]]
|align="center"|[[Image:Lorenz Ro13-200px.png]]
|-
|align="center"|{{math|1=''ρ'' = 14, ''σ'' = 10, ''β'' = {{sfrac|8|3}}}} [[:Image:Lorenz Ro14 20 41 20.png|(Enlarge)]]
|align="center"|{{math|1=''ρ'' = 13, ''σ'' = 10, ''β'' = {{sfrac|8|3}}}} [[:Image:Lorenz Ro13.png|(Enlarge)]]
|-
|align="center"|[[Image:Lorenz Ro15-200px.png]]
|align="center"|[[Image:Lorenz Ro28-200px.png]]
|-
|align="center"|{{math|1=''ρ'' = 15, ''σ'' = 10, ''β'' = {{sfrac|8|3}}}} [[:Image:Lorenz Ro15.png|(Enlarge)]]
|align="center"|{{math|1=''ρ'' = 28, ''σ'' = 10, ''β'' = {{sfrac|8|3}}}} [[:Image:Lorenz Ro28.png|(Enlarge)]]
|-
|align="center" colspan=2| For small values of {{mvar|ρ}}, the system is stable and evolves to one of two fixed point attractors. When {{math|''ρ'' > 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 {{math|1=''t'' = 1}} [[:Image:Lorenz caos1.png|(Enlarge)]]
|align="center"|Time {{math|1=''t'' = 2}} [[:Image:Lorenz caos2.png|(Enlarge)]]
|align="center"|Time {{math|1=''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 {{math|1=''ρ'' = 28}}, {{math|1=''σ'' = 10}} and {{math|1=''β'' = {{sfrac|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 {{mvar|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 [[Software:Mathematica|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 {{mvar|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|tent map]]. Lorenz also found that when the maximum {{mvar|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.

== A Generalized Lorenz System ==
Over the past several years, a series of papers regarding high-dimensional Lorenz models have yielded a generalized Lorenz model,<ref name=":0">{{Cite journal |last=Shen |first=Bo-Wen |date=2019-03-01 |title=Aggregated Negative Feedback in a Generalized Lorenz Model |journal=International Journal of Bifurcation and Chaos |volume=29 |issue=3 |pages=1950037–1950091 |doi=10.1142/S0218127419500378 |bibcode=2019IJBC...2950037S |s2cid=132494234 |issn=0218-1274|doi-access=free }}</ref> which can be simplified into the classical Lorenz model  for three state variables or the following five-dimensional Lorenz model for five state variables:<ref name=":1">{{Cite journal |last=Shen |first=Bo-Wen |date=2014-04-28 |title=Nonlinear Feedback in a Five-Dimensional Lorenz Model |url=http://dx.doi.org/10.1175/jas-d-13-0223.1 |journal=Journal of the Atmospheric Sciences |volume=71 |issue=5 |pages=1701–1723 |doi=10.1175/jas-d-13-0223.1 |bibcode=2014JAtS...71.1701S |s2cid=123683839 |issn=0022-4928}}</ref>
<math display="block"> \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 - x y_1 - \beta z, \\[6pt]
\frac{\mathrm{d}y_1}{\mathrm{d}t} &= x z - 2 x z_1 - d_0 y_1, \\[6pt]
\frac{\mathrm{d}z_1}{\mathrm{d}t} &= 2 x y_1 - 4\beta z_1.
\end{align} </math>

A choice of the parameter ''d''<sub>0</sub> = 19/3 has been applied to be consistent with the choice of the other parameters. See details in.<ref name=":0" /><ref name=":1" />

==Simulations==

[[File:Lorenz attractor in Julia.gif|thumb|alt=Lorenz System simulation in Julia|Julia Simulation]]

=== Julia simulation ===
<source lang="julia">
using Plots
# define the Lorenz attractor
@kwdef mutable struct Lorenz
    dt::Float64 = 0.02
    σ::Float64 = 10
    ρ::Float64 = 28
    β::Float64 = 8/3
    x::Float64 = 2
    y::Float64 = 1
    z::Float64 = 1
end

function step!(l::Lorenz)
    dx = l.σ * (l.y - l.x);         l.x += l.dt * dx
    dy = l.x * (l.ρ - l.z) - l.y;   l.y += l.dt * dy
    dz = l.x * l.y - l.β * l.z;     l.z += l.dt * dz
end

attractor = Lorenz()

# initialize a 3D plot with 1 empty series
plt = plot3d(
    1,
    xlim = (-30, 30),
    ylim = (-30, 30),
    zlim = (0, 60),
    title = "Lorenz Attractor",
    marker = 2,
)

# build an animated gif by pushing new points to the plot, saving every 10th frame
@gif for i=1:1500
    step!(attractor)
    push!(plt, attractor.x, attractor.y, attractor.z)
end every 10
</source>

===Maple simulation===
<source lang="mupad">
deq := [diff(x(t), t) = 10*(y(t) - x(t)), diff(y(t), t) = 28*x(t) - y(t) - x(t)*z(t), diff(z(t), t) = x(t)*y(t) - 8/3*z(t)]:
with(DEtools):
DEplot3d(deq, {x(t), y(t), z(t)}, t = 0 .. 100, x(0) = 10, y(0) = 10, z(0) = 10, stepsize = 0.01, x = -20 .. 20, y = -25 .. 25, z = 0 .. 50, linecolour = sin(t*Pi/3), thickness = 1, orientation = [-40, 80], title = `Lorenz Chaotic Attractor`);
</source>

===Maxima simulation===
<source lang="maxima">
[sigma, rho, beta]: [10, 28, 8/3]$
eq: [sigma*(y-x), x*(rho-z)-y, x*y-beta*z]$
sol: rk(eq, [x, y, z], [1, 0, 0], [t, 0, 50, 1/100])$
len: length(sol)$
x: makelist(sol[k][2], k, len)$
y: makelist(sol[k][3], k, len)$
z: makelist(sol[k][4], k, len)$
draw3d(points_joined=true, point_type=-1, points(x, y, z), proportional_axes=xyz)$
</source>

===MATLAB simulation===
<source 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))
</source>

=== Mathematica simulation ===
Standard way:

<source 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}]
</source>

Less verbose:

<source 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}]
</source>

===Python simulation===
<source lang="python">
import matplotlib.pyplot as plt
import numpy as np

def lorenz(xyz, *, s=10, r=28, b=2.667):
    """
    Parameters
    ----------
    xyz : array-like, shape (3,)
       Point of interest in three-dimensional space.
    s, r, b : float
       Parameters defining the Lorenz attractor.

    Returns
    -------
    xyz_dot : array, shape (3,)
       Values of the Lorenz attractor's partial derivatives at *xyz*.
    """
    x, y, z = xyz
    x_dot = s*(y - x)
    y_dot = r*x - y - x*z
    z_dot = x*y - b*z
    return np.array([x_dot, y_dot, z_dot])

dt = 0.01
num_steps = 10000

xyzs = np.empty((num_steps + 1, 3))  # Need one more for the initial values
xyzs[0] = (0., 1., 1.05)  # Set initial values
# Step through "time", calculating the partial derivatives at the current point
# and using them to estimate the next point
for i in range(num_steps):
    xyzs[i + 1] = xyzs[i] + lorenz(xyzs[i]) * dt

# Plot
ax = plt.figure().add_subplot(projection='3d')

ax.plot(*xyzs.T, lw=0.5)
ax.set_xlabel("X Axis")
ax.set_ylabel("Y Axis")
ax.set_zlabel("Z Axis")
ax.set_title("Lorenz Attractor")

plt.show()
</source>

==Applications==
=== Model for atmospheric convection===

As shown in Lorenz's original paper,<ref>{{harvtxt|Lorenz|1963}}</ref> the Lorenz system is a reduced version of a larger system studied earlier by Barry Saltzman.<ref>{{harvtxt|Saltzman|1962}}</ref> The Lorenz equations are derived from the [[Physics: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>{{harvtxt|Lorenz|1963}}</ref> This fluid circulation is known as [[Physics:Rayleigh–Bénard convection|Rayleigh–Bénard convection]]. The fluid is assumed to circulate in two dimensions (vertical and horizontal) with periodic rectangular boundary conditions.<ref>{{harvtxt|Lorenz|1963}}</ref>

The partial differential equations modeling the system's [[Physics:Stream function|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 from {{harvtxt|Hilborn|2000}}, Appendix C;  {{harvtxt|Bergé|Pomeau|Vidal|1984}}, Appendix D; or Shen (2016),<ref>{{Cite journal |last=Shen |first=B.-W. |date=2015-12-21 |title=Nonlinear feedback in a six-dimensional Lorenz model: impact of an additional heating term |url=https://npg.copernicus.org/articles/22/749/2015/ |journal=Nonlinear Processes in Geophysics |language=en |volume=22 |issue=6 |pages=749–764 |doi=10.5194/npg-22-749-2015 |bibcode=2015NPGeo..22..749S |issn=1607-7946|doi-access=free }}</ref> Supplementary Materials.

=== Model for the nature of chaos and order in the atmosphere ===
The scientific community accepts that the chaotic features found in low-dimensional Lorenz models could represent features of the Earth's atmosphere (<ref>{{Cite journal |last1=Ghil |first1=Michael |last2=Read |first2=Peter |last3=Smith |first3=Leonard |date=2010-07-23 |title=Geophysical flows as dynamical systems: the influence of Hide's experiments |url=http://dx.doi.org/10.1111/j.1468-4004.2010.51428.x |journal=Astronomy & Geophysics |volume=51 |issue=4 |pages=4.28–4.35 |doi=10.1111/j.1468-4004.2010.51428.x |bibcode=2010A&G....51d..28G |issn=1366-8781}}</ref><ref>{{Cite book |last=Read |first=P. |title=Application of Chaos to Meteorology and Climate. In The Nature of Chaos; Mullin, T., Ed |publisher=Oxford Science Publications |year=1993 |isbn=0198539541 |location=Oxford, UK |pages=220–260}}</ref><ref name=":2">{{Cite journal |last1=Shen |first1=Bo-Wen |last2=Pielke |first2=Roger |last3=Zeng |first3=Xubin |last4=Cui |first4=Jialin |last5=Faghih-Naini |first5=Sara |last6=Paxson |first6=Wei |last7=Kesarkar |first7=Amit |last8=Zeng |first8=Xiping |last9=Atlas |first9=Robert |date=2022-11-12 |title=The Dual Nature of Chaos and Order in the Atmosphere |journal=Atmosphere |language=en |volume=13 |issue=11 |pages=1892 |doi=10.3390/atmos13111892 |bibcode=2022Atmos..13.1892S |issn=2073-4433|doi-access=free }}</ref>), yielding the statement of “weather is chaotic.” By comparison, based on the concept of attractor coexistence within the generalized Lorenz model<ref name=":0" /> and the original Lorenz model (<ref>{{Cite journal |last1=Yorke |first1=James A. |last2=Yorke |first2=Ellen D. |date=1979-09-01 |title=Metastable chaos: The transition to sustained chaotic behavior in the Lorenz model |url=https://doi.org/10.1007/BF01011469 |journal=Journal of Statistical Physics |language=en |volume=21 |issue=3 |pages=263–277 |doi=10.1007/BF01011469 |bibcode=1979JSP....21..263Y |s2cid=12172750 |issn=1572-9613}}</ref><ref>{{Citation |last1=Shen |first1=Bo-Wen |date=2021 |url=https://link.springer.com/10.1007/978-3-030-70795-8_57 |work=13th Chaotic Modeling and Simulation International Conference |pages=805–825 |editor-last=Skiadas |editor-first=Christos H. |place=Cham |publisher=Springer International Publishing |language=en |doi=10.1007/978-3-030-70795-8_57 |isbn=978-3-030-70794-1 |access-date=2022-12-22 |last2=Pielke |first2=R. A. |last3=Zeng |first3=X. |last4=Baik |first4=J.-J. |last5=Faghih-Naini |first5=S. |last6=Cui |first6=J. |last7=Atlas |first7=R. |last8=Reyes |first8=T. A. L. |title=Is Weather Chaotic? Coexisting Chaotic and Non-chaotic Attractors within Lorenz Models |series=Springer Proceedings in Complexity |s2cid=245197840 |editor2-last=Dimotikalis |editor2-first=Yiannis}}</ref>), Shen and his co-authors <ref name=":2" /><ref>{{Cite journal |last1=Shen |first1=Bo-Wen |last2=Pielke |first2=Roger A. |last3=Zeng |first3=Xubin |last4=Baik |first4=Jong-Jin |last5=Faghih-Naini |first5=Sara |last6=Cui |first6=Jialin |last7=Atlas |first7=Robert |date=2021-01-01 |title=Is Weather Chaotic?: Coexistence of Chaos and Order within a Generalized Lorenz Model |journal=Bulletin of the American Meteorological Society |language=EN |volume=102 |issue=1 |pages=E148–E158 |doi=10.1175/BAMS-D-19-0165.1 |bibcode=2021BAMS..102E.148S |s2cid=208369617 |issn=0003-0007|doi-access=free }}</ref> proposed a revised view that “weather possesses both chaos and order with distinct predictability”. The revised view,  which is a build-up of the conventional view, is used to suggest that “the chaotic and regular features found in theoretical Lorenz models could better represent features of the Earth's atmosphere”.

===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]]?'. The problem was answered affirmatively by Warwick Tucker in 2002.<ref name="Tucker 2002"/> To prove this result, Tucker used rigorous numerics methods like [[Interval arithmetic|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>.<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',<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.

== Gallery ==
<gallery>
File:Lorenz system r28 s10 b2-6666.png|A solution in the Lorenz attractor plotted at high resolution in the {{mvar|xz}} 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 {{math|1=''ρ'' = 0}} to {{math|1=''ρ'' = 28}} <!-- prevent line break after left parenthesis -->{{awrap|({{math|1=''σ'' = 10}},}} {{math|1=''β'' = {{sfrac|8|3}}}}).
File:Lorenz(rho).gif|Animation of a Lorenz System with rho-dependence.
File:Lorenz Attractor Brain Dynamics Toolbox.gif|Animation of the Lorenz attractor in the Brain Dynamics Toolbox.<ref>Heitmann, S., Breakspear, M (2017-2022) Brain Dynamics Toolbox. [https://bdtoolbox.org bdtoolbox.org] [https://doi.org/10.5281/zenodo.5625923 doi.org/10.5281/zenodo.5625923]</ref>
</gallery>

== See also ==
* [[Eden's conjecture]] on the Lyapunov dimension
* [[Lorenz 96 model]]
* [[List of chaotic maps]]
* Takens' theorem

== Notes ==
{{Reflist|30em}}

== References ==
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| 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| s2cid=119614076 
}}
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* Shen, B.-W. (2015-12-21). "Nonlinear feedback in a six-dimensional Lorenz model: impact of an additional heating term". ''Nonlinear Processes in Geophysics''. '''22''' (6): 749–764. doi:10.5194/npg-22-749-2015. ISSN 1607-7946.
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* {{cite arXiv |last=Tzenov |first=Stephan |eprint=1406.0979v1 |title=Strange Attractors Characterizing the Osmotic Instability |class=physics.flu-dyn |year=2014 }}
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== 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&ndash;343
 | url = http://www.ee.cityu.edu.hk/~gchen/pdf/LN2015.pdf
 | doi = 10.1016/j.amc.2014.12.132
 | doi-access = free
 }}
* {{cite journal 
| last1=Pchelintsev | first1=A.N.
| title=On a high-precision method for studying attractors of dynamical systems and systems of explosive type
| journal=Mathematics
| year = 2022 | volume = 10 | issue = 8 | pages = 1207
| doi = 10.3390/math10081207 | arxiv=2206.08195 | url=https://www.mdpi.com/2227-7390/10/8/1207/pdf 
| doi-access=free
}}

== External links ==
* {{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.today/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}}

[[Category:Chaotic maps]]

{{Sourceattribution|Lorenz system}}