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{{Short description|System of ordinary differential equations with chaotic solutions}}{{Technical|date=December 2023}}{{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 {{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]]s first studied by mathematician and meteorologist [[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. The term "[[butterfly effect]]" in [[popular media]] may stem from the real-world implications of the Lorenz attractor, namely that tiny changes in initial conditions evolve to completely different [[trajectories]]. This underscores that chaotic systems can be completely [[Determinism|deterministic]] and yet still be inherently impractical or even impossible to predict over longer periods of time. For example, even the small flap of a [[butterfly]]'s wings could set the [[Weather_forecasting|earth's atmosphere]] on a vastly different trajectory, in which for example a [[hurricane]] occurs where it otherwise would have not (see [[Saddle_point|Saddle points]]). The shape of the Lorenz attractor itself, when plotted in [[phase space]], may also be seen to resemble a butterfly.

==Overview==
{{See also|Chaos theory#Lorenz's pioneering contributions to chaotic modeling}}In 1963, [[Edward Norton Lorenz|Edward Lorenz]], with the help of [[Ellen Fetter]] who was responsible for the numerical simulations and figures,<ref name="lorenz"/> and [[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 [[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 [[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 [[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> Interestingly, the same Lorenz equations were also derived in 1963 by Sauermann and Haken <ref> 
 {{cite journal | last1=Sauermann | first1=H. | last2=Haken | first2=H. | title=Nonlinear Interaction of Laser Modes | year=1963 | journal=[[Z. Phys.]] | volume=173 | issue=3 | pages=261–275 | doi=10.1007/BF01377828 | bibcode=1963ZPhy..173..261H }}</ref> for a single-mode laser. In 1975, Haken realized <ref>{{harvtxt|Haken|1975}}</ref> that their equations derived in 1963 were mathematically equivalent to the original Lorenz equations. Haken's paper <ref>{{harvtxt|Haken|1975}}</ref> thus started a new field called laser chaos or optical chaos. The Lorenz equations are often called Lorenz-Haken equations in optical literature. Later on, it was also shown the complex version of Lorenz equations also had laser equivalent ones. <ref> {{cite journal | last1=Ning | first1=C.Z. | last2=Haken | first2=H. | title= Detuned lasers and the complex Lorenz equations: Subcritical and supercritical Hopf bifurcations | year=1990 | journal=[[Phys. Rev. A]] | volume=41 | issue=7 | pages=3826–3837 | doi=10.1103/PhysRevA.41.3826 | pmid=9903557 | bibcode=1990PhRvA..41.3826N }}</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]], aperiodic, 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 {{mvar|σ}}, {{mvar|ρ}}, and {{mvar|β}} are positive. Lorenz used the values {{math|1=''σ'' = 10}}, {{math|1=''ρ'' = 28}}, and {{math|1=''β'' = {{sfrac|8|3}}}}. 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]], when {{math|''ρ'' < 1}}.<ref>{{harvtxt|Hirsch|Smale|Devaney|2003}}, pp. 306+307</ref>

A [[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]], 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 |bibcode=2020NonDy.102..713K }}</ref> and the [[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]].

{|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.
|}

{| class="wikitable" style=width:500px
|-
! Divergence of nearby trajectories.
|-
| [[File:Lorenz diverging trajectories small.gif|frameless|center]] Evolution of three initially nearby trajectories of the Lorenz system.  In this animation the equation is numerically integrated using a Runge-Kutta routine  — made using starting from three initial conditions {{math|(0.9,0,0)}} (green), {{math|(1.0,0,0)}} (blue) and {{math|(1.1,0,0)}} (red).  Produced with [[WxMaxima]].
|-
| The parameters are: <math>\rho= 28</math>, <math>\sigma= 10</math> and <math>\beta=8/3</math> .  Significant divergence is seen at around <math>t=24.0</math>, beyond which the trajectories become uncorrelated.  The full-sized graphic can be accessed [https://upload.wikimedia.org/wikipedia/commons/1/1d/Lorenz_diverging_trajectories.gif here].
|}

==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 {{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]]. 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 <math display="inline">d_0=\dfrac{19}{3}</math> 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]]

import numpy as np
import matplotlib.pyplot as plt

# Define the function and domain
x = np.linspace(0, 2 * np.pi, 400)
y = np.sin(np.exp(x))

# Plot the function
plt.plot(x, y)
plt.xlabel('x')
plt.ylabel('y = sin(e^x)')
plt.title('Plot of y = sin(e^x) for x in [0, 2π]')
plt.grid(True)
plt.show()

===Maple simulation===
<syntaxhighlight 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`);
</syntaxhighlight>

===Maxima simulation===
<syntaxhighlight 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)$
</syntaxhighlight>

===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="numpy">
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=0.6)
ax.set_xlabel("X Axis")
ax.set_ylabel("Y Axis")
ax.set_zlabel("Z Axis")
ax.set_title("Lorenz Attractor")

plt.show()
</syntaxhighlight>

[[File:Lorenz System simulation in R.png|thumb|alt=Lorenz System simulation in R|R Simulation]]

=== R simulation ===
<syntaxhighlight lang="R">
library(deSolve)
library(plotly)
# parameters
prm <- list(sigma = 10, rho = 28, beta = 8/3)
# initial values
varini <- c(
  X = 1,
  Y = 1, 
  Z = 1
)

Lorenz <- function (t, vars, prm) {
  with(as.list(vars), {
    dX <- prm$sigma*(Y - X)
    dY <- X*(prm$rho - Z) - Y
    dZ <- X*Y - prm$beta*Z
    return(list(c(dX, dY, dZ)))
   })
}

times <- seq(from = 0, to = 100, by = 0.01)
# call ode solver
out <- ode(y = varini, times = times, func = Lorenz,
           parms = prm)

# to assign color to points
gfill <- function (repArr, long) {
  rep(repArr, ceiling(long/length(repArr)))[1:long]
}

dout <- as.data.frame(out)

dout$color <- gfill(rainbow(10), nrow(dout))

# Graphics production with Plotly:
plot_ly(
  data=dout, x = ~X, y = ~Y, z = ~Z,
  type = 'scatter3d', mode = 'lines',
  opacity = 1, line = list(width = 6, color = ~color, reverscale = FALSE)
)
</syntaxhighlight>

==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 [[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 [[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 [[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]] 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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* {{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}}
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* {{cite journal | last1=Grassberger | first1=P. | last2=Procaccia | first2=I. | title=Measuring the strangeness of strange attractors | journal=Physica D | year = 1983 | volume = 9 | issue=1–2 | pages=189–208 | bibcode = 1983PhyD....9..189G | doi = 10.1016/0167-2789(83)90298-1}}
* {{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 }}
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* {{cite journal | last1=Hemati | first1=N. | title=Strange attractors in brushless DC motors | doi=10.1109/81.260218 | year=1994 | journal=[[IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications]] | issn=1057-7122 | volume=41 | issue=1 | pages=40–45}}
* {{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| bibcode=1992PhRvA..45..626K }}
* {{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| s2cid=119614076 
}}
* {{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| s2cid=123023929 }}
* {{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 }}
* 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 (identifier)|doi]]:10.5194/npg-22-749-2015. [[ISSN (identifier)|ISSN]] 1607-7946.
* {{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 | s2cid=353254 }}
* {{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 |s2cid=121427433 }}
* {{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|access-date=2020-09-16|archive-date=2019-05-23|archive-url=https://web.archive.org/web/20190523190103/http://eaps4.mit.edu/research/Lorenz/The_Statistical_Prediction_of_Solutions_1962.pdf|url-status=dead}}

== 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
 | doi = 10.1016/j.amc.2014.12.132
 | doi-access = free
 | arxiv = 1409.8649
 }}
* {{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 ==
{{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] {{Webarchive|url=https://web.archive.org/web/20090607080247/http://planetmath.org/encyclopedia/LorenzEquation.html |date=2009-06-07 }} 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://alpha.iodide.io/notebooks/34/?viewMode=report Interactive web based Lorenz Attractor] made with Iodide
{{Chaos theory}}
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