Lorenz system

{{See also|Chaos theory#Lorenz's pioneering contributions to chaotic modeling}}In 1963, Edward Lorenz, with the help of Ellen Fetter, who was responsible for the numerical simulations and figures, and Margaret Hamilton, who helped in the initial, numerical computations leading up to the findings of the Lorenz model,{{harvtxt|Lorenz|1960}} developed a simplified mathematical model for atmospheric convection.{{harvtxt|Lorenz|1963}} The model is a system of three ordinary differential equations now known as the Lorenz equations:

: $\backslash begin\{align\}$

\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}

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.{{harvtxt|Sparrow|1982}} 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.

The Lorenz equations can arise in simplified models for lasers,{{harvtxt|Haken|1975}} dynamos,{{harvtxt|Knobloch|1981}} thermosyphons,{{harvtxt|Gorman|Widmann|Robbins|1986}} brushless DC motors,{{harvtxt|Hemati|1994}} electric circuits,{{harvtxt|Cuomo|Oppenheim|1993}} chemical reactions{{harvtxt|Poland|1993}} and forward osmosis.{{harvtxt|Tzenov|2014}}{{citation needed|date=June 2017}} Interestingly, the same Lorenz equations were also derived in 1963 by Sauermann and Haken

{{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 }} for a single-mode laser. In 1975, Haken realized {{harvtxt|Haken|1975}} that their equations derived in 1963 were mathematically equivalent to the original Lorenz equations. Haken's paper {{harvtxt|Haken|1975}} 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. {{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 }}

The Lorenz equations are also the governing equations in Fourier space for the Malkus waterwheel.{{harvtxt|Kolář|Gumbs|1992}}{{harvtxt|Mishra|Sanghi|2006}} 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.

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.{{harvtxt|Hirsch|Smale|Devaney|2003}}, pp. 303–305

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}}.{{harvtxt|Hirsch|Smale|Devaney|2003}}, pp. 306+307

A pitchfork bifurcation occurs at {{math|1=ρ = 1}}, and for {{math|ρ > 1}} two additional critical points appear at$$\backslash left(\; \backslash sqrt\{\backslash beta(\backslash rho-1)\},\; \backslash sqrt\{\backslash beta(\backslash rho-1)\},\; \backslash rho-1\; \backslash right)\; \backslash quad\backslash text\{and\}\backslash quad\; \backslash left(\; -\backslash sqrt\{\backslash beta(\backslash rho-1)\},\; -\backslash sqrt\{\backslash beta(\backslash rho-1)\},\; \backslash rho-1\; \backslash right).$$ These correspond to steady convection. This pair of equilibrium points is stable only if

:

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.{{harvtxt|Hirsch|Smale|Devaney|2003}}, pp. 307–308

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 strange attractor, a fractal, and a self-excited attractor with respect to all three equilibria. Its Hausdorff dimension is estimated from above by the Lyapunov dimension (Kaplan-Yorke dimension) as {{val|2.06|0.01}},{{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 }} and the correlation dimension is estimated to be {{val|2.05|0.01}}.{{harvtxt|Grassberger|Procaccia|1983}} The exact Lyapunov dimension formula of the global attractor can be found analytically under classical restrictions on the parameters:{{harvtxt|Leonov|Kuznetsov|Korzhemanova|Kusakin|2016}}{{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}}

:$3\; -\; \backslash frac\{2\; (\backslash sigma\; +\; \backslash beta\; +\; 1)\}\{\backslash sigma\; +\; 1\; +\; \backslash sqrt\{\backslash left(\backslash sigma-1\backslash right)^2\; +\; 4\; \backslash sigma\; \backslash rho\}\}$

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.{{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/}} 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.{{harvtxt|Tucker|2002}}

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.

File:Lorenz_Map.png. Points above the red line correspond to the system switching lobes.|upright=1.3]]

In Figure 4 of his paper, Lorenz plotted the relative maximum value in the {{mvar|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.

Over the past several years, a series of papers regarding high-dimensional Lorenz models have yielded a generalized Lorenz model,{{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 }} which can be simplified into the classical Lorenz model for three state variables or the following five-dimensional Lorenz model for five state variables:{{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}}$$\backslash begin\{align\}$$

\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}

A choice of the parameter $d\_0=\backslash dfrac\{19\}\{3\}$ has been applied to be consistent with the choice of the other parameters. See details in.

File:Lorenz attractor in Julia.gif

using Plots

1. 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)

dy = l.x * (l.ρ - l.z) - l.y

dz = l.x * l.y - l.β * l.z

l.x += l.dt * dx

l.y += l.dt * dy

l.z += l.dt * dz

end

attractor = Lorenz()

1. 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,

)

1. 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

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`);

[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)$

% 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))

Standard way:

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}]

Less verbose:

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}]

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

1. Step through "time", calculating the partial derivatives at the current point
2. and using them to estimate the next point

for i in range(num_steps):

xyzs[i + 1] = xyzs[i] + lorenz(xyzs[i]) * dt

1. 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()

File:Lorenz System simulation in R.png

library(deSolve)

library(plotly)

1. parameters

prm <- list(sigma = 10, rho = 28, beta = 8/3)

1. 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)

1. call ode solver

out <- ode(y = varini, times = times, func = Lorenz,

parms = prm)

1. 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))

1. 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)

)

File:Hamed sage1.png

File:Hamed sage2.png

File:Hamed sage3.png

File:Hamed sage4.png

File:Hamed sage5.png

We try to solve this system of equations for $\backslash rho\; =\; 28$, $\backslash sigma\; =\; 10$, $\backslash beta\; =\; \backslash frac\{8\}\{3\}$, with initial conditions $y\_1(0)\; =\; 0$, $y\_2(0)\; =\; 0.5$, $y\_3(0)\; =\; 0$.

1. we solve the Lorenz system of the differential equations.
2. Runge-Kutta's method y_{n+1}= y_n + h*(k_1 + 2*k_2+2*k_3+k_4)/6; x_{n+1}=x_n+h
3. k_1=f(x_n,y_n), k_2=f(x_n+h/2, y_n+hk_1/2), k_3=f(x_n+h/2, y_n+hk_2/2), k_4=f(x_n+h, y_n+hk_3)
4. differential equation

def Runge_Kutta(f,v,a,b,h,n):

tlist = [a+i*h for i in range(n+1)]

y = [[0,0,0] for _ in range(n+1)]

# Taking length of f (number of equations).

m=len(f)

# Number of variables in v.

vm=len(v)

if m!=vm:

return("error, number of equations is not equal with the number of variables.")

for r in range(vm):

y[0][r]=b[r]

# making a vector and component will be a list

# main part of the algorithm

k1=[0 for _ in range(m)]

k2=[0 for _ in range(m)]

k3=[0 for _ in range(m)]

k4=[0 for _ in range(m)]

for i in range(1,n+1): # for each t_i, i=1, ... , n

# k1=h*f(t_{i-1},x_1(t_{i-1}),...,x_m(t_{i-1}))

for j in range(m): # for each f_{j+1}, j=0, ... , m-1

k1[j]=f[j].subs(t==tlist[i-1])

for r in range(vm):

k1[j]=k1[j].subs(v[r]==y[i-1][r])

k1[j]=h*k1[j]

for j in range(m): # k2=h*f(t_{i-1}+h/2,x_1(t_{i-1})+k1/2,...,x_m(t_{i-1}+k1/2))

k2[j]=f[j].subs(t==tlist[i-1]+h/2)

for r in range(vm):

k2[j]=k2[j].subs(v[r]==y[i-1][r]+k1[r]/2)

k2[j]=h*k2[j]

for j in range(m): # k3=h*f(t_{i-1}+h/2,x_1(t_{i-1})+k2/2,...,x_m(t_{i-1})+k2/2)

k3[j]=f[j].subs(t==tlist[i-1]+h/2)

for r in range(vm):

k3[j]=k3[j].subs(v[r]==y[i-1][r]+k2[r]/2)

k3[j]=h*k3[j]

for j in range(m): # k4=h*f(t_{i-1}+h,x_1(t_{i-1})+k3,...,x_m(t_{i-1})+k3)

k4[j]=f[j].subs(t==tlist[i-1]+h)

for r in range(vm):

k4[j]=k4[j].subs(v[r]==y[i-1][r]+k3[r])

k4[j]=h*k4[j]

for j in range(m): # Now x_j(t_i)=x_j(t_{i-1})+(k1+2k2+2k3+k4)/6

y[i][j]=y[i-1][j]+(k1[j]+2*k2[j]+2*k3[j]+k4[j])/6

return(tlist,y)

1. (Figure 1) Here, we plot the solutions of the Lorenz ODE system.

a=0.0 # t_0

b=[0.0,.50,0.0] # x_1(t_0), ... , x_m(t_0)

t=var('t')

x = var('x', n=3, latex_name='x')

v=[x[ii] for ii in range(3)]

f= [10*(x1-x0),x0*(28-x2)-x1,x0*x1-(8/3)*x2];

n=1600

h=0.0125

tlist,y=Runge_Kutta(f,v,a,b,h,n)

1. print(tlist)
2. print(y)

T=point3d(y[i][0],y[i][1],y[i][2 for i in range(n)], color='red')

S=line3d(y[i][0],y[i][1],y[i][2 for i in range(n)], color='red')

show(T+S)

1. (Figure 2) Here, we plot every y1, y2, and y3 in terms of time.

a=0.0 # t_0

b=[0.0,.50,0.0] # x_1(t_0), ... , x_m(t_0)

t=var('t')

x = var('x', n=3, latex_name='x')

v=[x[ii] for ii in range(3)]

Lorenz= [10*(x1-x0),x0*(28-x2)-x1,x0*x1-(8/3)*x2];

n=100

h=0.1

tlist,y=Runge_Kutta(Lorenz,v,a,b,h,n)

1. Runge_Kutta(f,v,0,b,h,n)
2. print(tlist)
3. print(y)

P1=list_plot(tlist[i],y[i][0 for i in range(n)], plotjoined=True, color='red');

P2=list_plot(tlist[i],y[i][1 for i in range(n)], plotjoined=True, color='green');

P3=list_plot(tlist[i],y[i][2 for i in range(n)], plotjoined=True, color='yellow');

show(P1+P2+P3)

1. (Figure 3) Here, we plot the y and x or equivalently y2 and y1

a=0.0 # t_0

b=[0.0,.50,0.0] # x_1(t_0), ... , x_m(t_0)

t=var('t')

x = var('x', n=3, latex_name='x')

v=[x[ii] for ii in range(3)]

f= [10*(x1-x0),x0*(28-x2)-x1,x0*x1-(8/3)*x2];

n=800

h=0.025

tlist,y=Runge_Kutta(f,v,a,b,h,n)

vv=y[i][0],y[i][1 for i in range(n)];

1. print(tlist)
2. print(y)

T=points(vv, rgbcolor=(0.2,0.6, 0.1), pointsize=10)

S=line(vv,rgbcolor=(0.2,0.6, 0.1))

show(T+S)

1. (Figure 4) Here, we plot the z and x or equivalently y3 and y1

a=0.0 # t_0

b=[0.0,.50,0.0] # x_1(t_0), ... , x_m(t_0)

t=var('t')

x = var('x', n=3, latex_name='x')

v=[x[ii] for ii in range(3)]

f= [10*(x1-x0),x0*(28-x2)-x1,x0*x1-(8/3)*x2];

n=800

h=0.025

tlist,y=Runge_Kutta(f,v,a,b,h,n)

vv=y[i][0],y[i][2 for i in range(n)];

1. print(tlist)
2. print(y)

T=points(vv, rgbcolor=(0.2,0.6, 0.1), pointsize=10)

S=line(vv,rgbcolor=(0.2,0.6, 0.1))

show(T+S)

1. (Figure 5) Here, we plot the z and x or equivalently y3 and y2

a=0.0 # t_0

b=[0.0,.50,0.0] # x_1(t_0), ... , x_m(t_0)

t=var('t')

x = var('x', n=3, latex_name='x')

v=[x[ii] for ii in range(3)]

f= [10*(x1-x0),x0*(28-x2)-x1,x0*x1-(8/3)*x2];

n=800

h=0.025

tlist,y=Runge_Kutta(f,v,a,b,h,n)

vv=y[i][1],y[i][2 for i in range(n)];

1. print(tlist)
2. print(y)

T=points(vv, rgbcolor=(0.2,0.6, 0.1), pointsize=10)

S=line(vv,rgbcolor=(0.2,0.6, 0.1))

show(T+S)

As shown in Lorenz's original paper,{{harvtxt|Lorenz|1963}} the Lorenz system is a reduced version of a larger system studied earlier by Barry Saltzman.{{harvtxt|Saltzman|1962}} The Lorenz equations are derived from the Oberbeck–Boussinesq approximation to the equations describing fluid circulation in a shallow layer of fluid, heated uniformly from below and cooled uniformly from above.{{harvtxt|Lorenz|1963}} 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.{{harvtxt|Lorenz|1963}}

The partial differential equations modeling the system's stream function and temperature are subjected to a spectral 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),{{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 }} Supplementary Materials.

The scientific community accepts that the chaotic features found in low-dimensional Lorenz models could represent features of the Earth's atmosphere,{{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}}{{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}}{{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 |hdl=10150/673501 |hdl-access=free }} yielding the statement of “weather is chaotic.” By comparison, based on the concept of attractor coexistence within the generalized Lorenz model and the original Lorenz model,{{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}}{{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}} Shen and his co-authors proposed a revised view that “weather possesses both chaos and order with distinct predictability”.{{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 |bibcode=2021BAMS..102E.148S |doi=10.1175/BAMS-D-19-0165.1 |issn=0003-0007 |s2cid=208369617 |doi-access=free}} 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”.

Smale's 14th problem asks, 'Do the properties of the Lorenz attractor exhibit that of a strange attractor?'. The problem was answered affirmatively by Warwick Tucker in 2002. To prove this result, Tucker used rigorous numerics methods like interval arithmetic and normal forms. First, Tucker defined a cross section $\backslash Sigma\backslash subset\; \backslash \{x\_3\; =\; r\; -\; 1\; \backslash \}$ that is cut transversely by the flow trajectories. From this, one can define the first-return map $P$, which assigns to each $x\backslash in\backslash Sigma$ the point $P(x)$ where the trajectory of $x$ first intersects $\backslash Sigma$.

Then the proof is split in three main points that are proved and imply the existence of a strange attractor.{{harvtxt|Viana|2000}} The three points are:

• There exists a region $N\backslash subset\backslash Sigma$ invariant under the first-return map, meaning $P(N)\backslash subset\; N$.
• The return map admits a forward invariant cone field.
• Vectors inside this invariant cone field are uniformly expanded by the derivative $DP$ of the return map.

To prove the first point, we notice that the cross section $\backslash Sigma$ is cut by two arcs formed by $P(\backslash Sigma)$. Tucker covers the location of these two arcs by small rectangles $R\_i$, the union of these rectangles gives $N$. Now, the goal is to prove that for all points in $N$, the flow will bring back the points in $\backslash Sigma$, in $N$. To do that, we take a plan $\backslash Sigma\text{'}$ below $\backslash Sigma$ at a distance $h$ small, then by taking the center $c\_i$ of $R\_i$ and using Euler integration method, one can estimate where the flow will bring $c\_i$ in $\backslash Sigma\text{'}$ which gives us a new point $c\_i\text{'}$. Then, one can estimate where the points in $\backslash Sigma$ will be mapped in $\backslash Sigma\text{'}$ using Taylor expansion, this gives us a new rectangle $R\_i\text{'}$ centered on $c\_i$. Thus we know that all points in $R\_i$ will be mapped in $R\_i\text{'}$. The goal is to do this method recursively until the flow comes back to $\backslash Sigma$ and we obtain a rectangle $Rf\_i$ in $\backslash Sigma$ such that we know that $P(R\_i)\backslash subset\; Rf\_i$. The problem is that our estimation may become imprecise after several iterations, thus what Tucker does is to split $R\_i\text{'}$ into smaller rectangles $R\_\{i,j\}$ and then apply the process recursively. Another problem is that as we are applying this algorithm, the flow becomes more 'horizontal', leading to a dramatic increase in imprecision. To prevent this, the algorithm changes the orientation of the cross sections, becoming either horizontal or vertical.

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 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}} {{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.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]

• Eden's conjecture on the Lyapunov dimension
• Lorenz 96 model
• List of chaotic maps
• Takens' theorem

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{{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

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