The Lorenz system is a system of ordinary differential equations first studied by Edward Lorenz. It is notable for having chaotic solutions for certain parameter values and initial conditions. In particular, the Lorenz attractor is a set of chaotic solutions of the Lorenz system. In popular media the 'butterfly effect' stems from the realworld implications of the Lorenz attractor, i.e. that in any physical system, in the absence of perfect knowledge of the initial conditions (even the minuscule disturbance of the air due to a butterfly flapping its wings), our ability to predict its future course will always fail. This underscores that physical systems can be completely deterministic and yet still be inherently unpredictable even in the absence of quantum effects. The shape of the Lorenz attractor itself, when plotted graphically, may also be seen to resemble a butterfly.
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Transcription
Contents
Overview
In 1963, Edward Lorenz developed a simplified mathematical model for atmospheric convection.^{[1]} The model is a system of three ordinary differential equations now known as the Lorenz equations:
The equations relate the properties of a twodimensional 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: is proportional to the rate of convection, to the horizontal temperature variation, and to the vertical temperature variation.^{[2]} The constants , , and are system parameters proportional to the Prandtl number, Rayleigh number, and certain physical dimensions of the layer itself.^{[3]}
The Lorenz equations also arise in simplified models for lasers,^{[4]} dynamos,^{[5]} thermosyphons,^{[6]} brushless DC motors,^{[7]} electric circuits,^{[8]} chemical reactions^{[9]} and forward osmosis.^{[10]}
From a technical standpoint, the Lorenz system is nonlinear, nonperiodic, threedimensional and deterministic. The Lorenz equations have been the subject of hundreds of research articles, and at least one booklength study.^{[11]}
Analysis
One normally assumes that the parameters , , and are positive. Lorenz used the values , and . The system exhibits chaotic behavior for these (and nearby) values.^{[12]}
If 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 .^{[13]}
A pitchfork bifurcation occurs at , and for two additional critical points appear at: and These correspond to steady convection. This pair of equilibrium points is stable only if
which can hold only for positive if . At the critical value, both equilibrium points lose stability through a subcritical Hopf bifurcation.^{[14]}
When , , and , the Lorenz system has chaotic solutions (but not all solutions are chaotic). Almost all initial points will tend to an invariant set – the Lorenz attractor – a strange attractor, a fractal, and a selfexcited attractor with respect to all three equilibria. Its Hausdorff dimension is estimated to be 2.06 ± 0.01^{[citation needed]}, and the correlation dimension is estimated to be 2.05 ± 0.01.^{[15]} The exact Lyapunov dimension (KaplanYorke dimension) formula of the global attractor can be found analytically under classical restrictions on the parameters^{[16]}:
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.^{[17]} 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.^{[18]}
For other values of , the system displays knotted periodic orbits. For example, with it becomes a T(3,2) torus knot.
Example solutions of the Lorenz system for different values of ρ  

ρ = 14, σ = 10, β = 8/3 (Enlarge)  ρ = 13, σ = 10, β = 8/3 (Enlarge) 
ρ = 15, σ = 10, β = 8/3 (Enlarge)  ρ = 28, σ = 10, β = 8/3 (Enlarge) 
For small values of ρ, the system is stable and evolves to one of two fixed point attractors. When ρ is larger than 24.74, the fixed points become repulsors and the trajectory is repelled by them in a very complex way. 
Sensitive dependence on the initial condition  

Time t = 1 (Enlarge)  Time t = 2 (Enlarge)  Time t = 3 (Enlarge) 
These figures — made using ρ = 28, σ = 10 and β = 8/3 — show three time segments of the 3D 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^{−5} in the xcoordinate. 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. 
MATLAB simulation
% 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]); % RungeKutta 4th/5th order ODE solver
plot3(a(:,1),a(:,2),a(:,3))
Mathematica simulation
a = 10; b = 8/3; r = 28;
x = 1; y = 1; z = 1;points = {{1,1,1}};
i := AppendTo[points, {x = N[x + (a*y  a*x)/100], y = N[y + (x*z + r*x  y)/100], z = N[z + (x*y  b*z)/100]}]
Do[i, {3000}]
ListPointPlot3D[points, PlotStyle > {Red, PointSize[Tiny]}]
An alternative with more Mathematica style:
Clear[LorenzSystemPoints, p, s, pts]
LorenzSystemPoints[parameters_List, steps_Integer] :=
Module[{σ, ρ, β, updates, δ = 1.*^3, pt0 = {1., 1., 1.}},
{σ, ρ, β} = parameters;
updates = x \[Function] {{1  δ σ, δ σ, 0},
{δ ρ, 1  δ, δ #},
{0, δ #, 1  δ β}}.{##} & @@ x;
NestList[updates, pt0, steps]
];
p = {10., 28., 8/3.};
s = 30000;
pts = LorenzSystemPoints[p, s];
ListPointPlot3D[pts, PlotRange > All, PlotTheme > "Scientific", ImageSize > Large]
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}]
Python simulation
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint
from mpl_toolkits.mplot3d import Axes3D
rho = 28.0
sigma = 10.0
beta = 8.0 / 3.0
def f(state, t):
x, y, z = state # unpack the state vector
return sigma * (y  x), x * (rho  z)  y, x * y  beta * z # derivatives
state0 = [1.0, 1.0, 1.0]
t = np.arange(0.0, 40.0, 0.01)
states = odeint(f, state0, t)
fig = plt.figure()
ax = fig.gca(projection='3d')
ax.plot(states[:,0], states[:,1], states[:,2])
plt.show()
Modelica simulation
model LorenzSystem
parameter Real sigma = 10;
parameter Real rho = 28;
parameter Real beta = 8/3;
parameter Real x_start = 1 "Initial xcoordinate";
parameter Real y_start = 1 "Initial ycoordinate";
parameter Real z_start = 1 "Initial zcoordinate";
Real x "xcoordinate";
Real y "ycoordinate";
Real z "zcoordinate";
initial equation
x = x_start;
y = y_start;
z = z_start;
equation
der(x) = sigma*(yx);
der(y) = rho*x  y  x*z;
der(z) = x*y  beta*z;
end LorenzSystem;
Julia simulation
using DifferentialEquations, ParameterizedFunctions, Plots
lorenz = @ode_def begin # define the system
dx = σ * (y  x)
dy = x * (ρ  z)  y
dz = x * y  β*z
end σ ρ β
u0 = [1.0,0.0,0.0] # initial conditions
tspan = (0.0,100.0) # timespan
p = [10.0,28.0,8/3] # parameters
prob = ODEProblem(lorenz, u0, tspan, p) # define the problem
sol = solve(prob) # solve it
plot(sol, vars = (1, 2, 3)) # plot solution in phase space  variables ordered with 1 based indexing
Derivation of the Lorenz equations as a model of atmospheric convection
The Lorenz equations are derived from the OberbeckBoussinesq approximation to the equations describing fluid circulation in a shallow layer of fluid, heated uniformly from below and cooled uniformly from above.^{[1]} This fluid circulation is known as RayleighBénard convection. The fluid is assumed to circulate in two dimensions (vertical and horizontal) with periodic rectangular boundary conditions.
The partial differential equations modeling the system's stream function and temperature are subjected to a spectral Galerkin 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.^{[19]} The Lorenz system is a reduced version of a larger system studied earlier by Barry Saltzman.^{[20]}
Resolution of Smale's 14th problem
Smale's 14th problem says 'Do the properties of the Lorenz attractor exhibit that of a strange attractor?', it was answered affirmatively by Warwick Tucker in 2002.^{[21]} To prove this result, Tucker used rigorous numerics methods like Interval arithmetic and Normal Forms. First, Tucker defined a cross section that is cut transversely by the flow trajectories. From this, one can define the firstreturn map , which assigns to each the point where the trajectory of first intersects .
Then the proof is split in three main points that are proved and imply the existence of a strange attractor^{[22]}. The three points are:
 There exists a region invariant under the firstreturn map, meaning
 The return map admits a forward invariant cone field
 Vectors inside this invariant cone field are uniformly expanded by the derivative of the return map.
To prove the first point, we notice that the cross section is cut by two arcs formed by (see ^{[23]}). Tucker covers the location of these two arcs by small rectangles , the union of these rectangles gives . Now, the goal is to prove that for all points in , the flow will bring back the points in , in . To do that, we take a plan below at a distance small, then by taking the center of and using Euler integration method, one can estimate where the flow will bring in which gives us a new point . Then, one can estimate where the points in will be mapped in using Taylor expansion, this gives us a new rectangle centered on . Thus we know that all points in will be mapped in . The goal is to do this method recursively until the flow comes back to and we obtain a rectangle in such that we know that . The problem is that our estimation may become imprecise after several iterations, thus what Tucker does is to split into smaller rectangles and then apply the process recursively. Another problem is that as we are applying this algorithm, the flow becomes more 'horizontal' (see ^{[24]}), 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
See also
Notes
 ^ ^{a} ^{b} Lorenz (1963)
 ^ Sparrow (1982)
 ^ Sparrow (1982)
 ^ Haken (1975)
 ^ Knobloch (1981)
 ^ Gorman, Widmann & Robbins (1986)
 ^ Hemati (1994)
 ^ Cuomo & Oppenheim (1993)
 ^ Poland (1993)
 ^ Tzenov (2014)^{[citation needed]}
 ^ Sparrow (1982)
 ^ Hirsch, Smale & Devaney (2003), pp. 303–305
 ^ Hirsch, Smale & Devaney (2003), pp. 306+307
 ^ Hirsch, Smale & Devaney (2003), pp. 307+308
 ^ Grassberger & Procaccia (1983)
 ^ Leonov et al. (2016)
 ^ Guckenheimer, John; Williams, R. F. (19791201). "Structural stability of Lorenz attractors". Publications Mathématiques de l'Institut des Hautes Études Scientifiques. 50 (1): 59–72. doi:10.1007/BF02684769. ISSN 00738301.
 ^ Tucker (2002)
 ^ Hilborn (2000), Appendix C; Bergé, Pomeau & Vidal (1984), Appendix D
 ^ Saltzman (1962)
 ^ Tucker (2002)
 ^ Viana (2000)
 ^ Viana (2000)
 ^ Viana (2000)
References
 Bergé, Pierre; Pomeau, Yves; Vidal, Christian (1984). Order within Chaos: Towards a Deterministic Approach to Turbulence. New York: John Wiley & Sons. ISBN 9780471849674.
 Cuomo, Kevin M.; Oppenheim, Alan V. (1993). "Circuit implementation of synchronized chaos with applications to communications". Physical Review Letters. 71 (1): 65–68. Bibcode:1993PhRvL..71...65C. doi:10.1103/PhysRevLett.71.65. ISSN 00319007. PMID 10054374.
 Gorman, M.; Widmann, P.J.; Robbins, K.A. (1986). "Nonlinear dynamics of a convection loop: A quantitative comparison of experiment with theory". Physica D. 19 (2): 255–267. Bibcode:1986PhyD...19..255G. doi:10.1016/01672789(86)900229.
 Grassberger, P.; Procaccia, I. (1983). "Measuring the strangeness of strange attractors". Physica D. 9 (1–2): 189–208. Bibcode:1983PhyD....9..189G. doi:10.1016/01672789(83)902981.
 Haken, H. (1975). "Analogy between higher instabilities in fluids and lasers". Physics Letters A. 53 (1): 77–78. Bibcode:1975PhLA...53...77H. doi:10.1016/03759601(75)903539.
 Hemati, N. (1994). "Strange attractors in brushless DC motors". IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications. 41 (1): 40–45. doi:10.1109/81.260218. ISSN 10577122.
 Hilborn, Robert C. (2000). Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers (second ed.). Oxford University Press. ISBN 9780198507239.
 Hirsch, Morris W.; Smale, Stephen; Devaney, Robert (2003). Differential Equations, Dynamical Systems, & An Introduction to Chaos (Second ed.). Boston, MA: Academic Press. ISBN 9780123497031.
 Lorenz, Edward Norton (1963). "Deterministic nonperiodic flow". Journal of the Atmospheric Sciences. 20 (2): 130–141. Bibcode:1963JAtS...20..130L. doi:10.1175/15200469(1963)020<0130:DNF>2.0.CO;2.
 Knobloch, Edgar (1981). "Chaos in the segmented disc dynamo". Physics Letters A. 82 (9): 439–440. Bibcode:1981PhLA...82..439K. doi:10.1016/03759601(81)902747.
 Pchelintsev, A.N. (2014). "Numerical and Physical Modeling of the Dynamics of the Lorenz System". Numerical Analysis and Applications. 7 (2): 159–167. doi:10.1134/S1995423914020098.
 Poland, Douglas (1993). "Cooperative catalysis and chemical chaos: a chemical model for the Lorenz equations". Physica D. 65 (1): 86–99. Bibcode:1993PhyD...65...86P. doi:10.1016/01672789(93)90006M.
 Tzenov, Stephan (2014). "Strange Attractors Characterizing the Osmotic Instability". arXiv:1406.0979v1 [physics.fludyn].
 Saltzman, Barry (1962). "Finite Amplitude Free Convection as an Initial Value Problem—I". Journal of the Atmospheric Sciences. 19 (4): 329–341. Bibcode:1962JAtS...19..329S. doi:10.1175/15200469(1962)019<0329:FAFCAA>2.0.CO;2.
 Sparrow, Colin (1982). The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors. Springer.
 Tucker, Warwick (2002). "A Rigorous ODE Solver and Smale's 14th Problem" (PDF). Foundations of Computational Mathematics. 2 (1): 53–117. CiteSeerX 10.1.1.545.3996. doi:10.1007/s002080010018.
 Viana, Marcelo (2000). "What's new on Lorenz strange attractors?". The Mathematical Intelligencer. 22 (3): 6–19. doi:10.1007/BF03025276.
 Leonov, G.A.; Kuznetsov, N.V.; Korzhemanova, N.A.; Kusakin, D.V. (2016). "Lyapunov dimension formula for the global attractor of the Lorenz system". Communications in Nonlinear Science and Numerical Simulation. 41: 84–103. arXiv:1508.07498. Bibcode:2016CNSNS..41...84L. doi:10.1016/j.cnsns.2016.04.032.
Further reading
 G.A. Leonov & N.V. Kuznetsov (2015). "On differences and similarities in the analysis of Lorenz, Chen, and Lu systems" (PDF). Applied Mathematics and Computation. 256: 334–343. doi:10.1016/j.amc.2014.12.132.
External links
Wikimedia Commons has media related to Lorenz attractors. 
 Hazewinkel, Michiel, ed. (2001) [1994], "Lorenz attractor", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 9781556080104
 Weisstein, Eric W. "Lorenz attractor". MathWorld.
 Lorenz attractor by Rob Morris, Wolfram Demonstrations Project.
 Lorenz equation on planetmath.org
 Synchronized Chaos and Private Communications, with Kevin Cuomo. The implementation of Lorenz attractor in an electronic circuit.
 Lorenz attractor interactive animation (you need the Adobe Shockwave plugin)
 3D Attractors: Mac program to visualize and explore the Lorenz attractor in 3 dimensions
 Lorenz Attractor implemented in analog electronic
 Lorenz Attractor interactive animation (implemented in Ada with GTK+. Sources & executable)
 Web based Lorenz Attractor (implemented in JavaScript/HTML/CSS)
 Interactive web based Lorenz Attractor made with Iodide