Lorenz system

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 which, when plotted, resemble a butterfly or figure eight.

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:

Here , , and make up the system state, is time, and , , are the system parameters. The Lorenz equations also arise in simplified models for lasers,[2] dynamos,[3] thermosyphons,[4] brushless DC motors,[5] electric circuits,[6] chemical reactions[7] and forward osmosis.[8]

From a technical standpoint, the Lorenz system is nonlinear, non-periodic, three-dimensional and deterministic. The Lorenz equations have been the subject of hundreds of research articles, and at least one book-length study.[9]

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.[10]

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 .[11]

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 Hopf bifurcation.[12]

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 and a fractal. Its Hausdorff dimension is estimated to be 2.06 ± 0.01, and the correlation dimension is estimated to be 2.05 ± 0.01.[13] The exact Lyapunov dimension (Kaplan-Yorke dimension) formula of the global attractor can be found analytically under classical restrictions on the parameters

[14]
.

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.[15] 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.[16]

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.
Java animation showing evolution for different values of ρ
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 3-D evolution of 2 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 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.
Java animation of the Lorenz attractor shows the continuous evolution.

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]);     % Runge-Kutta 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]}]

fancier 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()

Derivation of the Lorenz equations as a model of atmospheric convection

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.[1] This fluid circulation is known as Rayleigh-Bénard convection. The fluid is assumed to circulate in two dimensions (vertical and horizontal) with periodic rectangular boundary conditions.

The partial differential equations modeling the system's stream function and temperature are subjected to a spectral Galerkin 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.[17] The Lorenz system is a reduced version of a larger system studied earlier by Barry Saltzman.[18]

See also

Notes

References

Further reading

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