The Lorenz attractor, named for Edward N. Lorenz, is an example of a non-linear dynamic system corresponding to the long-term behavior of the Lorenz oscillator. The Lorenz oscillator is a 3-dimensional dynamical system that exhibits chaotic flow
Chaos theory
Chaos theory is a field of study in mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions, an effect which is popularly referred to as the...
In algebraic geometry, a lemniscate refers to any of several figure-eight or ∞ shaped curves. It may refer to:*The lemniscate of Bernoulli, often simply called the lemniscate, the locus of points whose product of distances from two foci equals the square of half the interfocal distance*The...
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a...
(the three variables of a three-dimensional system) evolves over time in a complex, non-repeating pattern.
Overview
The attractor itself, and the equations from which it is derived, were introduced in 1963 by Edward Lorenz
Edward Norton Lorenz
Edward Norton Lorenz was an American mathematician and meteorologist, and a pioneer of chaos theory. He discovered the strange attractor notion and coined the term butterfly effect.-Biography:...
, who derived it from the simplified equations of convection
Convection
Convection is the movement of molecules within fluids and rheids. It cannot take place in solids, since neither bulk current flows nor significant diffusion can take place in solids....
The atmosphere of Earth is a layer of gases surrounding the planet Earth that is retained by Earth's gravity. The atmosphere protects life on Earth by absorbing ultraviolet solar radiation, warming the surface through heat retention , and reducing temperature extremes between day and night...
.
In addition to its interest to the field of non-linear mathematics, the Lorenz model has important implications for climate and weather prediction. The model is an explicit statement that planetary and stellar atmospheres may exhibit a variety of quasi-periodic regimes that are, although fully deterministic, subject to abrupt and seemingly random change.
From a technical standpoint, the Lorenz oscillator is nonlinear
Nonlinearity
In mathematics, a nonlinear system is one that does not satisfy the superposition principle, or one whose output is not directly proportional to its input; a linear system fulfills these conditions. In other words, a nonlinear system is any problem where the variable to be solved for cannot be...
In mathematics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will thus always produce the same output from a given starting condition or initial state.-Examples:...
. For a certain set of parameters, the system exhibits chaotic behavior and displays what is today called a strange attractor. The strange attractor in this case is a fractal
Fractal
A fractal has been defined as "a rough or fragmented geometric shape that can be split into parts, each of which is a reduced-size copy of the whole," a property called self-similarity...
thumb|450px|Estimating the Hausdorff dimension of the coast of Great BritainIn mathematics, the Hausdorff dimension is an extended non-negative real number associated with any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space...
between 2 and 3. Grassberger (1983) has estimated the Hausdorff dimension to be 2.06 ± 0.01 and the correlation dimension
Correlation dimension
In chaos theory, the correlation dimension is a measure of the dimensionality of the space occupied by a set of random points, often referred to as a type of fractal dimension....
to be 2.05 ± 0.01.
The system also arises in simplified models for laser
Laser
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of photons. The term "laser" originated as an acronym for Light Amplification by Stimulated Emission of Radiation...
In electricity generation, an electric generator is a device that converts mechanical energy to electrical energy. A generator forces electric charge to flow through an external electrical circuit. It is analogous to a water pump, which causes water to flow...
.
Equations
The equations that govern the Lorenz oscillator are:
In fluid mechanics, the Rayleigh number for a fluid is a dimensionless number associated with buoyancy driven flow...
. All , , , but usually , and is varied. The system exhibits chaotic behavior for but displays knotted periodic orbits for other values of . For example, with it becomes a T(3,2) torus knot
Torus knot
In knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integers p and q. A torus link arises if p and q...
In the mathematical area of bifurcation theory a saddle-node bifurcation, tangential bifurcation or fold bifurcation is a local bifurcation in which two fixed points of a dynamical system collide and annihilate each other. The term 'saddle-node bifurcation' is most often used in reference to...
occurs at . When σ ≠ 0 and β (ρ-1) ≥ 0, the equations generate three critical points. The critical points at (0,0,0) correspond to no convection, and the critical points at correspond to steady convection. This pair is stable only if , which can hold only for positive if .
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.
For small values of ρ, the system is stable and evolves to one of two fixed point attractors. When ρ is larger than 24.28, the fixed points become repulsors and the trajectory is repelled by them in a very complex way, evolving without ever crossing itself.
The source code to simulate the Lorenz attractor in GNU Octave
GNU Octave
GNU Octave is a high-level language, primarily intended for numerical computations. It provides a convenient command-line interface for solving linear and nonlinear problems numerically, and for performing other numerical experiments using a language that is mostly compatible with MATLAB...
% Using LSODE to solve the ODE system.
clear all
close all
lsode_options("absolute tolerance",1e-3)
lsode_options("relative tolerance",1e-4)
t = linspace(0,25,1e3); X0 = [0,1,1.05];
[X,T,MSG]=lsode(@lorenzatt,X0,t);
T
MSG
plot3(X(:,1),X(:,2),X(:,3))
view(45,45)
% A simple Lorenz Solver in MatLab code
function dxdt=myLorenz(t,x)
% The RHS of the Lorenz attractor
% Save this function in a separate file 'myLorenz.m'
sigma = 10;
r = 28;
b = 8/3;
dxdt=[ sigma*(x(2)-x(1));
(1+r)*x(1)-x(2)-x(1)*x(3);
x(1)*x(2)-b*x(3)];
end
%% Main program: Save the program in a separate .m file and run it.
clear all; % clear all variables
t=linspace(0,50,3000)'; % time variables
y0=[-1;3;4]; % Initial conditions
[t,Y] = ode45(@myLorenz,t,y0); %Invoking built-in solver 'ode45'
plot3(Y(:,1),Y(:,2),Y(:,3)); % Plot results
grid on;
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The attractor itself, and the equations from which it is derived, were introduced in 1963 by Edward Lorenz
The equations that govern the Lorenz oscillator are:
is called the Prandtl number and 
is called the Rayleigh number
, 
, 
, but usually 
,
and 
is varied. The system exhibits chaotic behavior for 
but displays knotted periodic orbits for other values of 
. For example, with 
it becomes a T(3,2) torus knot
. When σ ≠ 0 and β (ρ-1) ≥ 0, the equations generate three critical points. The critical points at (0,0,0) correspond to no convection, and the critical points at 
correspond to steady convection. This pair is stable only if 
, which can hold only for positive 
if 
.
