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<html>
<head>
<title>
LORENZ_ODE - The Lorenz System
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
LORENZ_ODE <br> The Lorenz System
</h1>
<hr>
<p>
<b>LORENZ_ODE</b>
is a C++ program which
approximates solutions to the Lorenz system,
creating output files that can be displayed by Gnuplot.
</p>
<p>
The Lorenz system, originally intended as a simplified model of
atmospheric convection, has instead become a standard example of
sensitive dependence on initial conditions; that is, tiny differences
in the starting condition for the system rapidly become magnified.
The system also exhibits what is known as the "Lorenz attractor",
that is, the collection of trajectories for different starting points
tends to approach a peculiar butterfly-shaped region.
</p>
<p>
The Lorenz system includes three ordinary differential equations:
<pre>
dx/dt = sigma ( y - x )
dy/dt = x ( rho - z ) - y
dz/dt = xy - beta z
</pre>
where the parameters beta, rho and sigma are usually assumed to be
positive. The classic case uses the parameter values
<pre>
beta = 8 / 3
rho = 28
sigma - 10
</pre>
</p>
<p>
The initial conditions for this system are not often specified; rather,
investigators simply note that the trajectories associated with two very
close starting points will eventually separate. However, simply to
get started, we can suggest the following starting values at t=0:
<pre>
x = 8
y = 1
z = 1
</pre>
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this
web page are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>LORENZ_ODE</b> is available in
<a href = "../../c_src/lorenz_ode/lorenz_ode.html">a C version</a> and
<a href = "../../cpp_src/lorenz_ode/lorenz_ode.html">a C++ version</a> and
<a href = "../../f77_src/lorenz_ode/lorenz_ode.html">a FORTRAN77 version</a> and
<a href = "../../f_src/lorenz_ode/lorenz_ode.html">a FORTRAN90 version</a> and
<a href = "../../m_src/lorenz_ode/lorenz_ode.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/gnuplot/gnuplot.html">
GNUPLOT</a>,
C++ programs which
illustrate the use of the gnuplot graphics program.
</p>
<p>
<a href = "../../examples/graphics_examples_gnuplot/graphics_examples_gnuplot.html">
GRAPHICS_EXAMPLES_GNUPLOT</a>,
gnuplot scripts which
illustrate how various kinds of data can be displayed and analyzed graphically
using the interactive executable graphics program GNUPLOT.
</p>
<p>
<a href = "../../cpp_src/spring_ode2/spring_ode2.html">
SPRING_ODE2</a>,
a C++ program which
shows how gnuplot graphics can be used to illustrate
a solution of the ordinary differential equation (ODE) that describes
the motion of a weight attached to a spring.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Edward Lorenz,<br>
Deterministic Nonperiodic Flow,<br>
Journal of the Atmospheric Sciences,<br>
Volume 20, Number 2, 1963, pages 130-141.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "lorenz_ode.cpp">lorenz_ode.cpp</a>, the source code.
</li>
<li>
<a href = "lorenz_ode.sh">lorenz_ode.sh</a>,
BASH commands to compile the source code.
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "lorenz_ode_output.txt">lorenz_ode_output.txt</a>,
the output file.
</li>
<li>
<a href = "lorenz_ode_data.txt">lorenz_ode_data.txt</a>,
the graphics data file.
</li>
<li>
<a href = "lorenz_ode_commands.txt">lorenz_ode_commands.txt</a>,
the graphics command file.
</li>
<li>
<a href = "xyz_3d.png">xyz_3d.png</a>,
a plot, created by gnuplot, of (X(t),Y(t),Z(t)).
</li>
<li>
<a href = "xyz_time.png">xyz_time.png</a>,
a plot, created by gnuplot, of (t,X(t)), (t,Y(t), and (t,Z(t)).
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>LORENZ_RHS</b> evaluates the right hand side of the Lorenz ODE.
</li>
<li>
<b>R8VEC_LINSPACE_NEW</b> creates a vector of linearly spaced values.
</li>
<li>
<b>RK4VEC</b> takes one Runge-Kutta step for a vector ODE.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../cpp_src.html">
the C++ source codes</a>.
</p>
<hr>
<i>
Last revised on 10 October 2013.
</i>
<!-- John Burkardt -->
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