# Strange (or itinerant) attractors

An attractor is an invariant subset of a phase space towards which
dynamical systems tend to evolve in discrete time regardless of their
starting conditions. Simple attractors can be fixed points, sets of
points, limit cycles or manifolds. More interesting attractors are
"strange", "chaotic" or "itinerant" attractors, which span an array of
possible states in which a dynamical system can roam around without
repeating itself.

The following interactive model presents some examples of "strange
attractors".

## The Lorenz attractor

The famous Lorenz attractor is named after meteorologist Edward Lorenz
who, in efforts to model weather changes with the help of a computer,
put three equations together and by iterating them found that small
differences in initial conditions can have huge consequences after
some time - the so called butterfly
effect. The following three difference equations determine
the Lorenz attractor as presented in the interactive model above:

\[x_{n+1}=x_n + (-a*x_n+a*y_n)*dt\] \[y_{n+1}=y_n +
(b*x_n-y_n-x_n*z_n)*dt\] \[z_{n+1}=z_n + (-c*z_n+x_n*y_n)*dt\]

## The Peter de Jong attractor

This system provides a nice example of a truly "strange attractor"
being determined by the iterated interaction of four parameters. The
system is defined by the equations:

\[x_{n+1}=sin(a*y_n)-cos(b*x_n)\] \[y_{n+1}=sin(c*x_n)-cos(d*y_n)\]

## The Hénon attractor

One of the most studied dynamical systems is the Hénon attractor or
Hénon map, introduced by the astronomer Michel Hénon as a simplified
model of the so called Poincaré section of the Lorenz equations. It is
defined by the equations:

\[x_{n+1}=y_n + 1 - a*x_n^2\] \[y_{n+1}=b*x_n\]

## The Rössler attractor

This attractor, named after the German biochemist Otto Rössler, is
determined by the iterated interaction of the following three
equations:

\[x_{n+1}= x_n + (-y_n - z_n)*dt\] \[y_{n+1}=y_n + (x_n + a*y_n)*dt\]
\[z_{n+1}=z_n + (b + z_n*(x_n - c))*dt\]

## The Standard attractor

This last one of the above examples is determined by one parameter in
the following two equations:

\[x_{n+1}=x_n + (y_n mod 2\Pi)*dt\] \[y_{n+1}=y_n + (a * sin x_n mod
2\Pi)*dt\]

The following shows how the Lorenz attractor can be modeled using the
modeling platform VENSIM (which is introduced here):

In the control panel, indicate X as the values plotted to the X-Axis,
and Z as the dependent variable.