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".

Lorenz De Jong Henon Rössler Standard

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):

Vensim Lorenz
Lorenz butterfly
Lorenz Vensim-input

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