Deterministic Chaos

Deterministic Chaos - Examples and Simulations

The Lorenz butterfly effect
occurs even though two
points are initially very close.
Click in the screen where it
says CHAOS to see two
trajectories of evolution--
one blue and the other yellow--
starting from 2 different very
close starting points (they
differ only by 0.00001 in the
x-coordinate).

The Lorenz butterfly is a prime example of deterministic chaos. Double click at two nearby points 10 times and watch the separation of the orbits diverge exponentially.

A clearer idea of the geometry of the Lorenz butterfly can be seen is this image which contains three separate trajectories rather than the more typical two trajectories. Even more trajectories are seen is this applet, which shows each trajectory in a different color.

More detail about the Lorenz butterfly can be found at this enthusiast's website.

More 3-D representations of the Lorenz butterfly:

http://local.wasp.uwa.edu.au/~pbourke/fractals/lorenz/lorenz.m4v

http://en.wikipedia.org/wiki/Lorenz_attractor

http://mathworld.wolfram.com/LorenzAttractor.html

http://www.mandelbrot-dazibao.com/Chaos/Chaos.htm

http://www.physics.emory.edu/~weeks/research/tseries5.html#lorenz

http://plus.maths.org/latestnews/sep-dec04/lorenz/index.html

The Lorenz waterwheel is another example of deterministically chaotic behavior. A video of a real Lorenz waterwheel can be found on the multimedia page.

There is a relationship between the Lorenz butterfly and the Lorenz watetwheel (besides the name Lorenz!). A Georgia Tech student put together a web page that not only shows the relationship but describes his adventures in building one.

Strange attractors can be stunningly beautiful; here's some great eye candy.

Double pendulums are another form of strange attractor. Changing the initial conditions of two pendulums slightly results in chaotic behavior. This applet starts running with identical initial conditions. Watch it evolve for some time until you are convinced that the two oscillators evolve identically given identical initial conditions. Then pause the simulation and change the value of the angle phi for one of the oscillators. Watch it evolve for some time until you are convinced that the tiny change you made in phi has produced enormous differences between the two oscillators.

To get somewhat of a better idea of what a double pendulum is doing, check out this simulation of a double pendulum versus energy. Set E total = 1, 2, 5, 10, 20, 50, 100, 200, 500, and 1000 and observe the trajectories. Can you correlate/relate the pattern of the trajectories with the motion of the pendulum? Is the behavior of the system sensitive to the initial conditions at any of these energies? (Hint: to get a much faster graph, disable the pendulum display.)

Cobweb diagrams graphically display deterministically chaotic behavior. They can often be used to more quickly determine where chaotic behavior occurs in a system than the more analytical approach, so they are a useful tool for modeling. Here are two applets that show how to make cobweb diagrams [ 1 | 2 ]

A variety of applets [ 1 | 2 | 3 ] show the construction of a bifurcation diagram. It's worth changing the parameters from a range of 1 - 4 to something like 3.4 - 3.6--or an even smaller range--to see the details of the bifurcation. Can you find the Feigenbaum constant? Use these pages [ 1 | 2 | 3 ] as guides.

Most of the bifurcation diagrams above are based on quadratic maps. Other functions such as cubic and quartic functions also lend themselves to bifurcation diagrams as well.