DOUBLE PENDULUM SENSITIVE DEPENDENCE

     Theory

A double pendulum consists of one simple pendulum
suspended from another, with no friction and with motion
confined to a single plane.  The Double Pendulum Sensitive
Dependence simulation demonstrates chaos by simultaneously
determining the motion of two identical double-pendulum
systems with slightly different initial conditions.  The
simulation varies the initial angular velocity of the upper
pendulum, but any other system parameter could have been
chosen to demonstrate the "butterfly effect" (sensitive
dependence upon initial conditions).

The two double pendulums are independent systems (they are not
coupled in any fashion), but to emphasize the exponentially
rapid divergence of the two systems, the simulation superimposes
their images.  Therefore, until the pendulums achieve
significantly different configurations the two images will
appear as one.

To the right of the animation is a graph of the difference in
angular configuration of the two systems as a function of time.
This difference is defined, somewhat arbitrarily, as the square
root of the sum of the squares of the differences between the
two angles between corresponding pendulums of the two systems.
This quantity is small only when the two double pendulums are
in nearly identical configurations.


     Parameters

The upper pendulum is #1; the lower pendulum is #2.
Ratio of length 2 to length 1 [default = 2.0]
Ratio of mass 2 to mass 1 [default = 0.5]
Initial velocity of mass 1 of first double pendulum
                            [default = 7.0 radians/second]
Initial velocity of mass 1 of second double pendulum
                            [default = 7.01 radians/second]
Integration stepsize [default = 0.02 sec]

     Commands

-- Simulation --

The simulation starts automatically.
To pause until any key is pressed, press the "p" key.
To display the violation of the conservation of
energy (as a percentage of the original energy), press "t"
(press any key to resume the simulation).
To enter new parameter values, press "n".
To double the integration stepsize, press "2".
To halve the integration stepsize, press "5".
To exit the simulation, press the "Esc" key.

-- Display --

To clear the screen and continue calculating, press "c".
To speed up the display (without changing the
integration stepsize), press "+" one or more times.
To slow down the display (without changing the
integration stepsize), press "-" one or more times.
To reset the default display speed, press "r".
To toggle drawing the spheres, press "o" (omitting
the spheres increases the simulation speed).
To display the help screen, press "h".

-- Output --

To print the current screen image, press "g".

     Things to Try

Study the energy dependence of chaos in the system. To
detect chaos look for a rapid divergence in the behavior
of the two systems in the animation or in the angular
difference graph.  At what point in the motion (that is,
in what configuration) is this divergence most likely to
occur?  Vary the difference in initial velocities and
make a graph of time to diverge (choose a threshold)
versus initial velocity difference.  With this graph you
can confirm that sensitive dependence grows exponentially
fast.

Vray construction parameters and initial velocities for
the double pendulum and find chaotic and nonchaotic
states.